## 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.

Properties | Steel | Aluminium | Brass |

C/S Area (mm^{2}) |
200 | 370 | 370 |

E (N/mm^{2}) |
2×10^{3} |
7×10^{4} |
8.8×10^{4} |

(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 cm^{2} 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 T_{0}=20°C. Convection heat transfer takes place on the inner surface of the wall with T_{?}=800°C and h=30 W/m^{2}°C. Determine temperature distribution in the wall.

K_{1}=25 W/m-°C

K_{2}=30 W/m-°C

K_{3}=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 N_{1}=0.3. Determine N_{2}, N_{3} 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 20mm^{2} and length 1m. Take E=2×10^{5} N/mm^{2} and ρ=8000 kg/m^{3}. Take two linear elements.(10 marks)