i66
DOCK ENGINEERING.
of the back of the wall. In fig. 89, the point X is determined by drawing
MX at an angle, O M X = 2ß.
Fig. 89.
Then the total thrust is measured graphically
by
wh2 OX
~ ON?
or analytically by
p Wh2 ^ + sin/p - 2 sin p cos 2 /I
2 1 + sin p
• (18)
When the back of the wall is vertical, ß = 0, and the equation reduces
to
wh2 1 - sin (5
2 1 +sinp’
which agrees with Rankine’s result for similar conditions. The direction
of the resultant is constant at an angle 7 to the horizontal, such that
y= ß+ X, the last-named angle being deduced from the equation—
sin X =
sin y sin 2 ß _________________
?1 + sin2 p - 2 sin p cos 2 /?
• (19)
It will be observed that in none of the foregoing expressions is any
account taken of the friction exerted by the particles against the back of
the wall—a factor which tends to resist displacement. In faet, the assumed
conditions only hold good at a suitable distance from the wall beyond the
range of its frictional influence.
A formula has been devised by Professor Boussinesq to cover this defect.
If ^ be the angle of friction between the wall and the earth, and x the
horizontal distance from the face of the wall, the following expressions are
given by him for the intensity of horizontal and vertical pressure for values
of .r less than ----sinp _—
V1 + sin p
, 1 - sin p
w (n + xtan A)- ----:—
. ’'1+sinp
Horizontal! prcssurø = . == — ^ •
1 + - ^ tan4
V 1 + sin p
Vertical pressure =
w (/i + x tan ^)
1 + /LE^tan^
yl + sin p
(21)
At the face of the wall æ = O, and the expressions become—
Horizontal pressure =
1 +
7 1 - sinp
wn- ----:--
1 + sin p
/l - sinp ,
\\ ----; — - tan A
V 1 + sin p
(22)