FOUNDATIONS AND RETAINING WALLS 209 



Substituting this value of y in the expression for P', the latter becomes 



A A 



or, replacing A, B, C, D by their values, 

 t ^vh ^ sin 2 (a a>) 



a asin(a + ) / | 



V " ' N 



which is the general formula for the maximum inclined earth pressure 

 against retaining walls.* 



The various standard theories as to the maximum earth pressure 

 may now be obtained as special cases of the above general formula 

 by making the following assumptions. 



1. Weyrauch's formula. Assume that the pressure is normal to 

 the back of the wall. Then = 0, and formula (107) becomes 



f wh 2 sin 2 (a co) 



/., I sin (a> 

 2sm 8 a(l + J- 



\ M sin (a 



fi) sin a 



2. Rankine's formula. Assume that the angle of repose of earth 

 on masonry is equal to the angle of repose of earth on earth. Then 

 f = a), and formula (107) becomes 



wh 2 sm z (a a) 1 



, 



max ~ 



2sm 2 asin(a 



l 



I sin (&> 

 \siua - 



3. Poncelet's formula. In Rankine's formula assume that the earth 

 surface is horizontal and the back of the wall is vertical. Then 

 = and a = 90, and the preceding formula becomes 



f wli 2 cos co 



~2(l+V2sino>) 2 



4. Coulomb's formula.! Assume, as in 3, that the earth surface is 

 horizontal and the back of the wall vertical, and make the further 



* Cf. Merriman, Watts and Dams, p. 35. 

 t Deduced in 1773. 



