Page 185 - Buried Pipe Design

Design of Gravity Flow Pipes 159

Small-displacement theory. If the pipe material remains linear elastic
during the loading process, then beam and rod element stiffness matri-
ces are constant since their elastic properties are not strain-dependent
and it is assumed that the nodal coordinates do not change apprecia-
bly during the analysis (small-displacement theory). The elastic por-
tion of the soil element stiffness matrix is strain-dependent. During an
individual iteration, the elastic matrix is evaluated for each soil ele-
ment and is combined with the strain-displacement matrix during the
numerical integration. As each element stiffness matrix is formed, it is
summed with the global stiffness matrix at common nodes. A solution
procedure is then followed, as discussed previously, where the nodal
displacements are evaluated based on the incremental load vector and
where the incremental load vector is the nodal force vector due to con-
struction loads or external loads.

Large-displacement theory. The finite element analysis, which does eval-
uate the stiffness matrix based on deformed nodal coordinates, is defined
as a geometric nonlinear analysis. Thus, one which includes both non-
linear stress-strain properties and large-displacement theory performs
material and geometric nonlinear analysis. In a solution using multiple
construction or load steps, a geometric nonlinear analysis can be accom-
plished by simply updating the nodal coordinates using displacements
computed during the previous load step. The stiffness matrix for the next
load step utilizes these updated nodal coordinates. A limitation of this
approach is that the displacements of each individual load step must be
small. PIPE5 was recently modified with a corotational algorithm in
forming element stiffness matrices to allow accurate predictions of large
displacements (but small strains) in a single load step.
There has been some concern that the small-strain theory that has
been used in the FEA of flexible pipes was inducing some inaccuracies in
the results. For example, as an initially round pipe has a vertical load
applied, it becomes elliptical in shape. When one accounts for this change
in shape, it is shown that there is a reduction in pipe stiffness. The stiff-
ness matrices of the structural elements (i.e. beam and rod element) are
developed partly on the basis of the element length and its inclination.
Using the deformed coordinates to compute the stiffness matrices allows
stiffness changes due to a change in inclination to be simulated.
The geometric nonlinear analysis has been used to help determine
initial deflections by means of compaction simulation. Also, modifica-
tions have given the program the ability to model internal pressure
loads and rerounding effects with incremental loading.

Iteration procedure. The iteration procedure accommodates the changes
in elastic moduli when they occur. The soil elements are monitored to