Abstract:
An adjustable structural connection—a connection which can be adjusted to alter the location and angle of member attachment—transforms the traditional notion of connections as a static means of connecting members at specific angles to a dynamic means of assembling a variety of structurally efficient forms. The adjustable connection connects a diagonal member to a vertical or horizontal member using either curved plate mounted to both of the structural members or a temporarily rotatable link. The adjustable connection can be used in a kit-of-parts type system, using multiple versions of the adjustable connection, which can join members at a wide variety of angles using a small number of unique components.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
       [0001]    This application is a non-provisional application claiming priority from U.S. Provisional Application Ser. No. 62/343,185 filed May 31, 2016, entitled “Adjustable Connection For Structural Members” and is a continuation of U.S. Non-Provisional patent application Ser. No. 15/292,801 filed Oct. 13, 2016 entitled “Adjustable Modules for Variable Depth Structures” which in turns claims priority from U.S. Provisional Application Ser. No. 62/240,776 filed Oct. 13, 2015, entitled “Adjustable Module and Structure” and Ser. No. 62/286,678, filed Jan. 25, 2016, entitled “Adjustable Module for Variable Depth Arch Bridges.” All of which are incorporated herein by reference in their entirety. 
     
    
     GOVERNMENT LICENSE RIGHTS 
       [0002]    This invention was made with government support under CMMI-1351272 awarded by the National Science Foundation. The government has certain rights in the invention. 
     
    
     FIELD OF THE DISCLOSURE 
       [0003]    The present description relates generally to a unique approach for joining structural members at a range of angles. This approach can be implemented for any joint between angled structural members. Applications include, but are not limited to, buildings (e.g., apex connections of portal frames) and bridges (e.g., angled connections of arch and truss bridges) for temporary or permanent construction. 
       BACKGROUND OF RELATED ART 
       [0004]    In architecture, structural engineering, and construction, connections between structural members are typically individually designed for each joint in each structure. A variety of means are known in the art for affixing structural members together, including bolts, rivets, and welds, sometimes also incorporating plates. 
         [0005]    In these methods, there is significant inefficiency in design, fabrication, and erection as each connection can be different in a structure and each structure is typically designed as one-of-a-kind. Alternatively, there can be major gains in efficiency and economy by using prefabricated connections that can join members at different angles in a wide variety of structures. Accordingly, there is a demonstrated need for an improved connection as declared herein. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0006]      FIG. 1A  is an example adjustable connection according to the teachings of this disclosure at 30° from the horizontal. 
           [0007]      FIG. 1B  is an example adjustable connection according to the teachings of this disclosure at 40° from the horizontal. 
           [0008]      FIG. 1C  is an example adjustable connection according to the teachings of this disclosure at 50° from the horizontal. 
           [0009]      FIG. 1D  is an example adjustable connection according to the teachings of this disclosure at 60° from the horizontal. 
           [0010]      FIG. 1E  is an example adjustable connection according to the teachings of this disclosure at 30° from the horizontal. 
           [0011]      FIG. 1F  is an example adjustable connection according to the teachings of this disclosure at 40° from the horizontal. 
           [0012]      FIG. 1G  is an example adjustable connection according to the teachings of this disclosure at 50° from the horizontal. 
           [0013]      FIG. 1H  is an example adjustable connection according to the teachings of this disclosure at 60° from the horizontal. 
           [0014]      FIG. 2  is a depiction of the geometric scribing for the adjustable connection. 
           [0015]      FIG. 3  is a depiction of the example adjustable connection using the scribing method shown in  FIG. 2 . 
           [0016]      FIG. 4  is a depiction of a mechanism based adjustable connection according to the teachings of this disclosure applied to a joint between a diagonal and a vertical member. 
           [0017]      FIG. 5  is a depiction of a mechanism based adjustable connection according to the teachings of this disclosure applied to a joint between diagonal and horizontal member. 
           [0018]      FIG. 6  shows the original and mirrored angles of the mechanism based adjustable connection of  FIG. 4 . 
           [0019]      FIG. 7  is a table of θ 4  for different values of θ 3  and α. 
           [0020]      FIG. 8  is a graph showing the relation of α, θ 3 , and θ 4 . 
           [0021]      FIG. 9A  is a depiction of the location of Point B of  FIG. 4  in a leftward configuration at its initial angle. 
           [0022]      FIG. 9B  is a depiction of the location of Point B of  FIG. 4  in a leftward configuration at its initial angle. 
           [0023]      FIG. 9C  is a depiction of the location of Point B of  FIG. 4  in a leftward configuration at its mirrored angle. 
           [0024]      FIG. 9D  is a depiction of the location of Point B of  FIG. 4  in a rightward configuration at its mirrored angle. 
           [0025]      FIG. 10  is a graph showing the value of A while varying α, with x=0.2 and s=5. 
           [0026]      FIG. 11  is a graph showing the value of A while varying x, with α=1.1 and s=5. 
           [0027]      FIG. 12  is a graph showing the value of A while varying s, with α=1.1 and x=0.2. 
           [0028]      FIG. 13  is a graph showing the maximum force in diagonal members for example panelized bridges using 30°, 40°, 50°, and/or 60° angles, approximately 100 ft in span. 
           [0029]      FIG. 14  is a graph showing the maximum force in diagonal members for example panelized bridges using 30° and/or 40° angles, approximately 100 ft in span. 
           [0030]      FIG. 15  is a graph showing the maximum force in diagonal members for example panelized bridges using 30°, 40°, 50°, and/or 60° angles, approximately 200 ft in span. 
           [0031]      FIG. 16  is a graph showing the maximum force in diagonal members for example panelized bridges using 30°, 40°, 50°, and/or 60° angles, approximately 300 ft in span. 
           [0032]      FIG. 17A  is an elevation of an example simply supported panelized bridge, approximately L=100 ft in span, showing only half of the bridge with symmetry assumed. A pin restraint is shown. A roller restraint would be on the other end. 
           [0033]      FIG. 17B  is an elevation of an example simply supported panelized bridge, approximately L=200 ft in span, showing only half of the bridge with symmetry assumed. A pin restraint is shown. A roller restraint would be on the other end. 
           [0034]      FIG. 17C  is an elevation of an example simply supported panelized bridge, approximately L=300 ft in span, showing only half of the bridge with symmetry assumed. A pin restraint is shown. A roller restraint would be on the other end. 
           [0035]      FIG. 18  is an example simply supported variable depth truss bridge (solid line), approximately L=300 ft in span. Only half of the bridge is shown, with symmetry assumed. A pin restraint is shown. A roller restraint would be on the other end. Dashed line indicates shape of moment diagram for simply supported truss under a uniform load. 
           [0036]      FIG. 19  is an example three-span continuous variable depth truss bridge (solid line). Only half of the bridge is shown, with symmetry assumed. A pin and a roller restraint are shown. Additional roller restraints would be on the other end. Dashed line indicates shape of the envelope of the moment and shear diagrams for the continuous truss under multiple load cases. 
           [0037]      FIG. 20A  is an example articulated linkage in a configuration that would be amenable to the adjustable connection of the present disclosure. 
           [0038]      FIG. 20B  is a depiction of the configuration of rapidly erectable bridge modules. 
           [0039]      FIG. 20C  is a depiction of another configuration of rapidly erectable bridge modules. 
           [0040]      FIG. 20D  is a graph showing the deflection of the systems shown in  FIGS. 20B and 20C . 
       
    
    
     DETAILED DESCRIPTION 
       [0041]    The following description of example methods and apparatus is not intended to limit the scope of the description to the precise form or forms detailed herein. Instead the following description is intended to be illustrative so that others may follow its teachings. 
         [0042]    In the following description, the term “kit-of-parts” is used. One of ordinary in the art skill will appreciate in context that this is used to mean that the adjustable connection herein disclosed could be used with a combination of other like adjustable connections adapted for different angle ranges and scales of the structural members. 
         [0043]    The goal of the adjustable connection as disclosed is to join 2 or more structural members at a variety of angles using a small number of unique components. This facilitates the joining of members to form varying geometric structures using the same adjustable connection. Such adjustable connections can be used for many different joints in a one-of-a-kind type structures and/or connecting modules in a modular structure. 
         [0044]    The adjustable connection as disclosed offers significant design, construction and fabrication advantages. Design can be simplified as the same adjustable connection could be used for one or more joints in a structure. The adjustable connection can be prefabricated and mass-produced, thereby simplifying fabrication. Construction could be accelerated as the connection can be repeated in the same structure and is capable of joining standard sections together. 
         [0045]    This adjustable connection could apply to a wide variety of materials (e.g., steel, aluminum, advanced composites, wood). Applications include, but are not limited to, conventional structural design and construction (e.g., bridges, buildings), modular construction (e.g., panelized rapidly erectable bridges), or special structures (e.g., grid shells). 
         [0046]    An example adjustable connection for joining structural members is comprised of one or more cold bent plates with specific bend angles and curvature and a universal gusset plate. Turning to  FIG. 1A-H , an example adjustable connection  10  joins three structural members, including a diagonal structural member  12  to be joined to a first vertical structural member  14  and a second horizontal structural member  16 . In an example adjustable connection in  FIG. 1 , the diagonal structural member  12  will be positioned at a desired angle relative to the first vertical structural member  14  and second horizontal structural member  16  using the adjustable connection  10 . For any given angle, the diagonal member  12  is connected to the horizontal member  16  and the vertical member  14  by two curved plates  34  and the one universal gusset plate  32 . The vertical member  14  could also be directly connected to the horizontal member  16 , using the one universal gusset plate  32  or other means. A set of curved plates  34  can be prefabricated to serve different angles. An advantage of this connection is that the centerlines of all three members are intersecting at one point (O), thereby eliminating eccentric loading and additional bending. 
         [0047]    In the example adjustable connection in  FIG. 1A-H , structural members can be joined at angles θ. This disclosure shows example adjustable connections  10  with angles θ=30°, 40°, 50°, and 60°, but other angles could be considered.  FIG. 1A  joins the diagonal member  12  to the vertical member  14  and the horizontal member  16  at an angle of θ=−30°, measured relative to the horizontal member  16 .  FIG. 1D  joins the diagonal member  12  to the vertical member  14  and the horizontal member  16  at an angle of θ=60°, measured relative to the horizontal member  16 . The same set of curved or bent plates can be used to achieve both angles. More specifically, the bent plate  34  joining the horizontal member  16  and the diagonal member  12  shown in  FIG. 1A  can be used to join the vertical member  14  and the diagonal member  12  shown in  FIG. 1D . Similarly, the bent plate  34  joining the vertical member  14  and the diagonal member  12  shown in  FIG. 1A  can be used to join the in horizontal member  16  and the diagonal member  12  shown in  FIG. 1D . This same approach is used to achieve the example embodiment shown in  FIG. 1B  and  FIG. 1C , where the example embodiment in  FIG. 1B  joins the diagonal member  12  to the vertical member  14  and the horizontal member  16  at an angle of θ=40°, measured relative to the horizontal member  16  and the example embodiment in  FIG. 1C  joins the diagonal member  12  to the vertical member  14  and the horizontal member  16  at an angle of θ=50°, measured relative to the horizontal member  16 . 
         [0048]    While  FIG. 1  shows a connection between three structural members, the adjustable connection  10  could join 2 or more structural members. In the example adjustable connection  10  shown in  FIG. 1A-D , the diagonal, horizontal, and vertical members,  12 ,  16 , and  14 , respectively, are shown as wide flange (I-shaped) members. The diagonal and vertical members,  12  and  14 , respectively, are connected to the universal gusset plate  32  by plates  40 . 
         [0049]    In the example adjustable connection  10  shown in  FIG. 1E-H , the diagonal and vertical members,  12  and  14 , respectively, are shown as two back-to-back channel sections. In this case, plates are not needed to connect the diagonal and vertical members,  12  and  14 , respectively, to the universal gusset plate  32  as the sections could connect directly to the universal gusset plate  32  in double shear. The horizontal member  16  could be either a wide flange or back-to-back channel section, or other section. 
         [0050]    In the example embodiment in  FIG. 1 , the cold bent curved plates  34  are connected to the flanges of the members. This creates a moment-resisting joint between members that can be achieved through splice-type connections using bolts. This provides a stronger, more durable, and reliable connection between structural members as compared to a typical gusset plate connection. 
         [0051]    In the example embodiment in  FIG. 1 , the shape and size of the gusset plate  32  depends on the cross section of the various members and on where the diagonal  12  intersects with the vertical and horizontal members  14  and  16 , respectively. It is not necessarily drawn to scale in  FIG. 1 . Its size should be limited to avoid buckling. In the example embodiment in  FIG. 1 , the gusset plate  32  features a flange which is connected to the horizontal member  16  and a web which joins diagonal member  12  and vertical member  14 . 
         [0052]    As shown in  FIG. 2 , a linkage (i.e., an assembly of rigid structural members connected by joints) can be used to scribe the geometry for the adjustable connection. This concept is demonstrated here for an RRRP linkage (where R refers to revolute joints and P refers to prismatic joints) connecting a horizontal, vertical, and diagonal member,  16 ,  14 , and  12 , respectively. In  FIG. 2 , the linkage includes a first rigid link of length (l) connecting points A and C. A revolute joint at C, connects this first rigid link to a second rigid link of length (l) connecting points C and B. Another revolute joint is located at point B. This is then connected to slider AB. This is shown for the connection between the diagonal member  12  and horizontal member  16  with the subscript 1. It is shown for the connection between the diagonal member  12  and the vertical member  14  with the subscript 2. Note that many different linkages could be used, and this concept could be applied to many different types of structural forms. While this example uses geometry related to a linkage, the geometry of the adjustable connection  10  can be developed without any relationship to a linkage. 
         [0053]    In the RRRP linkage as shown in  FIG. 2 , the link length (l) is the same for link AC and CB. It is beneficial to have this constant distance (l) along which the members are connected by curved plates  34  for all angles. This minimizes the number of different connection locations (i.e., bolt holes) along the various structural members to achieve different angles, thereby facilitating prefabrication and erection of the members. 
         [0054]    As implemented in  FIG. 3 , detailed geometry related to this linkage is discussed below. The rigid links (l) are tangent lines to a circle with radius (r)—where r is the bend radius of the plates—at points A and B. The conceptual slider AB connects points A and B and is a chord of this circle. The length (l) and the radius (r) are related by: 
         [0000]    
       
         
           
             
               
                 
                   l 
                   = 
                   
                     r 
                     
                       tan 
                        
                       
                         ( 
                         
                           θ 
                            
                           
                             / 
                           
                            
                           2 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     1 
                     ) 
                   
                 
               
             
           
         
       
     
         [0055]    The same link length (l) is used for both the connection of the diagonal  12  to the horizontal  16  and the diagonal  12  to the vertical  14 . Several angles are considered for the purpose of this disclosure: θ=30°, 40°, 50°, 60° as shown in the example embodiment in  FIG. 1 . If the length of the links (l) and the angle (θ) are known, then the position of the plates can be determined. 
         [0056]    The location of the plates  34  relative to the origin (O) is now defined as shown in  FIG. 3 . The X and Y coordinates of points A 1  and B 1  on the horizontal member  16  and diagonal member  12 , respectively, can be determined by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       X 
                       
                         A 
                          
                         
                             
                         
                          
                         1 
                       
                     
                     = 
                     
                       
                         l 
                         + 
                         
                           
                             l 
                             1 
                           
                            
                           
                               
                           
                            
                           
                             Y 
                             
                               A 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                       
                       = 
                       
                         h 
                         2 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     2 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     X 
                     
                       B 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   = 
                   
                     
                       
                         l 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                       
                       + 
                       
                         
                           l 
                           1 
                         
                          
                         
                             
                         
                          
                         
                           Y 
                           
                             B 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                     
                     = 
                     
                       
                         l 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                       
                       + 
                       
                         h 
                         2 
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     3 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where l 1  is the horizontal distance from the origin (O) to the location where the linkage begins in  FIG. 3 , and it is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     l 
                     1 
                   
                   = 
                   
                     
                       d 
                       
                         2 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                       
                     
                     + 
                     
                       h 
                       
                         2 
                          
                         
                             
                         
                          
                         tan 
                          
                         
                             
                         
                          
                         θ 
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     4 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where, d is the depth of the diagonal member  12  and h is the depth of the horizontal member  16 . The X and Y coordinates of A 2  and B 2  on the vertical members  14  and diagonal member  12 , respectively, can be determined by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       X 
                       
                         A 
                          
                         
                             
                         
                          
                         2 
                       
                     
                     = 
                     
                       
                         
                           v 
                           2 
                         
                          
                         
                             
                         
                          
                         
                           Y 
                           
                             A 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                       = 
                       
                         l 
                         + 
                         
                           l 
                           2 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     5 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     X 
                     
                       B 
                        
                       
                           
                       
                        
                       2 
                     
                   
                   = 
                   
                     
                       
                         l 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                       
                       + 
                       
                         
                           v 
                           2 
                         
                          
                         
                             
                         
                          
                         
                           Y 
                           
                             B 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                     
                     = 
                     
                       
                         l 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                       
                       + 
                       
                         l 
                         2 
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where l 2  is the vertical distance from the origin (O) to the location where the linkage begins in  FIG. 3 , and it is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     l 
                     2 
                   
                   = 
                   
                     
                       
                         v 
                         2 
                       
                        
                       tan 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       d 
                       
                         2 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     7 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where v is the depth of the vertical member  14 . 
         [0057]    As identified in  FIG. 2 , the distance from the origin (O) to the point where the diagonal crosses the horizontal, measured along the centerline of the diagonal is defined as l 3  and is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     l 
                     3 
                   
                   = 
                   
                     
                       d 
                       
                         2 
                          
                         
                             
                         
                          
                         tan 
                          
                         
                             
                         
                          
                         θ 
                       
                     
                     + 
                     
                       h 
                       
                         2 
                          
                         
                             
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         θ 
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     8 
                     ) 
                   
                 
               
             
           
         
       
     
         [0058]    Following the same idea, the distance from the origin to the intersection point of the diagonal and the vertical is defined as l 4 , shown in  FIG. 2 , and is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     l 
                     4 
                   
                   = 
                   
                     
                       
                         d 
                         2 
                       
                        
                       tan 
                        
                       
                           
                       
                        
                       θ 
                     
                     + 
                     
                       v 
                       
                         2 
                          
                         
                             
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         θ 
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     9 
                     ) 
                   
                 
               
             
           
         
       
     
         [0059]    To avoid buckling, it is recommended that the size of the universal gusset plate  32  in  FIG. 1A-H  and the plates in  FIG. 1A-D  be as small as possible to achieve the desired geometry. 
         [0060]    Another embodiment of this disclosure is shown in  FIGS. 4-12  showing an articulated linkages approach in which another example adjustable connection  10 ′ is comprised of a linkage which alters the location and angle of the structural joint. During erection, the connection as a whole would remain a mechanism. When the desired position is achieved, the linkage would be fixed or locked into place. A wide variety of types and forms of linkages could be used for this embodiment of an adjustable connection. 
         [0061]    As shown in  FIG. 4 , another example embodiment of the adjustable connection  10 ′ joins a diagonal structural member  12  at an angle to both the vertical and horizontal members  14 ,  16 . This angle θ 3  is measured relative to the horizontal member  16  in  FIG. 4 . The diagonal structural member  12  is connected by a link  70  to the vertical member  14  in the example shown in  FIG. 4 . In the example shown, O is the point where the centerlines of the vertical member  14 , the horizontal member  16 , and the diagonal member  12  coincide. The diagonal member  12  would be rotatably connected to the vertical member  14  and the horizontal member  16  at point O. A is the point at which link  70  and diagonal member  12  meet. B is the point at which the link  70  meets the vertical member  14 . The connection points at O, A, and B shown in  FIG. 4  are initially rotatable joints, such as pin joints, allowing substantially free rotational motion. This motion allows the adjustable connection  10 ′ to become a functional linkage and be positioned into the correct angle desired by the user (angle θ 3 ). When the angle θ 3  is achieved, the connections of the link  70  and diagonal member  12  are made static or locked into place by adding bolts, welding, or any other suitable means of securing members as would be understood by one of ordinary skill in the art. It is also contemplated that this static connection could be lockable and use a secure but removable mechanism. 
         [0062]    As shown in  FIG. 5 , another example embodiment of the adjustable connection  10 ′ joins a diagonal structural member  12  and horizontal member  16  at an angle to the horizontal member  16 . This angle θ 3  is measured relative to the horizontal member  16  in  FIG. 5 . The diagonal structural member  12  is connected by a link  70  to the horizontal member  16  in the example shown in  FIG. 5 . In the example shown, O is the point where the centerlines of the horizontal member  16  and the diagonal member  12  coincide. The diagonal member  12  would be rotatably connected to the horizontal member  16  at point O. A is the point at which link  70  and diagonal member  12  meet. B is the point at which the link  70  meets the horizontal member  16 . The connection points at O, A, and B shown in  FIG. 4  are initially rotatable joints, such as pin joints, allowing substantially free rotational motion. This motion allows the adjustable connection  10 ′ to become a functional linkage and be positioned into the correct angle desired by the user (angle θ 3 ). When the angle θ 3  is achieved, the connections of the link  70  and diagonal member  12  are made static or locked into place by adding bolts, welding, or any other suitable means of securing members as would be understood by one of ordinary skill in the art. It is also contemplated that this static connection could be lockable and use a secure but removable mechanism. 
         [0063]    The concept is demonstrated here using geometry scribed by an RRRP linkage (where R refers to revolute joints and P refers to prismatic joints). The RRRP linkage is considered for two orientations: Vertical RRRP in  FIG. 4  (joining three structural members) and Horizontal RRRP in  FIG. 5  (joining two structural members). Only the Vertical RRRP configuration is discussed in detail here, but analogous procedures and equations could be used for the Horizontal RRRP configuration. The objective is to create a joint which can connect variable angle diagonal truss members  12 . 
         [0064]    In the vertical configuration shown in  FIG. 4 , the rigid links of the RRRP linkage connect points O, A, and B. The slider represents the location of point B along the vertical member  14 . As point B is translated along the vertical member  14 , the connection angle between the horizontal member  16  and the vertical member  14  (θ 3 ) changes. If it is mirrored, a second set of angles is possible: θ 4  as shown in  FIG. 6 . 
         [0065]    The system is defined by the ratio of the lengths of the links (α): 
         [0000]    
       
         
           
             
               
                 
                   α 
                   = 
                   
                     a 
                     b 
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     1 
                     ) 
                   
                 
               
             
           
         
       
     
         [0066]    and the angle θ 3 . It is required in this example that 0&lt;θ 3 &lt;90°. θ 4  is related to θ 3  by: 
         [0000]      θ 4 =arccos(α cos θ 3 )  eq. (2)
 
         [0067]    The slider length (s) (i.e., the distance between O and B) can be determined by: 
         [0000]        s=a  sin θ 3 ±√{square root over ( a   2  sin 2 θ 3   −a   2   +b   2 )}  eq. (3)
 
         [0068]    There are therefore two possibilities for slider length (s). Following the above, if θ 4  is restricted to 0&lt;θ 4 &lt;90°, only: 
         [0000]        s=a  sin θ 3 +√{square root over ( a   2  sin 2 θ 3   −a   2   +b   2 )}  eq. (4)
 
         [0069]    is possible. For slider length (s) to be real, the following must also be satisfied: 
         [0000]    
       
         
           
             
               
                 
                   α 
                   &lt; 
                   
                     1 
                     
                       cos 
                        
                       
                         ( 
                         
                           θ 
                           3 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     5 
                     ) 
                   
                 
               
             
           
         
       
     
         [0070]    Therefore, for each θ 3 , there is a corresponding maximum value for α:α max . For the range of 30°&lt;θ 3 &lt;70° considered here, α max =1.1547. 
         [0071]    Using typical desirable angles (30°≧θ 3 &lt;70°), α is selected such that the θ 4  values are as different as possible from θ 3 . This would enable a single configuration to result in a large number of different angles. In this case, five discretized angles are considered for θ 3 :θ 3 =30°, 40°, 50°, 60°, 70°.  FIG. 7  shows a chart variations of θ 4  for these values of θ 3  given different values of α. 
         [0072]    To further investigate this, a study was performed which varies α from 0 to 1.15 in increments of 0.00001. As a metric to determine the difference between each θ 3  and θ 4  for a given α, the following value is computed: 
         [0000]    
       
         
           
             
               
                 
                   A 
                   = 
                   
                     ave 
                      
                     
                       { 
                       
                         
                           
                             
                               min 
                                
                               
                                 [ 
                                 
                                   
                                     abs 
                                      
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           4 
                                         
                                         - 
                                         30 
                                       
                                       ) 
                                     
                                   
                                   ; 
                                   
                                     ∀ 
                                     
                                       θ 
                                       4 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               min 
                                
                               
                                 [ 
                                 
                                   
                                     abs 
                                      
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           4 
                                         
                                         - 
                                         40 
                                       
                                       ) 
                                     
                                   
                                   ; 
                                   
                                     ∀ 
                                     
                                       θ 
                                       4 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               min 
                                
                               
                                 [ 
                                 
                                   
                                     abs 
                                      
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           4 
                                         
                                         - 
                                         50 
                                       
                                       ) 
                                     
                                   
                                   ; 
                                   
                                     ∀ 
                                     
                                       θ 
                                       4 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               min 
                                
                               
                                 [ 
                                 
                                   
                                     abs 
                                      
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           4 
                                         
                                         - 
                                         60 
                                       
                                       ) 
                                     
                                   
                                   ; 
                                   
                                     ∀ 
                                     
                                       θ 
                                       4 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               min 
                                
                               
                                 [ 
                                 
                                   
                                     abs 
                                      
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           4 
                                         
                                         - 
                                         70 
                                       
                                       ) 
                                     
                                   
                                   ; 
                                   
                                     ∀ 
                                     
                                       θ 
                                       4 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
           
         
       
     
         [0073]    This metric is shown in  FIG. 8 . The standard deviation is also shown. It is desirable to have a high value of the metric A, indicating that there is a greater difference between each θ 3  and θ 4 . It is also desirable to have a low standard deviation of this metric. The value for α which features a value for A and a low standard deviation is: α=1.09682. 
         [0074]    As shown in  FIGS. 9A-D , if point B is also allowed to translate horizontally, additional connection angles are possible. In  FIGS. 4, 6-9 , the slider point B was considered to be on the centerline of the vertical member. In this study, point B is moved perpendicular to the center line (y-axis), either on the left or right side by a distance x, so that the slide position s remains the same, as shown in  FIGS. 9A-D . This new configuration of the linkage gives another set of angles: θ 3   x . If this configuration is then mirrored, another set of angles is possible: θ 4   x . 
         [0075]    When the pivot point B is moved left of the y-axis, the angle θ 3   x  is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     3 
                     x 
                   
                   = 
                   
                     90 
                     - 
                     
                       arccos 
                        
                       
                         
                           
                             a 
                             2 
                           
                           + 
                           
                             x 
                             2 
                           
                           + 
                           
                             s 
                             2 
                           
                           - 
                           
                             b 
                             2 
                           
                         
                         
                           2 
                            
                           a 
                            
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 s 
                                 2 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       arccos 
                        
                       
                         s 
                         
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               s 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     7 
                     ) 
                   
                 
               
             
           
         
       
     
         [0076]    And the mirrored angle θ 4   x  is computed: 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     4 
                     x 
                   
                   = 
                   
                     90 
                     - 
                     
                       arccos 
                        
                       
                         
                           
                             b 
                             2 
                           
                           + 
                           
                             x 
                             2 
                           
                           + 
                           
                             s 
                             2 
                           
                           - 
                           
                             a 
                             2 
                           
                         
                         
                           2 
                            
                           b 
                            
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 s 
                                 2 
                               
                             
                           
                         
                       
                     
                     - 
                     
                       arccos 
                        
                       
                         s 
                         
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               s 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     8 
                     ) 
                   
                 
               
             
           
         
       
     
         [0077]    When the pivot point B is moved right to the y-axis, the angle θ 3   x  is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     3 
                     x 
                   
                   = 
                   
                     90 
                     + 
                     
                       arccos 
                        
                       
                         s 
                         
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               s 
                               2 
                             
                           
                         
                       
                     
                     - 
                     
                       arccos 
                        
                       
                         
                           
                             a 
                             2 
                           
                           + 
                           
                             x 
                             2 
                           
                           + 
                           
                             s 
                             2 
                           
                           - 
                           
                             b 
                             2 
                           
                         
                         
                           2 
                            
                           a 
                            
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 s 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     9 
                     ) 
                   
                 
               
             
           
         
       
     
         [0078]    And the mirrored angle θ 4   x  is computed: 
         [0000]    
       
         
           
             
               
                 
                   
                     θ 
                     4 
                     x 
                   
                   = 
                   
                     90 
                     - 
                     
                       arccos 
                        
                       
                         s 
                         
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               s 
                               2 
                             
                           
                         
                       
                     
                     - 
                     
                       arccos 
                        
                       
                         
                           
                             b 
                             2 
                           
                           + 
                           
                             x 
                             2 
                           
                           + 
                           
                             s 
                             2 
                           
                           - 
                           
                             a 
                             2 
                           
                         
                         
                           2 
                            
                           b 
                            
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 s 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     10 
                     ) 
                   
                 
               
             
           
         
       
     
         [0079]    Another parametric study was performed to investigate the parameters: slide position s, ratio α=a/b, and x. The objective was to select parameters for which the values of θ 3 , θ 4 , θ 3   x , and θ 4   x  differ from each other to obtain a wide range of angles. As a metric to determine the difference between each θ 3 , θ 4 , θ 3   x , and θ 4   x , the following value is computed: 
         [0000]    
       
         
           
             
               
                 
                   
                     A 
                     2 
                   
                   = 
                   
                     ave 
                      
                     
                       { 
                       
                         
                           
                             
                               min 
                               [ 
                               
                                 abs 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       4 
                                     
                                     - 
                                     
                                       θ 
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               min 
                               [ 
                               
                                 abs 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       3 
                                       x 
                                     
                                     - 
                                     
                                       θ 
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               min 
                               [ 
                               
                                 abs 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       4 
                                       x 
                                     
                                     - 
                                     
                                       θ 
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               min 
                               [ 
                               
                                 abs 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       4 
                                     
                                     - 
                                     
                                       θ 
                                       4 
                                       x 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               min 
                               [ 
                               
                                 abs 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       4 
                                       x 
                                     
                                     - 
                                     
                                       θ 
                                       3 
                                       x 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     11 
                     ) 
                   
                 
               
             
           
         
       
     
         [0080]    To compute this metric, the following relations are computed. The length of each link can be computed by: 
         [0000]    
       
         
           
             
               
                 
                   a 
                   = 
                   
                     s 
                     
                       
                         sin 
                          
                         
                             
                         
                          
                         
                           θ 
                           3 
                         
                       
                       + 
                       
                         
                           
                             
                               ( 
                               
                                 1 
                                 α 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             
                               cos 
                               2 
                             
                              
                             
                               θ 
                               3 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     12 
                     ) 
                   
                 
               
             
             
               
                 
                   b 
                   = 
                   
                     a 
                     α 
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     13 
                     ) 
                   
                 
               
             
           
         
       
     
         [0081]    For θ 3   x , θ 4   x  to exist, the triangle OA′B′ must exist, for which the following must be satisfied: 
         [0000]    
       
         
           
             
               
                 
                   s 
                   ≥ 
                   
                     x 
                     
                       
                         
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   1 
                                   α 
                                 
                               
                               ) 
                             
                             2 
                           
                            
                           
                             
                               ( 
                               
                                 1 
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       1 
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             1 
                                             α 
                                           
                                           ) 
                                         
                                         2 
                                       
                                       - 
                                       
                                         
                                           cos 
                                           2 
                                         
                                          
                                         
                                           θ 
                                           1 
                                         
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     14 
                     ) 
                   
                 
               
             
           
         
       
     
         [0082]    The following possible ranges of values for the parameters are considered: 
         [0000]    θ 3 =30°; 40°; 50°; 60°; 70°;
 
0.1≦α≦α max , where
 
         [0000]    
       
         
           
             
               α 
               max 
             
             = 
             
               1 
               
                 cos 
                  
                 
                     
                 
                  
                 
                   θ 
                   3 
                 
               
             
           
         
       
     
         [0000]    0.1≦x≦0.5;
 
s min ≦s≦10;
 
         [0083]    where s min  is given by Equation 14. Some of the results of this study are charted in  FIGS. 10, 11, and 12 . It would be desirable to select parameters for which the metric A 2  is high and the standard deviation is low. 
         [0084]      FIGS. 13-17  demonstrate how the adjustable connections could be used to form panelized bridges where the angle of the diagonal can be θ=30°, 40°, 50°, or 60°, measured with respect to the horizontal. These angles could be achieved using the example adjustable connection  10  or example adjustable connection  10 ′. Varying the angle of the diagonal changes the amount of force in the diagonal. It can be advantageous to keep the force in the diagonals similar throughout the structure as the same section could then be used throughout. The following details one method to arrive at forms for panelized bridges using diagonals with angles of θ=30°, 40°, 50°, or 60°, measured with respect to the horizontal. Other methods for determining geometries of panelized bridges using the adjustable connection are also possible. 
         [0085]    Parametric studies were performed to select potential geometries of simply supported panelized bridges for which the forces in the diagonals are as similar as possible. Span lengths of 100, 200, and 300 ft with span to depth ratios of 20 were considered. More specifically, for 100 ft span, the depth is 5 ft. For the 200 ft span, the depth is 10 ft. For the 300 ft span, the depth is 15 ft. However, other span lengths and span to depth ratios are also possible. Diagonals were considered at angles of θ=30°, 40°, 50°, or 60°, measured with respect to the horizontal. Every possible permutation of this angle θ was considered for each diagonal in each span length. For each permutation, the method of joints was used to calculate the force in each diagonal under a uniformly distributed vehicular lane load of 0.64 kip/ft, as given by bridge design code.  FIGS. 13-16  graph the maximum force in any diagonal and the standard deviation for different span lengths. Selected options are drawn (shown as only half the span). Symmetry is assumed. FIG.  13  shows all of the options for approximately 100 ft span. Note that the actual span length varies from 100 ft to achieve an integer number of panels.  FIG. 14  shows the options for approximately 100 ft span, with the angle restricted to just θ=30° or 40°. This was considered since the panel lengths become short for the higher angles.  FIG. 15  shows the options for approximately 200 ft span.  FIG. 16  shows the options for approximately 300 ft span. Note that for  FIGS. 13, 15, and 16 , the first angle (the one nearest the support) was required to be θ=60° as the shear force is the highest near the support in this example.  FIG. 17A-C  shows elevation views of panelized bridges selected based on this parametric study, for the 100 ft, 200 ft, and 300 ft span, respectively. 
         [0086]      FIG. 18  shows an example simply supported variable depth truss bridge for an approximate 300 ft span. While this form features vertical members, the variable depth truss could also be achieved using a different topology (e.g., warren type truss) The variable depth form was selected based on the scaled moment diagram (dashed line; scaled to achieve a depth at midspan of 50 ft) for a simply supported beam under a uniform load (i.e., 0.64 kip/ft distributed vehicular lane load as given by bridge design code). The geometry of the truss (solid line) was scribed to approximate this shape. It was also required that each member not be longer than 60 ft. All angles between members are required to be θ=30°, 40°, 50°, 60°, 70°, 80°, or 90°. This form could be achieved using the example adjustable connection  10  or example adjustable connection  10 ′, if the set of possible angles of the example adjustable connection  10  or example adjustable connection  10 ′ is expanded. This represents the ability for the adjustability connection to form variable depth simply supported truss bridges. Other forms are also possible. Other methods for determining the geometry of the form are also possible. 
         [0087]      FIG. 19  shows an example three-span continuous variable depth truss bridge for an approximate 800 ft total span. In this example, the middle span is approximately 300 ft and the side spans are approximately 80% of this length. The variable depth form was selected based on the moment and shear diagrams for a three-span continuous beam under a number of load cases. These load cases include a uniform load (i.e., 0.64 kip/ft distributed vehicular lane load as given by bridge design code) over (1) the whole bridge, (2) half of the bridge, (3) only one side span, (4) only the middle span, (5) one side span and the middle span, and (6) the two side spans. The moment and shear diagrams were calculated for each load case. The highest value for the moment and the highest value of the shear over all load cases were then found. Each was then scaled to achieve a depth at the inner support of 60 ft. The higher value of both was then taken as the desired form (dashed line). The geometry of the truss (solid line) was scribed to approximate this shape. For feasibility, it was also required that each member not be longer than 60 ft and that each panel length must be at least 30 ft. All angles between members are required to be even increments of 10° to be achievable with the example adjustable connection  10  or the example adjustable connection  10 ′ (with an expanded set of possible angles). An exception is at midspan when other angles are allowed. Other forms are also possible as would be understood by one of ordinary skill in the art. Other methods for determining the geometry of the form are also possible. 
         [0088]    Another embodiment of this disclosure is shown in  FIG. 20 . As depicted in  FIGS. 20A-D , it is contemplated that an example adjustable connection  10 ″ can be used for joining rapidly erectable bridge modules. To demonstrate the potential for adjustable connections  10 ″ to increase the efficiency of a rapidly erectable bridging system, an investigation was performed for a 100 ft span comprised of Bailey panels ( FIG. 20B-C ). As will be appreciated, the structural members could be truss elements in the panel sections. 
         [0089]    A preliminary concept for an adjustable connection  10 ″ features a four-bar (4R) linkage (connecting abcd), as shown in  FIG. 20A , which permits a change in vertical alignment between panels. The linkage is comprised of two sides of panels, vertical members  24 ,  24 ′, and short link elements  22  (scale exaggerated), connected to gusset plates  26 . The 4R mechanism can be moved until the desired change in vertical alignment is achieved. Once the desired position is found, the linkage is fixed by bolting one of the links in its corresponding curved gusset plate  32  which features a series of bolt holes (point “e” in  FIG. 20A ). 
         [0090]    This change in vertical alignment would permit alternate configurations compared to the standard simply supported beam configuration of the Bailey System (panels are connected by pins at the upper and lower chords,  FIG. 20B ). One example of an alternate configuration is the arch  60  shown in  FIG. 20C  (panels are offset vertically by one half foot per panel for an arch depth of 2 ft over the 100 ft span). To demonstrate the effect of this difference in form, finite element analyses were performed for both configurations. Bailey panels were modeled as one-dimensional beam elements with approximated section properties. In some embodiments, the chords are 2, 4 in deep channels, assumed to be C4×7.25. A representative uniform load was applied to represent the panel self-weight. Deflections of the arch ( FIG. 20C ) with adjustable connections were 15-20% less than the standard configuration ( FIG. 20B ), as shown in  FIG. 20D . While it is no surprise to one of ordinary skill in the art that an arch has lower deflections than a simply supported beam, this preliminary study indicates the potential for increased efficiency through adjustable connections. 
         [0091]    Although certain example methods and apparatus have been described herein, the scope of coverage of this patent is not limited thereto. On the contrary, this patent covers all methods, apparatus, and articles of manufacture fairly falling within the scope of the appended claims either literally or under the doctrine of equivalents.