Environmental Engineering Reference
In-Depth Information
r
r
p
0
0.9
1
0.7
0.5
2
0.3
0.2
0.15
3
0.1
4
σ
2
/p = 0.05
5
FIGURE 3.68
The “pressure bulb” of stress distribution: contours of
vertical normal stress beneath a uniformly loaded
circular area on a linear elastic half-space.
6
3
2
1
0
1
2
3
r
Elastic Moduli
Young's modulus, shear modulus, bulk modulus, and constrained modulus are defined
in
Figure 3.69.
In general, settlement may be expressed crudely in terms of elastic mod-
uli as
∆
∆
S
H
p
(
H
/
E
) (in, cm)
(3.58)
where
p
is the pressure increase over effective overburden pressure at the midpoint of the
“pressure bulb” (zone of influence or stress distribution zone beneath the loaded area as
shown in Figure 3.68,
H
the depth of pressure bulb upto some arbitrary boundary such as
σ
z
/
p
∆
σ
0.05 (insignificant stress), or the stratum thickness, if less and
E
/
ε
as deter-
ε
∆
mined by
in situ
or laboratory testing with
L
is measured but
L
is not known; it is estimated, on the basis of Boussinesq stresses, as the depth to which
the significant stress occurs.
Note:
If
L
/
L
; in
in situ
testing
∆
p
is applied over a large area compared with the stratum thickness, the con-
strained modulus (
D
) is used to account for the effect of confining pressures (Figure 3.69d).
∆
Dynamic Elastic Moduli
Compression and shear wave velocities of sonic waves (see
Section 2.3.2)
are functions
of the elastic properties and mass density of the transmitting medium. Measurements
of
V
p
and
V
s
, therefore, provide the basis for computing the dynamic properties
of Poisson's ratio, Young's modulus, shear modulus, and bulk modulus as given in
Moduli from the Stress-Strain Curve
Derived from the triaxial compression or uniaxial compression test, these moduli include
the initial tangent modulus, tangent modulus, and secant modulus defined in
Figure 3.70.
The initial tangent modulus
E
i
is usually reported as the elastic modulus for rock (given
as
E
r
), whereas the secant modulus
E
se
is usually reported as the elastic modulus for soils,
E
s
, to be used for calculating deformations. The secant modulus
E
se
is used also for defin-
ing a stress or limiting deformation.
