1 (a)
Derive Lagrange's linear shape functions. What are the characteristics of shape functions? State the difference between shape functions and interpolation functions also plot the shape functions along length of the element.

10 M

1 (b)
Explain followings:

i) Compatibility

ii) Band width

iii) Convergence criteria

iv) Node numbering scheme

v) Aspect ratio.

i) Compatibility

ii) Band width

iii) Convergence criteria

iv) Node numbering scheme

v) Aspect ratio.

10 M

2 (a)
Using Newton Cote's formula find values of M

_{11}and M_{12}. Where \( M = \int^{he}_0 x^2 \phi 1\phi_1 dx \). \( \phi 1 = [1- (x/h_c)] \text{ and } \phi_2=(x/h_c) \) Compare your answer with exact.
10 M

2 (b)
Solve the following governing differential equation. \[ \dfrac{d^2y}{dx^2}- 10 x^2 = 5 \ \ BCs: 0\le x \le 1; \ y(0)= y(1)=0 \] using-

i) Galerkin method with cubic polynomial for approximate function

ii) Rayleigh Ritz method mapped over entire domain using one parameter.

Compare answers with exact.

i) Galerkin method with cubic polynomial for approximate function

ii) Rayleigh Ritz method mapped over entire domain using one parameter.

Compare answers with exact.

10 M

3 (a)
Following data is given for one-dimensional, steady state, conduction heat transfer through a composite wall: Global P.V.s. T

_{in}=800 °C. T_{out}=300 °C, K_{1}=10 W/m °C, K_{2}=50 W/m °C, K_{3}=5 W/m °C, L_{1}=0.10m, L_{2}=0.20 m, L_{3}=0.05m. Find the upknowns.

10 M

3 (b)
Derive shape functions for nine noded quadrilateral elements in natural coordinates.

10 M

4 (a)
Analyse following fluid network completely using EME directly. Where Q, b are constants, R=branch resistance constant ΔP=Rq, ΔP = pressure drop. Establish Equation for steady state pressure and flow distribution.

10 M

4 (b)
Find natural frequency of axial vibration of a bar of uniform cross section of 30 mm

^{2}and length 2m. Take, E=2×10^{5}N/mm^{2}and ρ=8000 kg/m^{3}. Take two linear elements.
10 M

5 (a)
Analyse following problem completely. Assume, E=200 Gpa.

10 M

5 (b)
Calculate the linear interpolation functions for the linear triangular element shown in figure below -

10 M

6 (a)
Define:

i) Primary variables

ii) Element

iii) Nodes

iv) Natural coordinates

v) Convergence

i) Primary variables

ii) Element

iii) Nodes

iv) Natural coordinates

v) Convergence

5 M

6 (b)
Develop element matrix equation for the most general using Rayleigh Ritz method. \( \dfrac {d}{dx} \left ( AE \dfrac {du}{dx} \right )+ f=0; \ 0\le x \le 1 \) Where, AE and f are constants. Use Lagrange's linear shape function to derive EME and use following data to solve global matrix equation.

Take 3 linear elements, A=0.2 m

At x=0, u=0

At x=L=12 cms, P=external force = 10 KN

f=0

Find displacements at nodes and the reactions.

Take 3 linear elements, A=0.2 m

^{2}for each element, E=100GPa for each element.At x=0, u=0

At x=L=12 cms, P=external force = 10 KN

f=0

Find displacements at nodes and the reactions.

15 M

7 (a)
Solve for complete analysis. Where E=2×10

^{7}N/cm^{2}ρ=75×10^{-3}N/cm^{2}.

10 M

7 (b)
Explain in short:

i) Serendipity approach

ii) Jacobian matrix

iii) Consistent and lumped mass matrices

iv) Boundary conditions

v) Iso-parametric elements.

i) Serendipity approach

ii) Jacobian matrix

iii) Consistent and lumped mass matrices

iv) Boundary conditions

v) Iso-parametric elements.

10 M

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