# Reanalysis of Structures: A Unified Approach for Linear, Nonlinear, Static and Dynamic Systems

## Reanalysis of Structures: A Unified Approach for Linear, Nonlinear, Static and Dynamic Systems

Structural analysis is a most exciting field of activity, but it is only a support
activity in the field of structural design. Analysis is a main part of the
formulation and the solution of any design problem, and it often must be
repeated many times during the design process. The analysis process helps
to identify improved designs with respect to performance and cost.
Referring to behavior under working loads, the objective of the analysis
of a given structure is to determine the internal forces, stresses and displacements
evaluate the response of the structure it is necessary to establish an analytical
model, which represents the structural behavior under application of
the structure adequately, and yet be simple to analyze. That is, the basic
assumptions of the analysis will ensure that the model represents the problem
under consideration and that the idealizations and approximations used
result in a simplified solution. This latter property is essential particularly
in the design of complex or large systems.

The overall effectiveness of an analysis depends to a large degree on the
numerical procedures used for the solution of the equilibrium equations
[1]. The accuracy of the analysis can, in general, be improved if a more refined
model is used. In practice, there is a tendency to employ more and
more refined models to approximate the actual structure. This means that
the cost of an analysis and its practical feasibility depend to a considerable
degree on the algorithms available for the solution of the resulting equations.
The time required for solving the equilibrium equations can be a
high percentage of the total solution time, particularly in nonlinear analysis
or in dynamic analysis, when the solution must be repeated many times.
An analysis may not be possible if the solution procedures are too costly.
Because of the requirement to solve large systems, much research effort

has been invested in equation solution algorithms.

In elastic analysis we refer to behavior under working loads. The forces
must satisfy the conditions of equilibrium, and produce deformations compatible
with the continuity of the structure and the support conditions. That
is, any method must ensure that both conditions of equilibrium and compatibility
are satisfied. In linear analysis we assume that displacements
(translations or rotations) vary linearly with the applied forces. That is, any
increment in a displacement is proportional to the force causing it. This assumption
is based on the following two conditions:
The material of the structure is elastic and obeys Hooke's law.
All deformations are assumed to be small, so that the displacements do
not significantly affect the geometry of the structure and hence do not
alter the forces in the members. Thus, the changes in the geometry are

small and can be neglected.

Repeated analysis, or reanalysis, is needed in various problems of structural
analysis, design and optimization. In general, the structural response
cannot be expressed explicitly in terms of the structure properties, and
structural analysis involves solution of a set of simultaneous equations.
Reanalysis methods are intended to analyze efficiently structures that are
modified due to various changes in their properties. The object is to evaluate
the structural response (e.g. displacements, forces and stresses) for such
changes without solving the complete set of modified simultaneous equations.
The solution procedures usually use the response of the original
structure. Some common problems, where multiple repeated analyses are
needed, are described in the following.

In structural optimization the solution is iterative and consists of
repeated analyses followed by redesign steps. The high computational
cost involved in repeated analyses of large-scale problems is one of the
main obstacles in the solution process. In many problems the analysis
part will require most of the computational effort, therefore only
methods that do not involve numerous time-consuming implicit analyses
might prove useful. Reanalysis methods, intended to reduce the
computational cost, have been motivated by some typical difficulties
involved in the solution process. The number of design variables is
usually large, and various failure modes under each of several load
conditions are often considered. The constraints are implicit functions of
the design variables, and evaluation of the constraint values for any
assumed design requires the solution of a set of simultaneous analysis
equations.

In structural damage analysis it is necessary to analyze the structure for
various changes due to deterioration, poor maintenance, damage, or
accidents. In general many hypothetical damage scenarios, describing
various types of damage, should be considered. These include partial or
complete damage in various elements of the structure and changes in the
support conditions. Numerous analyses are required to assess the
adequacy of redundancy and to evaluate various hypothetical damage
scenarios for different types of damage.