1(a)
What is the difference between shape function and interpolation function?Derive the shape function for cubic bar element in local co-ordinates and sketch the same along the length of the element.

10 M

1(b)
A quadrilateral element is defined by the co-ordinates (1,4),(4,2),(5,6) and (2,7).The temperature at the nodes are 20°C,30°C,40°C, and 25°C respectively.Determine the temperature at apoint P(3,4)

10 M

2(a)
The following differential equation arises in connection neat transfer through a rectangular fig

\[-KA \dfrac{d^{2}T}{dx^{2}} +hp(T-T_{infty})=0; 0 \leq x \leq 1

BCS: T(0) =T(0)\]

\[ KA \dfrac{dT}{dx}+hA(T-T_{infty})]_{x=L}=0\]

Develop an element matrix equation for the same linear element.

After getting element matrix equation use following data to find nodal temperature,K=385 W/m.k, h =25 W/m

Take 4 elements (linear)

\[-KA \dfrac{d^{2}T}{dx^{2}} +hp(T-T_{infty})=0; 0 \leq x \leq 1

BCS: T(0) =T(0)\]

\[ KA \dfrac{dT}{dx}+hA(T-T_{infty})]_{x=L}=0\]

Develop an element matrix equation for the same linear element.

After getting element matrix equation use following data to find nodal temperature,K=385 W/m.k, h =25 W/m

^{2}-k to =100°C T=20°C,L=100mm,b=5mm and t=1mmTake 4 elements (linear)

15 M

2(b)
Explain different types of nodes with suitable examples.

5 M

3(a)
Determine the displacement at the nodes principle of minimum potential energy and reaction at the supports

Use; K

Use; K

_{1}=100N/mm, K_{3}=150N/mm and K_{4}=200 N/mm.

8 M

3(b)
Analyse completely the problem given below; Assume,E =200 Gpa

8 M

3(c)
What do you mean by pre-processing and post-processing related to FEA software?

4 M

4(a)
Analyse the following truss for displacements reaction and stresses

For each element

E=100 Gpa

A=50cm

For each element

E=100 Gpa

A=50cm

^{2}

14 M

4(b)
Explain different types of boundary condition with suitable examples.

6 M

5(a)
Derive shape function for nine noded quadrilateral element in natural co-ordinates.

10 M

5(b)
Find the natural frequencies of axial vibration for a bar as shown in figure below using one linear element by using consistent and lumped mass matrices.

take E = 2 × 10

take E = 2 × 10

^{11}N/m^{2 }, ρ = 7800 kg/m^{3}

10 M

6(a)
Solve the following differential equation by

Galerkin method

Rayleigh-ritz method over entire domain

\[\dfrac{d^{2}u}{dx^{2}}-9u=x^{3}; 0\pm x\pm 1

BCS; U(0)=0 and u(1)=2

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

Galerkin method

Rayleigh-ritz method over entire domain

\[\dfrac{d^{2}u}{dx^{2}}-9u=x^{3}; 0\pm x\pm 1

BCS; U(0)=0 and u(1)=2

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

14 M

6(b)
Explain subparametric, isoparamertric and superparametric elements.

6 M

7(a)
Evaluate for following integral by using 2 × 2 Guass Quadratic rule

\[I= \int_{y=4}^{6} \int_{x=2}^{2} (1-x)^{2}(4-2)^{2}dx dy

Compare your answer with exact solution:

\[I= \int_{y=4}^{6} \int_{x=2}^{2} (1-x)^{2}(4-2)^{2}dx dy

Compare your answer with exact solution:

No.of Sampling Points | ϵi | Wi |

1 | 0 | 2 |

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

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

10 M

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

5 M

7(c)
Compare FEM with classical methods.

5 M

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