## Finite Element Analysis - May 2012

### 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.
** 2(b) ** Solve the following differential equation using Rayleigh-Ritz method
$$3\dfrac{d^{2}y}{dx^{2}}-\dfrac{dy}{dx}+8=0\ \cdots\ 0≤x≤1$$ with boundary condition y(0) =1 and y(1) =2.assume cubic Polynomial for trial solution. Find the value at y(2,3) and y(0,8)(10 marks)
** 3(a) ** Evaluate the following integral using Gauss Quadrature.
Compare your answer with exact

$$I=\int_{-1}^{1} \int_{-1}^{1}(r^{3}-1)(s-1)^{2}dr ds$$

n | ϵ | w |

1 | 0.0 | 2 |

2 | ± 0.5774 | 1 |

3 |
± 0.0 ± 0.7746 |
0.8889 0.5556 |

**3(b)**Explain the following :

Convergence requirements

Global, local and natural co - ordinate system (8 marks)

**4(a)**For the bar truss shown in figure, determine the nodal displacement, stresses in each element and reaction at support. Take $$E =2\times 10^{5} \dfrac{N}{mm^{2}}, A= 200mm^{2}$$

(15 marks)

**4(b)**Explain Band width.(8 marks)

**5(a)**Using Direct Stiffness method,determine the nodal displacements of stepped bar shown in

(12 marks)

**5(b)**Derive the shape function for a Quadratic bar element [3 noded 1 dimensional bar] using Lagrangian polynomial in,

Global co - ordinates and

Natural co - ordinates.(8 marks)

**6(a)**Find the shape function for two dimensional Nine rectangular elements mapped into natural coordinates.(12 marks)

**6(b)**The nodal co-ordinates of a triangular element are as shown in figure.The x co-ordinate of interior point P is 3.3 and shape function N

_{1}=0.3.Determine N

_{2}N

_{3}and y co-ordinates of point P

(8 marks)

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

^{2}and length 1m. Take $$E=2\times 10^{5} \dfrac{N}{mm^{2}}$$ and $$\rho =8000\dfrac{kg}{m^{3}}$$

Take 2 linear elements.(10 marks)

**7(b)**Discuss briefly higher order and iso - parametric elements with suitable sketches.(10 marks)

**1(a)**Attempt any four of the following:

Briefly explain application of FEM in Various fields.(5 marks)

**1(b)**Explain principle of minimum Potential Energy.(5 marks)

**1(c)**Explain different sources of error in a typical F.E.M solution.(5 marks)

**1(d)**Briefly explain Node Numbering Scheme.(5 marks)

**1(e)**Explain properties of Global Matrix.(5 marks)

**2(a)**Solve the following differential equation using Galerkins method $$3\dfrac{d ^ {2}u}{dx ^ {2}} - 3u + 4x ^ {2} = 0$$ with boundary condition u(o) = u(1) = 0.Assume Cubic polynomial for approximate solution.(10 marks)