1. a) Discuss how finite element is evolved in engineering field. (8) b) Explain the finite element idealization of structures with examples.


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1 M.TECH. DEGREE EXAMINATION Branch: Civil Engineering Specialization Geomechanics and structures Model Question Paper  I MCEGS FINITE ELEMENT ANALYSIS Time: 3 hours Maximum: 100 Marks Answer ALL questions 1. a) Discuss how finite element is evolved in engineering field. (8) b) Explain the finite element idealization of structures with examples. (8) c) Find the displacement in the free ends and forces in the members of the truss shown in figure. (9) 45 L Q P 2. a) A circular plate of radius R supported on a semi infinite space is loaded by a uniform pressure q. For a stress analysis of the problem, what are the component displacements,strains and stresses which are significant? Write down the relevant strain displacement and stress strain relations. (8) b) Explain plane stress and plane strain problems. (8) c) Using Rayleigh Ritz method obtain an approximate solution for the following problem d 2 u  u + x 2 = 0 0<x<1 dx 2 With the boundary conditions u (0) =0, du (1) = u (1) = 1 (9) dx
2 (Page 1/3) 3. a) Describe interpolation functions. (7) b) Derive shape function for the five shape functions for the transition element in (x,y) system. If the coordinates are transformed to a nondimensional system (ξ,ή) with node 1 as (1,1) and node 3 as (1,1) rewrite the shape functions. y (9) 4 3 x c) Briefly describe generalized coordinates and natural coordinates. (9) 4. a) Derive the relationship between natural (area) coordinates and Cartesian coordinates of a triangular element. (8) b) The two nodes 1 and 2 of the isosceles plane stress CST element in fig. are fixed If poisons ratio = 0.Determine the stiffness matrix associated with the dof at the unrestrained node 3. (8) 3 a 1 2 a a c) Explain weighted residual method in detail. (9)
3 (Page 2/3) 5. For a three bar truss shown in figure determine the nodal displacements and the stress in each member. Find support reactions also. Take modulus of elasticity as 200GPa. Assume area of cross section for element 1 =1500 mm 2,elements 2, 3 =2000 mm 2. (25) C 150 kn 400 mm A B 6. Assemble element stiffness matrices for the rigid frame shown in figure. Explain how you proceed further to solve the problem. E=200GPa, I 0 =40x106 m 4,A=4x103 m 2 (25) 20kN/m 30kN 400 mm 400 mm 2I o 3m I o I o 4 m 7. a) Write short notes on Numerical Integration. Evaluate the integral I= cos п/ 2 x dx using twopoint and threepoint Gauss rule. (13) b) Discuss different types of plate bending elements used in FEM. (12) 8. a) Discuss Kirchhoff s and Mindlin s plate bending theories. (13) b) Write short note on finite elements for shell analysis. (12)
4 M.TECH. DEGREE EXAMINATION Branch: Civil Engineering Specialization Geomechanics and structures Model Question Paper  II (For admission 2011 onwards) (Page 3/3) MCEGS FINITE ELEMENT ANALYSIS Time: 3 hours Maximum: 100 Marks Answer ALL questions 9. a) Discuss the various steps in finite element analysis of a elastic static problem. (8) b) In a plane strain problem, we have σ x = 12000MPa, σ y = 1000 MPa, E= 210 GPa and γ = Determine the value of the stress σ z. (5) c) An axial bar shown in figure 1 is subjected to a load P at a distance of L/2 from one of the fixed ends. Use the principle of stationary potential energy to calculate the reactions R 1 and R 2. (12) P/2 R 1 R 2 Fig1 P/2 10. a) What is meant by variational method? Discuss its applications in finite element analysis. (9) b) Figure 2 shows a truss subjected to a vertical load 50kN at joint 1. Using the principle of virtual displacement, compute the deflections u and v. Also find the forces in the members of the truss. Fig 2 50 kn (10)
5 (Page1/3) c) Explain mathematical modeling in finite element analysis. (6) 11. a) Derive stiffness matrix for CST element. (7) b) Assemble equations of equilibrium for the spring system by direct approach. Show that minimization of potential energy also yields the same result. (9) F 1 F 2 F 3 k 1 k 2 k 3 u 1 u 2 u 3 Fig 3 c) Using Galerkins method obtain an approximate solution for the following problem d 2 u / dx 2  u + x 2 = 0 0<x<1 With the boundary conditions u(0)=0, du(1)/dx =u (1)=1 (9) 12. a) Calculate the Jacobian ( J ) for the element shown in Figure 4 at the sampling point 3. (10) 4(20,100) Y 1(10,20) 3(80,60) 2(100,20) 3 2 η 1 4 ξ X Fig 4 b) Using Rayleigh Ritz method determine the expressions for deflection in a simply supported beam subjected to uniformly distributed load w over the entire span. (9) c) Discuss compatibility and convergence criteria in FEA. (6) (Page2/3)
6 13. Find deflections, moments and shears in a continuous beam shown in figure 5. Take modulus of elasticity as 200GPa. Assume I =10 5 mm 4. (25) 20 N/mm 400 mm 400 mm 400 mm Fig Assemble element stiffness matrices for the rigid frame shown in figure 6. Explain how you proceed further to solve the problem. E=200GPa,I 0 =1000cm 4,A=100cm 2 (25) 20 kn/cm 30 cm 30 cm 40 cm Fig a) Write short notes on Numerical Integration. Evaluate the integral I= e x x dx x using ( + 2) 1 onepoint and twopoint Gauss rule. Compare with exact solution. (13) b) Discuss applications of soil and rock mechanics in FEM. (12) 16. a) Derive the stiffness matrix for Mindlin s plate element. (13) b) Write short notes on finite elements for plate analysis. (12) (Page 3/3)
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