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o , o

whence tan. (6 — 90) = cot. (9 + — 90) = i? ;

or cot. 6 = tan. (B + 0);

and cot. B — tan. B — -2 tan. = 0;

but, cot. & = — p \

therefore, p^ + 2 j5 tan. — 1 = ;

whence, ^) = — tan. + v tan.^ 0+1



sin.



= Vr,



1 + sin.

which is the same as for Rankine Fig. 18. It also may be shown that
P„ and Pv (Cain) are equal in amount to P and W, as shown in Fig. 18.
For Fig. 23 it may be shown in a similar way that p = V ^, but. in this
case, Pff and Py are greater in amount than for the similar forces of
Fig. 22. (They are four times as great when cf) = 30°.) There-
fore, an analysis of Fig. 23 shows that the surcharge merely produces
greater compressive stresses in the wall, and the equations for p



EARTH PRESSURES 199

indicate that surcharging a wall does not increase its tendency to
overturn so long as the unit compression stress remains within safe
limits. Such an indication is so contrary to one's preconceived ideas
of the effect of surcharge that the theory from which it results can
scarcely be accepted until it is substantiated by experimental proof.

It may be noted that for Figs. 18, 19, 22, and 23, the equations
for p show that the width of the base is independent of the weight of
either the wall or the back-fill.

The equations for Figs. 16 and 17 give identical values of }) for

— = — "- — and = 0°, 30°, or 90° ; but p for Fig. 17 is the lesser for
w^ 140

values of ^ between 30° and 90° and greater for values less than 30°,
although the difference is not marked, as will be shown hereinafter.
The narrow base resulting from the Rankine surcharge formula is
due to the assumption that the resultant earth thrust is parallel to
the inclined surface, as this assumption introduces a vertical com-
ponent which produces a resisting moment sufficient to balance the
excess overturning moment due to the surcharge. This virtually
means the assumption of frictional resistance on the back of the wall,
an assiunption which Cain makes for all cases in his solution of
the sliding-wedge theory.

The figures and equations for p on Figs. 16 to 23 show inadequately
the inconsistencies of both theories when applied to various cases,
but they show that neither theory gives results for surcharged walla
which accord with the inference drawn from wall failures or from
the experiments of Leygue, namely, that for any type of wall, if
the surcharge is increased, the section must also be increased, pro-
vided the walls must satisfy the requirement that the resultant shall
cut the outer-third point, or any other common point within the base.

The difference between the cases shown by Figs 16 to 23 is shovm
more clearly by Fig. 24, on which the p curves are plotted with ref-
erence to the rectangular co-ordinates, and p. The reference
numbers on the curves refer to Figs. 16 to 23. The equations of the
curves are either shown or reference is made to Figs. 16 to 23 on
which they may be found, and, for all equations including the weight

w 100

of the wall and the earth, it was assumed that — = -tttt-

■Jt'i 140



200



EARTH PRESSURES



Curves 1 to 8 of Fig. 24 show the differences between the Rankine
and Cain formulas and also between the formulas for the extreme
conditions of no surcharge and maximum surcharge, but give no indi-
cation of the peculiar results obtained by the Rankine formula for
Fig. 12 which is so much snaaller than the corresponding wall,
Fig. 11, which is not surcharged. This, however, is shown by the
curves, 2a and 2h, in the equations for which ) is rl, as shown on Fig. 26, where

• After the methods outlined by the writer in "Cohesion in Earth", or methods
based on the same principles. Transactions, Am. Soc. C. E., Vol. LXXX (1916),
p. 1315.



DISCUSSION ON EARTH PRESSURES 209

ON = nl, NL = nl tan.



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