From the Timoshenko beam theory of natural vibrations, using cubic Hermite polynomials approximation, what is the 1^st element of the mass matrix?

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From the Timoshenko beam theory of natural vibrations, using cubic Hermite polynomials approximation, what is the 1^st element of the mass matrix?

(a) $\frac{\rho A}{3}$

(b) $\frac{\rho A}{6}$

(c) 0

(d) $\frac{\rho I}{3}$

The above asked question is from Eigen Value and Time Dependent Problems topic in section Single Variable Problems of Finite Element Method

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The correct option is (a) $\frac{\rho A}{3}$

The explanation is: Using the Timoshenko beam theory applied to natural vibrations, mode shape is approximated using the cubic Hermite polynomials $\psi_i^e$  and $\psi_j^e$. The first element of a mass matrix is $M_{ij}^{11} = \int_{x_a}^{x_b} \rho A \psi_i^e \psi_j^e$ dx, where x is the length of the element.  For the 1^st element, using appropriate values of $\psi_i^e$  and $\psi_j^e$, the term $M_{ij}^{11}$ reduces to $\frac{\rho A}{3}$, where ρ is the density of the beam material, and A is the cross-section area of the beam.

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