Patent ID: 11970830
Assignee: HAINAN UNIVERSITY
Field: Measurement (Instruments)
Classification: CPC E  G | IPC E  G

Claim 3:
4. The method according to claim 1, wherein: the mathematical expressions in step 3 are as follows to obtain two positive values pz3 and pz4, and one smaller value of the two positive values is selected as the bearing capacity of the foundation:, p
      
       z
       ⁢
       3
      
     
     =
     
      
       [
       
        
         
          6
          ⁢
          μγ
          ⁢
          
           h
           ⁡
           (
           
            
             μ
             2
            
            -
            
             7
             ⁢
             μ
            
            +
            1
           
           )
          
         
         
          
           
            (
            
             1
             +
             μ
            
            )
           
           2
          
          ⁢
          
           
            (
            
             1
             -
             μ
            
            )
           
           2
          
          ⁢
          
           (
           
            
             2
             ⁢
             μ
            
            -
            1
           
           )
          
         
        
        +
        
         
          8
          
           1
           -
           μ
          
         
         ⁢
         c
         ⁢
         
          tan
          ⁡
          (
          
           
            π
            4
           
           +
           
            
             φ
             1
            
            2
           
          
          )
         
        
       
       ]
      
      +
      
       
        
         
          [
          
           
            
             
              6
              ⁢
              
               μγ
               ⁡
               (
               
                
                 μ
                 2
                
                -
                
                 7
                 ⁢
                 μ
                
                +
                1
               
               )
              
             
             
              
               
                (
                
                 1
                 +
                 μ
                
                )
               
               2
              
              ⁢
              
               
                (
                
                 1
                 -
                 μ
                
                )
               
               2
              
              ⁢
              
               (
               
                
                 2
                 ⁢
                 μ
                
                -
                1
               
               )
              
             
            
            ⁢
            h
           
           +
           
            
             
              
               8
               
                1
                -
                μ
               
              
              ⁢
              c
              ⁢
              
               tan
               ⁡
               (
               
                
                 π
                 4
                
                +
                
                 
                  φ
                  1
                 
                 2
                
               
               )
              
             
             ]
            
            2
           
           -
           
            
             
              1
              ⁢
              2
              ⁢
              
               μ
               ⁡
               (
               
                2
                -
                μ
               
               )
              
             
             
              
               
                (
                
                 1
                 +
                 μ
                
                )
               
               2
              
              ⁢
              
               
                (
                
                 1
                 -
                 μ
                
                )
               
               2
              
             
            
            ⁢
            
             
              2
              ⁢
              7
              ⁢
              
               μ
               2
              
              ⁢
              
               γ
               2
              
              ⁢
              
               h
               2
              
             
             
              
               
                (
                
                 1
                 +
                 μ
                
                )
               
               2
              
              ⁢
              
               
                (
                
                 1
                 -
                 μ
                
                )
               
               2
              
              ⁢
              
               (
               
                
                 2
                 ⁢
                 μ
                
                -
                1
               
               )
              
             
            
           
           -
           
            
             4
             [
             
              
               
                
                 
                  6
                  ⁢
                  μ
                 
                 -
                 1
                
                
                 2
                 ⁢
                 
                  (
                  
                   
                    2
                    ⁢
                    μ
                   
                   -
                   1
                  
                  )
                 
                
               
               ⁢
               γ
               ⁢
               
                R
                1
               
              
              +
              
               
                
                 μ
                 2
                
                
                 
                  (
                  
                   1
                   -
                   μ
                  
                  )
                 
                 ⁢
                 
                  (
                  
                   
                    2
                    ⁢
                    μ
                   
                   -
                   1
                  
                  )
                 
                
               
               ⁢
               γ
               ⁢
               
                R
                1
               
              
             
             ]
            
            2
           
           -
           
            
             
              24
              ⁢
              μγ
             
             
              
               (
               
                1
                -
                μ
               
               )
              
              ⁢
              
               (
               
                
                 2
                 ⁢
                 μ
                
                -
                1
               
               )
              
             
            
            ⁢
            h
            ⁢
            c
            ⁢
            
             tan
             ⁡
             (
             
              
               π
               4
              
              +
              
               
                φ
                1
               
               2
              
             
             )
            
           
           +
           
            16
            ⁢
            
             c
             2
            
            ⁢
            
             
              tan
              2
             
             (
             
              
               π
               4
              
              +
              
               
                φ
                1
               
               2
              
             
             )
            
           
          
         
         }
        
       
       
        
         6
         ⁢
         
          μ
          (
          
           2
           -
           μ
          
         
        
        
         
          
           (
           
            1
            +
            μ
           
           )
          
          2
         
         ⁢
         
          
           (
           
            1
            -
            μ
           
           )
          
          2
         
        
       
      
     
    
   
   
    
     (
     2.1
     )
    
   
  
 

 
  
   
    
     
      p
      
       z
       ⁢
       4
      
     
     =
     
      
       [
       
        
         
          6
          ⁢
          
           μγh
           ⁡
           (
           
            
             μ
             2
            
            -
            
             7
             ⁢
             μ
            
            +
            1
           
           )
          
         
         
          
           
            (
            
             1
             +
             μ
            
            )
           
           2
          
          ⁢
          
           
            (
            
             1
             -
             μ
            
            )
           
           2
          
          ⁢
          
           (
           
            
             2
             ⁢
             μ
            
            -
            1
           
           )
          
         
        
        +
        
         
          8
          
           1
           -
           μ
          
         
         ⁢
         c
         ⁢
         
          tan
          ⁡
          (
          
           
            π
            4
           
           +
           
            
             φ
             1
            
            2
           
          
          )
         
        
       
       ]
      
      -
      
       
        
         
          [
          
           
            
             
              6
              ⁢
              
               μγ
               ⁡
               (
               
                
                 μ
                 2
                
                -
                
                 7
                 ⁢
                 μ
                
                +
                1
               
               )
              
             
             
              
               
                (
                
                 1
                 +
                 μ
                
                )
               
               2
              
              ⁢
              
               
                (
                
                 1
                 -
                 μ
                
                )
               
               2
              
              ⁢
              
               (
               
                
                 2
                 ⁢
                 μ
                
                -
                1
               
               )
              
             
            
            ⁢
            h
           
           +
           
            
             
              
               8
               
                1
                -
                μ
               
              
              ⁢
              c
              ⁢
              
               tan
               ⁡
               (
               
                
                 π
                 4
                
                +
                
                 
                  φ
                  1
                 
                 2
                
               
               )
              
             
             ]
            
            2
           
           -
           
            
             
              1
              ⁢
              2
              ⁢
              
               μ
               ⁡
               (
               
                2
                -
                μ
               
               )
              
             
             
              
               
                (
                
                 1
                 +
                 μ
                
                )
               
               2
              
              ⁢
              
               
                (
                
                 1
                 -
                 μ
                
                )
               
               2
              
             
            
            ⁢
            
             
              2
              ⁢
              7
              ⁢
              
               μ
               2
              
              ⁢
              
               γ
               2
              
              ⁢
              
               h
               2
              
             
             
              
               
                (
                
                 1
                 +
                 μ
                
                )
               
               2
              
              ⁢
              
               
                (
                
                 1
                 -
                 μ
                
                )
               
               2
              
              ⁢
              
               (
               
                
                 2
                 ⁢
                 μ
                
                -
                1
               
               )
              
             
            
           
           -
           
            
             4
             [
             
              
               
                
                 
                  6
                  ⁢
                  μ
                 
                 -
                 1
                
                
                 2
                 ⁢
                 
                  (
                  
                   
                    2
                    ⁢
                    μ
                   
                   -
                   1
                  
                  )
                 
                
               
               ⁢
               γ
               ⁢
               
                R
                1
               
              
              +
              
               
                
                 μ
                 2
                
                
                 
                  (
                  
                   1
                   -
                   μ
                  
                  )
                 
                 ⁢
                 
                  (
                  
                   
                    2
                    ⁢
                    μ
                   
                   -
                   1
                  
                  )
                 
                
               
               ⁢
               γ
               ⁢
               
                R
                1
               
              
             
             ]
            
            2
           
           -
           
            
             
              24
              ⁢
              μγ
             
             
              
               (
               
                1
                -
                μ
               
               )
              
              ⁢
              
               (
               
                
                 2
                 ⁢
                 μ
                
                -
                1
               
               )
              
             
            
            ⁢
            h
            ⁢
            c
            ⁢
            
             tan
             ⁡
             (
             
              
               π
               4
              
              +
              
               
                φ
                1
               
               2
              
             
             )
            
           
           +
           
            1
            ⁢
            6
            ⁢
            
             c
             2
            
            ⁢
            
             
              tan
              2
             
             (
             
              
               π
               4
              
              +
              
               
                φ
                1
               
               2
              
             
             )
            
           
          
         
         }
        
       
       
        
         6
         ⁢
         
          μ
          ⁡
          (
          
           2
           -
           μ
          
          )
         
        
        
         
          
           (
           
            1
            +
            μ
           
           )
          
          2
         
         ⁢
         
          
           (
           
            1
            -
            μ
           
           )
          
          2
         
        
       
      
     
    
   
   
    
     (
     2.2
     )
    
   
  
 

where symbols in the mathematical expressions are defined as follows:
φ1—an internal friction angle of the rock-soil mass, in rad;
c—cohesion of the rock-soil mass, in kPa;
γ—gravity density of the overlying rock-soil layer on a top surface of a columnar cavity, in kN/m3;
μ—the Poisson's ratio of the rock-soil mass;
h—the thickness of the overlying stratum, in m; and
R1—sectional radius of the shallow-hidden spherical cavity, in m.