# Which of the following expressions must be added to the element stiffness matrix for beams supported on elastic supports, where l is length of beam element, and s is the stiffness per unit length of element?

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Which of the following expressions must be added to the element stiffness matrix for beams supported on elastic supports, where l is length of beam element, and s is the stiffness per unit length of element?

(a) $\frac{sl}{420}\begin{bmatrix}156&22l&54&-13l\\22l&4l^2&13l&-3l^2\\54&13l&156&-22l\\-13l&-3l^2&-22l&4l^2\end{bmatrix}$

(b) $\frac{EI}{l^2}\begin{bmatrix}12&-6l&-12&6l\\-6l&4l^2&6l&2l^2\\-12&6l&12&6l\\6l&2l^2&6l&4l^2\end{bmatrix}$

(c) $\frac{EI}{l^3}\begin{bmatrix}12&-6l&-12&6l\\-6l&4l^2&6l&2l^2\\-12&6l&12&6l\\6l&2l^2&6l&4l^2\end{bmatrix}$

(d) $\frac{EI}{420}\begin{bmatrix}156&22l&54&-13l\\22l&4l^2&13l&-3l^2\\54&13l&156&-22l\\-13l&-3l^2&-22l&4l^2\end{bmatrix}$

I have been asked this question during a job interview.

This interesting question is from Beams on Elastic Support topic in section Beams and Frames of Finite Element Method

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The correct choice is (a) $\frac{sl}{420}\begin{bmatrix}156&22l&54&-13l\\22l&4l^2&13l&-3l^2\\54&13l&156&-22l\\-13l&-3l^2&-22l&4l^2\end{bmatrix}$

Best explanation: The value of stiffness matrix to be added for beams with elastic support is

$\frac{sl}{420}\begin{bmatrix}156&22l&54&-13l\\22l&4l^2&13l&-3l^2\\54&13l&156&-22l\\-13l&-3l^2&-22l&4l^2\end{bmatrix}$

Here l is length of beam element, and s is the stiffness per unit length of element. The stiffness matrix of the beam element affects the displacement of the nodes and their interpolation in the beam element.

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