1887.] Mr A. C. Elliott on Formula for Retaining Wall. 93 
s= ’0853 — = 11-723 
s 
To find the principal axes for this stress system, put a = 0 in (5), 
and let 0 a he the corresponding value of 0. This gives, on solution 
of the resulting quadratic, 
]{fzl . 
2 sin J3 si 4 sm 2 ft 
where / is written for 1 js. 
■ (15) 
Hence, 
tan 0 a = *046 or -21 ‘760 
6 a = 2° 38' or -87° 22' 
Assigning to p v for simplicity the value 10, and making use of 
(3) or (4), 
/ = 10-02 r"= '718 
where r' and r" denote respectively the greatest and least principal 
stresses. 
Now from Eankine’s ellipse of stress it is seen at once that the 
obliquity has a maximum value on two sections symmetrically 
situated with respect to the axes of amount 
sin -1 
t rr 
ty* ^ ry* 
r' + r 7 ’ 
which for the case in point becomes — 
sin ~ Ho-738 = sin_l 866 = 60 ° ; 
and this agrees with the original assumption. The relative positions 
of the two planes for which the obliquity is maximum, the principal 
axes, and the vertical and horizontal directions are shown in the 
diagram, upon which, also, the ellipse of stress for the system in 
question has been represented. 
Suppose, for simplicity, that the cross section of the wall is 
rectangular. The oblique pressure y? 0 , inclined to the normal to 
the inner face of the wall at the angle f3, is found, for any point, 
simply by multiplying the ratio s by the pressure due to the column 
of earth at that point. Let the earth have the uniform density p e . 
Then the oblique pressure on the face at the depth h from the 
surface is 
Po — sJlp e . 
