Civil Engineering Reference
In-Depth Information
It is also necessary to recognize that soil stiffness is highly non-linear - it changes
with stress and with strain - and it is necessary to select values appropriate to the
strains in the soil in the ground. Non-linear soil stiffness was discussed in Chapter 13.
The characteristic variation of stiffness with strain is illustrated in Fig. 13.8. At very
small strain the value of Young's modulus is
E
0
. This is found from
G
0
which can be
measured in dynamic laboratory or in situ tests and it varies with stress and state, as
given by Eq. (13.8) and shown in Fig. 13.9. Analyses can be done in one step using
secant values or in several steps using tangent moduli. Figure 18.6(a) shows a non
linear stress-strain curve for a triaxial test on a soil sample: this is similar to Fig. 3.2 in
Chapter 3. The diagram has axes
q
a
and the gradient is
Young's Modulus: if the test is drained the gradient is
E
and if it is undrained it is
E
u
.
At the point A at some stage of the test the secant modulus is
=
(
σ
−
σ
r
) and axial strain
ε
a
=
q
ε
a
E
sec
(18.10)
and the tangent modulus is
dq
d
E
tan
=
(18.11)
ε
a
where
represents the change of stress and strain from the start of the test. For simple
analyses the secant modulus method would normally be used and the step taken as
the whole foundation loading. Figure 18.6(b) shows the variations of tangent and
secant modulus with strain corresponding to the stress-strain curve in Fig. 18.6(a).
The stiffnesses have been normalized by dividing by
E
0
. At the critical state at F the
strain is about 10% and
E
tan
=
0. At the peak state at P the strain is about 1% and,
again,
E
tan
0. As discussed in Sec. 13.4 the average strains in the ground near a
typical foundation at working load are about 0.1% but locally they are often in the
range 0.01% to 1%. This range is shown in Fig. 18.6(b) and this demonstrates that
=
Figure 18.6
Tangent and secant moduli.
