## Finite Element Methods - Dec 2014

### 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)** Obtain an equilibrium equation of a 3-D elastic body subjected to a body force.(8 marks)
**1 (c)** Explain the general description of finite element method.(6 marks)
**1(b)** Discuss the types of elements based on geometry.(6 marks)
**2 (a)** Derive an expression for Total potential energy of an elastic body subjected to body force, traction force and a point force(8 marks)
**2 (b)** Using Raleigh's Ritz method find a deflection of a simply supported beam of length L subjected to a uniformly distributed load of P_{0} N/m.(12 marks)
**3 (a)** Write an interpolation polynomial for liner quadratic and cubine element.(6 marks)
**3 (b)** Obtain an expression for a strain displacement matrix of a rectangular element.(14 marks)
**4 (a)** Determine the nodal displacements, reactions and stresses for the Fig. Q4 (a) using penalty approach. Take E =210GPa, Area=250mm^{2}.
(12 marks)
**4 (b)** Find the nodal displacement stress and strain of the system shown in fig Q4(b).Take E=70GPa, Area -1m^{2}.
(8 marks)
**5 (a)** Find the shape functions of a 2-D quadrilateral quadratic (9 noded) element.(14 marks)
**5 (b)** With a sketch define ISO, Sub and super parametric elements(6 marks)
**6 (a)** Obtain an expression for stiffness matrix of a truss element.(8 marks)
**6 (b)** Find the nodal displacement, stress and reaction of truss element shown in fig Q6(b). take E=70GPa, Area =200mm^{2}.
(12 marks)
**7 (a)** Derive the Hermine shape function of a n beam element(8 marks)
**7 (b)** For the beam and loading shown in fig Q7(b) determine the slopes at 2 and 3 and the vertical deflection at the midpoints of the distributed load. Take E=200 Gpa, I=4×10^{6} mm^{4}.
(12 marks)
**8 (a)** Discuss the derivation of one dimensional heat transfer in thin films(8 marks)
**8 (b)** A composite wall consists of 3 material shown in fig Q8(b). the outer temperature is T_{0}=20°C, determine the temperature distribution in the wall. Convection heat transfer takes place at inner surface with T_{&infty;}=800°C. Take h=25 w/m^{2}°C area=1m^{2}.
(12 marks)