Numerical methods in bridge structure analysis - Computational Analysis and Design of Bridge Structures - page 58

However, these methods are based on particular mathematical models
of creep development. The implementation of these methods involves
different element stiffness computations. Automatic incremental creep
analysis method, developed by the authors and presented here, relies
only on the linear assumption of creep effects and separates the time
domain nonlinearity away from FEM itself. As long as the creep factor,
a coefficient scalar to describe the proportion of creep strain to elastic
strain, is not coupled with loads, this method is suitable for any creep
development model, and its implementation can be separated from any
FEM system.
The third topic discussed in this chapter is the influence line/surface
live loading method. Searching for the extreme live load positions where
internal forces or displacements at a particular point of interest are
maximal or minimal is a unique analysis problem to bridge analysis and
design. For some simple vehicle patterns defined by certain specifications,
the extreme positions can simply be identified from influence lines. For
some complex vehicle patterns in which only minimum vehicle spacing is
defined, simple enumeration may not work. Dynamic planning is com-
monly used as a generic influence line live loading analysis method. Based
on the longitudinal influence line live loading method, influence surface
live loading can be further developed, with certain assumptions on traffic
movements.
3.2 FINIte elemeNt method
3.2.1 Basics
FEM is an approximate approach to solve a global equilibrium problem with
a continuum domain by a discrete system that contains a finite number of
well-defined components or elements. With the fast computing power and
large memory capacity of a modern digital computer, the discrete system
can be used to solve a very large and complicated continuum problem. Due
to the complexity of real engineering structural problems, often the con-
tinuous close-form solution is absent or impossible. With more advanced
modern computer hardware and software technologies, the application of
FEM becomes the obvious choice in structural analyses.
The principle of FEM is based on the minimization of total potential
energy, which states that the sum of the internal strain energy and external
works must be stationary when equilibrium is reached. In elastic problems,
the total potential energy is not only stationary but also minimal. The sta-
tionary of total potential energy is equivalent to its variation over admis-
sible displacements being zero and can be expressed as (Zienkiewicz et al.
1977)