1(a)
Derive transformation matrix for a plane truss element,relating displacement in the local(element)coordinates and global coordinates.

5 M

1(b)
Explain different types of boundary condition giving examples.

5 M

1(c)
What is the difference between shape function and interpolation function?Derive and sketch the same for linear bar element.

5 M

1(d)
What do you mean by consistent and lumped mass matrices?Derive the same for linear bar element.

5 M

2(a)
Solve the following differential equation by:

Collocation method

Collocation method

5 M

2(b)
Galerkin method.

5 M

2(c)
Rayleigh ritz method mapped over entire domain

\[\dfrac{d^{2}u}{dx^{2}}+u=x^{2}\\] 0&leq ;x ≤ 1

BcS; u(0)=0 \[\dfrac{du}{dx}|_{x=1}=1\\]

Compare the answer with exact solution at x =0.25,0.5,0.75 and 1.

\[\dfrac{d^{2}u}{dx^{2}}+u=x^{2}\\] 0&leq ;x ≤ 1

BcS; u(0)=0 \[\dfrac{du}{dx}|_{x=1}=1\\]

Compare the answer with exact solution at x =0.25,0.5,0.75 and 1.

10 M

3(a)
The following differential equation arises in connection with heat transfer in an insulated rod:

\[\dfrac{d}{dx} \left(-k\dfrac{dT}{dx}\right)=q\] for 0≤x≤L

BcS;T(0)=To\left[k\dfrac{dT}{dx}+?(T-T_{&infty;})+q\right]_{x=L}=0\]

Where T is the temperature, K is the thermal conductivity and q the heat generation.Take the following values for the data;

L=0.1m,K=0.01 w/m -°C,β=25 W/m-°C,q=q=0

To =50° and T

\[\dfrac{d}{dx} \left(-k\dfrac{dT}{dx}\right)=q\] for 0≤x≤L

BcS;T(0)=To\left[k\dfrac{dT}{dx}+?(T-T_{&infty;})+q\right]_{x=L}=0\]

Where T is the temperature, K is the thermal conductivity and q the heat generation.Take the following values for the data;

L=0.1m,K=0.01 w/m -°C,β=25 W/m-°C,q=q=0

To =50° and T

_{∞}=5°C. Solve the problem using two linear finite elements for temperature values at x =L/2 and X=L. Derive the equation which you use.
14 M

3(b)
Explain the principle of minimum total potential with suitable example.

6 M

4(a)
Find the deflection and slopes at nodes and reaction at support for the beam as shown in figure below.Take EI=4000KN-m

^{2}

10 M

4(b)
Find the heat transfer per unit area through the composite wall as shown in figure below.

10 M

5(a)
What do you understand by Jacobin matrix? Derive the same for four-noded quadrilateral element.

15 M

5(b)
The triangular element has nodes coordinates (10,10),(40,20)and (30,50) for nodes 1,2,and 3 respectively.For the point P located inside the triangle,determine the x and y coordinates if the shape function N

_{1}and N_{2}are 0.15 and 0.25 respectively.
5 M

6(a)
Analyse the following truss completely,i.e for displacement,reaction and stresses
The nodes where load is applied settles by an amount &delta =50mm as shown in figure.

14 M

6(b)
Write any one algorithm for the solution of Eigen values problem.

6 M

7(a)
Evaluate the following integral by Newton-cotes and Gauss-quadrature formula.

\[I =\int_{0}^{le}\dfrac{dN1}{dx}0\frac{dN2}{dx}\bar{dx}\\]

Where,\[N1 =\left(1-\dfrac{\bar{x}}{le}\right)\left(1-\dfrac{\bar{2x}}{le}\right)\\]

\[N_{2}=\dfrac{\bar{4x}}{le}\left(1-\dfrac{\bar{x}}{le}\right)\\],br>Check your answer with exact value.

Use following table

For Newton-cotes.

For Gauss-quadrature

\[I =\int_{0}^{le}\dfrac{dN1}{dx}0\frac{dN2}{dx}\bar{dx}\\]

Where,\[N1 =\left(1-\dfrac{\bar{x}}{le}\right)\left(1-\dfrac{\bar{2x}}{le}\right)\\]

\[N_{2}=\dfrac{\bar{4x}}{le}\left(1-\dfrac{\bar{x}}{le}\right)\\],br>Check your answer with exact value.

Use following table

For Newton-cotes.

No.of sampling points | W1 | W2 | W3 | W4 |

1 | 1 | - | - | - |

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

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

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

For Gauss-quadrature

No.of sampling points | ? i | Wi |

1 | 0 | 2 |

2 | \[\pm \dfrac{1}{\sqrt{3}} | 1 |

3 | \[\dfrac{0}{\pm\sqrt{0.6}}\\] |
8/9 5/9 |

12 M

7(b)
Compare FEM with classical methods.

4 M

7(c)
What are the sources of error in FEM.

4 M

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