EQUATION VERIFICATION 



Equation Compatibility 



To verify this analytical development, the equations presented will be compared to 

 those developed by other investigators. Martin (1961) has shown that his solution for 

 active and passive pressures on a vertical plane agrees with developments by various 

 other investigators. Because this paper is an extension of Martin's work, demonstrat- 

 ing that the equations presented herein agree with Martin's equations would also imply 

 agreement with developments by other investigators. 



Compatibility is proved by showing that equation 35, for the normal stress d Q , 

 reduces to Martin's equation for normal stress; therefore the restrictions (no seepage 

 forces and stresses on a vertical plane) on Martin's development must be applied to 

 equation 35. 



? 1 



Recalling the trigonometric identity, (l+tan^<*) = ( 2(I ) > equation 35 can be 



rewritten : 



a = 2a cos 2cc - 2t, sin a cos a + 2a, sin 2 * - a,. (38) 

 co b b b 



The normal stress on a vertical plane is found by setting « equal to 90° in equation 38: 



a = a, . (39) 

 c b 



Therefore, the normal stress on a vertical plane is equal to the normal stress at the 

 origin of planes, a^ in equation 29. a D can be adapted to the no seepage force condi- 

 tion by setting S equal to zero in equation 29: 



2a 



l+tan 2 i 



(40) 



Substituting the expression for a from equation 28 into equation 40, and recalling that 

 a c equals a^ for this special case: 



a = 

 c 



2a + 2a tan 2 * + 2ctanc 

 a a 



l+tan 2 G 



(-2a -2a tan 2 <(>-2ctan<}>) 2 



3. 3. 



(l+tan z 6) 



2cn2 



(41) 



4 [a 2 (l+tan 2 <j>+tan 2 9+tan 2 <f)tan 2 e) - c 2 ] 



3. 



(l+tan 2 9) 2 



17 



