## 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

under application of the given loading conditions. In order to

evaluate the response of the structure it is necessary to establish an analytical

model, which represents the structural behavior under application of

the loadings. An acceptable model must describe the physical behavior 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.

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

under application of the given loading conditions. In order to

evaluate the response of the structure it is necessary to establish an analytical

model, which represents the structural behavior under application of

the loadings. An acceptable model must describe the physical behavior 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.

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.

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