# Finite Element Method Questions and Answers – Elastic Plates – Eigen Values and Time Dependent Problems

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Elastic Plates – Eigen Values and Time Dependent Problems”.

1. Which of the following is the general representation of eigenvalue problems that arise in bending of elastic plates?
a) ([Ke] – ω2[Me0) = 0
b) ([Ke] + ω2[Me0) = 0
c) ([Ke] * ω2[M]Δ0) = 0
d) ([Ke] / ω2[Me]) = 0

Explanation: The general representation of eigenvalue problems that arise in bending of elastic plates is
([Ke] – ω2[Me0) = 0
where,
Ke = element matrix
Me = mass matrix
Δ0 = inertia term

2. Which term is replaced in the eigenvalue equation in case of buckling analysis?
a) Element matrix
b) Mass matrix
c) Inertia term
d) All terms are same, no replacement

Explanation: In order to determine the solution of the compression force at which buckling takes place (buckling analysis), the generalized equation has one small change. The Mass matrix is replaced with the stability matrix and ω2 is replaced with the buckling load.

3. What is the expression for the stability matrix? Answer in accordance with the classical theory of plates.
a) Gij = ∫[N|x dφi / dx dφj / dx + N|y dφi / dy dφj / dy + N|xy (dφi / dx dφj / dy + dφi / dy dφj / dx )]dxdy
b) Gij = ∫[N|x dφi / dx dφj / dx – N|y dφi / dy dφj / dy + N|xy dφi / dx dφj / dy + dφi / dy dφj / dx ]dxdy
c) Gij = ∫[N|x dφi / dx dφj / dx + N|y dφi / dy dφj / dy – N|xy dφi / dx dφj / dy + dφi / dy dφj / dx ]dxdy
d) Gij = ∫[N|x dφi / dx dφj / dx * N|y dφi / dy dφj / dy / N|xy dφi / dx dφj / dy + dφi / dy dφj / dx ]dxdy

Explanation: According to the classical theory of plates, the expression for the stability matrix is given by
Gij = ∫[N|x dφi / dx dφj / dx + N|y dφi / dy dφj / dy + N|xy dφi / dx dφj / dy + dφi / dy dφj / dx ]dxdy
where,
N|x, N|y and N|xy = applied in – plane force
λ = Nx / N|x = Ny / N|y = Nxy / N|xy (Ratio of actual buckling loads and the applied in – plane forces)

4. To solve a time dependant problem, we must approximate the space variables.
a) True
b) False

Explanation: The given statement is false. To solve a time dependant problem, we must approximate the time variables. This helps us to obtain algebraic equations relating Δ at time (t + Δt) to Δ at time (t), where Δt is the time step. Once the solution is obtained, values of velocity and acceleration can be computed.

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