VALLIAMMAI ENGINEERING COLLEGEDEPARTMENT OF CIVIL ENGINEERING M.E STRUCTURAL ENGINEERING ST7201 – FINITE ELEMENT ANALYSIS QUESTION BANK YEAR/SEM: I/II UNIT-I - INTRODUCTION PART-A 1. What is the basic concept of finite element analysis? BT 2 2. Explain ‘discretization’? BT 6 3. What is the requirement of displacement field to be satisfied in the use of Rayleigh–Ritz method? BT 3 4. Give examples of Eigen value problems in structural mechanics. BT 5 5. How will you classify essential and non-essential boundary condition? BT 4 6. Derive thermal load vector for 1D bar element. BT 5 7. Explain body force and surface force with examples. BT 2 8. What do you mean by convergence in finite element analysis? BT 4 9. What is the significance of weak formulation? BT4 10. What is quadratic shape function? BT 1 11.Why polynomial shape functions are preferred? BT 2 12. What are the limitations of Galerkin formulation? BT 3 13. Write down the stiffness matrix for 2D beam element. BT 1 14. What is shape function? BT 1 15. Discuss displacement and shape function? BT 2 16. What is Ritz technique? BT 1 17. Illstrate the potential energy for beam of span ‘L’ simply supported at ends, subjected to a concentrated ‘P’ at midspan. Assume EI constant. BT3 18. What are the advantages of FEA? BT6 19. Define banded structure? BT 1 20. Define the modulus of resilience? BT 1 PREPARED BY A.KAVITHA AP/SG PART-B 1. Solve the following equations by Gauss elimination method? i) ii) iii) iv) BT3 28r1+6r2 =1 6r1+ 24r2+6r3=0 6r2+28r3+8r4= -1 8r3+16r4 = 10 2. A simply supported beam is subjected to uniformly distributed load over entire span. Find the bending moment and deflection at the mid span using Rayleigh –Ritz method and compare with exact solution. Use a two term trial function y=a1sin(πx/l)+ a2sin(3πx/l). BT1 3. Discuss Rayleigh –Ritz and Galerkin methods of formation by taking an example. BT2 4. Show from first principle that the stiffness matrix of a general finite element can be evaluated in the form of K=ʃ BTCBdr BT1 V 5. Find stiffness matrix for a one dimensional bar subjected to both distributed load and point loads. (Note: Differential equation should be formed and then apply basic Galerkin method on differential equation to form element stiffness matrix.) BT1 6. Solve the following differential equation using Ritz method. BT3 d2y/dx2 = -sin (πx) boundary conditions u(0) = 0 and u(1) = 0. 7.i) Taking a differential equation , explain the process of weak formulation. ii) Explain any two methods of weighted residuals with examples. BT2 8. i)Derive the element stiffness matrix and element force matrices for a one dimensional line element. ii) A simply supported beam of span L, young’s modulus, moment of inertia I is subjected to a uniformly distributed load of P/unit length. Determine the deflection W at the midspan. Use Rayleigh Ritz method. BT6 9.For the bar shown in fig, evaluate the nodal displacement, stress in each material and reaction forces. BT5 P 1. Aluminum 2.Steel PREPARED BY A.KAVITHA AP/SG A1= 2400 mm2 E1= 70GPa A2= 600mm2 E2= 200GPa P= 200kN 10. Analyze the frame shown in fig. EI= constant. B 10kN/m BT4 C 3m 3m A PART-C 1. List and briefly describe the general steps of finite element method. BT1 2. Discuss the importance of FEA in assisting design process. BT2 3.An alloy bar 1m long and 200mm2 in crosssection is fixed at one end is subjected to a compressive load of 20Kn.if the modulus of elasticity for the alloy is 100GPa, find the decrease in length of the bar. Also determine the stress developed and the decrease in length at 0.25m,0.5m and 0.75m.Solve by collocation method. BT3 4.a)Describe the historical background of FEM b)Explain the relevance of FEA for solving design problems with the aid of examples BT6 UNIT – II – FINITE ELEMENT ANALYSIS OF ONE DIMENSIONAL PROBLEMS PART-A 1. Differentiate scalar variable and vector variable problems. BT2 2. Define ‘Natural coordinate system’? BT1 3. What are the properties of shape functions? BT1 4. What are the advantages of expressing displacement field in Natural co-ordinates than generalized co-ordinates? BT6 5. Specify stress and strain tensors for plane stress case. Give suitable examples for plane stress problems. BT3 PREPARED BY A.KAVITHA AP/SG 6. Derive transformation equation used in Gaussian integration BT3 7. What is natural or intrinsic coordinate? BT1 8. What do you mean by isoparametric formulation? BT2 9. Explain plane strain problem with an example. BT2 10. Show any two examples where the weak formulations are adopted. BT3 11. Explain natural coordinates? BT6 12. What are equivalent nodal forces? BT1 13. What are the types of non-linearity in structural analysis? BT4 14. Write down the shape functions for four noded rectangular elements? BT4 15. What are incompatible displacement models? BT5 16. Write the natural co-ordinates for the point P of the triangular element. The point P is the C.G of the triangle. BT1 17. Examine the properties of stiffness matrix? BT4 18. Discuss the steps involved in finite element modeling. BT2 19. Write the shape function for constant strain triangle by using polynomial function? BT1 20. What are the conditions for a problem to be axisymmetric? BT5 PART-B 1. Derive shape functions and stiffness matrix for a 2D rectangular element. BT3 2. Evaluate the nodal load vector due to self-weight of a four noded rectangular element with two degrees of freedom (translations) at each node. Use Gauss quadrature method of numerical integration. BT5 3. Find the shape functions N1, N2,N3 at the interior point P(3.85,4.8) for the triangular element shown in fig . BT1 y 3(4,7) 2(7,3.5) 1(1.5,2) x PREPARED BY A.KAVITHA AP/SG 4. Discuss the convergence requirements of interpolation polynomials. BT4 5. The nodal coordinates of a quadrilateral element as given as (0,7), (9,4),(7,9) and (2,8). Find the integral ʃ (x2 +y2 -3xy) dA over the area of the element of second order Gaussian quadrature. BT1 6. Derive shape functions for constant strain triangle element. BT2 7. Integrate the following function using Gaussian integration. Proper Gauss points should be specified. The x limit is varying from 0 to 2 and y limit is varying from 1 to 3 ʃʃ(xy) dxdy. BT5 8. i) Derive the weights and Gauss points of Gauss one point formula and two point formula. ii) Derive the shape functions of a four noded quadrilateral element. BT2 11 9. i) Evaluate ʃ ʃ (x2 +xy2)dx dy by Gauss numerical integration. BT6 -1 -1 ii) Derive the element strain displacement matrix of a triangle element. BT3 10. Triangular elements are used for the stress analysis of a plate subjected to in plane loads. The components of displacements parallel to (x, y) axis at the modes 1, 2, 3 are found to be (-0.001, 0.01), (-0.002, 0.01) and (-0.002, 0.02) cm respectively. If the (x, y) coordinates of the nodes shown in Fig are in cm, find the components of BT1 displacement of the point (30, 25) cm. y 3(40,40) 1(20,20) 2(40,20) x PART-C 1.A cantilever beam of length 3.4m has an elastic spring support of stiffness 230kn/m at Its free end where a point load of 13kn acts.Take young’s modulus as 200GPa and area Moment of inertia of the crosssection as 1x10-4 m4.Evaluate the displacement and slope At the node and reactions. BT5 2.A two noded truss element is shown in fig.The nodal displacements are u1=5mm PREPARED BY A.KAVITHA AP/SG BT3 U2=8mm.Solve the displacement at x=l/4,l/3 and l/2. 1 u1=5mm 2 u2=8mm l 3.A concentrated load P=50KN is applied at the centre of a fixed beam of length 3m, Depth 200mm and width 120mm.Calculate the deflection and slope at the mid point. Assume E=2x105 N/mm2 BT4 P 1500 1500 4. Discuss the generation of stiffness matrix and load vector for a beam element. BT2 UNIT-III- FINITE ELEMENT ANALYSIS OF TWO DIMENSIONAL PROBLEMS PART-A 1. State the conditions to be satisfied in order to use axisymmetric elements? BT5 2. What is meant by error evaluation in FEM? BT1 3. What is meant by an Isoparametric element? BT1 4.Explain the Lagrange interpolation polynomials used for higher order elements. BT2 5.Write shape functions for 1 D linear strain element. BT1 6. What is the difference between h and p methods? BT4 7.Brief the application of higher order elements. BT4 8.Explain an ill conditioned element? BT2 9. Define a plane stress problem with a suitable example. BT4 10.What are the methods used for numerical integration in finite element method? BT6 11.How the discretization error is evaluated? BT5 PREPARED BY A.KAVITHA AP/SG 12. Describe lumped mass system? BT6 13. What is dynamic condensation? BT1 14. What do you mean by higher order element? BT2 15. What is geometric isotropy? BT1 16. In an element the geometry is defined using 4 nodes and the displacement is defined using 8 nodes. What is this element called? BT3 17. When Hermite interpolations functions are used? BT3 18. What are the ways in which 3D problem can be reduced to a 2D approach? BT3 19. What are the types of meshes? BT2 20. Write down the displacement equation for an axisymmetric element? BT1 PART-B 1. Determine the stiffness for the axisymmetric element shown in fig. Take E as 2.1x 10 5 N/mm2 and Poisson’s ratio as 0.3. BT3 0,50 3 50,0 0,0 1 2 2. Explain the isoparametric elements and its types. BT6 3.Derive the displacement interpolation matrix, strain displacement interpolation matrix B, and Jacobian operator J for the three node truss element shown in Fig . Also sketch the interpolation functions. BT3 r =-1 r=0 r=+1 x,u X1 L/2 L/2 PREPARED BY A.KAVITHA AP/SG 4. State the need for mesh refinement. Discuss the methods of mesh refinement. BT1 5. Derive the shape functions for element shown in fig. Shape functions should be specified in natural coordinate system.. BT4 7 8 9 η η £ 4 555555556 1 3 6. Derive the shape functions for ID cubic element. Shape functions should specified in both natural and global coordinate systems. BT4 7. i)Obtain the shape functions of a nine noded quadrilateral element. ii) Describe auto and adaptive mesh generation techniques. BT2 8. i) Comment on discretization error with an example. ii) Discuss p and h methods of refinement and give applications of each method. BT2 9. Determine the shape functions of six noded triangular elements. BT5 10. What is adoptive meshing? Explain any one algorithm for auto meshing. BT1 PART-C 1.Derive constitutive matrix for axisymmetric analysis. BT3 2.Explain with an example of each of the following (a) sub parametric element (b) iso parametric element (c) Super parametric element BT6 3.Analyse the element characteristics of a four node quadrilateral element. BT4 4.What are the nonzero strain and stress components of axisymmetric element? Explain? BT2 UNIT-IV – MESH GENERATION AND SOLUTION PROBLEMS PART-A 1. List out the meshing techniques? BT1 2. Give two examples of geometric nonlinear problems? BT1 PREPARED BY A.KAVITHA AP/SG 3. List the sources of errors in finite element analysis. BT1 4. List the methods used for evaluation of Eigen values and Eigen vectors. BT1 5. Zero stress gradient is observed along x direction from the following structural model. How many elements you will consider for your analysis? σo BT3 σo 6. How is geometry nonlinearity taken care in finite element analysis? BT4 7. What do you mean by material non linearity? BT2 8. What is a mass index? BT2 9. Give examples of thermal analysis problems. BT1 10. Explain normal modes? BT6 11. Why higher order elements are necessary? BT3 12. What are serendipity elements? BT2 13. What is discretization error? BT2 14. Explain weak formulation? BT6 15. How error is evaluated in finite element analysis? BT5 16. What are the types of non-linearity? BT1 17. Form the consistent and lumped mass matrix for a truss element. Length = 3m, Area = 20 x 10-4 m2 and mass density = 2.5x104 kgm-3 BT5 18. When the equilibrium equations are established with respect to the deformed shape, then the system is analyzed as ---------- nonlinear one. BT4 19. Explain isoparametric elements? BT4 20. Show the stiffness matrix for an axisymmetric triangular element. BT3 PART-B PREPARED BY A.KAVITHA AP/SG 1. Explain the automatic mesh generation technique. BT2 2. Write a detailed note on Matrix solution techniques and Natural coordinate systems. BT1 3. Explain the problems involved in the analysis of material non linearity and explain how a solution procedure for search problems may be established for structures made of ductile materials. BT2 4. Explain how the consistent mass matrix for a pinpointed bar element is obtained. BT6 5. Discuss the vector iteration method of Eigen value problems. BT3 6. The cross section of a bar of length 10cm is rectangular in section of width 3cm and depth 1 cm. The bar is subjected to forced convection over its length due to flow of fluid at temperature of 25OC. The convection coefficient is 5 W/cm2o C. Compute the thermal load vector due to convection. BT4 7. Determine Pcritical for a pinned –pinned column of length, L. Consider 3 elements for the analysis. And also draw mode shape. BT4 8.i) Explain characteristic polynomial technique of Eigen value- Eigen vector evaluation.BT1 ii) Explain any one method of handling geometric non- linearity. BT1 9. Determine the Eigen values and Eigen vectors for the stepped bar shown in fig. Take E= 2 x 105 N/mm2 specific weight = 78.5 kN/m3. BT5 250mm 125mm A1=600mm2 A2=300mm2 10. Using lumped mass approach obtain natural frequencies and shapes of flexural modes of a fixed end beam of span 900mm. A= 500mm2, I= 400mm2, p=7840 kg/m3 and E=200 kN/mm2 BT5 PART-C 1. Summarize Auto and adaptive Mesh generation techniques. BT6 2. Analyze the p and h methods of mesh refinement. BT4 PREPARED BY A.KAVITHA AP/SG 3.Discuss discretization errors with example BT2 4.Integrate in detail the error evaluation BT5 UNIT-V – SOFTWARE APPLICATION PART-A 1. List out the two advantages of post processing. BT1 2. Derive the constitutive matrix for a noded element. BT5 3. Write the equation for calculating element mass matrix in terms of shape functions. BT1 4. State the functions of a preprocessor in FEA software package. BT2 5. Name few software packages used for finite element analysis. BT1 6. Specify degrees of freedom for SOLID 45 element analysis. BT1 7. Give one dimensional heat flow equation. BT1 8. Give examples of thermal analysis problem. BT1 9. What is dynamic condensation? BT2 10. Why is the 3 noded element calledas CST element? BT4 11. What is the governing differential equation for a one dimensional heat transfer? BT5 12. What is natural coordinate system? BT2 13. What is lumped mass matrix? BT3 14. How thermal loads are input in finite element analysis? BT4 15. In buckling analysis the Eigen value and Eigen vectors are calculated. Actually what do they represent? BT6 16. Discuss Radiation of heat transfer. BT2 17.Brief the applications of higher order element. BT3 18.What is the difference between h and p methods. BT4 19. Exlpain an ill conditioned element. BT6 20.Write discretization errors. BT3 PART-B PREPARED BY A.KAVITHA AP/SG 1. Explain the generation of node numbers in FE analysis using soft wares. BT2 2. Write a detailed note on mesh plotting. BT1 3.Discuss the modeling procedure using soft wares by taking an example of a plate bending problems. BT2 4. Write short notes on: BT1 i) One dimensional heat transfer problems in finite element analysis. ii) Error evaluation in FEA 5.Plate with small centre hole (3mm diameter) is subjected to 50 N tensile load( refer fig). Thickness of the plate is 6mm and width of the plate is 28mm. Take E= 210 GPa and Poisson ratio = 0.3. How will you solve this problem using finite element software (ANSYS)? Determine steps should be provided. BT3 1m 6. How will you solve this problem (refer fig) using finite element software? Detailed steps should be clearly specified. BT4 P L 7.Write the step by step procedure of solving a structural problem by using any finite element software. PREPARED BY A.KAVITHA AP/SG BT6 8. Obtain the element matrices for the one dimensional heat conduction equation. d/dx (KdT/dx)+Q=0 subject to boundary conditions BT5 T/(x=0) = To q/(x=c) =n (T L-T∞) 9. Find the forces developed at the fixed end A due to rise of 60 oC. BT4 A= 800mm2 E= 200 kN/mm2 α =10x 10-6/oC A 2m B 10. Find the forces in the members due to rise in temperature of 30oC. BT3 A= 400mm2, E= 200kN/mm2 and α = 70x 10-6/oC. C 2m A B PART-C 1.Write the detailed requirements of pre and post processing. BT1 2.Discuss in detail about region and block representation. BT2 3.Explain in detail about preprocessing, solution phase and post processing with respect To an engineering structure BT5 4.Create a finite element model and analysis procedure for a design of retaining wall. PREPARED BY A.KAVITHA AP/SG BT6 PREPARED BY A.KAVITHA AP/SG