**B.E. (Mechanical Engineering )FINITE ELEMENT ANALYSIS(2015 Pattern) (End sem) ( Semester-I) (Elective-I)**

**Time : 2½ Hours Max. Marks : 70Instructions to the candidates:1) All questions carry equal marks.2) Neat diagrams must be drawn wherever necessary.3) Assume suitable data if required.**

**Q.1)** a) Explain in brief Rayleigh-Ritz method and relation between FEM and Rayleigh-Ritz method.**[6M]**

b) List different types of analysis and explain thermal analysis.**[4M]**

**OR**

**Q.2)** For truss given in figure 1, determine: **[10M]**

a) Nodal displacements

b) Stresses in each element and

c) Reaction at support. Take E = 200 x 10^{3} N/mm^{2} and A = 200 mm^{2}.

**Q.3)** a) Explain pascal’s triangle with neat sketch, show that pascal’s triangle satisfies all convergence conditions required for deciding interpolation function formulation.**[6M]**

b) Write element stiffness matrix for plane stress/strain problems and explain each terms of it.**[4M]**

**OR**

**Q.4)** a) The nodal coordinate of triangular element are as shown in figure 2. The x coordinate of interior point P is 3.3 and shape function N1 = 0.3. Determine N2,N3 and y coordinate of point P. (Use co-factor method).**[4M]**

b) Write a short note on: Element stiffness matrix and load vector for Beam element.**[6M]**

**Q.5)** a) Write short note on: **[8M]**

i) Natural coordinates of element and its significance in FEA.

ii) mesh refinements.

b) For the element shown in the figure 3 assemble Jacobin matrix for the Gauss point (0.7,0.5) also Check the element for uniqueness mapping.**[8M]**

**OR**

**Q.6)** a) Write a short note on Jacobian matrix for ”4-node Quadrilateral element”.**[8M]**

b) Evaluate Integral: I\;=\;\int_{y=4}^{y=6}\;\int_{x=-2}^{x=2}\;{(1-x)}^2\;{(4-y)}^2dxdy **[8M]**

**Q.7)** a) Write a note ”Heat transfer through pin-Fin” Explain with appropriate governing equation.**[8M]**

b) A metallic fin, with thermal conductivity 70W/m°K, 1 cm radius and 5 cm long extends from a plane wall whose temperature is 413°K. Determine the temperature distribution along the fin if heat is transfer to ambient air at 293 °K with heat transfer coefficient of 5 W/m²K. Take two elements along the fin.**[10M]**

**OR**

**Q.8)** a) Write Element thermal conductivity(stiffness) matrix and Element thermal load sector compare them Element stiffness matrix and element load vector of Bar element respectively.**[8M]**

b) A composite wall as shown in figure 4 is composed of two homogeneous slabs in contact. Let thermal conductivities be K =1W/m °C for firebrick slab 1 and K=0.3 W/m °C for insulating slab2. The left side is exposed to an ambient temperature of T_{fL} = 1000 °C inside the furnace with heat transfer coefficient of h_{1} = 10W/m² °C. The right side ambient temperature is T_{fR}=25 °C outside the furnace with heat transfer coefficient of h_{R} = 3 W/m² °C. The thickness of the slabs are L_{1} = 0.2 m and L_{2} = 0.1 m. Determine the temperature at the left edge, point between the two slabs and right of the composite wall.**[10M]**

**Q.9)** a) List different between consistent and lumped mass matrix technique for modal analysis of structure.**[10M]**

b) Find the natural frequency of longitudinal vibrations using consistent and lumped mass matrix method with one element of bar as shown in figure 5. Take E = 2 x10^{11} N/m^{2}, **ρ** = 7800 kg/m^{3}, L =1m. **[6M]**

**Q.10)** a) Find the natural frequency of longitudinal vibration of the stepped bar of cross-sectional areas A1 = 2500 mm^{2} and A2 = 1200 mm^{2} having equal stepped length L= 1 m as shown in figure 6. (Use consistent mass matrix). **[10M]**

b)Write a dynamic equation and explain each term. Convert this into a Eigen value problem and explain its significance.**[6M]**

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