Stiffness of soil - The Mechanics of Soils and Foundations - page 198

Table 13.1 Typical values for rigidity and degree of non-linearity of some common materials

Material E 0 MPa Strength MPa Rigidity n 1 ε r % ε p %

Concrete 28,000 40 700 0.15 0.35 2

Glass 70,000 1000 70 1.5 1.5 1

Mild steel 210,000 430 500 0.2 30 150

Copper 120,000 200 600 0.15 35 250

Aluminium 70,000 100 700 0.15 10 70

Rubber 10 20 0.5 200 800 4

Timber 10,000 20 500 0.2 5 25

Soft soil 100 0.05 2000 0.05 10 200

Stiff soil 300 0.3 1000 0.1 1 10

(Comparing Eqs. (13.12) and (13.15) ε = 1/R) the strain at the peak ε

p is larger than

r

the reference strain and the degree of non-linearity n 1 is defined as

= ε

p
ε r

n 1 (13.16)

Note that, as shown above, the peak strength q p is equal to the area below the
stiffness-strain curve from the origin to the peak state. As a consequence, the non-
linear stress-strain and stiffness-strain curves, shown as solid lines in Fig. 13.12, must
have shapes such that the area below the solid and broken stiffness-strain curves in
Fig. 13.13(b) are the same.
Table 13.1 gives typical values for rigidity and degree of non-linearity for some
common materials and for characteristic soft and stiff soils. (Some of the data in
Table 13.1 were given also in Table 3.1.) For soils, the degree of non-linearity varies
from about 10 to about 200. This is a large range, almost as big as for all other
materials. Notice that the rigidity of soil is relatively large, largely because of the
relatively low strength compared with other materials and the rigidity of soft soil is
larger than the rigidity of stiff soil which is a surprising result. The reasons for this
were discussed by Atkinson (2000).

13.8 Numerical modelling of soil stiffness
Equations (13.8) and (13.10) are convenient expressions relating the shear modulus
to the current state and to the stress history and there will be similar expressions
for the bulk modulus K . However, to be of practical use for design, soil behaviour
must be represented by mathematical expressions similar to those developed for Cam
clay in Chapter 12, although these are likely to be more complex to take account of
the non-linear behaviour for states inside the boundary surface. One possibility is to
regard soil behaviour inside the state boundary surface as essentially elastic, but non-
linear, and to use curve-fitting techniques to obtain an empirical expression relating
shear modulus G and bulk modulus K to strain. This is the approach followed by