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Ref. No. Ex/ML42D3/60/2006

B.Mech Engineering Final Examination, 20062

ndsemester

Subject: Elective III (Finite Element Analysis III)

Time: Three hours

Answer anyfive questions

(All questions carry equal marks)

Question 1

Consider a two-dimensional quasi-harmonic equation. On one part of the boundary

the dependent variable is specified. On the other,n

is specified.

a. Using Galerkins principle find out the finite element equations for an element

which has one of its edge on the boundary, wheren

is specified.

b. What would be the form of the above finite element equations if a 3-node

triangular element is used?

Question 2

Consider an axisymmetric quasi-harmonic equation.

a. Describe in details the finite element formulation for such a problem.

b. How do you compute the finite element matrices for a 3-node triangular element?

Please mention about the integration rule you propose to use.

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Question 3

Consider a set of ordinary differential equation resulted from finite element discretization

of a heat transfer problem as follows: -

[C]{ }+ [K]{}={R(t)}

a. What do you understand by explicit and implicit time integration schemes?

b. Describe the popular family of implicit and explicit time-integration schemes

in details along with the steps for their implementation.

Question 4

a. Write down the shape functions for a 4-node quadrilateral isoparametric finite

element

b. Show that a straight edge of the above element is mapped into a straight edge

in the object space

c. Show that if a four node quadrilateral element is mapped into a square in the

object space, the Jacobian matrix is a diagonal one with two equal entries.

Question 5

a. Write down the Helmholtz equation with homogenous boundary conditions

b. Consider free vibration of an elastic string of length l, which is fixed at two ends.

The string is subjected to a constant tension T. Write down the variational

formulation for the problem.

c. Take two equal linear elements and find out an estimate of the first natural

frequency.

Question 6

Consider finite element formulation for a Kirchoff plate-bending element.

a. What are the relevant stress and strain components for this problem?

b. Explain why curvature terms and stress-resultant terms are considered in place of

strains and stresses respectively. In the process of explanation also construct the

matrix relating stress-resultants to curvature.

c. Sketch a rectangular Kirchoff plate-bending element and show the degrees of

freedom at a node.