Finite Element Analysis - May 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 Pre and post processing in FEM.(5 marks) 1 (b) Derive shape function for ID quadratic element in natural co-ordinates.(5 marks) 1 (c) Explain the significance of Jacobian matrix.(5 marks) 1 (d) Explain Convergence results.(5 marks) 2 (a) Solve the following differential Equating using Galerkin Method. $$ \dfrac {d^2y}{dx^2} + 3x \dfrac {dy}{dx}- G_y=0 \ \ \ \ 0\ltx\lt1 $$="" boundary="" conditions="" are:="" y(0)="1," y'(1)="0.1" <br=""> Find y(0.2) and compare with exact solution.</x<1>(10 marks) 2 (b) For the given, steel blocks supporting rigid plates shown in figure, determine displacement matrix and stresses in each element.

(10 marks) 3 (a) What do you mean by consistent and jumped mass matrices? Driven the same for linear bar element.(10 marks) 3 (b) Consider the truss shown in figure. Given E=210 GPa and cross section area A=1 cm² for each element. Determine
i) Displacement at each node.
ii) Stresses induced in each element
iii) Reaction at supports

(10 marks) 4 (a) It is required to carry out one dimensional structural analysis of a circular bar of length 'L', fixed at one and carries a point load 'P' at other end. Find the suitable differential equation with required boundary condition (justify) and solve it by using Rayleigh-Ritz method for two linear element.(10 marks) 4 (b) A composite wall consists of three materials, as shown in figure. The outer temperature T₀=20°C. Convection heat transfer takes place on the inner surface of the wall with T_?=800°C and h=30 W/m²°C. Determine temperature distribution in the wall.
K₁=25 W/m-°C
K₂=30 W/m-°C
K₃=70 W/m-°C

(10 marks) 5 (a) The nodal coordinate of the triangular element are as shown in figure. At the interior point P, the x-coordinate is (4,5) and N₁=0.3. Determine N₂, N₃ and y-coordinate of point P.

(10 marks) 5 (b) For a CST element the nodal displacement vector Q^T=[0,0,0,0,2,-0,1] mm. Find the element stress. Take E=200GPa, plate thickness t=5mm and Poisson's ratio=0.3.(10 marks) 6 (a) What are serendipity elements? Derive and graphically represent interpolation functions for 8 nodded Quadrilateral elements.(10 marks) 6 (b) Find the natural frequency of axial vibration of a bar of uniform cross section of 20mm² and length 1m. Take E=2×10⁵ N/mm² and ρ=8000 kg/m³. Take two linear elements.(10 marks)