1 (a) (i)
Explain different types of nodes with suitable examples.

5 M

1 (a) (ii)
Give any two fields of engineering problems governed by the second-order differential equation and its parameters.

5 M

1 (b)
Use Gaussin Legandre Quadrature Method and for the integral in two dimension (2×2) and verify exactness by an analytical integration. \[ I=\int^{+1}_{-1}\int^{+1}_{-1} (2x^4 +3y^2 +x^3y)dx \ dy \]

10 M

2 (a)
Solve the following differential equation by using Galerkin's method. \[ \dfrac {d^2u}{dx^2}=f_c \] where the domain is changing from (0≤x≤L) and B.C.'s of the problem: \[ \dfrac {du}{dx}(0)=0; \ \dfrac {du}{dx}(L)=0 \]

10 M

2 (b)
Solve the following differential equation using (i) subdomain (ii) collocation \[ \dfrac{d}{dx} \left ( x \dfrac {dy}{dx} \right ) -4x=10 \ \ \ \1\le x \le 2 \] subject to y(1)=y(2)=0

10 M

3 (a) (i)
Explain 'h' and 'p' method in FEM.

4 M

3 (a) (ii)
Explain at which are the conditions for the Rayleigh Ritz and Galerkin methods yield the same solution.

4 M

3 (b)
Develop the Finite Element equation for the most general element using Rayleigh-Ritz method for the mathematical model given \[ \dfrac {d}{dx} \left ( AE \dfrac {du}{dx} \right ) =0 \ \ \ \ 0\le x \le 12 \ cms \] Take linear elements. Use Lagrange's linear shape functions. Use the following data to solve Global equation. Use three elements. A=0.1 m

^{2}, for each element, E=100 GPa for each element, u(0)=0 and u(12), p=external force=10 kN. Find displacements at nodes and forces. Explain each step clearly.
12 M

4 (a)
Determine the shape functions for Euler Bernoulli Beam.

8 M

4 (b)
Analyse the following Truss completely for reactions, stress and strains.

12 M

5 (a)
Find the displacement, stresses and strain in the elements of stepped bar as shown in figure. Take E=200 Gpa.

10 M

5 (b)
Find the temperature at interfaces and heat transfer per unit area through the wall.

10 M

6 (a) (i)
Write down the polynomial expression for quadratic rectangular element for the following:-

Lagrange Element

Serendipity Element

Lagrange Element

Serendipity Element

4 M

6 (a) (ii)
Using serendipity concept, find the shape function for the eight nodes rectangular element.

4 M

6 (b)
Calculate the linear interpolation function for the linear rectangular elements shown in figure.

8 M

7 (a) (i)
Write a notes on PATCH Test and Jacobian Matrix.

5 M

7 (a) (ii)
Explain compatibility and convergence criteria.

5 M

7 (b)
Find the natural frequencies of the axial vibrations of a fixed free bar of uniform cross section of 50 mm

^{2}and length of 1 meter using consistent and lumped mass matrix and compare the natural frequencies with exact frequencies. Take E=200 GPa and density=7860 kg/m^{3}. Take two linear elements.
10 M

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