Finite Element Analysis of Steel and Steel-Concrete Composite Bridges - Finite Element Analysis and Design of Steel and Steel-Concrete Composite Bridges - page 516

during the application of loading on the finite element model. The finite
element equations correspond to linear analysis of a structural problem
because the displacement response is a linear function of the applied force
vector. This means that if the forces are increased with a constant factor,
the corresponding displacements will be increased with the same factor.
On the other hand, in nonlinear analysis, the aforementioned assumptions
are not valid. The assumption is that the displacement must be small
and the finite element stiffness matrix and force vector are constant and
independent on the element displacements, because all integrations have
been performed over the original volume of the finite elements and the
strain-displacement relationships. The assumption of a linear elastic material
was implemented in the use of constant stress-strain relationships. Finally,
the assumption that the boundary conditions remain unchanged was
reflected in the use of constant restraint relations for the finite element equi-
librium equation.
Recognizing the previous discussion, we can define three main non-
linear analyses commonly known as materially nonlinear analysis , geometrically
( large displacement and large rotation ) nonlinear analysis , and materially and geomet-
rically nonlinear analysis . In materially nonlinear analysis, the nonlinear effect
lies in the nonlinear stress-strain relationship, with the displacements and
strains infinitesimally small. Therefore, the usual engineering stress and strain
measurements can be employed. In geometrically nonlinear analysis, the
structure undergoes large rigid-body displacements and rotations. Majority
of geometrically nonlinear analyses were based on von Karman nonlinear
equations such as the analyses presented in [ 5.8-5.15 ] . The equations allow
coupling between bending and membrane behavior with the retention of
Kirchhoff normality constraint [1.16]. Finally, materially and geometrically
nonlinear analysis combines both nonlinear stress-strain relationship and
large displacements and rotations experienced by the structure.
Most available general-purpose finite element computer program divides
the problem history (overall finite element analysis) into different steps as
shown in Figure 5.29 . An analysis procedure can be specified for each step,
with prescribing loads, boundary conditions, and output requests specified
for each step. A step is a phase of the problem history, and in its simplest
form, a step can be just a static analysis of a load changing from one magni-
tude to another. For each step, modelers can choose an analysis procedure.
This choice defines the type of analysis to be performed during the step such
as static stress analysis, eigenvalue buckling analysis, or any other types of
analyses. Static analyses are used when inertia effects can be neglected.