Matrix Structural Analysis

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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. Stiffness method is based on the utilizing the a virtual displacement (perturbation) of a node and calculating the system response.

Principal of Virtual Work

Stiffness method is formulated by utilizing the principle of virtual work, similarly an energy method that utilizes strain energy and virtual displacements to calculate 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;

\begin{bmatrix} P \end{bmatrix} = \begin{bmatrix} K \end{bmatrix}*\begin{bmatrix} U \end{bmatrix}

\begin{bmatrix} Pxa\\Pya \\Pxb \\Pyb\\ \end{bmatrix} =\begin{bmatrix} k11 & k12 & k13 & k14 \\ k21 & k22 & k23 & k24\\ k31 & k32 & k33 & k34\\ k41 & k42 & k43 & k44 \end{bmatrix} *\begin{bmatrix} ux\\vx \\uy\\ vy \end{bmatrix}

Unknown displacements are derived by multiplying the inverting the stiffness (K) matrix by the force vector.

\begin{bmatrix} U \end{bmatrix} = \begin{bmatrix} K \end{bmatrix}^{-1}*\begin{bmatrix} P \end{bmatrix}

\begin{bmatrix} ux\\vx\\uy \\vy\\ \end{bmatrix} =\begin{bmatrix} k11 & k12 & k13 & k14 \\ k21 & k22 & k23 & k24\\ k31 & k32 & k33 & k34\\ k41 & k42 & k43 & k44 \end{bmatrix} ^{-1}*\begin{bmatrix} Pxa\\Pya \\Pxb\\ Pyb \end{bmatrix}

One such method is Gauss-Jordan elimination mathematics. Gauss Jordan Elimination is very efficient in the reduction of matrix to the identity 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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