PREVIOUS WORK 



The differential equation for the surface of rupture in an infinite cohesive 

 earth slope was derived by Resal (1910) in 1910. His derivation was based upon a 

 consideration of stress equilibrium on a triangular-shaped differential element with- 

 in the slope (fig. 1), the sides of which are: (1) vertical, (2) parallel to the 

 ground slope, and (3) parallel to the tangent of the rupture surface. In compliance 

 with the infinite slope theory, the stress, r, acting on the vertical side of the 

 element and the stress, p, acting on the side parallel to ground slope are conjugate. 

 Thus, Rdsal's derivation enables determination of the normal (n) and tangential (t) 

 stresses acting on the assumed plane of rupture in terms of the conjugate vertical 

 and lateral stresses. To assure compliance with the Coulomb assumption, it can be 

 written that 



t-ntan4) = c (1) 



where 4) is the angle of internal friction and c is the cohesive strength. To determine 

 the angle of the tangent to the rupture surface, it is a simple matter to maximize the 

 function, t - n tan <t>, by differentiating the Coulomb expression; thus 



-7—, r- tan 4) = (2) 



da' da' 



where a' is the angle between the ground surface and the tangent to the rupture surface, 

 and is related to the failure surface by the expression 



dx' = £2l4.9_l^dy' (3) 



sm a' 



where x' is the distance from the intersection of the rupture surface with the ground 

 surface to the point of consideration measured parallel to the ground surface, y' 

 is the vertical distance from ground surface to the point of consideration, and 9 

 is the slope of the ground surface. 



By substituting the values of t and n (determined from the stress equilibrium 

 considerations and expressed in terms of p and r) into the above equations, Rdsal was 

 led to the differential equation 



- cos (a' - 6) sin (a' -0 + 4)) cos (2a' + <}>) 



+ sin a' cos (a' + 4>) cos (2a' - 28 + (()) 



= — ^ cos (b cos (2a' + <p) (4) 



Y y 



where y is the unit weight of the soil. Rdsal concluded that this equation was not 

 integrable. 



2 



