Finite Element Methods - Jun 2015
Mechanical Engg. (Semester 6)
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) Write the stress- strain relationship for both plane stress and plane strain problems.(6 marks) 1 (b) Discuss the types of elements based on geometry.(6 marks) 1 (c) Explain the various application fields of finite method.(8 marks) 2 (a) Derive an expression for total potential energy of an elastic body subjected to body force, traction force and point force.(8 marks) 2 (b) Using Rayleigh's Ritz method, determine the displacement at mid-point and stress in linear one-dimensional rod as shown in fig Q2(b). Use second degree polynomial approximation for the displacement. (12 marks) 3 (a) Write an interpolation polynomial for linear, quadratic and cubic element.(6 marks) 3 (b) Explain simplex, complex and multiples elements using element shapes.(6 marks) 3 (c) Derive the shape functions for a CST element.(8 marks) 4 (a) Solve for nodal displacement and elemental stress for the following. FigQ4(a), show a thin plate of uniform thickness of 1 mm, Young's modulus =200GPa, weight density of the plate=76.6 × 10-6 N/mm3. In addition to its weight it is subjected to a point load of 100 N at its midpoint and model the plate with 2 bar elements (10 marks) 4 (b) Determine the nodal displacement, reactions and stresses for Fig Q4 (b) using Penalty approach. Take E=210GPa, area =250m2. (10 marks) 5 (a) Distinguish between lower and higher order elements.(8 marks) 5 (b) Explain the concept ISO, sub and super parametric elements and their uses.(6 marks) 5 (c) Write a note on 2- point integration rule for 1D and 2D problems.(6 marks) 6 (a) Derive an expression for stiffness matrix of truss element.(8 marks) 6 (b) For the pin-joined configuration shown in Fig Q6(b) formulate the stiffness matrix. Also determine nodal displacement and stress in each element. (12 marks) 7 (a) Derive the Hermite shape function for a beam element.(8 marks) 7 (b) For the beam and loading shown in Fig Q7(b), determine the slopes at 2 and 3, vertical deflection at the mid points of the distributed load. Take E=200GPa, I=4 × 106 mm4. (12 marks) 8 (a) Discuss the derivation of one dimensional heat transfer in thin fin.(8 marks) 8 (b) Determine the temperature distribution through the composite wall, subjected to convection heat transfer on the right side surface, with convective heat transfer co-efficient shown in Fig Q8(b). The ambient temperature is -5°C. Assume unit area. (12 marks)