Page 177 - Buried Pipe Design

Design of Gravity Flow Pipes 151

3. Incremental nodal displacements are computed for the incremental
load vector by solving the system of equations represented by Eq. (3.32).
4. The incremental element strains are computed from a strain-
displacement matrix using the nodal displacements. The strain-
displacement matrix is based on nodal coordinates of each element
and the shape functions used to describe the element behavior. The
element strains are then used to compute the element stresses
using Hooke’s law and the initial elastic parameters used in step 1
above. The total stresses, strains, and displacements in the ele-
ment, are computed by adding the incremental stresses, strains,
and displacements from the previous increments. With estimates of
soil stresses now available, the soil elastic properties are evaluated.
These properties will be different from the initially assumed values.
Thus, Eq. (3.32) must be formed and solved again using the updated
soil properties and new incremental displacements, strains, and
stresses computed. An iteration sequence is followed until conver-
gence is achieved. The total stresses are used as starting values to
evaluate new elastic parameters for the next loading increment.
5. Once convergence is achieved for a particular load construction
increment, a new incremental load vector is computed, and the pro-
cedure outlined in steps 2 through 4 is again followed. This method
of analysis is called the incremental loading method (or equivalent
linear method) and is very common to most soil mechanics finite
element analysis programs. The accuracy of the solution is depen-
dent on the assumptions used to derive the stiffness matrix (includ-
ing the mathematical representation of soil stress-strain response),
the size of the loading increment, and many other factors.

The Utah State University research program included the develop-
ment of a model and its calibration by comparing FEA results with
actual physical test data. This FEA research has aided in the enhance-
ment of the computer code. These enhancements have resulted in abil-
ities to better model the actual conditions and predict actual responses.

Soil model. The soil model that is used is commonly called the Duncan
soil model. This soil model assumes that the stress-strain properties of
soil can be modeled using a hyperbolic relationship. Figure 3.38 shows
a typical nonlinear stress-strain curve and the hyperbolic transforma-
tion that is used. The value of the initial tangent modulus E i is a func-
tion of the confining pressure. Also, the change in slope of the curve gives
the change in the tangent modulus that occurs as strain increases. For a
given constant value of confining pressure, the value of the elastic mod-
ulus is a function of the percent of mobilized strength of the soil, or the
stress level. As the stress level approaches unity (100 percent of the