Question Paper: Finite Element Analysis : Question Paper Dec 2015 - Mechanical Engineering (Semester 6) | Mumbai University (MU)
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## Finite Element Analysis - Dec 2015

### 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.
1 (a) Explain applications of FEA in various fields.(5 marks) 1 (b) State different types of Boundary conditions.(5 marks) 1 (c) Explain with sketches: type of elements.(5 marks) 1 (d) Explain Shape function graphically for one dimensional Linear and quadratic element.(5 marks) 1 (e) Explain Gauss Elimination Method using an example.(5 marks) 2 (a) Solve following differential equation $$\dfrac {d^2y}{dx^2}+3x \dfrac {dy}{dx}- 6y=0; \ \ 0\le x\le 1$$ Bcs: y(0)=0 and y'(1)=0.1: Find y(0.2): using variational method and compare with exact solution.(12 marks) 2 (b) Evaluate following integral $\int^1_{-1}(3^x - x) dx$ Using (a) Newton Cotes Method using 3 sampling points.

 r W1 W2 W3 W4 1 1 2 1/2 1/2 3 1/6 4/6 1/6 4 1/8 3/8 3/8 1/8

 r ε1 W1 1 0.00 2.00 2 0.5773 1.00 3 0.00 0.8889 0.7746 0.5556
(8 marks) 3 (a) Find the natural frequency of axial vibrations of a bar of uniform cross section of 20 mm2 and length 1m. Take, E=2×105 N/mm2 and ρ=8000 kg/m3. Consider two linear elements.(10 marks) 3 (b) Using Direct Stiffness method, determine the nodal displacements of stepped bar shown in figure. Take G=100 GPa. (10 marks) 4 (a) Explain Lumped and consistent mass matrix.(6 marks) 4 (b) Analysis the plane truss for nodal displacement, element stresses and strains. Take, P1=5 KN, P2=2 KN, E=180 GPa. A=6 cm2 for all elements. (14 marks) 5 (a) Solve following differential equation $\dfrac {d^2y}{dx^2}-10x^2 = 5 \ \ 0\le x \le 1$
BCs: y(0)=y(1)=0. Using Rayleigh-Ritz method, mapped over entire domain using one parameter method.
(12 marks)
5 (b) Find the shape function for two dimensional eight noded element.(8 marks) 6 (a) Coordinates of nodes of a quadrilateral element are as shown in the figure below. Temperature distribution at each node is computed as T1=100°C, T2=60°C, T3=50° C and T4=90°C. Compute temperature at point P{2.5, 2.5} (10 marks) 6 (b) What are the h and p version of finite element method?(7 marks) 6 (c) Convergence requirement.(3 marks)