THE THEORY OF CONJUGATE PRESSURES.
163
and the total pressure
Q=^.
Hence, since
Z=P
Q ?
p wh2 1 - sin p
2 ’ 1 + sin p’
(13)
The line of action of P is, as in the case of water pressure, at one-third
of the height of the wall above its base.
A simple graphical construction for obtaining the numerical value of
h2
j -;—- may advantageously be inserted here.
Take a vertical line, A B
(fig. 86), to represent li, the height of the wall, to any convenient scale, and
Ni M No
Fig. 87.
from B draw B C, making the angle p with A B. Draw A 0 horizontally,
and with centre, C, and radius, C A, describe the arc A D. Then B D is
the line whose length measures —^_^ to ^ game gca]e-
VI + sin p
For B D2 = (B C - C D)2 = (B C - A C)2
= - h tan p
\cos p/
=ä2 /z^n2
\ COS ^ /
sin P)2
1 - sin2 p
=Ä2sinZ
1 + sin p
The case of conjugate stresses—viz., that in which the stresses are not
mutually perpendicular—is perhaps not strictly essential to the present
purpose, as its application is confined to those retaining walls in which the
surface of the earth backing is not horizontal—a condition of such rare
occurrence in the practice of dock engineering as scarcely to warrant any-
thing in the nature of a lengthy démonstration.* It may be of interest,
* There is only the possibility of a river wall being surcharged by a sloping embank-