Page 181 - Buried Pipe Design

Design of Gravity Flow Pipes 155

By differentiating the above expression with respect to ε vol, one can
get the following relationship for bulk modulus.
⎛ σ ⎞ 2
B = B 1 + B ε ⎟ (3.39)
m
i ⎜
⎝
i u ⎠
,
, and ε are obtained from testing various soils.
Values of B i m u
The PIPE5 computer code uses Eq. (3.36) and (3.37) or (3.39) to eval-
uate elasticity parameters that are required in the stiffness matrix.
Poisson’s ratio is calculated using classical theory of elasticity.

v = 1 − E (3.40)
2 6 B

Limitations are put on the magnitudes of Poisson’s ratio in order to
remain within the allowable limits of the theory of elasticity. If
Poisson’s ratio is computed to be more than 0.495, it defaults to 0.495.
Likewise, if it is computed to be less than 0.0, it again defaults to its
lower limit, 0.0. Shear modulus is calculated using

G = E (3.41)
(
21 + ) υ
Shear failure is also tested by evaluating the stress level before the
modulus of elasticity is computed. The stress level is used in Eq. (3.36)
and is expressed as
(σ − σ )(1 − sin )
φ
1 3 (3.42)
2C cosφ + 2 σ sinφ
3
If the stress level is computed to be more than 0.95, the modulus of
elasticity is computed based on a stress level of 0.95. This results in a
low modulus of elasticity which is consistent with a shear failure in the
soil. The bulk modulus is unaffected, thus modeling a high resistance
to volumetric compression in shear. A test is also performed to evaluate
if tension failure has occurred when computing the elastic parameters.
If the confining pressure is negative, then the soil element is in tension
failure. The elastic parameters are then set to very small values, thus
simulating a tension condition. The bulk modulus is set to 0.01B i ,
where B i is the initial bulk modulus. Poisson’s ratio is set to 0.495, and
the shear modulus is set to 0.0001B i . These constraints appear to be a
reasonable approach to modeling soil under shear or tension condi-
tions. The resulting output has been set up to identify the failed ele-
ments as the analysis progresses through the incremental loading.