CMSCE, DEPARTMENT OF MECHANICAL ENGINEERINGFINITE ELEMENT ANALYSIS UNIT – I INTRODUCTION CAT-1 PART – A (3X2=6) Answer THREE questions. 1. What is meant by finite element analysis? 2. Name any four applications of FEA. 3. Define the concept of potential energy. 4. Why polynomial type interpolation functions are preferred over trignometric functions? 5. List out FEM software packages PART - B (2X12=24) Answer TWO questions. 6. A simply supported beam is subjected to uniformly distributed load over entire span. Determine 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) 7. A beam AB of span ‘l’ simply supported at the ends and carrying a concentrated load ‘W’ at the centre ‘C’ as shown in the following figure. Determine the deflection at the mid span by using Rayleigh-Ritz method and compare with exact solution. Use a suitable one term trigonometric trial function. 8. A cantilever beam of length ‘L’ is loaded with a point load at the free end. Find the maximum deflection and maximum bending moment using Rayleigh-Ritz method using the function y=a {1-Cos (8x/2L)}.Given EI is constant. CAT-2 PART – A (3X2=6) Answer THREE questions. What is meant by ‘discretization’? Briefly explain Gaussion elimination method. List out the various weighted-residual methods. Name any four applications of FEA What is the need for FEA? PART - B (2X12=24) Answer TWO questions. The following differential equation is available for a physical phenomenon. d2y/dx2 + 50 = 0, 0<x<10. The trial function is, y=ax (10-x). The boundary conditions are y (0) =0 and y (10) =0. Find the value of the parameter ‘a’ by (i) Point collocation method (ii) Sub-domain collocation method (iii) Least squares method (iv) Galerkin’s method Explain the process of discretization in detail. Explain the general procedure of finite element analysis. 1. 2. 3. 4. 5. 1. 2. 3. A stepped bar is subjected to an axial load of 200 KN at the place of change of cross section and material as shown in the following fig. 4. i) Derive the shape functions for a 2-D beam element (6) ii) Derive the stiffness matrix of a 2-D truss element (6) 2. Derive the stiffness matrix of a 3 noded bar element using the principle of potential energy. 4. What is a higher order element? Give an example Define ‘Natural coordinate system’ What are the different coordinate systems used in FEM? What are 1-Dimensional scalar and vector variable problems? What types of problems are treated as one-dimensional problems? 1. 2.B (2X12=24) Answer TWO questions. 3. Calculate the nodal displacements and forces for the bar loaded as shown in figure 3. 5. 5. DEPARTMENT OF MECHANICAL ENGINEERING FINITE ELEMENT ANALYSIS UNIT – II ONE DIMENSIONAL PROBLEMS CAT-3 PART – A (3X2=6) Answer THREE questions. 3. CAT-4 PART – A (3X2=6) Answer THREE questions.B (2X12=24) Answer TWO questions. 1. Find (a) The nodal displacements (b) the reaction forces (c) the induced stresses in each material. 1. State its significance Write down the expressions for the element stiffness matrix of a beam element. 6. Define aspect ratio. 2. PART .CMSCE. . List out the properties of stiffness matrix What are shape functions and what are their properties? Write down the expressions for shape functions of 1-D bar element. PART . For a tapered bar of uniform thickness t=10mm as shown in the following figure. the bar is subjected to a point load P= 1 KN at its centre. . the young’s modulus E = 2x105 MN/m2. It is given that E =200GPa and A= 500 mm2 for all the elements. The bar has a mass density ρ = 7800 Kg/M3. Determine (a) Nodal displacements (b) Support r e actions (c) Element stresses. Consider a 4-bar truss as shown in the following figure.CMSCE. Also determine the reaction forces at the support. 8. In addition to self weight. find the displacements at the nodes by forming into two element model. DEPARTMENT OF MECHANICAL ENGINEERING FINITE ELEMENT ANALYSIS 7. The element experiences a rise of 10°C. 6. 7. 3. CAT-6 PART – A (3X2=6) Answer THREE questions.B (2X12=24) Answer TWO questions. Write down the expression for the stress-strain relationship matrix for a 2D system. Differentiate between a CST and LST element. 5. PART . 1. Also determine the location of the 42°C contour line for the triangular element shown in the following fig. What is meant by a two dimensional vector variable problem PART . What are the differences between use of linear triangular element and bilinear rectangular element? 5. 4. DEPARTMENT OF MECHANICAL ENGINEERING FINITE ELEMENT ANALYSIS UNIT – III TWO DIMENSIONAL CONTINUUM CAT-5 PART – A (3X2=6) Answer THREE questions. State the expression for stiffness matrix for a bar element subjected to torsion 4. Write down the finite element equation for one-dimensional heat conduction. Derive the stiffness matrix and equations for a CST element. Define ‘Plane stress’ and ‘Plane strain’ with suitable example.B (2X12=24) Answer TWO questions. Derive the stiffness matrix and equations for a LST element . 1. Find the temperature at a point P (1. Calculate the element stiffness matrix and thermal force vector for the plane stress element shown in the following figure. 8.5) inside a triangular element shown with nodal temperatures given as Ti= 40°C. 6.CMSCE. Tj= 34°C and Tk= 46°C. 1. Write down the shape functions for a ‘Rectangular element State a two dimensional scalar variable problem with an example What is meant by a CST element? State its properties. 2. 3. 2. Also determine the three points on the 50°C contour line.B (2X12=24) Answer TWO questions. Derive an expression for the strain-displacement matrix for an axisymmetric triangular element. 6. T2= 54°C and T3= 56°C and T4= 46°C. 5. 2. Let E =2. What are the ways by which a 3-dimensional problem can be reduced to a 2-D problem? What is meant by axisymmetric solid? Write down the expression for shape functions for a axisymmetric triangular element. DEPARTMENT OF MECHANICAL ENGINEERING FINITE ELEMENT ANALYSIS 7. Derive the shape functions for a bilinear rectangular element 8. 3. PART . 4. The coordinates are in mm.CMSCE. UNIT – IV AXISYMMETRIC CONTINUUM CAT-7 PART – A (3X2=6) Answer THREE questions. Determine the temperature at the point (7.1X105 MN/m2 and µ=0.6.25. For the axisymmetric element shown in the following figure. Derive the shape functions for an axisymmetric triangular element 7.4). The nodal values of the temperatures are T1= 42°C. State the conditions to be satisfied in order to use axisymmetric elements State the expression used for ‘gradient matrix’ for axisymmetric triangular element 1. determine the stiffness matrix. . 8. For a 4-noded rectangular element shown in figure 3. 6. z2) = (50. 8.CMSCE.002. 1. (r2. Determine the strain displacement matrix for that element. z2) = (50. z1) = (30. All dimensions are in cm. w1= 0. 10).001. z3) = (40. 10). The nodal coordinates for an axisymmetric triangular element at its three nodes are (r1 . Distinguish between plane stress. 5. . 60). w3= 0. 2. u2= 0. plane strain and axisymmetric analysis in solid Mechanics Sketch ring shaped axisymmetric solid formed by a triangular and quadrilateral element PART . z3) = (40.001. (r2. z1) = (30.003. 3. Determine the strain displacement matrix for that element. 60). DEPARTMENT OF MECHANICAL ENGINEERING FINITE ELEMENT ANALYSIS CAT-8 PART – A (3X2=6) Answer THREE questions.B (2X12=24) Answer TWO questions. (r3.004.002. 7. (r3. Determine the element strains for an axisymmetric triangular element shown in the following figure. 4. State the constitutive law for axisymmetric problems Write down the expression for stiffness matrix for an axisymmetric triangular element Sketch an one-dimensional axisymmetric (shell) element and two-dimensional axisymmetric element. 10). The nodal displacements are u1= 0. 10). u3= 0. w2= 0. The nodal coordinates for an axi symmetric triangular element at its three nodes are (r1. 57735 and s= -0. What is meant by isoparametric formulation? How do you convert Cartesian coordinates into natural coordinates? Write down the expression for strain-displacement for a four-noded quadrilateral element using natural coordinates PART . 3. Compare this with exact solution.B (2X12=24) Answer TWO questions.57735 1. DEPARTMENT OF MECHANICAL ENGINEERING FINITE ELEMENT ANALYSIS UNIT – V ISO PARAMETRIC ELEMENTS FOR TWO DIMENSIONAL CONTINUUM CAT-9 PART – A (3X2=6) Answer THREE questions. PART – A (3X2=6) Answer THREE questions. 7. 4.B (2X12=24) Answer TWO questions. Using natural coordinates derive the shape function for a linear quadrilateral element . 5. 3. at Gauss point r= 0. Write down the shape functions for 4-noded linear quadrilateral element using natural coordinate system. 4. What is an ‘Iso-parametric element’? Differentiate between Isoparametric. super parametric and sub parametric elements. 2. 2. How do you calculate the number of Gaussian points in Gaussian quadrature method? Find out the number Gaussian points to be considered for _ (x4+3x3-x) dx. 8. 5. CAT-10 1. 6.quadrature. 8..CMSCE. Derive the stiffness matrix for a linear isoparametric element Establish the strain displacement matrix for the linear quadrilateral element as shown in the following figure. 7. 6. Use Gaussian quadrature rule Use Gaussian quadrature rule (n=2) to numerically integrate ∫ ∫ xy dx dy Evaluate the integral I = ∫ (3ex + x2 + 1/(x+2) dx using one point and two point Gauss. What is a ‘Jacobian transformation’? What is the Jacobian transformation fro a two nodded isoparametric element? PART . Integrate f(x) = 10 + 20x – (3x2/10) + (4x3/100) – (-5x4/1000) + (6x5/10000) between 8 and 12.