336 Design of surfuce excavations
In this diagram the force W represents the weight of the material
which overlies the foundation, and W1 and W2 represent the weight of
the two blocks 1 and 2, respectively.
Having drawn the free body dia-
gram, it is necessary to compute the re-
lations between the various virtual dis-
by accurately drawing the vectors on a
separate diagram.
One of these vectors must be given
yli
placements. This is most easily done VI
VI2
an arbitrary value in order to scale the 60"
remainder. Here, we set VI = 1.0, and t= VIH
4
sin 120 sin25
- v2 = 0.4880,
so we find VI
-- - " 3 v12 = 0.6623,
sin35 sin120
v12
u1v = u1 sin35 + vlV = 0.5736,
and
U~H = ~1 COS35 3 UIH = 0.8192.
We can now write down an expression for the external virtual work
P sin30. u.2 + Pcos30. U~V + W2 - YV - (W + Wl) e vlv = EVW
which, because U~V = 0.0, simplifies to
The internal virtual work is given by
P sin 30 1 v2 - (W + Wl) v1v = EVW.
5
sin 15
50 x 5 COS 30 * ~2 + 50 x 5 COS 30 * ~12 + - x 25 COS 20 = IVW.
For equilibrium we have EVW = IVW, and so
cos 20
sin 15
250 30 - (u2 + u12) + 125- = P sin 30. v2 - (W, + W). vIv,
from which we have
cos 20
sin 15
250.30 (~2 + ~12) + 125- + (Wi + W). viv
P=. (18.1)
sin 30 v2
We now need to compute W1 and W, the weight of block 1 and the
weight of the overburden:
x 25 = 1166.3 kN/m,
1 5
w1=- x5x -
2 tan 15
and
5
tan 15
WE- x 4 x 19 = 1418.2 kN/m.
Thus, substituting into Eq. (lS.l), we have
250 cos 30 x (0.4880 + 0.6623) + 453.8 + 2584.4 x 0.5736
sin 30 x 0.4880
P=
= 8956 kN/m.