Finite Element Analysis - May 2012

Mechanical Engineering (Semester 8)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full 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 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 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
$$I=\int_{-1}^{1} \int_{-1}^{1}(r^{3}-1)(s-1)^{2}dr ds$$

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)

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