COULOMB'S THEORY. 



227 



The pressure on an inclined retaining wall may also be calculated by means of the graphic solution 

 shown in Fig. 3 if the direction of the thrust be known. From Rankine's theory we know that 

 the resultant pressure on a vertical retaining wall is always parallel to the top surface where the 

 stirrhar^e is K-vel or is inclined upwards away from the wall. The pressure on a retaining wall 

 inclined away from the filling may then be calculated as follows: 



FIG. 3. PRESSURE ON AN INCLINED RETAINING WALL. 



In Fig. 3 the retaining wall A CDB sustains the pressure of a filling having an angle of repose 

 ^, and sloping up away from the top of the wall at an angle 5. Calculate P' the pressure on the 

 plane E-B by means of formula (2). P' acts at a point \EB above B and is parallel to the 

 top surface DE. Let the weight of the triangle of filling DBE be G, which acts through the 

 center of gravity of the triangle and intersects P' at point O. Then P t , the resultant of P' 

 and G, will be the resultant pressure at O, and makes an angle z with a normal to the back of the 

 wall, and an angle, X = + z 90 with the horizontal. 



(2) COULOMB'S THEORY. In this theory it is assumed that there is a wedge having 

 the wall as one side and a plane called the plane of rupture as the other side, which exerts a maxi- 

 mum thrust on the wall. The plane of rupture lies between the angle of repose of the filling and 

 the back of the wall. It may coincide with the plane of repose. For a wall without surcharge 

 (horizontal surface back of the wall) and a vertical wall the plane of rupture bisects the angle 

 between the plane of repose and the back of the wall. This theory does not determine the direc- 

 tion of the thrust, and leads to many other theories having assumed directions for the resultant 

 pressure. 



Algebraic Method. In Fig. 4, the wall with a height h, slopes toward the earth, being in- 

 clined to the horizontal at an angle Q, and the earth has a surcharge with slope S, which is not 

 greater than <, the angle of repose. It is required to find the pressure P against the retaining 

 wall, it being assumed that the resultant pressure makes an angle z with the back of the wall. 



It is assumed that the triangular prism of earth above some plane, the trace of which is the 

 line A E, will produce the maximum pressure on the wall and on the earth below the plane, and 

 that in turn the prism will be supported by the reactions of the wall and the earth. Let OW 

 represent the weight of the prism ABE, the length of the prism being assumed equal to unity, 

 let OP be the reaction of the wall, and OR be the reaction of the earth below. 



Now the forces OW, OP, and OR will be concurrent and will be in equilibrium; OP and OR 

 will therefore be components of OW. When the prism ABE is just on the point of moving OP 



