COULOMB’S THEOREM.
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Vertical pressure = -----wn ---------_ . , (23)
1 , /l - sin a
1 + \ ----= — — tan 4
V 1 + sin p
Coulomb’s Theorem.—What is practically the same formula as that
enunciated by Rankine has been developed by MM. Prony and Coulomb,
on somewhat different lines, as follows: —
In fig. 90, C E is the line of repose. Were the wedge of earth, DCE,
a solid mass it would have no tendency to slide down the plane, C E, the
frictional resistance between the two surfaces being sufficient to counteract
movement. Evidently, then, if the earth yield at all, it must do so by
fracturing along some other plane, the position of which remains to be
determined. Meanwhile, assume a position, 0 F.
Through the centre of gravity of the wedge, D C F, draw K O, vertically,
to represent its weight, W. Draw L O, making an angle, p, with the normal
to the plane, C F, to represent the ultimate reaction of the plane, and L K
a horizontal line through K. Then the pressure on the back of the wall is
measured by
P = LK = W tan tf = — — - tan 0 cot (tf + p). . (24)
It is now necessary to find the angle which gives the greatest possible
value to P. Take the variable factors in the preceding expression, differ-
entiate, and equate to zero.
d tan tf cot (tf + p) Ä „
--------------------— = sec2 6 cot (tf + p) - tan tf cosec2 (tf + p) = 0.
This reduces to
sin (2 tf + 2 p) = sin 2 tf, . . . (20)
and, therefore, since the sines of supplementary angles are equal,
2tf + 2p = w - 2 tf,
•■•2tf + p= J
2tf=2-p,
whence it is evident that the greatest thrust is obtained when the line of
rupture, C F, bisects the complément, D CE, of the angle of repose. In
this case,
P=^.tan2 tf,
which is a variant, in form only, of Rankine’s expression, since
1 + sinp 1 (4 2/
There are, in fact, several different methods of arriving at the same