226 



RETAINING WALLS. 



CHAP. V. 



more or less merit. All solutions based on the theory of the wedge assume that the resultant 

 thrust is applied at one-third the height for a wall with a level or inclined surcharge, as is given 

 by Rankine; but the resultant is assumed as making angles with a normal to the back of the 

 wall varying from zero to the angle of repose of the filling. In Rankine's solution the resultant 

 pressure is parallel to the plane of the surcharge for a vertical wall with a level or positive surcharge. 



(i) RANKINE'S THEORY. In this theory the filling is assumed to consist of an incom- 

 pressible, homogeneous, granular mass, without cohesion, the particles are held in position by 

 friction on each other; the mass being of indefinite extent, having a plane top surface, resting 

 on a homogeneous foundation, and being subjected to its own weight. The principal and conju- 

 gate stresses in the mass are calculated, thus leading to the ellipse of stress. In the analysis it 

 is proved (a) that the maximum angle between the pressure on any plane and the normal to 

 the plane is equal to the angle of internal friction, and (b) that there is no active upward component 

 of stress in a granular mass. Both of these laws have been verified by experiments on semi- 

 fluids. Ra'nkine deduced algebraic formulas for calculating the resultant pressure on a vertical 

 wall with a horizontal surcharge, and on a vertical wall with a surcharge equal to 5, an angle 

 equal to or less than the angle of repose. The general case is best solved by constructing the 

 ellipse of stress by graphics, or Weyrauch's algebraic solution may be used. The author has 

 extended Rankine's solution in "The Design of Walls, Bins and Grain Elevators," so that it is 

 perfectly general. 



Rankine's Formulas. With a vertical wall and a horizontal surcharge, Fig. i, the total 

 resultant pressure is 



, ,, i sin <t> , . 



P = \-W-W j r 7 (i) 



I + sin < 



where w is the weight of the filling in Ib. per cu. ft., h is the depth of the wall in feet, <f> is the angle 

 of repose of the filling, and P is the resultant pressure on the wall in pounds. The resultant 

 pressure, P, will be horizontal. 



D 



FIG. i. 



For a vertical wall with surcharge at an angle 5, Fig. 2, the pressure is given by the formula 



(2) 



P = %w-h?-cos d 

 Where 8 is equal to 4>, formula (2) becomes 



P = 



cos S \ cos 2 5 cos 2 



cos 5 + Vcos 2 5 cos 2 <f> 



cos<(> 



(3) 



The resultant pressure, P, is parallel to the inclined top surface for a vertical wall with a level 

 or a positive surcharge (many authors have incorrectly assumed that the resultant pressure is 

 always parallel to the top surface of the surcharged filling). 



Inclined Retaining Wall. The pressure on an inclined retaining wall may be calculated by 

 means of the ellipse of stress see the author's "The Design of Walls, Bins and Grain Elevators." 



