Question

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

I had been asked this question in an interview for job.

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

Answer 1

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.