Question

The element stiffness matrix (k) for beam element is given by which of the following expressions?

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

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

This question was posed to me by my school teacher while I was bunking the class.

Asked question is from Beams and Frames topic in portion Beams and Frames of Finite Element Method

Answer 1

Correct choice is (a) \(\frac{EI}{l^3}\begin{bmatrix}12&6l&-12&6l\\6l&4l^2&-6l&2l^2\\-12&-6l&12&-6l\\6l&2l^2&-6l&4l^2\end{bmatrix}\)

For explanation I would say: The value of stiffness matrix is given by

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

Here l is length of beam element, and E is the modulus of elasticity, and I moment of inertia of the beam element. The stiffness matrix of the beam element affects the displacement of the nodes and their interpolation in the beam element. The beam element stiffness matrix is a symmetric matrix.