MU Mechanical Engineering (Semester 8)
Finite Element Analysis
May 2012
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a) Attempt any four of the following:
Briefly explain application of FEM in Various fields.
5 M
1(b) Explain principle of minimum Potential Energy.
5 M
1(c) Explain different sources of error in a typical F.E.M solution.
5 M
1(d) Briefly explain Node Numbering Scheme.
5 M
1(e) Explain properties of Global Matrix.
5 M

2(a) Solve the following differential equation using Galerkin's method \[3\dfrac{d^{2}u}{dx^{2}}-3u+4x^{2}=0\] with boundary condition u(o) =u(1) =0.Assume Cubic polynomial for approximate solution.
10 M
2(b) Solve the following differential equation using Rayleigh-Ritz method \[3\dfrac{d^{2}y}{dx^{2}}-\dfrac{dy}{dx}+8=0\ \cdots\ 0≤x≤1\] with boundary condition y(0) =1 and y(1) =2.assume cubic Polynomial for trial solution. Find the value at y(2,3) and y(0,8)
10 M

3(a) Evaluate the following integral using Gauss Quadrature. Compare your answer with exact
\[I=\int_{-1}^{1} \int_{-1}^{1}(r^{3}-1)(s-1)^{2}dr ds\]
n ϵ w
1 0.0 2
2 ± 0.5774 1

± 0.0

± 0.7746



12 M
3(b) Explain the following:
Convergence requirements
Global,local and natural co-ordinate system
8 M

4(a) For the bar truss shown in figure, determine the nodal displacement, stresses in each element and reaction at support. Take \[E =2\times 10^{5} \dfrac{N}{mm^{2}}, A= 200mm^{2}\]
15 M
4(b) Explain Band width.
8 M

5(a) Using Direct Stiffness method,determine the nodal displacements of stepped bar shown in

12 M
5(b) Derive the shape function for a Quadratic bar element [3 noded 1 dimensional bar]using Lagrangian polynomial in,
Global co-ordinates and
Natural co-ordinates.
8 M

6(a) Find the shape function for two dimensional Nine rectangular elements mapped into natural coordinates.
12 M
6(b) The nodal co-ordinates of a triangular element are as shown in figure.The x co-ordinate of interior point P is 3.3 and shape function N1=0.3.Determine N2 N3 and y co-ordinates of point P

8 M

7(a) Find the natural frequency of axial vibration of a bar of uniform cross section of 20mm2 and length 1m. Take \[E=2\times 10^{5} \dfrac{N}{mm^{2}}\] and \[\rho =8000\dfrac{kg}{m^{3}}\]
Take 2 linear elements.
10 M
7(b) Discuss briefly higher order and iso-parametric elements with suitable sketches.
10 M

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