# Matrix Structural Analysis

## Energy Methods

Matrix structural analysis is a formulation of energy methods. Trusses, Beams, and Frames can represent by a system of equations. Energy methods formulate Stiffness and Flexibility methods. Flexibility or force method uses a unit force applied to the structure. In contrast, the stiffness method applies a unit displacement to calculate a response.

## Principal of Virtual Work

The principle of virtual work is used in the formulation of the stiffness method. Virtual work method is an energy method that utilizes strain energy and virtual displacements to calculuate a response, also is known as the Maxwell-Mohr method. The application of this method requires inducing a virtual displacement at a point other than a reaction. Or way say that we “perturb” the system and generate a response i.e all displacements for stable structure. Stiffness matrix (K) is generated and multiplied by the column vector of unknown displacements to equal to applied forces;

An inverted stiffnes (K) matrix is multipled it by a force vector to obtain unknow displacements.

Determinates is a way to obtain an inverse of a matrix, but more usefully by that of linear algebraic methods. One such method is Gauss-Jordan elimination mathematics. Guass Jordan Elimiation is very effeicient in the reducation of matrix to the identiy matrix and the direct solution of the displacements. For more information see Linear Algebra by Gilbert Strang. Or MIT Videos https://www.youtube.com/watch?v=ZK3O402wf1c

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