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 RayleighRitz 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
$$I=\int_{1}^{1} \int_{1}^{1}(r^{3}1)(s1)^{2}dr ds$$
n 
ϵ 
w 
1 
0.0 
2 
2 
± 0.5774 
1 
3 
± 0.0
± 0.7746

0.8889
0.5556

(12 marks)
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 coordinates of a triangular element are as shown in figure.The x coordinate of interior point P is 3.3 and shape function N_{1}=0.3.Determine N_{2} N_{3} and y coordinates 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)