Page 119 - Buried Pipe Design

94 Chapter Three

Hooke’s law for
z in three dimensions is as follows:

E

z z ( x y z )
1 1 2
where E is the elastic soil modulus (Young’s modulus) and is
Poisson’s ratio.
For the constrained compression test (Fig. 3.11), both x and y are
assumed to be zero. In this case, Hooke’s law above takes on the fol-
lowing form:

E (1 )

z z
(1 ) (1 2 )
The term in brackets is the effective modulus and is called the con-
strained modulus M s .
Thus, M s is related to Young’s modulus for the soil E s and Poisson’s
ratio by the following equation:

E s (1 )
M s (3.6)
(1 ) (1 2 )
where M s constrained soil modulus
E s Young’s modulus of soil, MPa, lb/in 2
Poisson’s ratio of soil

Typically, values for M s are computed as the slope of the secant from
the origin of the stress-strain curve to the stress level on the curve that
represents the free field soil stress at the side of the pipe (the average
modulus in Fig. 3.11).
Krizek et al. 22 reported that M s could vary from 0.7 to 1.5 times E .
Hartley and Duncan 10 and McGrath 28 proposed a direct substitution,
that is, E M s . In developing an elasticity model for a pipe embedded
in uniform soil mass, Burns and Richard used the constrained modulus
3
as the soil property most representative of soil behavior in the ground.
For purposes of buried pipe installations, the precision of the design mod-
els is sufficiently low that an approximate relationship is acceptable.
The constrained modulus can be derived directly from the hyperbolic
soil model. Two constants are required to define behavior of an elastic
material. The hyperbolic model uses Young’s modulus and the bulk
modulus as the parameters. The bulk modulus K in terms of E and
is as follows:

E
K
3 (1 2 )