Civil Engineering Reference
In-Depth Information
Note that for undrained or constant volume straining no work is done by the normal
stress
n
because there is no displacement normal to the slip surface. For an upper
bound calculation youmust evaluate Eq. (19.13) for all the slip planes in the compatible
mechanism.
σ
19.6 Simple upper bounds for a foundation
In order to illustrate the use of the bound theorems I shall obtain solutions for the
bearing capacity of a foundation subject to undrained loading. Figure 19.6 shows
a foundation with unit length out of the page so that the width
B
is equal to the
area
A
. The foundation itself is weightless so the bearing pressure
q
=
V
/
B
. As the
foundation load
V
and bearing pressure
q
are raised the settlement
will increase
until the foundation can be said to have failed at the collapse load
V
c
or the bearing
capacity
q
c
. The foundation is smooth so there are no shear stresses between the soil
and the foundation. I will obtain solutions using, firstly, a simple mechanism and,
secondly, two stress discontinuities, and later I will obtain more complex solutions
using a slip fan and a stress fan. The purpose here is to illustrate the principles of the
bound solutions; I will consider the bearing capacity of foundations in more detail in
Chapter 22.
Figure 19.7(a) shows a simple mechanism consisting of three triangular wedges and
Fig. 19.7(b) is the corresponding displacement diagram. The increments of work done
by the self-weight forces sum to zero since block B moves horizontally while the vertical
components of the displacements of blocks A and C are equal and opposite. Hence,
from Eq. (19.8), we have
ρ
δ
E
=
V
u
δ
w
f
(19.14)
In order to calculate the work done by the internal stresses on the slip planes, from
Eq. (19.13) it is easiest to tabulate
s
u
,
L
and
δ
w
for each slip plane. Hence, from
Table 19.1,
δ
W
=
6
s
u
B
δ
w
f
(19.15)
and, equating
δ
E
and
δ
W
, an upper bound for the collapse load is
V
u
=
6
Bs
u
(19.16)
Figure 19.6
Bearing capacity of a simple foundation.
