Keywords: structural re-analysis, structural variation, topological modification, plate bending elements, finite element for plate flexure.

Structural re-analysis becomes a necessity whenever there is a change in the cross sectional and/or material properties of one or members in a structure. There are number of areas in structural engineering where such changes are common. One of these occurs in the structural design where the cross sectional properties of members are modified from one design to another until all the design requirements are satisfied. In each design cycle re-analysis becomes a necessity to update the response of the structure under the external loads due to these changes. The other area where re-analysis is required is non- linear analysis of structures. In general there are two sources of non-linearity in structures. These are geometric non-linearity and material non-linearity. In some structures due to their geometry large deformations take place that makes it necessary to consider the equilibrium conditions on the deformed shape. The material non-linearity occurs when the stress-strain relationship of the material becomes non-linear. The analysis methods that obtain the responds of a structure when these non-linearities are present are not direct but iterative in nature. They converge to the solution through successive iterations. Linear structural analysis is carried out in each iteration. Hence structural re-analysis is the main tool of non-linear analysis methods. Another area where structural re-analysis becomes a requirement is the change that takes place due to deterioration or accidental impacts. Damage in the member itself may cause reduction in its stiffness or sometimes even total elimination is the member itself from the structure. In such cases in order to assess the severity of the damage and determine the adequacy of the remaining structure it becomes necessary to carry out several analysis.

Many studies were carried out in order to develop a technique that can replace the re- analysis. The method is expected to obtain the new response of the structure whenever there is a change in the cross-sectional properties of any member or members from the response of the old structure. The existing methods can be divided into two groups. These are exact methods and approximate methods.

The theorems of structural variation belong to exact methods that predict the forces and displacements throughout a structure without need of fresh analysis when the physical properties of one or more of its members are altered. It is shown that by means of these theorems a simple elastic analysis of a structure is sufficient to obtain the elastic, non- linear elastic and elastic-plastic response of a number of other structures. The theorems of structural variation carries out the linear elastic analysis of a parent structure under the applied loads and set of unit loading cases required to study the effect of changes in the cross-sectional properties of a member in the structure. It then calculates what is called as variation factors by making use of the member end forces obtained in the previous analysis. The joint displacements and member end forces of the parent structure are then simply multiplied by these variation factors to determine the new joint displacements and member end forces of the new structure with the member or members with modified cross-sectional properties. Later, these theorems are extended to cover finite element structures such as triangular, quadrilateral and cubic finite element structures.

In this study, the theorems of structural variation are extended to rectangular finite elements for plate flexure. The thickness of the plate is assumed to be small compare to its other dimensions and the deflection of the plate under load is assumed to be small compare to its thickness. The unit loading cases required to study the modification of the thickness of a flexural plate element are determined. These unit-loading cases are used to derive an expression from which the variation factors are computed. The joint displacements and nodal forces of the new finite element structures are then simply computed by multiplying the joint displacement and nodal forces of the original finite element structure. A number of numerical examples is considered to demonstrate the application of these theorems.

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