Patent Publication Number: US-2021181056-A1

Title: Method for determining temperature-induced sag variation of main cable and tower-top horizontal displacement of suspension bridges

Description:
CROSS REFERENCE TO THE RELATED APPLICATIONS 
     This application is based upon and claims priority to Chinese Patent Application No. 201911299463.1, filed on Dec. 16, 2019, the entire contents of which are incorporated herein by reference. 
     TECHNICAL FIELD 
     The present disclosure relates to the technical field of analysis and monitoring of bridge structures, and more particularly directed to a method for determining the variations in the main cable sag and tower-top horizontal displacement of suspension bridges with the ambient temperature changes. 
     BACKGROUND 
     Structural health monitoring systems in suspension bridges usually focus on the global deformation of the main cable, which can be characterized by the sag change of the main cable and the horizontal displacement of the tower top. Field measurement has shown that the shape of the main cable of a suspension bridge varies significantly with the variation of ambient temperature. The temperature-induced deformation can unfavorably mask abnormal deformations of the bridge structure caused by structural damage or degradation, and it needs to be excluded from the measured total deformation in order to highlight the abnormal deformation and subsequently evaluate the structural condition more accurately. Therefore, it is imperative to study the relationship between the temperature changes and the sag variation of the main cable and the horizontal displacement of the tower top of suspension bridges. 
     At present, the methods for calculating the temperature deformation of suspension bridges include: (1) regression analysis; (2) finite element analysis; and (3) physics-based formulas. The regression analysis does not reflect the causal relationship between the variables, and the obtained model is dedicated to specific bridges, which has poor generality. The finite element analysis requires detailed design information and necessary expertise, and as being case by case, a separate model is required for different bridges. In spite of an approximate estimate, the physics-based formulas have merits of clear concepts, general applicability, and great convenience for parametric analysis and field calculation, thus making it more advantageous than the other two methods. However, the physics-based formulas for the temperature deformation of suspension bridges are few and imperfect at best. The sag variation of the main-span cable is calculated by either the single-span cable model, or simplified formulas with ignorance of the sag effect of the side span cables, while the calculation formula of the tower-top horizontal displacement is even rarely reported. 
     SUMMARY 
     The objective of the present disclosure is to provide a method for determining the temperature-induced sag variation of the main cable and the tower-top horizontal displacement of suspension bridges. 
     The present disclosure first provides a calculation method for the temperature-induced variation in the main cable sag and the tower-top horizontal displacement of the two-tower ground-anchored suspension bridge, and then extends the calculation method to the self-anchored suspension bridge as well as other cable systems with any number of spans. The calculation process for the temperature deformation of the two-tower ground-anchored suspension bridge is as follows: 
     (1) according to the equilibrium condition that the horizontal tensions of the main cable on both sides of the tower top are always equal, establishing the following equation: 
     
       
         
           
             
               
                 
                   δ 
                    
                   
                     f 
                     i 
                   
                 
                 
                   f 
                   i 
                 
               
               - 
               
                 
                   δ 
                    
                   
                     l 
                     i 
                   
                 
                 
                   l 
                   i 
                 
               
             
             = 
             
               
                 
                   δ 
                    
                   
                     f 
                     
                       i 
                       + 
                       1 
                     
                   
                 
                 
                   f 
                   
                     i 
                     + 
                     1 
                   
                 
               
               - 
               
                 
                   δ 
                    
                   
                     l 
                     
                       i 
                       + 
                       1 
                     
                   
                 
                 
                   l 
                   
                     i 
                     + 
                     1 
                   
                 
               
             
           
         
       
     
     where i=1, 2; f i  is a sag (or mid-span deflection) of an i th  span main cable; δf i  is a variation of f i  caused by a temperature variation; l i  is a span (horizontal distance of supports at both ends) of the i th  span main cable; δl i  is a variation of l i  caused by the temperature variation; subscripts 1, 2, 3 of variables indicate a left side span, a main span, and a right side span, respectively; 
     (2) according to a geometric relationship between a shape and a length of the main cable, establishing the following equation: 
     
       
         
           
             
               
                 
                   
                     
                       c 
                       ni 
                     
                     
                       l 
                       i 
                     
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   f 
                   i 
                 
               
               - 
               
                 
                   
                     
                       
                         c 
                         ni 
                       
                       · 
                       
                         n 
                         i 
                       
                     
                     
                       l 
                       i 
                     
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   l 
                   i 
                 
               
               + 
               
                 
                   
                     c 
                     li 
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   l 
                   i 
                 
               
               - 
               
                 
                   
                     
                       
                         
                           c 
                           
                             α 
                              
                             
                                 
                             
                              
                             i 
                           
                         
                         · 
                         sin 
                       
                        
                       
                           
                       
                        
                       2 
                        
                       
                         α 
                         i 
                       
                     
                     
                       2 
                       · 
                       
                         l 
                         i 
                       
                     
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   l 
                   i 
                 
               
             
             = 
             
               
                 δ 
                  
                 
                     
                 
                  
                 
                   S 
                   i 
                 
               
               - 
               
                 
                   
                     
                       
                         c 
                         
                           α 
                            
                           
                               
                           
                            
                           i 
                         
                       
                       · 
                       
                         cos 
                         2 
                       
                     
                      
                     
                       α 
                       i 
                     
                   
                   
                     l 
                     i 
                   
                 
                 · 
                 
                   ( 
                   
                     
                       δ 
                        
                       
                           
                       
                        
                       
                         h 
                         Pi 
                       
                     
                     - 
                     
                       δ 
                        
                       
                           
                       
                        
                       
                         h 
                         
                           P 
                            
                           
                             ( 
                             
                               i 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     where i=1, 2, 3; n i  is a sag-to-span ratio of the i th  span main cable, i.e. n i =f i /l i ; α i  is a chord inclination (positive for a counterclockwise rotation from a horizontal line) of the i th  span main cable; coefficients c ni , c li , and c αi  are respectively: 
     
       
         
           
             
               c 
               ni 
             
             = 
             
               
                 l 
                 i 
               
               · 
               
                 [ 
                 
                   
                     
                       16 
                       3 
                     
                      
                     
                       n 
                       i 
                     
                      
                     
                         
                     
                      
                     
                       cos 
                       3 
                     
                      
                     
                         
                     
                      
                     
                       α 
                       i 
                     
                   
                   - 
                   
                     
                       128 
                       5 
                     
                      
                     
                       
                         n 
                         i 
                         3 
                       
                        
                       
                         ( 
                         
                           
                             5 
                              
                             
                                 
                             
                              
                             
                               cos 
                               7 
                             
                              
                             
                               α 
                               i 
                             
                           
                           - 
                           
                             4 
                              
                             
                                 
                             
                              
                             
                               cos 
                               5 
                             
                              
                             
                               α 
                               i 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               c 
               li 
             
             = 
             
               
                 sec 
                  
                 
                     
                 
                  
                 
                   α 
                   i 
                 
               
               + 
               
                 
                   8 
                   3 
                 
                  
                 
                   n 
                   i 
                   2 
                 
                  
                 
                   cos 
                   3 
                 
                  
                 
                   α 
                   i 
                 
               
               - 
               
                 
                   32 
                   5 
                 
                  
                 
                   
                     n 
                     i 
                     4 
                   
                    
                   
                     ( 
                     
                       
                         5 
                          
                         
                             
                         
                          
                         
                           cos 
                           7 
                         
                          
                         
                           α 
                           i 
                         
                       
                       - 
                       
                         4 
                          
                         
                             
                         
                          
                         
                           cos 
                           5 
                         
                          
                         
                           α 
                           i 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               c 
               
                 α 
                  
                 
                     
                 
                  
                 i 
               
             
             = 
             
               
                 l 
                 i 
               
               · 
               
                 [ 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         i 
                       
                     
                     
                       
                         cos 
                         2 
                       
                        
                       
                           
                       
                        
                       
                         α 
                         i 
                       
                     
                   
                   - 
                   
                     8 
                      
                     
                       n 
                       i 
                       2 
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     
                       α 
                       i 
                     
                      
                     
                       cos 
                       2 
                     
                      
                     
                         
                     
                      
                     
                       α 
                       i 
                     
                   
                   + 
                   
                     32 
                      
                     
                       n 
                       i 
                       4 
                     
                      
                     
                         
                     
                      
                     
                       cos 
                       4 
                     
                      
                     
                         
                     
                      
                     
                       α 
                       i 
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     
                       
                         α 
                         i 
                       
                        
                       
                         ( 
                         
                           
                             7 
                              
                             
                                 
                             
                              
                             
                               cos 
                               2 
                             
                              
                             
                               α 
                               i 
                             
                           
                           - 
                           4 
                         
                         ) 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     δS i  is a length variation of the i th  span main cable caused by the temperature variation; δh Pi  and δh P(i−1)  are an elevation change of the support points i and i=1 of the main cable, respectively, and δh P0 =δh P3 =0; 
     δS i  (i=1, 2, 3) and δh Pi  (i=1, 2) are estimated by the following equations: 
     
       
         
           
             
               δ 
                
               
                   
               
                
               
                 S 
                 i 
               
             
             = 
             
               
                 
                   
                     S 
                     i 
                   
                   · 
                   
                     θ 
                     C 
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   T 
                   C 
                 
               
               = 
               
                 
                   
                     l 
                     i 
                   
                   · 
                   
                     θ 
                     C 
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   
                     T 
                     C 
                   
                    
                   
                     [ 
                     
                       
                         sec 
                          
                         
                             
                         
                          
                         
                           α 
                           i 
                         
                       
                       + 
                       
                         
                           8 
                           3 
                         
                          
                         
                           n 
                           i 
                           2 
                         
                          
                         
                             
                         
                          
                         
                           cos 
                           3 
                         
                          
                         
                             
                         
                          
                         
                           α 
                           i 
                         
                       
                       - 
                       
                         
                           32 
                           5 
                         
                          
                         
                           
                             n 
                             i 
                             4 
                           
                            
                           
                             ( 
                             
                               
                                 5 
                                  
                                 
                                     
                                 
                                  
                                 
                                   cos 
                                   7 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   i 
                                 
                               
                               - 
                               
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 
                                   cos 
                                   5 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   i 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
       
         
           
             
               δ 
                
               
                   
               
                
               
                 h 
                 Pi 
               
             
             = 
             
               
                 
                   h 
                   Pi 
                 
                 · 
                 
                   θ 
                   P 
                 
                 · 
                 δ 
               
                
               
                   
               
                
               
                 T 
                 P 
               
             
           
         
       
     
     where θ C  is a linear expansion coefficient of the main cable, θ P  is a linear expansion coefficient of a tower of the suspension bridge, δT C  is a temperature variation of the main cable, δT P  is a temperature variation of the tower of the suspension bridge, and h Pi  is a height of the tower of the suspension bridge; 
     (3) according to a compatibility condition to be satisfied by a sum of spans of a left side span cable, a main span cable, and a right side span cable, that is, the distance between the anchorages at both ends is constant, establishing the following equation: 
     
       
         
           
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   3 
                 
                  
                 
                     
                 
                  
                 
                   δ 
                    
                   
                       
                   
                    
                   
                     l 
                     i 
                   
                 
               
               = 
               0 
             
             ; 
           
         
       
     
     (4) according to the following linear system of equations consisting of the equations in steps (1), (2), and (3), simultaneously obtaining the sag variation δf i  and the span variation δl i  of the left side span cable, the main span cable, and the right side span cable: 
     
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         - 
                         
                           1 
                           
                             f 
                             1 
                           
                         
                       
                     
                     
                       
                         1 
                         
                           f 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         
                           l 
                           1 
                         
                       
                     
                     
                       
                         - 
                         
                           1 
                           
                             l 
                             2 
                           
                         
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           
                             f 
                             2 
                           
                         
                       
                     
                     
                       
                         1 
                         
                           f 
                           3 
                         
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         
                           l 
                           2 
                         
                       
                     
                     
                       
                         - 
                         
                           1 
                           
                             l 
                             3 
                           
                         
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       1 
                     
                     
                       1 
                     
                   
                   
                     
                       
                         
                           c 
                           
                             n 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                         
                           l 
                           1 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         M 
                         1 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         
                           c 
                           
                             n 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                         
                           l 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         M 
                         2 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         
                           c 
                           
                             n 
                              
                             
                                 
                             
                              
                             3 
                           
                         
                         
                           l 
                           3 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         M 
                         3 
                       
                     
                   
                 
                 ] 
               
               · 
               
                 [ 
                 
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           f 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           f 
                           2 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           f 
                           3 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           l 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           l 
                           2 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           l 
                           3 
                         
                       
                     
                   
                 
                 ] 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                   
                   
                     
                       
                         Δ 
                         1 
                       
                     
                   
                   
                     
                       
                         Δ 
                         2 
                       
                     
                   
                   
                     
                       
                         Δ 
                         3 
                       
                     
                   
                 
                 ] 
               
                
               
                   
               
                
               where 
             
           
         
       
       
         
           
             
               
                 M 
                 i 
               
               = 
               
                 
                   - 
                   
                     
                       
                         c 
                         ni 
                       
                       · 
                       
                         n 
                         i 
                       
                     
                     
                       l 
                       i 
                     
                   
                 
                 + 
                 
                   c 
                   li 
                 
                 - 
                 
                   
                     
                       
                         c 
                         
                           α 
                            
                           
                               
                           
                            
                           i 
                         
                       
                       · 
                       sin 
                     
                      
                     
                         
                     
                      
                     2 
                      
                     
                       α 
                       i 
                     
                   
                   
                     2 
                     · 
                     
                       l 
                       i 
                     
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 Δ 
                 i 
               
               = 
               
                 
                   δ 
                    
                   
                       
                   
                    
                   
                     S 
                     i 
                   
                 
                 - 
                 
                   
                     
                       
                         
                           c 
                           
                             α 
                              
                             
                                 
                             
                              
                             i 
                           
                         
                         · 
                         
                           cos 
                           2 
                         
                       
                        
                       
                           
                       
                        
                       
                         α 
                         i 
                       
                     
                     
                       l 
                       i 
                     
                   
                   · 
                   
                     ( 
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           Pi 
                         
                       
                       - 
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           
                             P 
                              
                             
                               ( 
                               
                                 i 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
             
             , 
             
               
 
             
              
             
               i 
               = 
               1 
             
             , 
             2 
             , 
             3. 
           
         
       
     
     The tower-top horizontal displacement of the left tower and the right tower are δl i  and δl 3 , respectively (positive for a movement toward the main span), and a horizontal distance variation between the tower top of the left tower and the tower top of the right tower is δl 2 . 
     When the higher-order terms of the sag-to-span ratio n i  in the coefficients c ni , c li , and c αi  are ignored (n i  of a suspension bridge is generally between 1/12 and 1/9), the analytical solutions of the sag variation δf i  and the span variation δl i  of the i th  span main cable are respectively: 
     
       
         
           
             
               δ 
                
               
                   
               
                
               
                 f 
                 i 
               
             
             = 
             
               
                 
                   
                     n 
                     i 
                   
                   
                     cos 
                      
                     
                         
                     
                      
                     
                       α 
                       i 
                     
                   
                 
                  
                 δ 
                  
                 
                     
                 
                  
                 
                   S 
                   i 
                 
               
               - 
               
                 
                   n 
                   i 
                 
                  
                 tan 
                  
                 
                     
                 
                  
                 
                   
                     α 
                     i 
                   
                    
                   
                     ( 
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           Pi 
                         
                       
                       - 
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           
                             P 
                              
                             
                               ( 
                               
                                 i 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   
                     
                       n 
                       i 
                     
                      
                     
                       ( 
                       
                         
                           3 
                            
                           
                               
                           
                            
                           
                             l 
                             i 
                           
                         
                         - 
                         
                           16 
                            
                           
                               
                           
                            
                           
                             r 
                             i 
                           
                         
                       
                       ) 
                     
                   
                   
                     16 
                     · 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         3 
                       
                        
                       
                         r 
                         j 
                       
                     
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         3 
                       
                        
                       
                           
                       
                        
                       
                         
                           δ 
                            
                           
                               
                           
                            
                           
                             S 
                             k 
                           
                         
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             α 
                             k 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         2 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             ( 
                             
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                               
                               - 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   k 
                                 
                               
                             
                             ) 
                           
                           · 
                           δ 
                         
                          
                         
                             
                         
                          
                         
                           h 
                           Pk 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               δ 
                
               
                   
               
                
               
                 l 
                 i 
               
             
             = 
             
               
                 
                   δ 
                    
                   
                       
                   
                    
                   
                     S 
                     i 
                   
                 
                 
                   cos 
                    
                   
                       
                   
                    
                   
                     α 
                     i 
                   
                 
               
               - 
               
                 tan 
                  
                 
                     
                 
                  
                 
                   
                     α 
                     i 
                   
                    
                   
                     ( 
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           Pi 
                         
                       
                       - 
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           
                             P 
                              
                             
                               ( 
                               
                                 i 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   
                     r 
                     i 
                   
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       3 
                     
                      
                     
                       r 
                       j 
                     
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         3 
                       
                        
                       
                           
                       
                        
                       
                         
                           δ 
                            
                           
                               
                           
                            
                           
                             S 
                             k 
                           
                         
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             α 
                             k 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         2 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             ( 
                             
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                               
                               - 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   k 
                                 
                               
                             
                             ) 
                           
                           · 
                           δ 
                         
                          
                         
                             
                         
                          
                         
                           h 
                           Pk 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     where r i =l i ·n i   2 ·cos 2 α i  (i=1, 2, 3), r j =l j ·n j   2 ·cos 2 α j  (j=1, 2, 3), and i, j, and k are all subscripts. 
     When the tower-top elevations of the left tower and the right tower of the suspension bridge are equal to each other, that is α 2 =0, and the conditions of α 1 &gt;0, α 3 &lt;0, h P1 ≈h 1 , h P2 ≈|h 3 |, θ C ·δT C ≈θ P ·δT P  are satisfied, the sag variation δf 2  of the main span cable can be estimated by the following equation: 
     
       
         
           
             
               
                 δ 
                  
                 
                     
                 
                  
                 
                   f 
                   2 
                 
               
               = 
               
                 
                   
                     
                       r 
                       2 
                     
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         3 
                       
                        
                       
                         r 
                         j 
                       
                     
                   
                   · 
                   
                     
                       3 
                        
                       
                           
                       
                        
                       
                         
                           θ 
                           C 
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         T 
                         C 
                       
                     
                     
                       16 
                        
                       
                         n 
                         2 
                       
                     
                   
                 
                  
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       1 
                     
                     3 
                   
                    
                   
                       
                   
                    
                   
                     l 
                     i 
                   
                 
               
             
             ; 
           
         
       
     
     When the sags of the left side span cable and the right side span cable are further ignored, the above equation is simplified as: 
     
       
         
           
             
               δ 
                
               
                   
               
                
               
                 f 
                 2 
               
             
             = 
             
               
                 
                   3 
                    
                   
                       
                   
                    
                   
                     
                       θ 
                       C 
                     
                     · 
                     δ 
                   
                    
                   
                       
                   
                    
                   
                     T 
                     C 
                   
                 
                 
                   16 
                    
                   
                     n 
                     2 
                   
                 
               
                
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   3 
                 
                  
                 
                     
                 
                  
                 
                   
                     l 
                     i 
                   
                   . 
                 
               
             
           
         
       
     
     When the tower-top elevations of the left tower and the right tower of the suspension bridge are equal to each other, that is α 2 =0, the sags of the left side span cable and the right side span cable are neglected, that is, r 1 =r 3 =0, and the conditions of α 1 &gt;0, α 3 &lt;0, h P1 ≈h 1 , h P2 ≈|h 3 |, θ C ·δT C ≈θ P ·δT P  are satisfied, the tower-top horizontal displacement δl i  (i.e. span variation) of the left tower, the tower-top horizontal displacement δl 3  (i.e. span variation) of the right tower, and the tower-top horizontal distance variation δl 2  (the span variation of the main span main cable) are calculated by the following equations: 
       δ l   1   =l   1 θ C   ·δT   C  
 
       δ l   2 =−( l   1   +l   3 )θ C   ·δT   C  
 
       δ l   3   =l   3 θ C   ·δT   C .
 
     When the suspension bridge is a two-tower self-anchored suspension bridge, the main cable of the two-tower self-anchored suspension bridge is directly anchored to the end of the main girder, and the thermal expansion and contraction of the main girder causes the distance change between both ends of the main cable. Thus, the calculation procedure of the temperature deformation of the two-tower self-anchored suspension bridge is the same as the calculation procedure for the two-tower ground-anchored suspension bridge, provided that the column vector on the right side of the linear system of equations changes from [0 0 0 Δ 1  Δ 2  Δ 3 ] T  to [0 0 Δ G  Δ 1  Δ 2  Δ 3 ] T , while the coefficient matrix remains unchanged. Δ G  is a variation of the sum of spans of the left side span cable, the main span cable, and the right side span cable. If the main girder is continuous at the girder-tower intersections with the total length of L G , Δ G  can be estimated by Δ G =L G θ G ·δT G  where θ G  is a linear expansion coefficient of the main girder and δT G  is a temperature variation of the main girder. Therefore, the temperature deformation of the two-tower self-anchored suspension bridge will be the solution of the following linear system of equations: 
     
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         - 
                         
                           1 
                           
                             f 
                             1 
                           
                         
                       
                     
                     
                       
                         1 
                         
                           f 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         
                           l 
                           1 
                         
                       
                     
                     
                       
                         - 
                         
                           1 
                           
                             l 
                             2 
                           
                         
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           
                             f 
                             2 
                           
                         
                       
                     
                     
                       
                         1 
                         
                           f 
                           3 
                         
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         
                           l 
                           2 
                         
                       
                     
                     
                       
                         - 
                         
                           1 
                           
                             l 
                             3 
                           
                         
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       1 
                     
                     
                       1 
                     
                   
                   
                     
                       
                         
                           c 
                           
                             n 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                         
                           l 
                           1 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         M 
                         1 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         
                           c 
                           
                             n 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                         
                           l 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         M 
                         2 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         
                           c 
                           
                             n 
                              
                             
                                 
                             
                              
                             3 
                           
                         
                         
                           l 
                           3 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         M 
                         3 
                       
                     
                   
                 
                 ] 
               
               · 
               
                 [ 
                 
                   
                     
                       
                         δ 
                          
                         
                           f 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                           f 
                           2 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                           f 
                           3 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                           l 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                           l 
                           2 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                           l 
                           3 
                         
                       
                     
                   
                 
                 ] 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                   
                   
                     
                       
                         Δ 
                         G 
                       
                     
                   
                   
                     
                       
                         Δ 
                         1 
                       
                     
                   
                   
                     
                       
                         Δ 
                         2 
                       
                     
                   
                   
                     
                       
                         Δ 
                         3 
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
     When the higher-order terms of the sag-to-span ratio n i  in the coefficients c ni , c li , and c αi  are ignored, the analytical solutions of the sag variation δf i  and the span variation δl i  of the two-tower self-anchored suspension bridge are respectively: 
     
       
         
           
             
               δ 
                
               
                   
               
                
               
                 f 
                 i 
               
             
             = 
             
               
                 
                   
                     n 
                     i 
                   
                   
                     cos 
                      
                     
                         
                     
                      
                     
                       α 
                       i 
                     
                   
                 
                  
                 δ 
                  
                 
                     
                 
                  
                 
                   S 
                   i 
                 
               
               - 
               
                 
                   n 
                   i 
                 
                  
                 tan 
                  
                 
                     
                 
                  
                 
                   
                     α 
                     i 
                   
                    
                   
                     ( 
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           Pi 
                         
                       
                       - 
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           
                             P 
                              
                             
                               ( 
                               
                                 i 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   
                     
                       n 
                       i 
                     
                      
                     
                       ( 
                       
                         
                           3 
                            
                           
                               
                           
                            
                           
                             l 
                             i 
                           
                         
                         - 
                         
                           16 
                            
                           
                               
                           
                            
                           
                             r 
                             i 
                           
                         
                       
                       ) 
                     
                   
                   
                     16 
                     · 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         3 
                       
                        
                       
                         r 
                         j 
                       
                     
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         3 
                       
                        
                       
                           
                       
                        
                       
                         
                           δ 
                            
                           
                               
                           
                            
                           
                             S 
                             k 
                           
                         
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             α 
                             k 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         2 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             ( 
                             
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                               
                               - 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   k 
                                 
                               
                             
                             ) 
                           
                           · 
                           δ 
                         
                          
                         
                             
                         
                          
                         
                           h 
                           Pk 
                         
                       
                     
                     - 
                     
                       Δ 
                       G 
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               δ 
                
               
                   
               
                
               
                 l 
                 i 
               
             
             = 
             
               
                 
                   δ 
                    
                   
                       
                   
                    
                   
                     S 
                     i 
                   
                 
                 
                   cos 
                    
                   
                       
                   
                    
                   
                     α 
                     i 
                   
                 
               
               - 
               
                 tan 
                  
                 
                     
                 
                  
                 
                   
                     α 
                     i 
                   
                    
                   
                     ( 
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           Pi 
                         
                       
                       - 
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           
                             P 
                              
                             
                               ( 
                               
                                 i 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   
                     r 
                     i 
                   
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       3 
                     
                      
                     
                       r 
                       j 
                     
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         3 
                       
                        
                       
                           
                       
                        
                       
                         
                           δ 
                            
                           
                               
                           
                            
                           
                             S 
                             k 
                           
                         
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             α 
                             k 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         2 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             ( 
                             
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                 
                               
                               - 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   k 
                                 
                               
                             
                             ) 
                           
                           · 
                           δ 
                         
                          
                         
                             
                         
                          
                         
                           h 
                           Pk 
                         
                       
                     
                     - 
                     
                       Δ 
                       G 
                     
                   
                   ] 
                 
               
             
           
         
       
     
     For the two-tower suspension bridge, a mid-span elevation variation δD 2  of the main span cable can be estimated from the sag variation. The mid-span elevation variation d 2  of the chord of the main span cable caused by the tower height variation is as follows: 
     
       
         
           
             
               d 
               2 
             
             = 
             
               
                 
                   
                     δ 
                      
                     
                       h 
                       
                         P 
                          
                         1 
                       
                     
                   
                   + 
                   
                     δ 
                      
                     
                       h 
                       
                         P 
                          
                         2 
                       
                     
                   
                 
                 2 
               
               = 
               
                 
                   
                     
                       
                         h 
                         
                           P 
                            
                           1 
                         
                       
                       + 
                       
                         h 
                         
                           P 
                            
                           2 
                         
                       
                     
                     2 
                   
                   · 
                   
                     θ 
                     P 
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   
                     T 
                     P 
                   
                   . 
                 
               
             
           
         
       
     
     Since the elevation takes the vertical upward direction as positive, the elevation variation δD 2  is equal to minus δf 2  plus d 2 , i.e., 
     
       
         
           
             
               δ 
                
               
                 D 
                 2 
               
             
             = 
             
               
                 
                   - 
                   δ 
                 
                  
                 
                   f 
                   2 
                 
               
               + 
               
                 
                   
                     
                       
                         h 
                         
                           P 
                            
                           1 
                         
                       
                       + 
                       
                         h 
                         
                           P 
                            
                           2 
                         
                       
                     
                     2 
                   
                   · 
                   
                     θ 
                     P 
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   
                     T 
                     P 
                   
                   . 
                 
               
             
           
         
       
     
     As the length of the suspender at mid span or the thickness of the central clamp of the suspension bridge are relatively small, and their thermal deformation can be ignored, the elevation variation of the main span girder at mid span can also be approximated by δD 2 . 
     The above analysis method for the temperature deformation of the two-tower suspension bridge can be extended to other cable systems with any number of spans (multi-span suspension bridges, transmission lines, ropeways, etc.). When the cable system has ti spans numbered as 1, 2, . . . , u−1, u and contains u+1 supports (including both ends) numbered as 0, 1, . . . , u−1, u, and u≥1, the calculation method for the temperature-induced sag variation of the main cable and the tower-top horizontal displacement of the multi-span suspension bridge is as follows: 
     (1) according to the equilibrium condition that the horizontal tensions of the main cables on both sides of the tower top are always equal, establishing the following u−1 equations: 
     
       
         
           
             
               
                 
                   δ 
                    
                   
                     f 
                     i 
                   
                 
                 
                   f 
                   i 
                 
               
               - 
               
                 
                   δ 
                    
                   
                       
                   
                    
                   
                     l 
                     i 
                   
                 
                 
                   l 
                   i 
                 
               
             
             = 
             
               
                 
                   δ 
                    
                   
                     f 
                     
                       i 
                       + 
                       1 
                     
                   
                 
                 
                   f 
                   
                     i 
                     + 
                     1 
                   
                 
               
               - 
               
                 
                   δ 
                    
                   
                     l 
                     
                       i 
                       + 
                       1 
                     
                   
                 
                 
                   l 
                   
                     i 
                     + 
                     1 
                   
                 
               
             
           
         
       
     
     where i=1, 2, . . . , u−1; f i  is the sag (or mid-span deflection) of the i th  span main cable; δf i  is the variation of f i  caused by the temperature variation; l i  is the span (horizontal distance of supports at both ends) of the i th  span main cable; δl i  is the variation of l i  caused by the temperature variation; 
     (2) according to the geometric relationship between the shape of the main cable and the length of the main cable, establishing the following u equations: 
     
       
         
           
             
               
                 
                   
                     
                       c 
                       ni 
                     
                     
                       l 
                       i 
                     
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   f 
                   i 
                 
               
               - 
               
                 
                   
                     
                       
                         c 
                         ni 
                       
                       · 
                       
                         n 
                         i 
                       
                     
                     
                       l 
                       i 
                     
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   l 
                   i 
                 
               
               + 
               
                 
                   
                     c 
                     li 
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   l 
                   i 
                 
               
               - 
               
                 
                   
                     
                       
                         
                           c 
                           
                             α 
                              
                             
                                 
                             
                              
                             i 
                           
                         
                         · 
                         sin 
                       
                        
                       
                           
                       
                        
                       2 
                        
                       
                           
                       
                        
                       
                         α 
                         i 
                       
                     
                     
                       2 
                       · 
                       
                         l 
                         i 
                       
                     
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   l 
                   i 
                 
               
             
             = 
             
               
                 δ 
                  
                 
                   S 
                   i 
                 
               
               - 
               
                 
                   
                     
                       
                         c 
                         
                           α 
                            
                           
                               
                           
                            
                           i 
                         
                       
                       · 
                       
                         cos 
                         2 
                       
                     
                      
                     
                       α 
                       i 
                     
                   
                   
                     l 
                     i 
                   
                 
                 · 
                 
                   ( 
                   
                     
                       δ 
                        
                       
                           
                       
                        
                       
                         h 
                         Pi 
                       
                     
                     - 
                     
                       δ 
                        
                       
                         h 
                         
                           P 
                            
                           
                             ( 
                             
                               i 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     where i=1, 2, . . . , u; n i  is the sag-to-span ratio of the i th  span main cable, i.e. n i =f i /l i ; α i  is the chord inclination (positive for the counterclockwise rotation from the horizontal line) of the i th  span main cable; the coefficients c ni , c li , and c αi  are respectively: 
     
       
         
           
             
               c 
               ni 
             
             = 
             
               
                 l 
                 i 
               
               · 
               
                 [ 
                 
                   
                     
                       
                         1 
                          
                         6 
                       
                       3 
                     
                      
                     
                       n 
                       i 
                     
                      
                     
                       cos 
                       3 
                     
                      
                     
                       α 
                       i 
                     
                   
                   - 
                   
                     
                       
                         1 
                          
                         2 
                          
                         8 
                       
                       5 
                     
                      
                     
                       
                         n 
                         i 
                         3 
                       
                        
                       
                         ( 
                         
                           
                             5 
                              
                             
                               cos 
                               7 
                             
                              
                             
                               α 
                               i 
                             
                           
                           - 
                           
                             4 
                              
                             
                               cos 
                               5 
                             
                              
                             
                               α 
                               i 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               c 
               
                 l 
                  
                 i 
               
             
             = 
             
               
                 sec 
                  
                 
                   α 
                   i 
                 
               
               + 
               
                 
                   8 
                   3 
                 
                  
                 
                   n 
                   i 
                   2 
                 
                  
                 
                   cos 
                   3 
                 
                  
                 
                   α 
                   i 
                 
               
               - 
               
                 
                   
                     3 
                      
                     2 
                   
                   5 
                 
                  
                 
                   
                     n 
                     i 
                     4 
                   
                    
                   
                     ( 
                     
                       
                         5 
                          
                         
                           cos 
                           7 
                         
                          
                         
                           α 
                           i 
                         
                       
                       - 
                       
                         4 
                          
                         
                           cos 
                           5 
                         
                          
                         
                           α 
                           i 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               c 
               
                 α 
                  
                 
                     
                 
                  
                 i 
               
             
             = 
             
               
                 l 
                 i 
               
               · 
               
                 [ 
                 
                   
                     
                       sin 
                        
                       
                         α 
                         i 
                       
                     
                     
                       
                         cos 
                         2 
                       
                        
                       
                         α 
                         i 
                       
                     
                   
                   - 
                   
                     8 
                      
                     
                       n 
                       i 
                       2 
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     
                       α 
                       i 
                     
                      
                     
                       cos 
                       2 
                     
                      
                     
                       α 
                       i 
                     
                   
                   + 
                   
                     3 
                      
                     2 
                      
                     
                       n 
                       i 
                       4 
                     
                      
                     
                       cos 
                       4 
                     
                      
                     
                       α 
                       i 
                     
                      
                     sin 
                      
                     
                       
                         α 
                         i 
                       
                        
                       
                         ( 
                         
                           
                             7 
                              
                             
                               cos 
                               2 
                             
                              
                             
                               α 
                               i 
                             
                           
                           - 
                           4 
                         
                         ) 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     δS i  is the length variation of the i th  span main cable caused by the temperature variation; δh Pi  is the elevation variation of the intermediate supports (tower tops), i=1, 2, . . . , u=1, and δh P0 =δh P3 =0; and δS i  and δh Pi  are calculated according to the following equations: 
     
       
         
           
             
               δ 
                
               
                   
               
                
               
                 S 
                 i 
               
             
             = 
             
               
                 
                   
                     S 
                     i 
                   
                   · 
                   
                     θ 
                     C 
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   T 
                   C 
                 
               
               = 
               
                 
                   
                     l 
                     i 
                   
                   · 
                   
                     θ 
                     C 
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   
                     T 
                     C 
                   
                    
                   
                     [ 
                     
                       
                         sec 
                          
                         
                             
                         
                          
                         
                           α 
                           i 
                         
                       
                       + 
                       
                         
                           8 
                           3 
                         
                          
                         
                           n 
                           i 
                           2 
                         
                          
                         
                           cos 
                           3 
                         
                          
                         
                           α 
                           i 
                         
                       
                       - 
                       
                         
                           32 
                           5 
                         
                          
                         
                           
                             n 
                             i 
                             4 
                           
                            
                           
                             ( 
                             
                               
                                 5 
                                  
                                 
                                     
                                 
                                  
                                 
                                   cos 
                                   7 
                                 
                                  
                                 
                                   α 
                                   i 
                                 
                               
                               - 
                               
                                 4 
                                  
                                 
                                   cos 
                                   5 
                                 
                                  
                                 
                                   α 
                                   i 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
       
         
           
             
               δ 
                
               
                   
               
                
               
                 h 
                 
                   P 
                    
                   
                       
                   
                    
                   i 
                 
               
             
             = 
             
               
                 
                   h 
                   Pi 
                 
                 · 
                 
                   θ 
                   P 
                 
                 · 
                 δ 
               
                
               
                   
               
                
               
                 T 
                 P 
               
             
           
         
       
     
     where θ C  is the linear expansion coefficient of the main cable, θ P  is the linear expansion coefficient of the tower of the suspension bridge, δT C  is the temperature variation of the main cable, δT P  is the temperature variation of the bridge tower, and h Pi  is the height of the bridge tower; 
     (3) according to the compatibility condition to be satisfied by the sum of all spans of the main cable, that is, the distance between the anchorages at both ends is constant, establishing the following equation: 
     
       
         
           
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   u 
                 
                  
                 
                   δ 
                    
                   
                     l 
                     i 
                   
                 
               
               = 
               0 
             
             ; 
           
         
       
     
     (4) according to the following linear system of equations consisting of the above equations in steps (1), (2), and (3), simultaneously obtaining the sag variation δf i  and the span variation of each span main cable: 
     
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         A 
                         
                           
                             ( 
                             
                               u 
                               - 
                               1 
                             
                             ) 
                           
                           × 
                           u 
                         
                       
                     
                     
                       
                         B 
                         
                           
                             ( 
                             
                               u 
                               - 
                               1 
                             
                             ) 
                           
                           × 
                           u 
                         
                       
                     
                   
                   
                     
                       
                         0 
                         
                           1 
                           × 
                           u 
                         
                       
                     
                     
                       
                         1 
                         
                           1 
                           × 
                           u 
                         
                       
                     
                   
                   
                     
                       
                         C 
                         
                           u 
                           × 
                           u 
                         
                       
                     
                     
                       
                         D 
                         
                           u 
                           × 
                           u 
                         
                       
                     
                   
                 
                 ] 
               
               · 
               
                 [ 
                 
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           F 
                           
                             u 
                             × 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                           L 
                           
                             u 
                             × 
                             1 
                           
                         
                       
                     
                   
                 
                 ] 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       0 
                       
                         u 
                         × 
                         1 
                       
                     
                   
                 
                 
                   
                     
                       Δ 
                       
                         u 
                         × 
                         1 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     where A, B, C, D, 0, 1, δF, δL, Δ represent a matrix or a vector, and the subscript represents the size of the matrix or vector. The elements in matrix A, B, C, D are as follows: 
     
       
         
           
             
               A 
               ij 
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           - 
                           1 
                         
                         / 
                         
                           f 
                           i 
                         
                       
                     
                     
                       
                         
                           when 
                            
                           
                               
                           
                            
                           i 
                         
                         = 
                         j 
                       
                     
                   
                   
                     
                       
                         1 
                         / 
                         
                           f 
                           j 
                         
                       
                     
                     
                       
                         
                           
                             when 
                              
                             
                                 
                             
                              
                             i 
                           
                           + 
                           1 
                         
                         = 
                         j 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       others 
                     
                   
                 
                 , 
                 
                   
 
                 
                  
                 
                   i 
                   = 
                   1 
                 
                 , 
                 2 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 
                   
                     u 
                     - 
                     1 
                   
                   ; 
                   
                     j 
                     = 
                     1 
                   
                 
                 , 
                 2 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 
                   u 
                   ; 
                   
                     
 
                   
                    
                   
                     
                       B 
                       ij 
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 1 
                                 / 
                                 
                                   l 
                                   i 
                                 
                               
                             
                             
                               
                                 
                                   when 
                                    
                                   
                                       
                                   
                                    
                                   i 
                                 
                                 = 
                                 j 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   1 
                                 
                                 / 
                                 
                                   l 
                                   j 
                                 
                               
                             
                             
                               
                                 
                                   
                                     when 
                                      
                                     
                                         
                                     
                                      
                                     i 
                                   
                                   + 
                                   1 
                                 
                                 = 
                                 j 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               others 
                             
                           
                         
                         , 
                         
                           
 
                         
                          
                         
                           i 
                           = 
                           1 
                         
                         , 
                         2 
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           
                             u 
                             - 
                             1 
                           
                           ; 
                           
                             j 
                             = 
                             1 
                           
                         
                         , 
                         2 
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           u 
                           ; 
                           
                             
 
                           
                            
                           
                             
                               C 
                               ij 
                             
                             = 
                             
                               { 
                               
                                 
                                   
                                     
                                       
                                         
                                           c 
                                           ni 
                                         
                                         / 
                                         
                                           l 
                                           i 
                                         
                                       
                                     
                                     
                                       
                                         
                                           when 
                                            
                                           
                                               
                                           
                                            
                                           i 
                                         
                                         = 
                                         j 
                                       
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       others 
                                     
                                   
                                 
                                 , 
                                 
                                   
 
                                 
                                  
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 , 
                                 2 
                                 , 
                                 … 
                                  
                                 
                                     
                                 
                                 , 
                                 
                                   u 
                                   ; 
                                   
                                     j 
                                     = 
                                     1 
                                   
                                 
                                 , 
                                 2 
                                 , 
                                 … 
                                  
                                 
                                     
                                 
                                 , 
                                 
                                   u 
                                   ; 
                                   
                                     
 
                                   
                                    
                                   
                                     
                                       D 
                                       ij 
                                     
                                     = 
                                     
                                       { 
                                       
                                         
                                           
                                             
                                               
                                                 M 
                                                 i 
                                               
                                             
                                             
                                               
                                                 
                                                   when 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   i 
                                                 
                                                 = 
                                                 j 
                                               
                                             
                                           
                                           
                                             
                                               0 
                                             
                                             
                                               others 
                                             
                                           
                                         
                                         , 
                                         
                                           
 
                                         
                                          
                                         
                                           i 
                                           = 
                                           1 
                                         
                                         , 
                                         2 
                                         , 
                                         … 
                                          
                                         
                                             
                                         
                                         , 
                                         
                                           u 
                                           ; 
                                           
                                             j 
                                             = 
                                             1 
                                           
                                         
                                         , 
                                         2 
                                         , 
                                         … 
                                          
                                         
                                             
                                         
                                         , 
                                         
                                           u 
                                           ; 
                                           
                                             
                                               where 
                                                
                                               
                                                 
 
                                               
                                                
                                               
                                                 M 
                                                 i 
                                               
                                             
                                             = 
                                             
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       c 
                                                       ni 
                                                     
                                                     · 
                                                     
                                                       n 
                                                       i 
                                                     
                                                   
                                                   
                                                     l 
                                                     i 
                                                   
                                                 
                                               
                                               + 
                                               
                                                 c 
                                                 li 
                                               
                                               - 
                                               
                                                 
                                                   
                                                     
                                                       c 
                                                       
                                                         α 
                                                          
                                                         
                                                             
                                                         
                                                          
                                                         i 
                                                       
                                                     
                                                     · 
                                                     sin 
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     α 
                                                     i 
                                                   
                                                 
                                                 
                                                   2 
                                                   · 
                                                   
                                                     l 
                                                     i 
                                                   
                                                 
                                               
                                             
                                           
                                           ; 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     0 or 1 represent a vector with all elements being 0 or 1. For example, 0 1×u  is a 1-by-u vector of zeros, and 1 1×u  is a 1-by-u vector of ones. The remaining vectors are: 
     
       
         
           
             
               δ 
                
               
                   
               
                
               
                 F 
                 
                   u 
                   × 
                   1 
                 
               
             
             = 
             
               
                 [ 
                 
                   δ 
                    
                   
                       
                   
                    
                   
                     f 
                     1 
                   
                    
                   
                       
                   
                    
                   δ 
                    
                   
                       
                   
                    
                   
                     f 
                     2 
                   
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   δ 
                    
                   
                       
                   
                    
                   
                     f 
                     u 
                   
                 
                 ] 
               
               T 
             
           
         
       
       
         
           
             
               δ 
                
               
                   
               
                
               
                 L 
                 
                   u 
                   × 
                   1 
                 
               
             
             = 
             
               
                 [ 
                 
                   δ 
                    
                   
                       
                   
                    
                   
                     l 
                     1 
                   
                    
                   
                       
                   
                    
                   δ 
                    
                   
                       
                   
                    
                   
                     l 
                     2 
                   
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   δ 
                    
                   
                       
                   
                    
                   
                     l 
                     u 
                   
                 
                 ] 
               
               T 
             
           
         
       
       
         
           
             
               Δ 
               
                 u 
                 × 
                 1 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       Δ 
                       1 
                     
                      
                     
                         
                     
                      
                     
                       Δ 
                       2 
                     
                      
                     
                         
                     
                      
                     … 
                      
                     
                         
                     
                      
                     
                       Δ 
                       u 
                     
                   
                   ] 
                 
                 T 
               
                
               
                   
               
                
               where 
             
           
         
       
       
         
           
             
               Δ 
               i 
             
             = 
             
               
                 δ 
                  
                 
                     
                 
                  
                 
                   S 
                   i 
                 
               
               - 
               
                 
                   
                     
                       
                         c 
                         
                           α 
                            
                           
                               
                           
                            
                           i 
                         
                       
                       · 
                       
                         cos 
                         2 
                       
                     
                      
                     
                       α 
                       i 
                     
                   
                   
                     l 
                     i 
                   
                 
                 · 
                 
                   
                     ( 
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           Pi 
                         
                       
                       - 
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           
                             P 
                              
                             
                               ( 
                               
                                 i 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                   . 
                 
               
             
           
         
       
     
     The advantages of the above-mentioned technical solution of the present disclosure are as follows. 
     The above solution provides a general method for determining the temperature-induced sag variation of the main cable and the tower-top horizontal displacement of the two-tower ground-anchored suspension bridge or the two-tower self-anchored suspension bridge and other cable systems with any number of spans (multi-span suspension bridges, transmission lines, ropeways, etc.). The method facilitates the temperature deformation calculation as it merely relies on the overall geometry of a suspension bridge rather than the finite element model or the regression model based on long-term measured data. Also, the derived formulas have merits of clear concepts, general applicability, and great convenience for parametric analysis and field calculation. The present disclosure can be used to guide the deployment of measurement points in the structural health monitoring system of suspension bridges, and to provide a priori knowledge for establishing a temperature-deformation baseline model. 
     The general calculation method for the sag variation of the main cable and the tower-top horizontal displacement of suspension bridges under the ambient temperature variation, according to the present disclosure, belongs to the physics-based formula method. The method takes the variation in the span and sag of each span cable of a suspension bridge as the unknown quantities, and constructs a linear system of equations to solve them by using the following three conditions: 1) the equilibrium condition that the horizontal tensions of the main cables on both sides of the tower top are always equal, 2) the geometric relationship between the shape and the length of the main cable, and 3) the compatibility condition to be satisfied by the sum of all spans of the main cable. For a three-span suspension bridge, the present disclosure not only gives accurate formulas of the solution to the above-mentioned linear system of equations, but also gives approximate calculation formulas which are convenient for field applications. As the main cable and the tower contributions as well as the sag effect of the side span cables are taken into consideration, the present disclosure provides a good estimation of the temperature deformation of suspension bridges with high accuracy. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a simplified analysis model of a two-tower ground-anchored suspension bridge according to an embodiment of the present disclosure. 
         FIG. 2  is a schematic diagram showing the deformation of a two-tower ground-anchored suspension bridge according to an embodiment of the present disclosure. 
         FIG. 3  is a simplified analysis model of a multi-span cable system according to an embodiment of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     In order to illustrate the technical problems, the technical solutions, and the advantages of the present disclosure, the present disclosure is described below in detail with reference to the drawings and the specific embodiments. 
     The present disclosure provides a method for determining the variations in the main cable sag and tower-top horizontal displacement of suspension bridges with the ambient temperature changes. 
     The method includes the following steps. 
     (1) According to the equilibrium condition that the horizontal tensions of the main cable on both sides of the tower top are always equal, the following equation is established: 
     
       
         
           
             
               
                 
                   δ 
                    
                   
                     f 
                     i 
                   
                 
                 
                   f 
                   i 
                 
               
               - 
               
                 
                   δ 
                    
                   
                       
                   
                    
                   
                     l 
                     i 
                   
                 
                 
                   l 
                   i 
                 
               
             
             = 
             
               
                 
                   δ 
                    
                   
                     f 
                     
                       i 
                       + 
                       1 
                     
                   
                 
                 
                   f 
                   
                     i 
                     + 
                     1 
                   
                 
               
               - 
               
                 
                   δ 
                    
                   
                     l 
                     
                       i 
                       + 
                       1 
                     
                   
                 
                 
                   l 
                   
                     i 
                     + 
                     1 
                   
                 
               
             
           
         
       
     
     where, i=1, 2; f i  is the sag of the i th  span main cable; δf i  is the variation of f i  caused by temperature variations; l i  is the span of the i th  span main cable; δl i  is the variation of l i  caused by temperature variations; the subscripts 1, 2, 3 of the variables indicate the left side span, the main span, and the right side span, respectively. 
     (2) According to the geometric relationship between the shape of the main cable and the length of the main cable, the following equation is established: 
     
       
         
           
             
               
                 
                   
                     
                       c 
                       ni 
                     
                     
                       l 
                       i 
                     
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   f 
                   i 
                 
               
               - 
               
                 
                   
                     
                       
                         c 
                         ni 
                       
                       · 
                       
                         n 
                         i 
                       
                     
                     
                       l 
                       i 
                     
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   l 
                   i 
                 
               
               + 
               
                 
                   
                     c 
                     li 
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   l 
                   i 
                 
               
               - 
               
                 
                   
                     
                       
                         
                           c 
                           
                             α 
                              
                             
                                 
                             
                              
                             i 
                           
                         
                         · 
                         sin 
                       
                        
                       
                           
                       
                        
                       2 
                        
                       
                           
                       
                        
                       
                         α 
                         i 
                       
                     
                     
                       2 
                       · 
                       
                         l 
                         i 
                       
                     
                   
                   · 
                   δ 
                 
                  
                 
                     
                 
                  
                 
                   l 
                   i 
                 
               
             
             = 
             
               
                 δ 
                  
                 
                     
                 
                  
                 
                   S 
                   i 
                 
               
               - 
               
                 
                   
                     
                       
                         c 
                         
                           α 
                            
                           
                               
                           
                            
                           i 
                         
                       
                       · 
                       
                         cos 
                         2 
                       
                     
                      
                     
                         
                     
                      
                     
                       α 
                       i 
                     
                   
                   
                     2 
                     · 
                     
                       l 
                       i 
                     
                   
                 
                 · 
                 
                   ( 
                   
                     
                       δ 
                        
                       
                           
                       
                        
                       
                         h 
                         
                           P 
                            
                           
                               
                           
                            
                           i 
                         
                       
                     
                     - 
                     
                       δ 
                        
                       
                           
                       
                        
                       
                         h 
                         
                           P 
                            
                           
                             ( 
                             
                               i 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     where: i=1, 2, 3; n i  is the sag-to-span ratio of the i th  span main cable, i.e. n i =f i /l i ; α i  is the chord inclination of the i th  span main cable; the coefficients c ni , c li , and c αi  are respectively: 
     
       
         
           
             
               c 
               ni 
             
             = 
             
               
                 l 
                 i 
               
               · 
               
                 [ 
                 
                   
                     
                       16 
                       3 
                     
                      
                     
                       n 
                       i 
                     
                      
                     
                       cos 
                       3 
                     
                      
                     
                       α 
                       i 
                     
                   
                   - 
                   
                     
                       128 
                       5 
                     
                      
                     
                       
                         n 
                         i 
                         3 
                       
                        
                       
                         ( 
                         
                           
                             5 
                              
                             
                                 
                             
                              
                             
                               cos 
                               7 
                             
                              
                             
                               α 
                               i 
                             
                           
                           - 
                           
                             4 
                              
                             
                                 
                             
                              
                             
                               cos 
                               5 
                             
                              
                             
                               α 
                               i 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               c 
               li 
             
             = 
             
               
                 sec 
                  
                 
                     
                 
                  
                 
                   α 
                   i 
                 
               
               + 
               
                 
                   8 
                   3 
                 
                  
                 
                   n 
                   i 
                   2 
                 
                  
                 
                   cos 
                   3 
                 
                  
                 
                   α 
                   i 
                 
               
               - 
               
                 
                   32 
                   5 
                 
                  
                 
                   
                     n 
                     i 
                     4 
                   
                    
                   
                     ( 
                     
                       
                         5 
                          
                         
                             
                         
                          
                         
                           cos 
                           7 
                         
                          
                         
                           α 
                           i 
                         
                       
                       - 
                       
                         4 
                          
                         
                             
                         
                          
                         
                           cos 
                           5 
                         
                          
                         
                           α 
                           i 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               c 
               
                 α 
                  
                 
                     
                 
                  
                 i 
               
             
             = 
             
               
                 l 
                 i 
               
               · 
               
                 [ 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         α 
                         i 
                       
                     
                     
                       
                         cos 
                         2 
                       
                        
                       
                         α 
                         i 
                       
                     
                   
                   - 
                   
                     8 
                      
                     
                       n 
                       i 
                       2 
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     
                       α 
                       i 
                     
                      
                     
                       cos 
                       2 
                     
                      
                     
                       α 
                       i 
                     
                   
                   + 
                   
                     32 
                      
                     
                       n 
                       i 
                       4 
                     
                      
                     
                       cos 
                       4 
                     
                      
                     
                       α 
                       i 
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     
                       
                         α 
                         i 
                       
                        
                       
                         ( 
                         
                           
                             7 
                              
                             
                                 
                             
                              
                             
                               cos 
                               2 
                             
                              
                             
                               α 
                               i 
                             
                           
                           - 
                           4 
                         
                         ) 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     δS i  is the length variation of the i th  span main cable caused by temperature variations. 
     δh Pi  and δh P(i−1)  are the elevation change of the supports i and i−1 of the main cable, respectively, and since the position of the anchorages are unchanged, δh P0 =δh P3 =0. 
     δS i  (i=1, 2, 3) and δh Pi  (i=1, 2) are estimated by the following equations: 
               δ                   S   i       =           S   i     ·     θ   C     ·   δ                     T   C       =         l   i     ·     θ   C     ·   δ                       T   C          [       sec                   α   i                  +       8   3          n   i   2          cos   3          α   i       -         3      2     5            n   i   4          (       5        cos   7          α   i       -     4        cos   5          α   i         )           ]                 δ h   Pi   =h   Pi ·θ P   ·δT   P  
 
     where θ C  and θ P  are respectively the linear expansion coefficients of the main cable and the tower, δT C  and δT P  are respectively the temperature variation of the main cable and the tower, and h Pi  is the height of the tower. 
     (3) According to the compatibility condition to be satisfied by the sum of spans of the left side span cable, the main span cable, and the right side span cable, that is, the distance between the anchorages at both ends is constant, the following equation is established: 
     
       
         
           
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   3 
                 
                  
                 
                   δ 
                    
                   
                     l 
                     i 
                   
                 
               
               = 
               0 
             
             ; 
           
         
       
     
     (4) According to the following linear system of equations consisting of the equations in steps (1), (2), and (3), the sag variation δf i  and the span variation δl i  of each span of the main cable can be simultaneously obtained: 
     
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         - 
                         
                           1 
                           
                             f 
                             1 
                           
                         
                       
                     
                     
                       
                         1 
                         
                           f 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         
                           l 
                           1 
                         
                       
                     
                     
                       
                         - 
                         
                           1 
                           
                             l 
                             2 
                           
                         
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         - 
                         
                           1 
                           
                             f 
                             2 
                           
                         
                       
                     
                     
                       
                         1 
                         
                           f 
                           3 
                         
                       
                     
                     
                       0 
                     
                     
                       
                         1 
                         
                           l 
                           2 
                         
                       
                     
                     
                       
                         1 
                         
                           l 
                           3 
                         
                       
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       1 
                     
                     
                       1 
                     
                   
                   
                     
                       
                         
                           c 
                           
                             n 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                         
                           l 
                           1 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         M 
                         1 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         
                           c 
                           
                             n 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                         
                           l 
                           2 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         M 
                         2 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         
                           c 
                           
                             n 
                              
                             
                                 
                             
                              
                             3 
                           
                         
                         
                           l 
                           3 
                         
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                         M 
                         3 
                       
                     
                   
                 
                 ] 
               
               · 
               
                 [ 
                 
                   
                     
                       
                         δ 
                          
                         
                           f 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                           f 
                           2 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                           f 
                           3 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                           l 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                           l 
                           2 
                         
                       
                     
                   
                   
                     
                       
                         δ 
                          
                         
                           l 
                           3 
                         
                       
                     
                   
                 
                 ] 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                   
                   
                     
                       
                         Δ 
                         1 
                       
                     
                   
                   
                     
                       
                         Δ 
                         2 
                       
                     
                   
                   
                     
                       
                         Δ 
                         3 
                       
                     
                   
                 
                 ] 
               
                
               
                   
               
                
               where 
                
               
                 : 
               
             
           
         
       
       
         
           
             
               
                 M 
                 i 
               
               = 
               
                 
                   - 
                   
                     
                       
                         c 
                         ni 
                       
                       · 
                       
                         n 
                         i 
                       
                     
                     
                       l 
                       i 
                     
                   
                 
                 + 
                 
                   c 
                   li 
                 
                 - 
                 
                   
                     
                       
                         c 
                         
                           α 
                            
                           
                               
                           
                            
                           i 
                         
                       
                       · 
                       sin 
                     
                      
                     
                         
                     
                      
                     2 
                      
                     
                         
                     
                      
                     
                       α 
                       i 
                     
                   
                   
                     2 
                     · 
                     
                       l 
                       i 
                     
                   
                 
               
             
             , 
             
                 
             
              
             
               
                 Δ 
                 i 
               
               = 
               
                 
                   δ 
                    
                   
                       
                   
                    
                   
                     S 
                     i 
                   
                 
                 - 
                 
                   
                     
                       
                         
                           c 
                           
                             α 
                              
                             
                                 
                             
                              
                             i 
                           
                         
                         · 
                         
                           cos 
                           2 
                         
                       
                        
                       
                         α 
                         i 
                       
                     
                     
                       l 
                       i 
                     
                   
                   · 
                   
                     ( 
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           Pi 
                         
                       
                       - 
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           h 
                           
                             P 
                              
                             
                               ( 
                               
                                 i 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
             
             , 
             
               i 
               = 
               1 
             
             , 
             2 
             , 
             3. 
           
         
       
     
     The above-mentioned calculation method is further described below with reference to the specific embodiments. 
     The equation based on the horizontal tension equilibrium on the tower top in step (1) is specifically derived as follows. 
     In the analysis model of a two-tower suspension bridge shown in  FIG. 1 , l i , f i , α i  and h i  represent the span, the sag, the chord inclination (positive for the counterclockwise rotation from the horizontal line), and the elevation difference of the supports of each span of the main cable, respectively. The subscripts 1, 2, 3 indicate the left side span, the main span, and the right side span, respectively. The heights (the length subjected to the temperature changes) of the left and right towers are respectively defined as h P1  and h P2 . The horizontal distance between the anchorages at both ends of the main cable is L. The variations in the span and the sag of each span cable are respectively δl i  and δf i  (i=1, 2, 3) as shown in  FIG. 2 . 
     When the i th  span main cable is subjected to the vertical load q i  uniformly distributed along the span l i , the cable curve is a parabola, and the horizontal components H i  of the tension in the cable is a constant. The total vertical load acting on the i th  span main cable is W i =q i ·l i , and the cable sag is 
     
       
         
           
             
               
                 
                   
                     f 
                     i 
                   
                   = 
                   
                     
                       
                         W 
                         i 
                       
                       · 
                       
                         l 
                         i 
                       
                     
                     
                       8 
                        
                       
                         H 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     As δW i =0, the differentiation of the above equation leads to: 
     
       
         
           
             
               
                 
                   
                     
                       δ 
                        
                       
                         f 
                         i 
                       
                     
                     
                       f 
                       i 
                     
                   
                   = 
                   
                     
                       
                         δ 
                          
                         
                           l 
                           i 
                         
                       
                       
                         l 
                         i 
                       
                     
                     - 
                     
                       
                         δ 
                          
                         
                           H 
                           i 
                         
                       
                       
                         H 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     The horizontal tension equilibrium on the tower top is assumed to be maintained, so 
     
       
         
           
             
               
                 
                   
                     
                       δ 
                        
                       
                         H 
                         1 
                       
                     
                     
                       H 
                       1 
                     
                   
                   = 
                   
                     
                       
                         δ 
                          
                         
                           H 
                           2 
                         
                       
                       
                         H 
                         2 
                       
                     
                     = 
                     
                       
                         δ 
                          
                         
                           H 
                           3 
                         
                       
                       
                         H 
                         3 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     By combining the equations (2) and (3), the following two equations can be obtained (i=1, 2): 
     
       
         
           
             
               
                 
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           f 
                           i 
                         
                       
                       
                         f 
                         i 
                       
                     
                     - 
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           l 
                           i 
                         
                       
                       
                         l 
                         i 
                       
                     
                   
                   = 
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           l 
                           
                             i 
                             + 
                             1 
                           
                         
                       
                       
                         f 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                     - 
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           l 
                           
                             i 
                             + 
                             1 
                           
                         
                       
                       
                         l 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     The equation based on the geometric relationship between the shape and the length of the main cable in step (2) is specifically derived as follows. 
     The formula for calculating the length of each span main cable is: 
     
       
         
           
             
               
                 
                   
                     S 
                     i 
                   
                   = 
                   
                     
                       l 
                       i 
                     
                      
                     
                       [ 
                       
                         
                           sec 
                            
                           
                             α 
                             i 
                           
                         
                         + 
                         
                           
                             8 
                             3 
                           
                            
                           
                             n 
                             i 
                             2 
                           
                            
                           
                             cos 
                             3 
                           
                            
                           
                             α 
                             i 
                           
                         
                         - 
                         
                           
                             
                               3 
                                
                               2 
                             
                             5 
                           
                            
                           
                             
                               n 
                               i 
                               4 
                             
                              
                             
                               ( 
                               
                                 
                                   5 
                                    
                                   
                                     cos 
                                     7 
                                   
                                    
                                   
                                     α 
                                     i 
                                   
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     cos 
                                     5 
                                   
                                    
                                   
                                     α 
                                     i 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     where n i  is the sag-to-span ratio, that is: 
     
       
         
           
             
               
                 
                   
                     n 
                     i 
                   
                   = 
                   
                     
                       f 
                       i 
                     
                     
                       l 
                       i 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Differentiating the equation (5) yields: 
       δ S   i   =c   ni   ·δn   i   +c   li ·δ li   +c   αi ·δα i   (7)
 
     where the coefficients c ni , c li , and c αi  are respectively: 
     
       
         
           
             
               
                 
                   
                     c 
                     ni 
                   
                   = 
                   
                     
                       l 
                       i 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               1 
                                
                               6 
                             
                             3 
                           
                            
                           
                             n 
                             i 
                           
                            
                           
                             cos 
                             3 
                           
                            
                           
                             α 
                             i 
                           
                         
                         - 
                         
                           
                             
                               1 
                                
                               2 
                                
                               8 
                             
                             5 
                           
                            
                           
                             
                               n 
                               i 
                               3 
                             
                              
                             
                               ( 
                               
                                 
                                   5 
                                    
                                   
                                     cos 
                                     7 
                                   
                                    
                                   
                                     α 
                                     i 
                                   
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     cos 
                                     5 
                                   
                                    
                                   
                                     α 
                                     i 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                     c 
                     li 
                   
                   = 
                   
                     
                       sec 
                        
                       
                         α 
                         i 
                       
                     
                     + 
                     
                       
                         8 
                         3 
                       
                        
                       
                         n 
                         i 
                         2 
                       
                        
                       
                         cos 
                         3 
                       
                        
                       
                         α 
                         i 
                       
                     
                     - 
                     
                       
                         
                           3 
                            
                           2 
                         
                         5 
                       
                        
                       
                         
                           n 
                           i 
                           4 
                         
                          
                         
                           ( 
                           
                             
                               5 
                                
                               
                                 cos 
                                 7 
                               
                                
                               
                                 α 
                                 i 
                               
                             
                             - 
                             
                               4 
                                
                               
                                 cos 
                                 5 
                               
                                
                               
                                 α 
                                 i 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     c 
                     
                       α 
                        
                       
                           
                       
                        
                       i 
                     
                   
                   = 
                   
                     
                       l 
                       i 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             sin 
                              
                             
                               α 
                               i 
                             
                           
                           
                             
                               cos 
                               2 
                             
                              
                             
                               α 
                               i 
                             
                           
                         
                         - 
                         
                           8 
                            
                           
                             n 
                             i 
                             2 
                           
                            
                           
                             α 
                             i 
                           
                            
                           
                             cos 
                             2 
                           
                            
                           
                             α 
                             i 
                           
                         
                         + 
                         
                           3 
                            
                           2 
                            
                           
                             n 
                             i 
                             4 
                           
                            
                           
                             cos 
                             4 
                           
                            
                           
                             α 
                             i 
                           
                            
                           sin 
                            
                           
                             
                               α 
                               i 
                             
                              
                             
                               ( 
                               
                                 
                                   7 
                                    
                                   
                                     cos 
                                     2 
                                   
                                    
                                   
                                     α 
                                     i 
                                   
                                 
                                 - 
                                 4 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     δn i  and δ αi  in the equation (7) can be replaced by the expressions containing δl i  and δf i . Differentiating the equation (6) with respect to f i  and  1   i  leads to: 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                       n 
                       i 
                     
                   
                   = 
                   
                     
                       δ 
                        
                       
                         ( 
                         
                           
                             f 
                             i 
                           
                           
                             l 
                             i 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               l 
                               i 
                             
                             · 
                             δ 
                           
                            
                           
                               
                           
                            
                           
                             f 
                             i 
                           
                         
                         - 
                         
                           
                             
                               f 
                               i 
                             
                             · 
                             δ 
                           
                            
                           
                               
                           
                            
                           
                             l 
                             i 
                           
                         
                       
                       
                         l 
                         i 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     The elevation difference between the two end supports of each span cable is: 
         h   i   =l   i ·tan α i   (12)
 
     By differentiating the equation (12) with respect to l i  and α i , the following equation is obtained: 
       δ h   i   =δl   i ·tan α i   +l   i ·sec 2 α i ·δα i   (13)
 
     δh i  is equal to the difference of the elevation changes at two end supports of each span cable: 
       δ h   i   =δh   Pi   −δh   P(i−1)   (14)
 
     where i=1, 2, 3, δh P0  and δh P3  correspond to the elevation variation of the left and right anchorages, respectively, so δh P0 =δh P3 =0. Substituting the equation (14) into the equation (13), δα i  is obtained: 
     
       
         
           
             
               
                 
                   
                     δα 
                     i 
                   
                   = 
                   
                     
                       
                         
                           - 
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 α 
                                 i 
                               
                             
                             
                               2 
                               · 
                               
                                 l 
                                 i 
                               
                             
                           
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         l 
                         i 
                       
                     
                     + 
                     
                       
                         
                           
                             cos 
                             2 
                           
                            
                           
                             α 
                             i 
                           
                         
                         
                           l 
                           i 
                         
                       
                        
                       
                         ( 
                         
                           
                             δ 
                              
                             
                               h 
                               Pi 
                             
                           
                           - 
                           
                             δ 
                              
                             
                               h 
                               
                                 P 
                                  
                                 
                                   ( 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     By substituting the equations (11) and (15) into the equation (7), three equations can be obtained as follows (i=1, 2, 3): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             ni 
                           
                           
                             l 
                             i 
                           
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         f 
                         i 
                       
                     
                     - 
                     
                       
                         
                           
                             
                               c 
                               ni 
                             
                             · 
                             
                               n 
                               i 
                             
                           
                           
                             l 
                             i 
                           
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         l 
                         i 
                       
                     
                     + 
                     
                       
                         
                           c 
                           li 
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         l 
                         i 
                       
                     
                     - 
                     
                       
                         
                           
                             
                               
                                 c 
                                 
                                   α 
                                    
                                   
                                       
                                   
                                    
                                   i 
                                 
                               
                               · 
                               sin 
                             
                              
                             
                                 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             
                               α 
                               i 
                             
                           
                           
                             2 
                             · 
                             
                               l 
                               i 
                             
                           
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         l 
                         i 
                       
                     
                   
                   = 
                   
                     
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         i 
                       
                     
                     - 
                     
                       
                         
                           
                             
                               c 
                               
                                 α 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                             
                             · 
                             
                               cos 
                               2 
                             
                           
                            
                           
                             α 
                             i 
                           
                         
                         
                           l 
                           i 
                         
                       
                       · 
                       
                         ( 
                         
                           
                             δ 
                              
                             
                                 
                             
                              
                             
                               h 
                               Pi 
                             
                           
                           - 
                           
                             δ 
                              
                             
                                 
                             
                              
                             
                               h 
                               
                                 P 
                                  
                                 
                                   ( 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     where δS i  (i=1, 2, 3) is the length variation of the i th  span main cable caused by temperature variations, and δh Pi  (1=1, 2) is the height variation of the towers. δS i  and δh Pi  can be estimated by the linear expansion coefficient as follows: 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                       S 
                       i 
                     
                   
                   = 
                   
                     
                       
                         
                           S 
                           i 
                         
                         · 
                         
                           θ 
                           C 
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         T 
                         C 
                       
                     
                     = 
                     
                       
                         
                           l 
                           i 
                         
                         · 
                         
                           θ 
                           C 
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         
                           T 
                           C 
                         
                          
                         
                           [ 
                           
                             
                               sec 
                                
                               
                                 α 
                                 i 
                               
                             
                             + 
                             
                               
                                 8 
                                 3 
                               
                                
                               
                                 n 
                                 i 
                                 2 
                               
                                
                               
                                 cos 
                                 3 
                               
                                
                               
                                 α 
                                 i 
                               
                             
                             - 
                             
                               
                                 
                                   3 
                                    
                                   2 
                                 
                                 5 
                               
                                
                               
                                 
                                   n 
                                   i 
                                   4 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       5 
                                        
                                       
                                         cos 
                                         7 
                                       
                                        
                                       
                                         α 
                                         i 
                                       
                                     
                                     - 
                                     
                                       4 
                                        
                                       
                                         cos 
                                         5 
                                       
                                        
                                       
                                         α 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     δ 
                      
                     
                       h 
                       Pi 
                     
                   
                   = 
                   
                     
                       
                         h 
                         Pi 
                       
                       · 
                       
                         θ 
                         P 
                       
                       · 
                       δ 
                     
                      
                     
                         
                     
                      
                     
                       T 
                       P 
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     where θ C  and θ P  are the linear expansion coefficients of the main cable and the tower, respectively; δT C  and δT P  are the temperature variations of the main cable and the tower, respectively; and h Pi  is the height of the tower. 
     In step (3), according to the compatibility condition to be satisfied by the sum of spans of the left side span, main span, and right side span main cables, that is, the distance between the anchorages at both ends is constant, the following equation is established: 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       3 
                     
                      
                     
                       δ 
                        
                       
                         l 
                         i 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     In step (4), the above equations (4), (16), and (19) constitute a linear system of equations with six unknowns δf 1 , δf 2 , δf 3 , δl 1 , δl 2  and δl 3 . 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               - 
                               
                                 1 
                                 
                                   f 
                                   1 
                                 
                               
                             
                           
                           
                             
                               1 
                               
                                 f 
                                 2 
                               
                             
                           
                           
                             0 
                           
                           
                             
                               1 
                               
                                 l 
                                 1 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   l 
                                   2 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   f 
                                   2 
                                 
                               
                             
                           
                           
                             
                               1 
                               
                                 f 
                                 3 
                               
                             
                           
                           
                             0 
                           
                           
                             
                               1 
                               
                                 l 
                                 2 
                               
                             
                           
                           
                             
                               1 
                               
                                 l 
                                 3 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             1 
                           
                           
                             1 
                           
                         
                         
                           
                             
                               
                                 c 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                               
                                 l 
                                 1 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               M 
                               1 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               
                                 c 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                               
                                 l 
                                 2 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               M 
                               2 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               
                                 c 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                 
                               
                               
                                 l 
                                 3 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               M 
                               3 
                             
                           
                         
                       
                       ] 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               δ 
                                
                               
                                 f 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               δ 
                                
                               
                                 f 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               δ 
                                
                               
                                 f 
                                 3 
                               
                             
                           
                         
                         
                           
                             
                               δ 
                                
                               
                                 l 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               δ 
                                
                               
                                 l 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               δ 
                                
                               
                                 l 
                                 3 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                         
                         
                           
                             
                               Δ 
                               1 
                             
                           
                         
                         
                           
                             
                               Δ 
                               2 
                             
                           
                         
                         
                           
                             
                               Δ 
                               3 
                             
                           
                         
                       
                       ] 
                     
                      
                     
                         
                     
                      
                     where 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                     i 
                   
                   = 
                   
                     
                       - 
                       
                         
                           
                             c 
                             ni 
                           
                           · 
                           
                             n 
                             i 
                           
                         
                         
                           l 
                           i 
                         
                       
                     
                     + 
                     
                       c 
                       li 
                     
                     - 
                     
                       
                         
                           
                             c 
                             
                               α 
                                
                               
                                   
                               
                                
                               i 
                             
                           
                           · 
                           sin 
                         
                          
                         
                             
                         
                          
                         2 
                          
                         
                             
                         
                          
                         
                           α 
                           i 
                         
                       
                       
                         2 
                         · 
                         
                           l 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     i 
                   
                   = 
                   
                     
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         i 
                       
                     
                     - 
                     
                       
                         
                           
                             
                               c 
                               
                                 α 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                             
                             · 
                             
                               cos 
                               2 
                             
                           
                            
                           
                             α 
                             i 
                           
                         
                         
                           l 
                           i 
                         
                       
                       · 
                       
                         ( 
                         
                           
                             δ 
                              
                             
                                 
                             
                              
                             
                               h 
                               Pi 
                             
                           
                           - 
                           
                             δ 
                              
                             
                                 
                             
                              
                             
                               h 
                               
                                 P 
                                  
                                 
                                   ( 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     By solving the equation (20), the sag variation δf i  (i=1, 2, 3) and the span variation δl i  (i=1, 2, 3) of each span main cable can be obtained. 
     The tower-top horizontal displacement of the left and right towers are δl 1  and δl 3 , respectively (positive for the movement toward the main span), and the horizontal distance variation between both tower tops is δl 2 . 
     The sag-to-span ratio of the main span cable of a suspension bridge is generally between 1/12 and 1/9, while the sag-to-span ratios of the side span cables are even smaller than that of the main span cable. Therefore, the higher-order terms of n i  in the equations (8) to (10) can be ignored, that is, 
     
       
         
           
             
               
                 
                   
                     c 
                     ni 
                   
                   = 
                   
                     
                       16 
                       3 
                     
                      
                     
                       
                         l 
                         i 
                       
                       · 
                       
                         n 
                         i 
                       
                     
                      
                     
                       cos 
                       3 
                     
                      
                     
                       α 
                       i 
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                     c 
                     li 
                   
                   = 
                   
                     sec 
                      
                     
                         
                     
                      
                     
                       α 
                       i 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   
                     c 
                     
                       α 
                        
                       
                           
                       
                        
                       i 
                     
                   
                   = 
                   
                     
                       l 
                       i 
                     
                     · 
                     
                       
                         sin 
                          
                         
                             
                         
                          
                         
                           α 
                           i 
                         
                       
                       
                         
                           cos 
                           2 
                         
                          
                         
                           α 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     By substituting the equations (23) to (25) into the equation (20) and replacing the sag f i  with n i ·l i , the solution of the equation (20) can be written as follows: 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       f 
                       1 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 n 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   
                                     3 
                                      
                                     
                                       l 
                                       1 
                                     
                                   
                                   - 
                                   
                                     16 
                                      
                                     
                                       r 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               16 
                                
                               
                                   
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 
                                   α 
                                   1 
                                 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     3 
                                   
                                    
                                   
                                     r 
                                     i 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               n 
                               1 
                             
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 1 
                               
                             
                           
                         
                         ] 
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         1 
                       
                     
                     + 
                     
                       
                         
                           
                             n 
                             1 
                           
                            
                           
                             ( 
                             
                               
                                 3 
                                  
                                 
                                   l 
                                   1 
                                 
                               
                               - 
                               
                                 16 
                                  
                                 
                                   r 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                         
                           16 
                            
                           
                               
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               2 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         2 
                       
                     
                     + 
                     
                       
                         
                           
                             n 
                             1 
                           
                            
                           
                             ( 
                             
                               
                                 3 
                                  
                                 
                                   l 
                                   1 
                                 
                               
                               - 
                               
                                 16 
                                  
                                 
                                   r 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                         
                           16 
                            
                           
                               
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               3 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         3 
                       
                     
                     + 
                     
                         
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       n 
                                       1 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           3 
                                            
                                           
                                             l 
                                             1 
                                           
                                         
                                         - 
                                         
                                           16 
                                            
                                           
                                             r 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           2 
                                         
                                       
                                       - 
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   16 
                                   · 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       3 
                                     
                                      
                                     
                                       r 
                                       i 
                                     
                                   
                                 
                               
                               - 
                               
                                 
                                   
                                     n 
                                     1 
                                   
                                   · 
                                   tan 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   1 
                                 
                               
                             
                             ] 
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         - 
                         
                           
                             
                               
                                 
                                   n 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       3 
                                        
                                       
                                         l 
                                         1 
                                       
                                     
                                     - 
                                     
                                       16 
                                        
                                       
                                         r 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       2 
                                     
                                   
                                   - 
                                   
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               16 
                               · 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   3 
                                 
                                  
                                 
                                   r 
                                   i 
                                 
                               
                             
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       f 
                       2 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             n 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 3 
                                  
                                 
                                   l 
                                   2 
                                 
                               
                               - 
                               
                                 16 
                                  
                                 
                                   r 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         
                           16 
                            
                           
                               
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               1 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         1 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               n 
                               2 
                             
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               
                                 n 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     3 
                                      
                                     
                                       l 
                                       2 
                                     
                                   
                                   - 
                                   
                                     16 
                                      
                                     
                                       r 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               16 
                                
                               
                                   
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 
                                   α 
                                   2 
                                 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     3 
                                   
                                    
                                   
                                     r 
                                     i 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         2 
                       
                     
                     + 
                     
                       
                         
                           
                             n 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 3 
                                  
                                 
                                   l 
                                   2 
                                 
                               
                               - 
                               
                                 16 
                                  
                                 
                                   r 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         
                           16 
                            
                           
                               
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               3 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         3 
                       
                     
                     + 
                     
                         
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       n 
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           3 
                                            
                                           
                                             l 
                                             2 
                                           
                                         
                                         - 
                                         
                                           16 
                                            
                                           
                                             r 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           2 
                                         
                                       
                                       - 
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   16 
                                   · 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       3 
                                     
                                      
                                     
                                       r 
                                       i 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     n 
                                     2 
                                   
                                   · 
                                   tan 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   2 
                                 
                               
                             
                             ] 
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         - 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       n 
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           3 
                                            
                                           
                                             l 
                                             2 
                                           
                                         
                                         - 
                                         
                                           16 
                                            
                                           
                                             r 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           2 
                                         
                                       
                                       - 
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           3 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   16 
                                   · 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       3 
                                     
                                      
                                     
                                       r 
                                       i 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     n 
                                     2 
                                   
                                   · 
                                   tan 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   2 
                                 
                               
                             
                             ] 
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       f 
                       3 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             n 
                             3 
                           
                            
                           
                             ( 
                             
                               
                                 3 
                                  
                                 
                                   l 
                                   3 
                                 
                               
                               - 
                               
                                 16 
                                  
                                 
                                   r 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         
                           16 
                            
                           
                               
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               1 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         1 
                       
                     
                     + 
                     
                       
                         
                           
                             n 
                             3 
                           
                            
                           
                             ( 
                             
                               
                                 3 
                                  
                                 
                                   l 
                                   3 
                                 
                               
                               - 
                               
                                 16 
                                  
                                 
                                   r 
                                   3 
                                 
                               
                             
                             ) 
                           
                         
                         
                           16 
                            
                           
                               
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               2 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         2 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               
                                 n 
                                 3 
                               
                                
                               
                                 ( 
                                 
                                   
                                     3 
                                      
                                     
                                       l 
                                       3 
                                     
                                   
                                   - 
                                   
                                     16 
                                      
                                     
                                       r 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               16 
                                
                               
                                   
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 
                                   α 
                                   3 
                                 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     3 
                                   
                                    
                                   
                                     r 
                                     i 
                                   
                                 
                               
                             
                           
                            
                           
                             
                               n 
                               3 
                             
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 3 
                               
                             
                           
                         
                         ] 
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         3 
                       
                     
                     + 
                     
                         
                       
                         
                           
                             
                               
                                 
                                   n 
                                   3 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       3 
                                        
                                       
                                         l 
                                         3 
                                       
                                     
                                     - 
                                     
                                       16 
                                        
                                       
                                         r 
                                         3 
                                       
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       2 
                                     
                                   
                                   - 
                                   
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               16 
                               · 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   3 
                                 
                                  
                                 
                                   r 
                                   i 
                                 
                               
                             
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         - 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       n 
                                       3 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           3 
                                            
                                           
                                             l 
                                             3 
                                           
                                         
                                         - 
                                         
                                           16 
                                            
                                           
                                             r 
                                             3 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           2 
                                         
                                       
                                       - 
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           3 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   16 
                                   · 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       3 
                                     
                                      
                                     
                                       r 
                                       i 
                                     
                                   
                                 
                               
                               - 
                               
                                 
                                   
                                     n 
                                     3 
                                   
                                   · 
                                   tan 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   3 
                                 
                               
                             
                             ] 
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       l 
                       1 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             1 
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 1 
                               
                             
                           
                           - 
                           
                             
                               r 
                               1 
                             
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 
                                   α 
                                   1 
                                 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     3 
                                   
                                    
                                   
                                     r 
                                     i 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         1 
                       
                     
                     - 
                     
                       
                         
                           r 
                           1 
                         
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               2 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         2 
                       
                     
                     - 
                     
                       
                         
                           r 
                           1 
                         
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               3 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         3 
                       
                     
                     + 
                     
                         
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     r 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           2 
                                         
                                       
                                       - 
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     3 
                                   
                                    
                                   
                                     r 
                                     i 
                                   
                                 
                               
                               - 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   1 
                                 
                               
                             
                             ] 
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         - 
                         
                           
                             
                               
                                 r 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       3 
                                     
                                   
                                   - 
                                   
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       l 
                       2 
                     
                   
                   = 
                   
                     
                       
                         
                           r 
                           2 
                         
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               1 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         1 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             1 
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 2 
                               
                             
                           
                           - 
                           
                             
                               r 
                               2 
                             
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 
                                   α 
                                   2 
                                 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     3 
                                   
                                    
                                   
                                     r 
                                     i 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         2 
                       
                     
                     - 
                     
                       
                         
                           r 
                           2 
                         
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               3 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         3 
                       
                     
                     + 
                     
                         
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     r 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           1 
                                         
                                       
                                       - 
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           3 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     3 
                                   
                                    
                                   
                                     r 
                                     i 
                                   
                                 
                               
                               + 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   2 
                                 
                               
                             
                             ] 
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         - 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     r 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           3 
                                         
                                       
                                       - 
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     3 
                                   
                                    
                                   
                                     r 
                                     i 
                                   
                                 
                               
                               + 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   2 
                                 
                               
                             
                             ] 
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       l 
                       3 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           
                             r 
                             1 
                           
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               
                                 α 
                                 1 
                               
                               · 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   3 
                                 
                                  
                                 
                                   r 
                                   i 
                                 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         1 
                       
                     
                     - 
                     
                       
                         
                           r 
                           3 
                         
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               2 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         2 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             1 
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 α 
                                 3 
                               
                             
                           
                           - 
                           
                             
                               r 
                               3 
                             
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 
                                   α 
                                   3 
                                 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     3 
                                   
                                    
                                   
                                     r 
                                     i 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         3 
                       
                     
                     + 
                     
                         
                       
                         
                           
                             
                               
                                 r 
                                 3 
                               
                                
                               
                                 ( 
                                 
                                   
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       1 
                                     
                                   
                                   - 
                                   
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         - 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     r 
                                     3 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           3 
                                         
                                       
                                       - 
                                       
                                         tan 
                                          
                                         
                                             
                                         
                                          
                                         
                                           α 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     3 
                                   
                                    
                                   
                                     r 
                                     i 
                                   
                                 
                               
                               - 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   3 
                                 
                               
                             
                             ] 
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
     where the parameter r i  (i=1, 2, 3) is as follows: 
         r   i   =l   i   ·n   i   2 ·cos 2 α i   (32)
 
     The equations (26) to (31) can be written in a compact form as follows (i=1, 2, 3): 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       f 
                       i 
                     
                   
                   = 
                   
                     
                       
                         
                           n 
                           i 
                         
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             α 
                             i 
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         i 
                       
                     
                     - 
                     
                       
                         n 
                         i 
                       
                        
                       tan 
                        
                       
                           
                       
                        
                       
                         
                           α 
                           i 
                         
                          
                         
                           ( 
                           
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 h 
                                 Pi 
                               
                             
                             - 
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 h 
                                 
                                   P 
                                    
                                   
                                     ( 
                                     
                                       i 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             n 
                             i 
                           
                            
                           
                             ( 
                             
                               
                                 3 
                                  
                                 
                                   l 
                                   i 
                                 
                               
                               - 
                               
                                 16 
                                  
                                 
                                   r 
                                   i 
                                 
                               
                             
                             ) 
                           
                         
                         
                           16 
                           · 
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 1 
                               
                               3 
                             
                              
                             
                               r 
                               j 
                             
                           
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               3 
                             
                              
                             
                               
                                 δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   S 
                                   k 
                                 
                               
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   k 
                                 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               2 
                             
                              
                             
                               
                                 
                                   ( 
                                   
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       
                                         k 
                                         + 
                                         1 
                                       
                                     
                                      
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       k 
                                     
                                   
                                   ) 
                                 
                                 · 
                                 δ 
                               
                                
                               
                                   
                               
                                
                               
                                 h 
                                 Pk 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       l 
                       i 
                     
                   
                   = 
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           S 
                           i 
                         
                       
                       
                         cos 
                          
                         
                             
                         
                          
                         
                           α 
                           i 
                         
                       
                     
                     - 
                     
                       tan 
                        
                       
                           
                       
                        
                       
                         
                           α 
                           i 
                         
                          
                         
                           ( 
                           
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 h 
                                 Pi 
                               
                             
                             - 
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 h 
                                 
                                   P 
                                    
                                   
                                     ( 
                                     
                                       i 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         
                           r 
                           i 
                         
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             3 
                           
                            
                           
                             r 
                             j 
                           
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               3 
                             
                              
                             
                               
                                 δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   S 
                                   k 
                                 
                               
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   k 
                                 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               2 
                             
                              
                             
                               
                                 
                                   ( 
                                   
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       
                                         k 
                                         + 
                                         1 
                                       
                                     
                                      
                                     tan 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       k 
                                     
                                   
                                   ) 
                                 
                                 · 
                                 δ 
                               
                                
                               
                                   
                               
                                
                               
                                 h 
                                 Pk 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     The sag variation δf 2  of the main span cable of a suspension bridge is usually more concerned, and the equation (27) can be rewritten as follows: 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       f 
                       2 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             n 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 3 
                                  
                                 
                                   l 
                                   2 
                                 
                               
                               - 
                               
                                 16 
                                  
                                 
                                   r 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         
                           16 
                            
                           
                               
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               1 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         1 
                       
                     
                     + 
                     
                       
                         
                           
                             n 
                             2 
                           
                           ( 
                           
                             
                               3 
                                
                               
                                 l 
                                 2 
                               
                             
                             + 
                             
                               16 
                                
                               
                                 r 
                                 1 
                               
                             
                             + 
                             
                               16 
                                
                               
                                 r 
                                 2 
                               
                             
                           
                         
                         
                           16 
                            
                           
                               
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               2 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         2 
                       
                     
                     + 
                     
                       
                         
                           
                             n 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 3 
                                  
                                 
                                   l 
                                   2 
                                 
                               
                               - 
                               
                                 16 
                                  
                                 
                                   r 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         
                           16 
                            
                           
                               
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             
                               α 
                               3 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 3 
                               
                                
                               
                                 r 
                                 i 
                               
                             
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         3 
                       
                     
                     + 
                     
                         
                       
                         
                           
                             
                               
                                 n 
                                 2 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             3 
                                              
                                             
                                               l 
                                               2 
                                             
                                           
                                           + 
                                           
                                             16 
                                              
                                             
                                               r 
                                               1 
                                             
                                           
                                           + 
                                           
                                             16 
                                              
                                             
                                               r 
                                               2 
                                             
                                           
                                         
                                         ) 
                                       
                                       · 
                                       tan 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       2 
                                     
                                   
                                   - 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             3 
                                              
                                             
                                               l 
                                               2 
                                             
                                           
                                           - 
                                           
                                             16 
                                              
                                             
                                               r 
                                               2 
                                             
                                           
                                         
                                         ) 
                                       
                                       · 
                                       tan 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       1 
                                     
                                   
                                 
                                 ] 
                               
                             
                             
                               16 
                               · 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   3 
                                 
                                  
                                 
                                   r 
                                   i 
                                 
                               
                             
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         - 
                         
                           
                             
                               
                                 n 
                                 2 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             3 
                                              
                                             
                                               l 
                                               2 
                                             
                                           
                                           + 
                                           
                                             16 
                                              
                                             
                                               r 
                                               1 
                                             
                                           
                                           + 
                                           
                                             16 
                                              
                                             
                                               r 
                                               3 
                                             
                                           
                                         
                                         ) 
                                       
                                       · 
                                       tan 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       2 
                                     
                                   
                                   - 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             3 
                                              
                                             
                                               l 
                                               2 
                                             
                                           
                                           - 
                                           
                                             16 
                                              
                                             
                                               r 
                                               2 
                                             
                                           
                                         
                                         ) 
                                       
                                       · 
                                       tan 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       3 
                                     
                                   
                                 
                                 ] 
                               
                             
                             
                               16 
                               · 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   3 
                                 
                                  
                                 
                                   r 
                                   i 
                                 
                               
                             
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     According to the equation (1), the sag-to-span ratio is n i =q i ·l i /(8H i ). Assuming that the horizontal tension H i  and the vertical distributed load q i  are respectively equal for different spans of the main cable, the sag-to-span ratio of each span cable is proportional to the span. The span ratio of each span cable to the main span cable is defined as ζ i  (i=1, 2, 3) then 
     
       
         
           
             
               
                 
                   
                     ζ 
                     i 
                   
                   = 
                   
                     
                       
                         l 
                         i 
                       
                       
                         l 
                         2 
                       
                     
                     = 
                     
                       
                         n 
                         i 
                       
                       
                         n 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     According to the equation (36), l i =ζ i ·l 2 , and the ratio is calculated as follows: 
     
       
         
           
             
               
                 
                   
                     
                       16 
                        
                       
                         r 
                         i 
                       
                     
                     
                       3 
                        
                       
                         l 
                         2 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           16 
                           · 
                           
                             ζ 
                             i 
                           
                         
                          
                         
                           
                             l 
                             2 
                           
                           · 
                           
                             n 
                             i 
                             2 
                           
                         
                          
                         
                           cos 
                           2 
                         
                          
                         
                           α 
                           i 
                         
                       
                       
                         3 
                          
                         
                           l 
                           2 
                         
                       
                     
                     = 
                     
                       
                         
                           16 
                           3 
                         
                          
                         
                           
                             n 
                             2 
                             2 
                           
                           · 
                           
                             ζ 
                             i 
                             3 
                           
                           · 
                           
                             cos 
                             2 
                           
                         
                          
                         
                           α 
                           i 
                         
                       
                       ≤ 
                       
                         
                           16 
                           3 
                         
                          
                         
                           
                             n 
                             2 
                             2 
                           
                           · 
                           
                             ζ 
                             i 
                             3 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     The sag-to-span ratio n 2  of the main span cable is usually between 1/12 and 1/9, and herein, the maximum of 1/9 is taken. Since the side span is usually no more than half of the main span, that is ζ i ≤0.5, 16r i /(3l 2 )≤0.8% for i=1, 3. Due to ζ 2 =1, 16r 2 /(3l 2 )≤6.6%. By introducing 3l 2 +16r 1 +16r 3 ≈3l 2  and 3l 2 −16r 2 ≈3l 2 , the equation (35) is simplified as follows: 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       f 
                       2 
                     
                   
                   = 
                   
                     
                       
                         3 
                          
                         
                           l 
                           2 
                         
                          
                         
                           n 
                           2 
                         
                       
                       
                         16 
                         · 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             3 
                           
                            
                           
                             r 
                             i 
                           
                         
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             δ 
                              
                             
                                 
                             
                              
                             
                               S 
                               1 
                             
                           
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               1 
                             
                           
                         
                         + 
                         
                           
                             δ 
                              
                             
                                 
                             
                              
                             
                               S 
                               2 
                             
                           
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               2 
                             
                           
                         
                         + 
                         
                           
                             δ 
                              
                             
                                 
                             
                              
                             
                               S 
                               3 
                             
                           
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               3 
                             
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   2 
                                 
                               
                               - 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   1 
                                 
                               
                             
                             ) 
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   2 
                                 
                               
                               - 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   3 
                                 
                               
                             
                             ) 
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     When the sags of the side span cables are ignored, that is r 1 =r 3 =0, the equation (38) becomes: 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       f 
                       2 
                     
                   
                   = 
                   
                     
                       3 
                       
                         16 
                          
                         
                           
                             n 
                             2 
                           
                           · 
                           
                             cos 
                             2 
                           
                         
                          
                         
                           α 
                           2 
                         
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             δ 
                              
                             
                                 
                             
                              
                             
                               S 
                               1 
                             
                           
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               1 
                             
                           
                         
                         + 
                         
                           
                             δ 
                              
                             
                                 
                             
                              
                             
                               S 
                               2 
                             
                           
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               2 
                             
                           
                         
                         + 
                         
                           
                             δ 
                              
                             
                                 
                             
                              
                             
                               S 
                               3 
                             
                           
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               α 
                               3 
                             
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   2 
                                 
                               
                               - 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   1 
                                 
                               
                             
                             ) 
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   2 
                                 
                               
                               - 
                               
                                 tan 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   3 
                                 
                               
                             
                             ) 
                           
                            
                           δ 
                            
                           
                               
                           
                            
                           
                             h 
                             
                               P 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     Most suspension bridges have both tower tops at the same elevation (α 2 =0), and α 1 &gt;0 and α 3 &lt;0. The higher-order terms in equations (17) and (18) can be ignored, that is, δS i =l i  sec α i ·θ C ·δT C  (i=1, 2, 3) and δh Pj =h Pj ·θ P ·δT P  (j=1, 2), and the structural geometric dimensions can be used to represent the trigonometric functions, that is, sec α i =√{square root over (l i   2 +h i   2 )}/l i  and tan α i =h i /l i  (i=1, 2, 3). Under the approximations that h P1 ≈h 1 , h P2 ≈|h 3 |, θ C ·δT C ≈θ P ·δT P , the equations (38) and (39) can be greatly simplified as follows: 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                       f 
                       2 
                     
                   
                   = 
                   
                     
                       
                         
                           r 
                           2 
                         
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             3 
                           
                            
                           
                             r 
                             j 
                           
                         
                       
                       · 
                       
                         
                           3 
                            
                           
                             
                               θ 
                               C 
                             
                             · 
                             δ 
                           
                            
                           
                               
                           
                            
                           
                             T 
                             C 
                           
                         
                         
                           1 
                            
                           6 
                            
                           
                             n 
                             2 
                           
                         
                       
                     
                      
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         3 
                       
                        
                       
                         l 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
             
               
                 
                   
                     δ 
                      
                     
                       f 
                       2 
                     
                   
                   = 
                   
                     
                       
                         3 
                          
                         
                           
                             θ 
                             C 
                           
                           · 
                           δ 
                         
                          
                         
                             
                         
                          
                         
                           T 
                           C 
                         
                       
                       
                         1 
                          
                         6 
                          
                         
                           n 
                           2 
                         
                       
                     
                      
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         3 
                       
                        
                       
                         l 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
     When the elevations of both tower tops of a suspension bridge are equal (α 2 =0), the side span cable sag is not considered (r 1 =r 2 =0), and the approximations of α 1 &gt;0, α 3 &lt;0, h P1 ≈h 1 , h P2 ≈|h 3 |, θ C ·δT C ≈θ P ·δT P  are adopted, the equations (29) to (31) to calculate the tower-top horizontal displacement δl i  (i=1, 2, 3) can be greatly simplified as: 
       δ l   1   =l   1 θ C   ·δT   C   (42)
 
       δ l   2 =−( l   1   +l   3 )θ C   ·δT   C   (43)
 
       δ l   3   =l   3 θ C   ·δT   C   (44)
 
     The field monitoring usually measures the elevation variation of the main span cable or girder by GPS technology. In order to facilitate the comparison with the measurement, it is necessary to give a formula to estimate the elevation variation. The thermal expansion and contraction of the tower will change the mid-span elevation of the chord of the main span cable, which is denoted as d 2 : 
     
       
         
           
             
               
                 
                   
                     d 
                     2 
                   
                   = 
                   
                     
                       
                         
                           δ 
                            
                           
                             h 
                             
                               P 
                                
                               1 
                             
                           
                         
                         + 
                         
                           δ 
                            
                           
                             h 
                             
                               P 
                                
                               2 
                             
                           
                         
                       
                       2 
                     
                     = 
                     
                       
                         
                           
                             
                               h 
                               
                                 P 
                                  
                                 1 
                               
                             
                             + 
                             
                               h 
                               
                                 P 
                                  
                                 2 
                               
                             
                           
                           2 
                         
                         · 
                         
                           θ 
                           P 
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         T 
                         P 
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     Since the elevation takes the vertical upward direction as positive, the elevation variation δD 2  of the main span cable at mid span is equal to minus δf 2  plus d 2 : 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                       D 
                       2 
                     
                   
                   = 
                   
                     
                       
                         - 
                         δ 
                       
                        
                       
                         f 
                         2 
                       
                     
                     + 
                     
                       
                         
                           
                             
                               h 
                               
                                 P 
                                  
                                 1 
                               
                             
                             + 
                             
                               h 
                               
                                 P 
                                  
                                 2 
                               
                             
                           
                           2 
                         
                         · 
                         
                           θ 
                           P 
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         T 
                         P 
                       
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
           
         
       
     
     As the length of the suspender at mid span or the thickness of the central clamp of the suspension bridge are relatively small, and their thermal deformation can be ignored, the elevation variation of the main span girder at mid span can also be approximated by δD 2 . 
     The main cable of a self-anchored suspension bridge is directly anchored to the end of the main girder, so the thermal expansion and contraction of the main girder cause the distance change between both ends of the main cable. It is assumed that the main girder is continuous at the girder-tower intersections with the total length of L G , and the linear expansion coefficient and the temperature variation of the girder are θ G  and δT G , respectively. As a result, the horizontal distance variation between both ends of the main cable is Δ G =L G θ G ·δT G , and then the equation (19) becomes: 
       δ l   1   +δl   2   +δl   3 =Δ G   (47)
 
     Assuming that the vertical load on each span cable is the same and the horizontal tension equilibrium at the tower top is always maintained, the column vector on the right side of the linear system of equations, i.e., the equation (20), should be changed from [0 0 0 Δ 1  Δ 2  Δ 3 ] T  to [0 0 Δ G  Δ 1  Δ 2  Δ 3 ] T , while the coefficient matrix remains unchanged. Therefore, the temperature deformation of the two-tower self-anchored suspension bridge will be the solution of the following linear system of equations: 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               - 
                               
                                 1 
                                 
                                   f 
                                   1 
                                 
                               
                             
                           
                           
                             
                               1 
                               
                                 f 
                                 2 
                               
                             
                           
                           
                             0 
                           
                           
                             
                               1 
                               
                                 l 
                                 1 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   l 
                                   2 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               - 
                               
                                 1 
                                 
                                   f 
                                   2 
                                 
                               
                             
                           
                           
                             
                               1 
                               
                                 f 
                                 3 
                               
                             
                           
                           
                             0 
                           
                           
                             
                               1 
                               
                                 l 
                                 2 
                               
                             
                           
                           
                             
                               1 
                               
                                 l 
                                 3 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             1 
                           
                           
                             1 
                           
                         
                         
                           
                             
                               
                                 c 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                               
                                 l 
                                 1 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               M 
                               1 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               
                                 c 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                               
                                 l 
                                 2 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               M 
                               2 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               
                                 c 
                                 
                                   n 
                                    
                                   
                                       
                                   
                                    
                                   3 
                                 
                               
                               
                                 l 
                                 3 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               M 
                               3 
                             
                           
                         
                       
                       ] 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               δ 
                                
                               
                                 f 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               δ 
                                
                               
                                 f 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               δ 
                                
                               
                                 f 
                                 3 
                               
                             
                           
                         
                         
                           
                             
                               δ 
                                
                               
                                 l 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               δ 
                                
                               
                                 l 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               δ 
                                
                               
                                 l 
                                 3 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           
                             Δ 
                             G 
                           
                         
                       
                       
                         
                           
                             Δ 
                             1 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   Δ 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   Δ 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
     If the parameters c ni , c li , and c αi  (i=1, 2, 3) are calculated by the equations (23) to (25), and f i  is replaced by n i ·l i , then the solution of the equation (48) is: 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       f 
                       i 
                     
                   
                   = 
                   
                     
                       
                         
                           n 
                           i 
                         
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             α 
                             i 
                           
                         
                       
                        
                       δ 
                        
                       
                           
                       
                        
                       
                         S 
                         i 
                       
                     
                     - 
                     
                       
                         n 
                         i 
                       
                        
                       tan 
                        
                       
                           
                       
                        
                       
                         
                           α 
                           i 
                         
                          
                         
                           ( 
                           
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 h 
                                 
                                   P 
                                    
                                   
                                       
                                   
                                    
                                   i 
                                 
                               
                             
                             - 
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 h 
                                 
                                   P 
                                    
                                   
                                     ( 
                                     
                                       i 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             n 
                             i 
                           
                            
                           
                             ( 
                             
                               
                                 3 
                                  
                                 
                                   l 
                                   i 
                                 
                               
                               - 
                               
                                 16 
                                  
                                 
                                   r 
                                   i 
                                 
                               
                             
                             ) 
                           
                         
                         
                           16 
                           · 
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 1 
                               
                               3 
                             
                              
                             
                               r 
                               j 
                             
                           
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               3 
                             
                              
                             
                               
                                 δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   S 
                                   k 
                                 
                               
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   k 
                                 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               2 
                             
                              
                             
                               
                                 
                                   ( 
                                   
                                     
                                       tan 
                                        
                                       
                                           
                                       
                                        
                                       
                                         α 
                                         
                                           k 
                                           + 
                                           1 
                                         
                                       
                                     
                                     - 
                                     
                                       tan 
                                        
                                       
                                           
                                       
                                        
                                       
                                         α 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                                 · 
                                 δ 
                               
                                
                               
                                   
                               
                                
                               
                                 h 
                                 Pk 
                               
                             
                           
                           - 
                           
                             Δ 
                             G 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     
                       l 
                       i 
                     
                   
                   = 
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           S 
                           i 
                         
                       
                       
                         cos 
                          
                         
                             
                         
                          
                         
                           α 
                           i 
                         
                       
                     
                     - 
                     
                       tan 
                        
                       
                           
                       
                        
                       
                         
                           α 
                           i 
                         
                          
                         
                           ( 
                           
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 h 
                                 
                                   P 
                                    
                                   
                                       
                                   
                                    
                                   i 
                                 
                               
                             
                             - 
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 h 
                                 
                                   P 
                                    
                                   
                                     ( 
                                     
                                       i 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         
                           r 
                           i 
                         
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             3 
                           
                            
                           
                             r 
                             j 
                           
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               3 
                             
                              
                             
                               
                                 δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   S 
                                   k 
                                 
                               
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   k 
                                 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               2 
                             
                              
                             
                               
                                 
                                   ( 
                                   
                                     
                                       tan 
                                        
                                       
                                           
                                       
                                        
                                       
                                         α 
                                         
                                           k 
                                           + 
                                           1 
                                         
                                       
                                     
                                     - 
                                     
                                       tan 
                                        
                                       
                                           
                                       
                                        
                                       
                                         α 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                                 · 
                                 δ 
                               
                                
                               
                                   
                               
                                
                               
                                 h 
                                 Pk 
                               
                             
                           
                           - 
                           
                             Δ 
                             G 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
           
         
       
     
     The above analysis method for the temperature deformation of the two-tower suspension bridge can be extended to other cable systems with any number of spans (multi-span suspension bridges, transmission lines, ropeways, etc.). The multi-span cable system in  FIG. 3  has u spans numbered as 1, 2, . . . , u−1, u and contains u+1 supports (including both ends) numbered as 0, 1, . . . , u−1, u, and u≥1. 
     Similar to the equation (2), for each span cable, the following equations are obtained: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           δ 
                            
                           
                               
                           
                            
                           
                             f 
                             i 
                           
                         
                         
                           f 
                           i 
                         
                       
                       - 
                       
                         
                           δ 
                            
                           
                               
                           
                            
                           
                             l 
                             i 
                           
                         
                         
                           l 
                           i 
                         
                       
                     
                     = 
                     
                       - 
                       
                         
                           δ 
                            
                           
                               
                           
                            
                           
                             H 
                             i 
                           
                         
                         
                           H 
                           i 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     ( 
                     
                       
                         i 
                         = 
                         1 
                       
                       , 
                       2 
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       
                         u 
                         - 
                         1 
                       
                       , 
                       u 
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
           
         
       
     
     It is assumed that the horizontal tension equilibrium at the tower top is always maintained, that is, 
     
       
         
           
             
               
                 
                   
                     
                       δ 
                        
                       
                           
                       
                        
                       
                         H 
                         1 
                       
                     
                     
                       H 
                       1 
                     
                   
                   = 
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           H 
                           2 
                         
                       
                       
                         H 
                         2 
                       
                     
                     = 
                     
                       … 
                        
                       
                           
                       
                       = 
                       
                         
                           δ 
                            
                           
                               
                           
                            
                           
                             H 
                             u 
                           
                         
                         
                           H 
                           u 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
           
         
       
     
     According to the equations (51) and (52), u−1 equations can be obtained as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           f 
                           i 
                         
                       
                       
                         f 
                         i 
                       
                     
                     - 
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           l 
                           i 
                         
                       
                       
                         l 
                         i 
                       
                     
                   
                   = 
                   
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           f 
                           
                             i 
                             + 
                             1 
                           
                         
                       
                       
                         f 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                     - 
                     
                       
                         δ 
                          
                         
                             
                         
                          
                         
                           l 
                           
                             i 
                             + 
                             1 
                           
                         
                       
                       
                         l 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
           
         
       
     
     The horizontal distance between the anchorages at both ends of the cable system is unchanged: 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       u 
                     
                      
                     
                       δ 
                        
                       
                         l 
                         i 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
           
         
       
     
     The length S i  of each span cable is as follows: 
     
       
         
           
             
               
                 
                   
                     S 
                     i 
                   
                   = 
                   
                     
                       l 
                       i 
                     
                      
                     
                       [ 
                       
                         
                           sec 
                            
                           
                             α 
                             i 
                           
                         
                         + 
                         
                           
                             8 
                             3 
                           
                            
                           
                             n 
                             i 
                             2 
                           
                            
                           
                             cos 
                             3 
                           
                            
                           
                             α 
                             i 
                           
                         
                         - 
                         
                           
                             
                               3 
                                
                               2 
                             
                             5 
                           
                            
                           
                             
                               n 
                               i 
                               4 
                             
                              
                             
                               ( 
                               
                                 
                                   5 
                                    
                                   
                                     cos 
                                     7 
                                   
                                    
                                   
                                     α 
                                     i 
                                   
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     cos 
                                     5 
                                   
                                    
                                   
                                     α 
                                     i 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
           
         
       
     
     where n i  is the sag-to-span ratio of the i th  span main cable: 
     
       
         
           
             
               
                 
                   
                     n 
                     i 
                   
                   = 
                   
                     
                       f 
                       i 
                     
                     
                       l 
                       i 
                     
                   
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
           
         
       
     
     Differentiating the equation (55) yields: 
       δ S   i   =c   ni   ·δn   i   +c   li   ·δl   i   +c   αi ·δα i   (57)
 
     where the coefficients c ni , c li , and c li  are respectively: 
     
       
         
           
             
               
                 
                   
                     c 
                     ni 
                   
                   = 
                   
                     
                       l 
                       i 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               1 
                                
                               6 
                             
                             3 
                           
                            
                           
                             n 
                             i 
                           
                            
                           
                             cos 
                             3 
                           
                            
                           
                             α 
                             i 
                           
                         
                         - 
                         
                           
                             
                               1 
                                
                               2 
                                
                               8 
                             
                             5 
                           
                            
                           
                             
                               n 
                               i 
                               3 
                             
                              
                             
                               ( 
                               
                                 
                                   5 
                                    
                                   
                                     cos 
                                     7 
                                   
                                    
                                   
                                     α 
                                     i 
                                   
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     cos 
                                     5 
                                   
                                    
                                   
                                     α 
                                     i 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
             
               
                 
                   
                     c 
                     li 
                   
                   = 
                   
                     
                       sec 
                        
                       
                         α 
                         i 
                       
                     
                     + 
                     
                       
                         8 
                         3 
                       
                        
                       
                         n 
                         i 
                         2 
                       
                        
                       
                         cos 
                         3 
                       
                        
                       
                         α 
                         i 
                       
                     
                     - 
                     
                       
                         
                           3 
                            
                           2 
                         
                         5 
                       
                        
                       
                         
                           n 
                           i 
                           4 
                         
                          
                         
                           ( 
                           
                             
                               5 
                                
                               
                                 cos 
                                 7 
                               
                                
                               
                                 α 
                                 i 
                               
                             
                             - 
                             
                               4 
                                
                               
                                 cos 
                                 5 
                               
                                
                               
                                 α 
                                 i 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   59 
                   ) 
                 
               
             
             
               
                 
                   
                     c 
                     
                       α 
                        
                       
                           
                       
                        
                       i 
                     
                   
                   = 
                   
                     
                       l 
                       i 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             sin 
                              
                             
                               α 
                               i 
                             
                           
                           
                             
                               cos 
                               2 
                             
                              
                             
                               α 
                               i 
                             
                           
                         
                         - 
                         
                           8 
                            
                           
                             n 
                             i 
                             2 
                           
                            
                           sin 
                            
                           
                             α 
                             i 
                           
                            
                           
                             cos 
                             2 
                           
                            
                           
                             α 
                             i 
                           
                         
                         + 
                         
                           3 
                            
                           2 
                            
                           
                             n 
                             i 
                             4 
                           
                            
                           
                             cos 
                             4 
                           
                            
                           
                             α 
                             i 
                           
                            
                           sin 
                            
                           
                             
                               α 
                               i 
                             
                              
                             
                               ( 
                               
                                 
                                   7 
                                    
                                   
                                     cos 
                                     2 
                                   
                                    
                                   
                                     α 
                                     i 
                                   
                                 
                                 - 
                                 4 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
           
         
       
     
     Differentiating the equation (56) with respect to f i  and l i  leads to: 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                       n 
                       l 
                     
                   
                   = 
                   
                     
                       δ 
                        
                       
                         ( 
                         
                           
                             f 
                             i 
                           
                           
                             l 
                             i 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               l 
                               i 
                             
                             · 
                             δ 
                           
                            
                           
                               
                           
                            
                           
                             f 
                             i 
                           
                         
                         - 
                         
                           
                             
                               f 
                               i 
                             
                             · 
                             δ 
                           
                            
                           
                               
                           
                            
                           
                             l 
                             i 
                           
                         
                       
                       
                         l 
                         i 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   61 
                   ) 
                 
               
             
           
         
       
     
     The elevation difference h i  between the two end supports of each span cable is: 
         h   i   =l   i ·tan α i   (62)
 
     By differentiating the equation (62) with respect to l i  and α i , the following equation is obtained: 
       δ h   i   =δl   i ·tan α i   +l   i ·sec 2 α i ·δα i   (63)
 
     The variation of h i  in the above equation is equal to the difference of the elevation changes at two end supports of the i th  span cable: 
       δ h   i   =δh   Pi   −δh   P(i−1)   (64)
 
     where δh P0  and δh Pu  correspond to the elevation variations of the anchorages at both ends, which are always equal to zero. 
     By substituting the equation (64) into the equation (63), Sa is obtained: 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                       α 
                       i 
                     
                   
                   = 
                   
                     
                       
                         
                           - 
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 α 
                                 i 
                               
                             
                             
                               2 
                               · 
                               
                                 l 
                                 i 
                               
                             
                           
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         l 
                         i 
                       
                     
                     + 
                     
                       
                         
                           
                             cos 
                             2 
                           
                            
                           
                             α 
                             i 
                           
                         
                         
                           l 
                           i 
                         
                       
                        
                       
                         ( 
                         
                           
                             δ 
                              
                             
                               h 
                               Pi 
                             
                           
                           - 
                           
                             δ 
                              
                             
                               h 
                               
                                 P 
                                  
                                 
                                   ( 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   65 
                   ) 
                 
               
             
           
         
       
     
     By substituting the equations (61) and (65) into the equation (57), u equations can be obtained: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             ni 
                           
                           
                             l 
                             i 
                           
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         f 
                         i 
                       
                     
                     - 
                     
                       
                         
                           
                             
                               c 
                               ni 
                             
                             · 
                             
                               n 
                               i 
                             
                           
                           
                             l 
                             i 
                           
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         l 
                         i 
                       
                     
                     + 
                     
                       
                         
                           c 
                           li 
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         l 
                         i 
                       
                     
                     - 
                     
                       
                         
                           
                             
                               
                                 c 
                                 
                                   α 
                                    
                                   
                                       
                                   
                                    
                                   i 
                                 
                               
                               · 
                               sin 
                             
                              
                             
                                 
                             
                              
                             2 
                              
                             
                               α 
                               i 
                             
                           
                           
                             2 
                             · 
                             
                               l 
                               i 
                             
                           
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         l 
                         i 
                       
                     
                   
                   = 
                   
                     
                       δ 
                        
                       
                         S 
                         i 
                       
                     
                     - 
                     
                       
                         
                           
                             
                               c 
                               
                                 α 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                             
                             · 
                             
                               cos 
                               2 
                             
                           
                            
                           
                             α 
                             i 
                           
                         
                         
                           l 
                           i 
                         
                       
                       · 
                       
                         ( 
                         
                           
                             δ 
                              
                             
                                 
                             
                              
                             
                               h 
                               Pi 
                             
                           
                           - 
                           
                             δ 
                              
                             
                               h 
                               
                                 P 
                                  
                                 
                                   ( 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
           
         
       
     
     It is noted that the equations (53), (54), and (66) constitute a linear system of equations with 2u unknowns δf i  and δl i  (i=1, 2, . . . , u): 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               A 
                               
                                 
                                   ( 
                                   
                                     u 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 × 
                                 u 
                               
                             
                           
                           
                             
                               B 
                               
                                 
                                   ( 
                                   
                                     u 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 × 
                                 u 
                               
                             
                           
                         
                         
                           
                             
                               0 
                               
                                 1 
                                 × 
                                 u 
                               
                             
                           
                           
                             
                               1 
                               
                                 1 
                                 × 
                                 u 
                               
                             
                           
                         
                         
                           
                             
                               C 
                               
                                 u 
                                 × 
                                 u 
                               
                             
                           
                           
                             
                               D 
                               
                                 u 
                                 × 
                                 u 
                               
                             
                           
                         
                       
                       ] 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 F 
                                 
                                   u 
                                   × 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 L 
                                 
                                   u 
                                   × 
                                   1 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             0 
                             
                               u 
                               × 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             Δ 
                             
                               u 
                               × 
                               1 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   67 
                   ) 
                 
               
             
           
         
       
     
     where A, B, C, D, 0, 1, δF, δL, Δ represent a matrix or a vector, and the subscript represents the size of the matrix or vector. The elements of matrix A, B, C, D are as follows: 
     
       
         
           
             
               
                 
                   
                     A 
                     ij 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   - 
                                   1 
                                 
                                 / 
                                 
                                   f 
                                   i 
                                 
                               
                                
                               
                                   
                               
                                
                               when 
                                
                               
                                   
                               
                                
                               i 
                             
                             = 
                             j 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   1 
                                   / 
                                   
                                     f 
                                     j 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 when 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                               + 
                               1 
                             
                             = 
                             j 
                           
                         
                       
                       
                         
                           
                             
                               0 
                                
                               
                                   
                               
                                
                               others 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   , 
                                   2 
                                   , 
                                   ⋯ 
                                    
                                   
                                       
                                   
                                   , 
                                   
                                     
                                       u 
                                       - 
                                       1 
                                     
                                     ; 
                                     
                                       j 
                                       = 
                                       1 
                                     
                                   
                                   , 
                                   2 
                                   , 
                                   ⋯ 
                                    
                                   
                                       
                                   
                                   , 
                                   u 
                                 
                                 ) 
                               
                             
                              
                             
                                 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   68 
                   ) 
                 
               
             
             
               
                 
                   
                     B 
                     ij 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 1 
                                 / 
                                 
                                   l 
                                   i 
                                 
                               
                                
                               
                                   
                               
                                
                               when 
                                
                               
                                   
                               
                                
                               i 
                             
                             = 
                             j 
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     - 
                                     1 
                                   
                                   / 
                                   
                                     l 
                                     j 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 when 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                               + 
                               1 
                             
                             = 
                             j 
                           
                         
                       
                       
                         
                           
                             
                               0 
                                
                               
                                   
                               
                                
                               others 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   , 
                                   2 
                                   , 
                                   ⋯ 
                                    
                                   
                                       
                                   
                                   , 
                                   
                                     
                                       u 
                                       - 
                                       1 
                                     
                                     ; 
                                     
                                       j 
                                       = 
                                       1 
                                     
                                   
                                   , 
                                   2 
                                   , 
                                   ⋯ 
                                    
                                   
                                       
                                   
                                   , 
                                   u 
                                 
                                 ) 
                               
                             
                              
                             
                                 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   69 
                   ) 
                 
               
             
             
               
                 
                   
                     C 
                     ij 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   c 
                                   ni 
                                 
                                 / 
                                 
                                   l 
                                   i 
                                 
                               
                                
                               
                                   
                               
                                
                               when 
                                
                               
                                   
                               
                                
                               i 
                             
                             = 
                             j 
                           
                         
                       
                       
                         
                           
                             
                               0 
                                
                               
                                   
                               
                                
                               others 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   , 
                                   2 
                                   , 
                                   ⋯ 
                                    
                                   
                                       
                                   
                                   , 
                                   
                                     u 
                                     ; 
                                     
                                       j 
                                       = 
                                       1 
                                     
                                   
                                   , 
                                   2 
                                   , 
                                   ⋯ 
                                    
                                   
                                       
                                   
                                   , 
                                   u 
                                 
                                 ) 
                               
                             
                              
                             
                                 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   70 
                   ) 
                 
               
             
             
               
                 
                   
                     D 
                     ij 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   M 
                                   i 
                                 
                                  
                                 
                                     
                                 
                                  
                                 when 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                               = 
                               j 
                             
                           
                         
                         
                           
                             
                               
                                 0 
                                  
                                 
                                     
                                 
                                  
                                 others 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     , 
                                     2 
                                     , 
                                     ⋯ 
                                      
                                     
                                         
                                     
                                     , 
                                     
                                       u 
                                       ; 
                                       
                                         j 
                                         = 
                                         1 
                                       
                                     
                                     , 
                                     2 
                                     , 
                                     ⋯ 
                                      
                                     
                                         
                                     
                                     , 
                                     u 
                                   
                                   ) 
                                 
                               
                                
                               
                                   
                               
                             
                           
                         
                       
                        
                       
                         
 
                       
                        
                       where 
                     
                   
                 
               
               
                 
                   ( 
                   71 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                     i 
                   
                   = 
                   
                     
                       
                         
                           c 
                           ni 
                         
                         · 
                         
                           n 
                           i 
                         
                       
                       
                         l 
                         i 
                       
                     
                     + 
                     
                       c 
                       li 
                     
                     - 
                     
                       
                         
                           
                             c 
                             
                               α 
                                
                               
                                   
                               
                                
                               i 
                             
                           
                           · 
                           sin 
                         
                          
                         
                             
                         
                          
                         2 
                          
                         
                           α 
                           i 
                         
                       
                       
                         2 
                         · 
                         
                           l 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   72 
                   ) 
                 
               
             
           
         
       
     
     0 or 1 represent a vector with all elements being 0 or 1. For example, 0 1×u  is a 1-by-u vector of zeros, and 1 1×u  is a 1-by-u vector of ones. The remaining vectors are: 
       δ F   u×1 =[δ f   1   δf   2    . . . δf   u ] T   (73)
 
       δ L   u×1 =[δ l   1   δl   2    . . . δl   u ] T   (74)
 
       Δ u×1 =[Δ 1 Δ 2 Δ u ] T   (75)
 
     where 
     
       
         
           
             
               
                 
                   
                     Δ 
                     i 
                   
                   = 
                   
                     
                       δ 
                        
                       
                         S 
                         i 
                       
                     
                     - 
                     
                       
                         
                           
                             
                               c 
                               
                                 α 
                                  
                                 
                                     
                                 
                                  
                                 i 
                               
                             
                             · 
                             
                               cos 
                               2 
                             
                           
                            
                           
                             α 
                             i 
                           
                         
                         
                           l 
                           i 
                         
                       
                       · 
                       
                         ( 
                         
                           
                             δ 
                              
                             
                                 
                             
                              
                             
                               h 
                               Pi 
                             
                           
                           - 
                           
                             δ 
                              
                             
                               h 
                               
                                 P 
                                  
                                 
                                   ( 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   76 
                   ) 
                 
               
             
           
         
       
     
     δS i  in Δ i  is the length variation of the i th  span main cable caused by temperature variations. When the elastic deformation caused by the thermal stress is ignored, the length variation of the main cable can be estimated by the one-dimensional thermal expansion and contraction calculation formula as follows: 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                       S 
                       i 
                     
                   
                   = 
                   
                     
                       
                         
                           S 
                           i 
                         
                         · 
                         
                           θ 
                           c 
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         T 
                         c 
                       
                     
                     = 
                     
                       
                         
                           l 
                           i 
                         
                         · 
                         
                           θ 
                           c 
                         
                         · 
                         δ 
                       
                        
                       
                           
                       
                        
                       
                         
                           T 
                           c 
                         
                          
                         
                           [ 
                           
                             
                               sec 
                                
                               
                                 α 
                                 i 
                               
                             
                             + 
                             
                               
                                 8 
                                 3 
                               
                                
                               
                                 n 
                                 i 
                                 2 
                               
                                
                               
                                 cos 
                                 3 
                               
                                
                               
                                 α 
                                 i 
                               
                             
                             - 
                             
                               
                                 
                                   3 
                                    
                                   2 
                                 
                                 5 
                               
                                
                               
                                 
                                   n 
                                   i 
                                   4 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       5 
                                        
                                       
                                         cos 
                                         7 
                                       
                                        
                                       
                                         α 
                                         i 
                                       
                                     
                                     - 
                                     
                                       4 
                                        
                                       
                                         cos 
                                         5 
                                       
                                        
                                       
                                         α 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   77 
                   ) 
                 
               
             
           
         
       
     
     where θ C  and δT C  are respectively the linear expansion coefficient and the temperature variation of the main cable. 
     δh Pi  in Δ i  is the elevation variation of the supports of the cable system, and Δh P0 =δh Pu =0. When the elastic deformation of the tower is ignored, δh Pi  can be estimated by the following equation: 
       δ h   Pi   =h   Pi ·θ P   ·δT   P   (78)
 
     where θ P  and δT P  are the linear expansion coefficient and the temperature variation of the tower, respectively, and h Pi  is the height of the tower. 
     In a specific application, the main span of a two-tower ground-anchored suspension bridge is l 2 =1990.796 m, and the spans of the left and right side span cables are l 1 =959.999 m and l 3 =960.295 m, respectively. Therefore, the horizontal distances between the two anchorages is L=3911.090 m. The sag-to-span ratios of the left side span, main span, and right side span cables are n 1 =1/21.55, n 2 =10.22, and n 3 =1/21.83; both towers are of the same height h P1 =h P2 =287.2 m; the chord inclinations of the left side span, main span, and right side span cables are α 1 =14.3°, α 2 =0, and α 3 =14.3°; the elevation difference of the supports of each span cable is taken as h 1 =245.159 m, h 2 =0.289 m, and h 3 =244.785 m; and the linear expansion coefficients of the steel main cable and the steel tower are θ C =1.2×10 −5 /° C. and θ P 1.2×10 −5 /° C., respectively. 
     Based on the field measurement from 2016 to 2018, the fitted sensitivity coefficient of the mid-span elevation of the main span cable with respect to the cable temperature variation is 0.07274° C. As a matter of experience, the temperature change of the tower is close to that of the main cable on the annual cycle, so δT P =δT C =1° C. is assumed. The relevant parameters of the case bridge are substituted into the equation (46), wherein δf 2  is calculated according to the equation (27). As a result, the calculated sensitivity coefficient of the mid-span elevation of the main span cable with the cable temperature is 0.07084° C., which is very close to the measured value with a relative error of about 2.5%. If the mid-span elevation variation of the main span cable is based on the equation (46) with δf 2  estimated by the most simplified equation (40), then the thermal sensitivity coefficient is −0.07494° C., which is still close to the fitted slope of the measured data. 
     It is worth noting that the traditional physics-based formulas have a significant calculation error compared with the present method. If only the thermal deformation of the main span cable is considered, the calculated sensitivity coefficient is only −0.03954° C.; meanwhile, if the sags of the side span cables are ignored, the calculated sensitivity coefficient is −0.08504° C. The relative errors of these two traditional calculation methods are approximately 46% and 17%, respectively, which indicate that the sag effects of the main span and side span cables should be taken into consideration in order to better estimate the thermal deformation of the suspension bridges with a large sag of the side span cables.