## Finite Element Analysis - Dec 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 applications of FEA in various fields.(5 marks)
**1 (b)** State different types of Boundary conditions.(5 marks)
**1 (c)** Explain with sketches: type of elements.(5 marks)
**1 (d)** Explain Shape function graphically for one dimensional Linear and quadratic element.(5 marks)
**1 (e)** Explain Gauss Elimination Method using an example.(5 marks)
**2 (a)** Solve following differential equation $$ \dfrac {d^2y}{dx^2}+3x \dfrac {dy}{dx}- 6y=0; \ \ 0\le x\le 1 $$ Bcs: y(0)=0 and y'(1)=0.1: Find y(0.2): using variational method and compare with exact solution.(12 marks)
**2 (b)** Evaluate following integral $ \int^1_{-1}(3^x - x) dx $ Using (a) Newton Cotes Method using 3 sampling points.

(b) Three points Gauss Quadrature

r |
W_{1} |
W_{2} |
W_{3} |
W_{4} |

1 | 1 | |||

2 | 1/2 | 1/2 | ||

3 | 1/6 | 4/6 | 1/6 | |

4 | 1/8 | 3/8 | 3/8 | 1/8 |

r |
ε_{1} |
W_{1} |

1 | 0.00 | 2.00 |

2 | 0.5773 | 1.00 |

3 | 0.00 | 0.8889 |

0.7746 | 0.5556 |

**3 (a)**Find the natural frequency of axial vibrations of a bar of uniform cross section of 20 mm

^{2}and length 1m. Take, E=2×10

^{5}N/mm

^{2}and ρ=8000 kg/m

^{3}. Consider two linear elements.(10 marks)

**3 (b)**Using Direct Stiffness method, determine the nodal displacements of stepped bar shown in figure. Take G=100 GPa. (10 marks)

**4 (a)**Explain Lumped and consistent mass matrix.(6 marks)

**4 (b)**Analysis the plane truss for nodal displacement, element stresses and strains. Take, P

_{1}=5 KN, P

_{2}=2 KN, E=180 GPa. A=6 cm

_{2}for all elements. (14 marks)

**5 (a)**Solve following differential equation $ \dfrac {d^2y}{dx^2}-10x^2 = 5 \ \ 0\le x \le 1 $

BCs: y(0)=y(1)=0. Using Rayleigh-Ritz method, mapped over entire domain using one parameter method.(12 marks)

**5 (b)**Find the shape function for two dimensional eight noded element.(8 marks)

**6 (a)**Coordinates of nodes of a quadrilateral element are as shown in the figure below. Temperature distribution at each node is computed as T

_{1}=100°C, T

_{2}=60°C, T

_{3}=50° C and T

_{4}=90°C. Compute temperature at point P{2.5, 2.5} (10 marks)

**6 (b)**What are the h and p version of finite element method?(7 marks)

**6 (c)**Convergence requirement.(3 marks)