# From the Euler-Bernoulli beam theory of natural vibrations, using cubic Hermite polynomials approximation, what is the 1^st element of the stiffness matrix?

From the Euler-Bernoulli beam theory of natural vibrations, using cubic Hermite polynomials approximation, what is the 1^st element of the stiffness matrix?

(a) $\frac{12EI}{h^3}$

(b) $\frac{12EA}{h^3}$

(c) $\frac{12EA}{h}$

(d) $\frac{12AI}{h^3}$

This question was posed to me during an internship interview.

Origin of the question is Eigen Value and Time Dependent Problems in chapter Single Variable Problems of Finite Element Method

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Right option is (a) $\frac{12EI}{h^3}$

Explanation: In the formulation of the Euler-Bernoulli beam theory, there are two degrees of freedom at a point, w and $\frac{dw}{dx}$. Typically, the finite element model of this theory uses cubic polynomial. The first element of the stiffness matrix is $\frac{12EI}{h^3}$, where E is Young’s modulus, I is the area moment of inertia and h is the length of the element.

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