Civil Engineering Reference
In-Depth Information
are inherently much safer. They suffer very large deformations and so give plenty
of warning before they fail. Tall steel-framed buildings wave around with wind or
earthquake loadings but they do not often fall down.
3.5 Stiffness
As discussed earlier in Sec. 3.1 and illustrated in Figs. 3.1 and 3.2 stiffness is the
relationship between stress and strain. The stiffness modulus, which is the gradient of
the stress-strain curve, may be a tangent or a secant: if the material is linear these are
the same.
For isotropic loading for which
q
remains constant we can define a bulk modulus
K
and for triaxial loading with
p
constant we can define a shear modulus
G
as:
d
p
d
K
=
(3.12)
ε
v
d
q
d
3
G
=
(3.13)
ε
s
For loading in a shear test illustrated in Fig. 3.3(b) with zero or constant shear stress
we can define a one-dimensional modulus
M
and for shearing with constant normal
stress the shear modulus is
G
where
σ
n
d
M
=
(3.14)
d
ε
v
τ
d
G
=
(3.15)
d
γ
ν
. These
are obtained directly from a uniaxial compression or extension test in which the radial
stress
Alternative stiffness parameters are Young's modulus
E
and Poisson's ratio
σ
r
is held constant (or zero) and are given by
σ
a
d
d
E
=
(3.16)
ε
a
d
ε
r
ν
=−
(3.17)
d
ε
a
In soil mechanics the shear and bulkmoduli
G
and
K
are often used instead of Young's
modulus and Poisson's ratio because it is important to consider shearing and change
of shape separately or decoupled from compression and change of size.
These parameters are often called elastic parameters because they are usually derived
in text books for elastic materials. In Eqs. 3.12 to 3.17 they have been defined as
tangent moduli and they are simply the gradients of the appropriate stress-strain
curves.
