1(a)
Attempt any four of the following:

Briefly explain application of FEM in Various fields.

Briefly explain application of FEM in Various fields.

5 M

1(b)
Explain principle of minimum Potential Energy.

5 M

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

5 M

1(d)
Briefly explain Node Numbering Scheme.

5 M

1(e)
Explain properties of Global Matrix.

5 M

2(a)
Solve the following differential equation using Galerkin's 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 M

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 M

3(a)
Evaluate the following integral using Gauss Quadrature.
Compare your answer with exact

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

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

n | ϵ | w |

1 | 0.0 | 2 |

2 | ± 0.5774 | 1 |

3 |
± 0.0 ± 0.7746 |
0.8889 0.5556 |

12 M

3(b)
Explain the following:

Convergence requirements

Global,local and natural co-ordinate system

Convergence requirements

Global,local and natural co-ordinate system

8 M

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 M

4(b)
Explain Band width.

8 M

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

12 M

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.

Global co-ordinates and

Natural co-ordinates.

8 M

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

12 M

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 M

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

Take 2 linear elements.

^{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 M

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

10 M

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