Engineering Rock Mechanics

41 4 Appendix A: Stress and strain analysis

For small rotations sin R = R, so that

du=-Rdy
dv = Rdx
or

(b) Normal strain

• 
  
  
  
  • 
    
    
    
    • 
      
      
      
      • 
        
        
        
        • 
          
          
          
          • 
            
            
            
            • 
              
              
              
              • 
                
                
                
                • Because we use compression
                  I
                  I I
                  I we have
                
                
                
                
              
              
              
              
            
            
            
            
          
          
          
          
        
        
        
        
      
      
      
      
    
    
    
    
  
  
  
  

(and hence contraction) positive,

I

----^__- J I

w+

from which we obtain

du

dx

& x =--

or

(c) Shear strain

This is negative shear strain: P'Q

is longer than PQ, and extension

is negative. For small angles, to a

good approximation we have

du =dysina!+dxcosa-dx

butforsmallanglessina=aandcosa=l,sothatdu= ady+dx-dx=

a dy and similarly dv = adx.

The definition of shear strain is the change in angle between two lines

originally perpendicular to each other, i.e. ywy = (p - V2).

q2 = p+2a =3 -2a= p-x/2 = r,,