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Finite Element Analysis : Question Paper May 2012 - Mechanical Engineering (Semester 6) | Mumbai University (MU)

## Finite Element Analysis - May 2012

### Mechanical Engineering (Semester 6)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
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 marks) 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 3 ± 0.0 ± 0.7746 0.8889 0.5556
(12 marks) 3(b) Explain the following :
Convergence requirements
Global, local and natural co - ordinate system
(8 marks)
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 marks)
4(b) Explain Band width.(8 marks) 5(a) Using Direct Stiffness method,determine the nodal displacements of stepped bar shown in

(12 marks)
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 marks)
6(a) Find the shape function for two dimensional Nine rectangular elements mapped into natural coordinates.(12 marks) 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 marks)
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 marks)
7(b) Discuss briefly higher order and iso - parametric elements with suitable sketches.(10 marks) 1(a) Attempt any four of the following:
Briefly explain application of FEM in Various fields.
(5 marks)
1(b) Explain principle of minimum Potential Energy.(5 marks) 1(c) Explain different sources of error in a typical F.E.M solution.(5 marks) 1(d) Briefly explain Node Numbering Scheme.(5 marks) 1(e) Explain properties of Global Matrix.(5 marks) 2(a) Solve the following differential equation using Galerkins 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 marks)