## 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/mm^{3}. 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 =250m^{2}.
(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 × 10^{6} mm^{4}.
(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)