ALPHA COLLEGE OF ENGINEERINGThirumazhisai, Chennai – 600124 DEPARTMENT OF MECHANICAL ENGINEERING ME 6603 - FINITE ELEMENT ANALYSIS (ANNA UNIVERSITY- IMPORTANT QUESTIONS) UNIT-I (PART-A ) (2 Marks) 1. What is meant by finite element? A small units having definite shape of geometry and nodes is called finite element. 2. What is meant by node or joint? Each kind of finite element has a specific structural shape and is inter- connected with the adjacent element by nodal point or nodes. At the nodes, degrees of freedom are located. The forces will act only at nodes at any others place in the element. 3. What is the basic of finite element method? Discretization is the basis of finite element method. The art of subdividing a structure in to convenient number of smaller components is known as discretization. 4. What are the types of boundary conditions? Primary or essential boundary conditions Secondary boundary conditions 5. State the methods of engineering analysis? Experimental methods Analytical methods Numerical methods or approximate methods 6. What are the types of element? 1D element (example- bar and beam element) 2D element(example- triangular and rectangular element) 3D element(example- tetrahedral and hexahedral element) 7. State the three phases of finite element method. Preprocessing Analysis Post Processing 8. What is structural problem? Displacement at each nodal point is obtained. By these displacements solution, stress and strain in each element can be calculated. 9. What is non structural problem? Temperature or fluid pressure at each nodal point is obtained. By using these values, properties such as heat flow, fluid flow for each element can be calculated. 10. What are the methods are generally associated with the finite element analysis? Force method Displacement or stiffness method. 11 . Explain stiffness method. Displacement or stiffness method, displacement of the nodes is considered as the unknown of the problem. Among them two approaches, displacement method is desirable. 12. What is meant by post processing? Analysis and evaluation of the solution result is referred to as post processing. Postprocessor computer program help the user to interpret the result by displaying them in graphical form. 13. Define FEA. It is numerical method for solving problems of engineering and mathematical physics. In this method, instead of solving the problem for entire body in one operation, we formulate the equations for each finite element and combine them to obtain the solution of the whole body. 14. What is meant by degrees of freedom? When the force or reaction act at nodal point node is subjected to deformation. The deformation includes displacement, rotation and or strains. These are collectively known as degrees of freedom 15. What is meant by discretization and assemblage? The art of subdividing a structure in to convenient number of smaller components is known as discretization. These smaller components are then put together. The process of uniting the various elements together is called assemblage. 16. What is Rayleigh-Ritz method? It is integral approach method which is useful for solving complex structural problem, encountered in finite element analysis. This method is possible only if a suitable function is available. 17. What is Aspect ratio? It is defined as the ratio of the largest dimension of the element to the smallest dimension. In many cases, as the aspect ratio increases the in accuracy of the solution increases. 18. What is truss element? The truss elements are the part of a truss structure linked together by point joint which transmits only axial force to the element. 19. What are the ‘h’ and ‘p’ versions of finite element method? It is used to improve the accuracy of the finite element method. In ‘h’ version, the order of polynomial approximation for all elements is kept constant and the numbers of elements are increased. In ‘p’ version, the numbers of elements are maintained constant and the order of polynomial approximation of element is increased. ALPHA COLLEGE OF ENGINEERING THIRUMAZHISAI, CHENNAI – 600124 DEPARTMENT OF MECHANICAL ENGINEERING ME 6603 - FINITE ELEMENT ANALYSIS (ANNA UNIVERSITY- IMPORTANT QUESTIONS) UNIT-I ( PART-A 2 MARKS) 1. Define FEA. List any four advantages of finite element method. 2. Name the weighted residual methods. 3. What is finite element? Name different types of elements. 4. What is meant by node or joint? List the types of nodes. 5. Mention the basic steps of Rayleigh - Ritz method. 6. Classify boundary conditions 7. What is the limitation of using a finite difference method? 8. List the various methods of solving boundary value problems. 9. Compare the Rayleigh - Ritz technique with nodal approximation method 10. What is Galerkin method of approximation? 11. Write down the boundary conditions of a cantilever beam AB of span L fixed at A and free at B subjected to a uniformly distributed load of P throughout the span. 12. Write the potential energy for beam of span ‘L’ simply support at ends, subjected to a concentrated load ‘P’ at mid span. Assume EI constant. 13. What is Raleigh-Ritz method? 14. Write down the strain – displacement relation. (PART-B) (16 MARKS) 1. The following differential equation is available for a physical phenomenon (d2y/dx2)+50=0, 0≤x≤10 Trail function is y=a1x (10-x) B.C’s are y (0) =0, y(10)=0 Find the value of the parameters a1 by the following methods (i) Point collocation (ii) Sub domain collocation (iii) Least squares and (iv) Galerkin. 2. List and briefly describe the general steps of Finite Element Analysis. 3. (i) Find the solution of the initial-value problem (d2y/dx2) + (dy/dx) - 2y = 0 With B.C’s are y (0) =2, y’ (0) =5. k1=k3=100 N/m. K2=200N/m. 6. (ii) Explain Rayleigh Ritz method with an example. displacement of nodes 2 and 3. Also calculate the forces in the spring 2.the deformation of the bar is governed by the differential equation given below.C’s u(0)=0 .C y(0)=0 and y(H)=0 (i) Variational method (ii) Collocation method 7. the reaction forces at node 1 and 4. For the spring system shown in fig. Determine the displacement using weighted residual method AE (d2u/dx2) +q0=0 With the B. Discuss the following method to solve the given differential equation : EI (d2y/dx2)-M(x)=0 With the B. du/dx(x=l)=0 5. Assume. calculate the global stiffness matrix. . Use Rayleigh Ritz method to find the displacement of the midpoint of the rod shown in fig. 4. u1=u4=0 and P3=500N. A uniform rod subjected to a uniform axial load is illustrated in fig . Using Raleigh-Ritz method determining the maximum deflection under the two loads and the maximum bending moment. find the solution of the problem using the Galerkin method . Given: EI is constant. 15. 12. y(1)=0. Use y=a1sin( x/1)+a2sin(3 x/l). The B. find the maximum displacement of the tapered rod as shown in Fig. Find the solution of the problem using Rayleigh-Ritz method by considering a two-term solution as y(x) =c1x (1-x) +c2x2 (1-x). A cantilever beam of length L is loaded with a point load at the free end.C are given by w(0)=w(1)=0 . 9.075 N / cm2 . (i) Derive the element level equation for one dimensional bar element based on the stationary of a functional. 10. γ = 0. A SSB (span L and flexural rigidity EI) carries a point load at centre with udl throught its length. 13. Find the deflection and bending moment at mid span using Raleigh-Ritz method and the compare with exact solution. A physical phenomenon is governed by the differential equation (d2w/dx2)-10x2=5 for 0≤x≤1. Determine the expression for the deflection and bending moment in a SSB subjected to uniformly distributed load over entire span.C y(0)=0.8. 11. By taking a two-term trail solution as w(x)= c1f1(x)+c2f2(x) with f1(x)=x(x-1) and f2(x)=x2(x-1) . Find the maximum bending moment using Raleigh-Ritz method using function Y=A {1-cos( x/2L)} . The functional corresponding to this problem to be extremized is given by I= x2y}dx . Derive the characteristic equations for the one dimensional bar element by using piece-wise defined interpolations and weak form of the weighted residual method? 14. Consider the differential equation (d2y/dx2) +400x2=0 for 0≤x≤1 subject to B. Using collocation method. Take E = 2 x 107 N/cm2. (ii) Explain the general procedure of FEA . 0 ≤ x ≤ 1 B.4λ y’ + 4λ2y = 0 B.16. Find the solution of the boundary-value problem y’’ + y = 0 B.C’s are y’ (1) =0.C’s are y (0) =0. 18. The following differential equation is available for a physical phenomenon (d2y/dx2) + y = 4x.C’s are y (0) =0. y (π/6) = 4. Obtain one term approximate solution by using Galerkin’s method of weighted residuals. . 17. y(1)=1. Find the eigen value and eigen function of y’’. y (2) + 2y’(2) = 0. and N3 are the interpolation functions. Since it is in the element level. What are the types of loading acting on the structure? Body force (f).IMPORTANT QUESTIONS) UNIT-II ( 2 MARKS) 1. Traction force (T).y) φ1 + N2 (x. Point load (P) 7. Local axes are established in an element. They are same in direction for all the elements even though the elements are differently oriented. Differentiate between global and local axes. Example: Self weight due to gravity 8. Define shape function. Define the body force A body force is distributed force acting on every elemental volume of the body Unit: Force per unit volume. and N3 are also called shape functions because they are used to express the geometry or shape of the element. The direction differs from element to element. What are the characteristic of shape function? It has unit value at one nodal point and zero value at other nodal points.y) = N1 (x. The sum of shape function is equal to one. N2. Define traction force Traction force is defined as distributed force acting on the surface of . 5. State the properties of stiffness matrix. 3. (ii) Sum of the elements in any column must be equal to zero.y) φ2 + N3 (x. ALPHA COLLEGE OF ENGINEERING Department of Mechanical Engineering (ANNA UNIVERSITY. (iii) It is an unstable element. φ2. 2. 6. Approximate relation φ (x. and φ3 are the values of the field variable at the nodes N1. N2.y) φ3 Where φ1. (i) It is the symmetric matrix. 4. What is the difference between static and dynamic analysis? Static analysis: The solution of the problem does not vary with time is known as static analysis Example: stress analysis on a beam Dynamic analysis: The solution of the problem varies with time is known as dynamic analysis. Global axes are defined for the entire system. N1. they change with the change in orientation of the element. Name any four FEA software’s. What is a truss? A truss is defined as a structure. 11. The accuracy of the result can be improved by increasing the order of the polynomial. What is natural coordinates? A natural coordinate system is used to define any point inside the element by a set of dimensionless number whose magnitude never exceeds unity. riveted or welded together. surface shear 9. the body. What is Global coordinates? The points in the entire structure are defined using coordinates system is known as global coordinate system. viscous drag. What is point load? Point load is force acting at a particular point which causes displacement. different material interjections and sudden change in point load 17. made up of several bars. 14. Write down the general finite element equation. Example: Frictional resistance. Unit: Force per unit area. Why polynomials are generally used as shape function? Differentiation and integration of polynomial are quite easy. mention the places where it is necessary to place a node? Concentrated load acting point. During discretization. 15. What are the classifications of coordinates? Global coordinates Local coordinates Natural coordinates 13. 10. ANSYS NASTRAN COSMOS DYNA 16. Cross-section changing point. 12. It . This system is very useful in assembling of stiffness matrices. T for one dimensional heat conduction element. then the vibrations are known as transverse vibrations 22. What are methods used for solving transient vibration problems? There are two methods for solving transient vibration. Write down the finite element equation for one dimensional heat conduction with free end convection . 24. What is meant by longitudinal vibrations? When the particles of the shaft or disc moves parallel to the axis of the shaft . What is meant by transverse vibrations? When the particles of the shaft or disc move approximately perpendicular to the axis of the shaft. Contour diagrams can be used to understand the solution easily and quickly. is easy to formulate and computerize the finite element equations 18. Required result can be obtained in graphical form. N and temperature function. List the two advantages of post processing.then the vibrations are known as longitudinal vibrations. 21. How do you calculate the size of the global stiffness matrix? Global stiffness matrix size = Number of nodes X Degrees of freedom per node. They are Mode superposition method Direct integration method 23. 20. 19. Write down the expression of shape function. 18. 21. (Dec-2009) 8. Write down the stiffness matrix equation for one dimensional heat conduction element.May-2011. Write the stiffness matrix for the simple beam element. List the characteristics of shape functions. What is discretization? 13. . ALPHA COLLEGE OF ENGINEERING Thirumazhisai. Write down the lumped mass matrix for the truss element. State the assumptions are made while finding the forces in a truss. Write down the finite element equation for 1D 2 noded bar element. (Dec-2009) 9. (May-2009) 10. (Dec-2008) 12. (May-2010) 7. (Dec-2011.2010) 5. (May-2009) 11. 20. 16. Determine the element mass matrix for 1D dynamic structural problem. Chennai – 600124 DEPARTMENT OF MECHANICAL ENGINEERING (ANNA UNIVERSITY. Why polynomials are generally used as shape function? (Dec-2011. What is truss? 15. Draw the shape function of a 1D line element with 3 nodes. State the properties of stiffness matrix. Write down the governing equation for longitudinal and transverse vibration analysis. Draw the shape function for a 2 nodded line element with one degree of freedom at each node. (Dec-2011) 4. Write the properties of global stiffness matrix. What is meant by dynamic analysis? (Dec-2011) 17.May2011) 2. Distinguish between 1D bar element and 1D beam element. What is meant by longitudinal& transverse vibration? 19. What are the difference between 2 dimensional scalar variable and vector variable elements? 22.May-2009) 6. (Dec-2011) 3. What is shape function? Write the shape functions of 1D linear element. (Dec-2010. 14. Assume 2 noded linear element. Write down the expression of stiffness matrix of a truss element.IMPORTANT QUESTIONS) UNIT-II ( 2 MARKS) 1. Name a few boundary condition involved in any heat transfer analysis. (May-2011) 5. Determine the nodal displacement.2 axial force P= 500KN is applied as shown. (Dec-2011) . Derive the shape function. Calculate nodal displacement and element stresses for the truss shown in fig. stiffness matrix and finite element equation for one dimensional two noded bar element. UNIT-II (1-D FINITE ELEMENT ANALYSIS) (PART-B) ( 16 MARKS) 1. Find the nodal displacement and element stresses for the bar shown in fig. Consider the bar shown in fig. May 2010) 3. element stresses in each element and reaction forces. A fixed beam length of 2L m carries a uniformly distributed load of a‘w’ (in N / m ) which run over a length of ‘L’ m from the fixed end. E=70GPa cross-sectional area A = 2 cm² for all truss members. Calculate the rotation at point B using FEA.(Dec-2011. as shown in Fig. (May-2011) E=2x105 N/mm2 4. 2. Derive the shape functions for a 2D beam element 11. ΔT = 80 C. generate the stiffness matrix correspond in to the three coordinates indicated. . 1 is subjected to an increase in temperature. Determine the force in the members of the truss shown in Fig.3. Derive the stiffness matrix for a 2D truss element.(May-12) 10. Use the following shape functions. element stresses in each element and reactions. The stepped bar shown in fig. Take P1 = 10 KN and P2 = 15 KN (Dec-2009) 8. Determine the nodal displacements. 12(b). For the prismatic bar shown in fig. Derive the shape functions and stiffness matrix for Quadratic bar element.6.(May-2009) N1 = (-1/2) r (1-r) N2 = (1/2) r (1-r) N3 = 1-r² 9. (May-2011) 7. u2 = 8 mm. determine the natural frequencies of solid circular shaft fixed at one end shown in fig. Using 2 equal. (Dec-2011) 15.length finite elements. For the given two bar truss element as shown in fig. Calculate the displacement at x = l/4. Take E = 70 GPa. Determine the natural frequencies for the truss shown in fig using finite element method. (i) displacement at node 1 (ii) stress in the element 1-3. The nodal displacements are u 1 = 5 mm. Compute natural frequencies of free longitudinal vibration of a stepped bar shown in fig. Find the following. A two noded truss element is shown in fig.12.(May-14) 13. l/3 and l/2. (May-2007) . 16. A = 200mm2 14. h = 3W/cm² ºC . (Dec-2011) 21. T = 20 ºC ( May 2011) 20. Determine the natural frequencies and mode shapes of transverse vibration for a beam fixed at both ends. . ρ = 7800 Kg/m³. 19. The beam may be modeled by 2 elements . The fin has rectangular cross section and is 8cm long 4cm wide and 1cm thick. the velocities in each pipe. A wall of 0.06 m having an average thermal conductivity of 0. Determine all natural frequencies using two types of mass matrices. Take K = 3W/cm ºC . Determine the potentials at the junctions. The potentials at the left end is10 m and that at the right end is 2m. For the smooth pipe of variable cross section shown in fig.2 W/m-k the wall is to be insulated with a material thickness of 0. each of length L and cross sectional area A. Consider lumped mass matrix approach (May-2011) 18. Determine the temperature distribution in one dimensional rectangular cross section as shown in fig. A= 300cm². Calculate the nodal temperature using FEA. E = 2e11 N/m². Assume that convection heat loss occurs from the end of the fin.3 W/m-k. The permeability coefficient is 1 m/s. Consider the simply supported beam whose length 1m. I = 100mm⁴.17.6m thickness having thermal conductivity of 1. The inner surface temperature is 1000 ºC and outside of the insulation is exposed to atmospheric air at 30 ºC with heat transfer coefficient of 35 W/m² k. 22. A steel rod of diameter d = 2 cm. length l = 5 cm and K = 50W/m ºC is exposed at one end to a constant temperature of 320 ºC. The other end is in ambient air of temperature 20 ºC with a convection coefficient of h = 100W/m² ºC. . Determine the temperature at the mid point of rod using FEA. 3. u3. . Define plane strain analysis. It has six unknown displacement degrees of freedom (u1. What meant by plane stress analysis? Plane stress is defined to be a state of stress in which the normal stress and shear stress directed perpendicular to the plane are assumed to be zero. 5. What is CST element? Three noded triangular elements are known as CST. u2. v1. It is also called as cubic displacement triangle. The element is called CST because it has a constant strain throughout it. v3).2D SCALAR VARIABLE PROBLEMS (Part –A 2Marks) 1. What is QST element? Ten nodded triangular elements are known as Quadratic strain triangle. What is LST element? Six noded triangular elements are known as LST. The displacement function for the elements are quadratic instead of linear as in the CST. 6. 2. Plane strain is defined to be state of strain normal to the xy plane and the shear strains are assumed to be zero. v2. 4. How do you define two dimensional elements? Two dimensional elements are define by three or more nodes in a two dimensional plane. The basic element useful for two dimensional analyses is the triangular element. It has twelve unknown displacement degrees of freedom. ALPHA COLLEGE OF ENGINEERING Department of Mechanical Engineering UNIT-III. Write down the stress . .7. Write down the stress. Write down the stiffness matrix equation for two dimensional CST elements. strain relationship matrix for plane strain condition 10. strain relationship matrix for plane stress condition 9. Write a strain displacement matrix for CST element? 8. 11. Write down the expression for the shape function for a CST element. . Write down the shape functions of a CST element. Obtain the shape functions of eight noded quadrilateral elements. (Dec-2011) 3.May 2011) 4. (Dec-2011. What is higher order elements? Why are they preferred? (Dec-2011. If a displacement field is described by u = (-3x² + 2 y² + 6xy) 10⁻⁴ v = (2x + 6y . What are the advantages of natural coordinates over global coordinates? (Dec-2008) 6.2008) 2. Write the shape functions of 4 node quadrilateral element. y = 0.2D SCALAR VARIABLE PROBLEMS ( Part-A) ( 2 MARKS) 1. (Dec-2011) 2. . Sketch a 4 node quadrilateral element along with nodal degree of freedom. Chennai – 600124 DEPARTMENT OF MECHANICAL ENGINEERING UNIT-III. Write the step procedure of finding stresses in the elements of a rectangular plate Problem by taking two triangular elements and assuming plane stress condition. (PART-B) (16 MARKS) 1. Distinguish between plane stress and plane strain problems.May 2011. (Dec-2010) 9.y²) 10⁻⁴ Determine the direct strains in x and y directions as well as the shear strain at the point x = 1.2009. Derive the element matrices and vector matrices of 2D heat transfer element. Differentiate CST and LST elements? (Dec-2009) 4. Write down the nodal displacement equations for a 2D triangular element (May-2010) 10. Explain the important properties of CST element? (Dec-2008) 5. Write down the expression for constitutive matrix for 2D element. May 2011.2008. 11. ALPHA COLLEGE OF ENGINEERING Thirumazhisai. Write down the stiffness matrix equation for two dimensional heat conduction and convection. (Dec-2011. (Dec-2011) 8. (Dec-2009) 7.2008) 3. Determine the element stresses and element strains.(4.2011). For the configuration shown in fig 5. Take E = 200 GPa and v = 0.3.Prove that the resulting stiffness matrix is singular.5. u₆= 0. t = 0.(3. u₄ = 0.2) and the nodal displacements are u₁ = 0. v= 0.3.002mm.Use unit thickness for plane strain.3. Use one element model.2).3 and N₁ = 0.2). Assume plane stress condition. Determine the N₂. Take E = 15e6 N/ cm². .(3.2). May-2008) 10. determine the deflection (displacements) at the point of load application.4). For the plane strain element whose coordinates are (2. Assume v = 0.005mm. (Dec-2010) 9.(4. N₃ and y coordinate of P. The element experiences 20 ºC increases in temperature. u₅ = 0.4). Determine the element stiffness matrix and the thermal load vector for the plane stress element whose coordinates are (2.004mm. u₂ = 0. Derive the shape functions and strain –displacement matrix of a CST element (May-2011. (May-2010) 6.5 cm and α = 6e-6/ ºC 7. The nodal coordinates of the triangular element are shown in fig 14 (b). u₃ = 0. (May.3 (Dec-2010) 8. Derive the stiffness matrix for the equilateral triangular element with plane stress conditions shown in fig 14 (a). At the interior point ‘P’ the x coordinate is 3. (Dec 2008) 14. obtain the strain. Take k = 60 W/cm k . Compute the element matrices and vectors for the element shown in fig. (ii) For the same triangular element. Evaluate the shape functions N1.11. N₂ & N₃ at an interior point P (2.12. Also determine the location of the 42ºC contour line for the triangular element shown in Figure. i. when the edge k-j experiences convection heat loss. (Dec 2009) 13. 2. j and k of a triangular element are given by (0. 12.0) and (1.5. Derive the expression for constitutive stress-strain relationship and also reduce it to the problem of plane stress and plane strain. T2 = 45ºC. (3.0).displacement matrix B.y) co-ordinates of nodes. 4) mm respectively. Find the temperature at a point P (10.5) mm for the element. and T3 = 40ºC .5) inside the triangular element shown with the nodal temperatures given as T1 = 35ºC. Q = 50 W/cm3 . (i) The (x. What do you mean by constitutive law and give constitutive law for axi- symmetric problems? (May-2008) 5. Name the 4 basic elasticity equations. Write down the shape functions for an axisymmetric triangular element. 13. The (r . What are the assumptions used in shell element? 15. The element experiences a 15°C increases in temperature. E = 2x105 N/mm2. What are the conditions for a problem to be axisymmetric. z1 = 15 mm .25. 10. (May-2011) 2.May 2011) 3. State the types and advantages of shell element PART-B (16 MARKS) 1. 11. (6. What are the assumptions used in thick and thin plate element? 14. r2 = 25 mm . v = 0. Write down the shape functions for an axisymmetric triangular element.5) and (5. Evaluate the temperature force vector for the axisymmetric triangular element whose coordinates are : r1 = 15 mm. Give the strain-displacement matrix equation for an axisymmetric triangular element 9.IMPORTANT QUESTIONS) UNIT-IV (TWO DIMENSIONAL VECTOR VARIABLE PROBLEMS) UNIT-IV ( 2 MARKS) 1. . r3 = 35 mm . z3 = 50 mm. Write down the stress-strain relationship matrix for an axisymmetric triangular element. 8. Write down the constitutive relationship for the plane stress problem(Dec- 2010) 4. Give 4 applications where axisymmetric elements can be used. ALPHA COLLEGE OF ENGINEERING Department of Mechanical Engineering (ANNA UNIVERSITY.4). determine the element stresses. Write short notes on axisymmetric problems? (Dec-2009. What is meant by thin and thick plate element? 12. 6. 8) cm respectively. 7. 2. Define shell element.z) co-ordinates of nodes i. Take α = 10x10-6/°C. j and k of an axisymmetric triangular element are given by (3. z2 = 15 mm . w1 = 0. 7. w2 = 0. 0. Determine [B] & [K] matrix for that element. 0.004. 6. u2 = 0. v = 0. The nodal co-ordinates for an axisymmetric triangular are given below: r1 = 15 mm.02 mm. Take E = 2x105 N/mm2. 5. Plane strain (ii) Node.z) co-ordinates of nodes .25.003.03 mm.5) and (5. Derive the shape functions. By using 2 elements on the 15 mm length the cylinder. z1 = 15 mm . (6.001.25. Let E = 210Gpa and v = 0.-0. Determine the element strains.05 mm.001. Derive 4 basic sets of elasticity equations (i) Stress-Displacement relationship equations (ii) Stress-Strain relationship equations (iii) Equlibrium Equations (iv) Compatibility Equations (May-2010) 9. w3 = 0 3. 0. i. 8) cm respectively. u3 = 0 .25. calculate the displacement at inner radius. Define the following terms with suitable examples: (i) Plane stress. Take E = 2x105 N/mm2. j and k of an axisymmetric triangular element are given by (3.4). The (r . r2 = 25 mm . Write short notes on plate and shell elements.007)T. The displacement are u 1 = 0. . (May-2010) 8. strain – displacement matrix [B] for axisymmetric triangular element 4. r3 = 35 mm . z3 = 50 mm. z2 = 15 mm . A hollow cylinder of inside diameter 100 mm and outside diameter 140 mm is subjected to an internal pressure of 4 N/mm2.002.02 mm. 0. Derive the stress-strain relationship matrix [D] for the axisymmetric triangular element. Element and Shape functions (iii) Axisymmetric analysis. (Dec 2009) 10. The element displacement (in cm) vector is given as u = (0. v = 0. 5. 3. Plane strain is defined to be state of strain normal to the xy plane and the shear strains are assumed to be zero. ALPHA COLLEGE OF ENGINEERING Department of Mechanical Engineering (ANNA UNIVERSITY. All loading conditions must be symmetric about the axis of revolution. The problem domian must be symmetric about the axis of revolution 2. 4. What is meant by plane stress analysis? Plane stress is defined to be a state of stress in which the normal stress and shear stress directed perpendicular to the plane are assumed to be zero.Such types of problems are solved by a special two dimensional element called as axisymmetric element. What are the conditions for a problem to be axisymmetric? 1. All boundary conditions must be symmetric about the axis of revolution 3. 2.IMPORTANT QUESTIONS) UNIT-IV ( 2 MARKS) 1. . Define plane strain analysis. Write down the displacement equation for an axisymmetric triangular element. What is axisymmetric element? Many three dimensional problems in engineering exhibit symmetry about an axis odf rotation . Give 4 applications where axisymmetric elements can be used. Radial. (i) Pressure vessels (ii) Rocket castings (iii) Cooling towers (iv) Submarine hills (v) Springs. 10. Circumferential. 7. 6. Longitudinal and Shear strains. . Write down the shape function for an axisymmetric triangular element. Longitudinal and Shear stresses. 9. What are the stresses need to be calculated in axisymmetric problems? Radial. Circumferential. What are the 4 basic elasticity equations? (i) Equlibrium equation (ii) Compatibility equation (iii) Strain-displacement relationship equation (iv) Stress-strain relationship equation 8. Name different strain available in axisymmetric elements. What are serendipity elements? 9. Write the Lagrangian shape functions for a 1D. What do you meant by isoparametric element? (Dec-2010. Define super-parametric elements 8. What is the function of post-processor? 13. 2noded element? (Dec-2008) 3.IMPORTANT QUESTIONS) UNIT-V. Evaluate the integral using one point and two point Gaussian quadrature to obtain an exact value of the integral. ALPHA COLLEGE OF ENGINEERING Department of Mechanical Engineering (ANNA UNIVERSITY.(Dec-2011) . PART-B (16 MARKS) 1. What are the stages in FEA? 10.ISOPARAMETRIC FORMULATION PART-A ( 2 MARKS) 1. State the applications of FEA.Dec 2011) 2. What is the function of pre-processor? 12. What is the significance of Jacobian matrix in 2D problems. Define sub-parametric elements 7. What is the function of processor? 11. (Dec-2011) 4. (May-2011) 2. Write down the Jacobian matrix (Dec -2010) 5. June 2009) 6. Evaluate the integral using three point Gaussian quadrature and compare with exact solution. (May-2011. Write the Gauss points and weights for 2 point formula of numerical integration. 003. Evaluate the Cartesian coordinates of point P which has the local coordinates of ε =0. (i) The Cartesian (global) co-ordinates of the corner nodes of an isoparametric quadrilateral element are given by (1. 0. (8. 8. (9. Explain in detail about FEA analysis software. (2.5. 11. (2. 5. 7. A 4 noded rectangular elements whose coordinates are given by (0. v = 0. 0).25.O. u = (0.8 for the isoparametric quadrilateral element which has the coordinates (3. 9) and (5.(ii) Strain-displacement matrix and (iii) Element stresses.5. 4).5) and (1. 0). ή=0. Establish the strain displacement matrix for the linear quadrilateral element whose coordinates are (1. Derive the shape functions of (i) 8 noded rectangular element and (ii) 6 noded triangular element.004.2011) 4.Find its Jacobian matrix. 1).57735. (Dec-2011. 1). 0.3.5) and (2. (5. Use Gauss quadrature rule (n=2) to numerically ∫ ∫ x2y dx dy with limits -1 to +1 (Dec 2008) 6. Derive the shape functions of isoparametric element. May . . 1). 5). Determine the following (i) Jacobian matrix .6.4) at ε =0. (2. 1. 1) and (0. Take E = 2x105 N/mm2.006.0)T and ε =ή=0.F system as shown in fig(i). 7).2). 0). 10. (ii) Distinguish between sub parametric and super parametric elements. Evaluate the integral I = ∫ ∫ (x² + y² + 6xy) dx dy with limits -1 to +1 using three point Gauss Numerical integration. ή=0.0. 12. 0.004. Consider the undamped 2 D. (2. 0.Find the response of the system when the first mass alone is given an initial displacement of unity and released from rest.Assume plane stress condition. (4.57735 9. 0. 0). a family of elements known as “ Isoperimetric elements “ are used.. 4.IMPORTANT QUESTIONS) UNIT-V ( 2 MARKS) 1. . 2.then it is known as isoparametric element. What is meant by sub parametric element? If the number of nodes used for defining the geometry is same as number of nodes used for defining the dispalcements. Write down the shape function for 4 noded rectangular elements using natural coordinate system.e. 5. A large number of finite elements may be used to obtain reasonable resemblance between original body and the assemblage . Write down the element force vector equation for four noded quadrilateral elements. for problems involving curved boundries. If the number of nodes used for defining the geometry is more than number of nodes used for defining the dispalcements.then it is known as super parametric element. What is the purpose of Isoparametric Elements? It is difficult to represent the curve boundaries by straight edges elements. 3.isoparametric elements are used i. Define super parametric element. ALPHA COLLEGE OF ENGINEERING Department of Mechanical Engineering (ANNA UNIVERSITY.In order to overcome this drawback. . fluid mechanics. 11. electrical and magnetic fields. 9. What is the function of post-processor? It computes the solution and its derivative at desired points of the domain and plot the results. 8. FEA involves solution of wide variety of problems in Solid mechanics. material data. 7. boundary conditions and initial conditions of the problem. State the applications of FEA. Write down the Gaussian quadrature expression for numerical integration. What is the function of processor? (i) Generate finite element mesh (ii) Calculate element matrices (iii) Assemble element equations (iv) Solve the equations.6. heat transfer. Write down the shape function for 4 noded rectangular elements using natural coordinate system. 10. What is the function of pre-processor? It read geometry. m2) are real and un-equal The Complementary function y(x) = C1em1x+ C2em2x (ii) When two roots (m1. CHENNAI – 600124 DEPARTMENT OF MECHANICAL ENGINEERING ME 6603 . m2) are real and equal (m1=m2=m) The Complementary function y(x) = (C1+ C2x) emx (iii) When two roots are having real and imaginary part (α±iβ) The Complementary function y(x) = eαx (C1Cosβx+ C2Sinβx) 2. ALPHA COLLEGE OF ENGINEERING THIRUMAZHISAI.FINITE ELEMENT ANALYSIS UNIT WISE IMPORTANT FORMULAE UNIT – I (INTRODUCTION) 1. Weighted Residual Methods (i) Point Collocation method Residual(R) = 0 (ii) Sub-domain Collocation method ʃ R dx = 0 (iii) Least Square Method ʃ R2 dx = 0 (or) ʃ R (dR/da)dx = 0 (iv) Galerkin Method ʃ wi R dx = 0 1 . Initial and Boundary value Problems: (i) When two roots (m1. 3. Rayleigh-Ritz Method Total potential energy = Strain Energy – Work done by external forces π=U–H (i) For beam problem.M (M)= EI (d2y/dx2) (ii) For bar Problem. U = (EA/2) ʃ (du/dx)2dx H = Fu (or) Pu y = a0+a1x+a2x2 (iii) For Spring Problem 2 . U = (EI/2) ʃ (d2y/dx2)2dx H = ʃ w y dx (udl load) H = W ymax (point load) y = a1 Sin (πx/l) + a2 Sin (3πx/l) B. [D] Elasticity matrix or Stress-strain relationship matrix. Shape functions N1 = N=2 3. Where. ��= u -u ) 2 1 H = Fu UNIT – II ONE DIMENSIONAL FINITE ELEMENT ANALYSIS 1-D BAR ELEMENT. Stiffness matrix. [B] Strain displacement relationship matrix. [D] = [E] = E = Young’s modulus. u = ao+a1x 2. =A 3 . U = ½ K �� 2 (Where. 1. Linear Polynomial Equation. 5. {F} = [K] {u} Where. E = Young’s modulus 4 . 7. 1D Displacement equation. Force vector due to self weight. Stiffness matrix 4. 6. Stress. Reaction force. {u} is a nodal displacement [Column matrix]. [K] is a stiffness matrix [Row matrix]. General FEA equation is. {F} is an element force vector [Column matrix]. 8. Where. Strain energy. ∆T = Temperature difference. α = Coefficient of thermal expansion. Stiffness matrix. 1. = 9. Stress. A = Area of the truss element E = Young’s modulus of element le = Equivalent length 2. Where. Temperature effect. A = Area of cross section of bar element. Force. Where. TRUSS ELEMENTS. 5 . Shape functions. 2. 3. Tensile force. Finite element general equation. Stress. [K] = stiffness matrix {U} = nodal displacement matrix 4. Stiffness matrix. Where. 6 . k = spring constant = change in deformation BEAMS 1. SPRINGS 1. Where. Finite element equation. Where. Stiffness matrix. 3. L = length of the beam element E = Young’s modulus I = Moment of inertia LONGITUDINAL & TRANSVERSE VIBRATION PROBLEMS 7 . 2. ONE DIMENSIONAL HEAT TRANSFER PROBLEMS 1. Convection and Internal Heat Generation 8 . Finite Element Equation For 1D Heat Conduction with free end Convection 2. Finite Element Equation For 1D Heat Conduction. Shape functions. 1. Where. UNIT III (2D SCALAR VARIABLE PROBLEMS CONSTANT TRIANGULAR ELEMENT (CST). 9 . 3.strain matrix in general 2D form. [K] = [B]T [D] [B] A t 4. 5. Stress . Strain displacement matrix. Displacement functions.2. Stiffness matrix. 10 . 6. Plane stress condition. Element stress. Plane strain condition. 11 . 7. 12 . Plane strain. 8. Maximum stress. 1. Initial strain. TEMPETATURE EFFECT OF CST ELEMENT. Element strain. Plane stress. Minimum stress. Principle angle. Shape function. 13 . 2D HEAT TRANSFER PROBLEMS Stiffness Matrix for both Conduction and Convection UNIT IV (2D VECTOR VARIABLE PROBLEMS AXI-SYMMETRIC ELEMENT. 2. 1. t = thickness A= area of the element. Where. Element temperature force. Stress strain relation 14 . Where. 3. 2. Strain displacement matrix. 15 . TEMPETATURE EFFECT OF Axisymmetric Element. Element temperature force. 4. 4. 3. t = thickness A= area of the element. Stiffness matrix. Where. Initial strain. UNIT V-ISOPARAMETRIC FORMULATION Iso Parametric Quadrilateral Element 1. 16 . 2. Strain displacement matrix. Shape functions. Displacement function. Rectangular element. 17 .3. Element stress. Jacobian matrix.4. = natural co-ordinates [B] = strain-displacement relationship matrix [D] = stress strain relationship matrix N = shape function = load or force on x direction = force on y direction 6. Where: 5. Where. Force vector. 18 . w2 = 8/9 and x1= √3/5. x2 = 0. x3 = -√3/5 (iii) For double Integration. w1 = w2 =1 and x1= √1/3.y1) + w22f(x2.y2) + w2w1f(x2. w12f(x1. w1 = w3 = 5/9.y1) + w1w2f(x1. x2 = -√1/3 (ii) For 3 point Quadrature Where.Gaussian Quadrature (Or) Numerical Integration (i) For 2 point Quadrature Where.y2) 19 .
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