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.
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.
9.
What is structural problem? Displacement at each nodal point is obtained. By these displacements solution, stress and strain in each element can be calculated. 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?
15.
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 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.
(ii) Explain Rayleigh Ritz method with an example.
4.
A uniform rod subjected to a uniform axial load is illustrated in fig .the deformation of the bar is governed by the differential equation given below. Determine the displacement using weighted residual method
AE (d2u/dx2) +q0=0
5.
6.
7.
With the B.C’s
u(0)=0 , du/dx(x=l)=0
Use Rayleigh Ritz method to find the displacement of the midpoint of the rod shown in fig.
Discuss the following method to solve the given differential equation : EI (d2y/dx2)-M(x)=0 With the B.C y(0)=0 and y(H)=0 (i) Variational method (ii) Collocation method
For the spring system shown in fig, calculate the global stiffness matrix, displacement of nodes 2 and 3, the reaction forces at node 1 and 4. Also calculate the forces in the spring 2. Assume, k1=k3=100 N/m, K2=200N/m, u1=u4=0 and P3=500N.
8.
Consider the differential equation (d2y/dx2) +400x2=0 for 0≤x≤1 subject to B.C y(0)=0, y(1)=0. The functional corresponding to this problem to be extremized is given by
I=
x2y}dx . 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).
9.
A physical phenomenon is governed by the differential equation (d2w/dx2)-10x2=5 for 0≤x≤1. The B.C are given by w(0)=w(1)=0 . 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 solution of the problem using the Galerkin method .
10.
A SSB (span L and flexural rigidity EI) carries a point load at centre with udl throught its length. Using Raleigh-Ritz method determining the maximum deflection under the two loads and the maximum bending moment.
11.
Determine the expression for the deflection and bending moment in a SSB subjected to uniformly distributed load over entire span. Find the deflection and bending moment at mid span using Raleigh-Ritz method and the compare with exact solution. Use y=a1sin( x/1)+a2sin(3 x/l).
12.
A cantilever beam of length L is loaded with a point load at the free end. Find the maximum bending moment using Raleigh-Ritz method using function Y=A {1-cos( x/2L)} . Given: EI is constant.
13.
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.
(i) Derive the element level equation for one dimensional bar element based on the stationary of a functional. (ii) Explain the general procedure of FEA . Using collocation method, find the maximum displacement of the tapered rod as shown in Fig. Take E = 2 x 107 N/cm2, γ = 0.075 N / cm2
15.
16.
The following differential equation is available for a physical phenomenon (d2y/dx2) + y = 4x, 0 ≤ x ≤ 1 B.C’s are y (0) =0, y(1)=1. Obtain one term approximate solution by using Galerkin’s method of weighted residuals.
17. Find the solution of the boundary-value problem y’’ + y = 0 B.C’s are y (0) =0, y (π/6) = 4. 18. Find the eigen value and eigen function of y’’- 4λ y’ + 4λ2y = 0 B.C’s are y’ (1) =0, y (2) + 2y’(2) = 0.
ALPHA COLLEGE OF ENGINEERING Department of Mechanical Engineering (ANNA UNIVERSITY- IMPORTANT QUESTIONS) UNIT-II ( 2 MARKS) 1. Define shape function. Approximate relation φ (x,y) = N1 (x,y) φ1 + N2 (x,y) φ2 + N3 (x,y) φ3 Where φ1, φ2, and φ3 are the values of the field variable at the nodes N1, N2, and N3 are the interpolation functions. N1, N2, and N3 are also called shape functions because they are used to express the geometry or shape of the element. 2. What are the characteristic of shape function? It has unit value at one nodal point and zero value at other nodal points. The sum of shape function is equal to one. 3. 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. 4. State the properties of stiffness matrix. (i) It is the symmetric matrix. (ii) Sum of the elements in any column must be equal to zero. (iii) It is an unstable element. 5. Differentiate between global and local axes. Local axes are established in an element. Since it is in the element level, they change with the change in orientation of the element. The direction differs from element to element. Global axes are defined for the entire system. They are same in direction for all the elements even though the elements are differently oriented. 6.
What are the types of loading acting on the structure? Body force (f), Traction force (T), Point load (P)
7.
Define the body force A body force is distributed force acting on every elemental volume of the body Unit: Force per unit volume. Example: Self weight due to gravity
8.
Define traction force Traction force is defined as distributed force acting on the surface of
the body. Unit: Force per unit area. Example: Frictional resistance, viscous drag, surface shear 9.
What is point load? Point load is force acting at a particular point which causes displacement.
10.
Write down the general finite element equation.
11.
What is a truss? A truss is defined as a structure, made up of several bars, riveted or welded together.
12.
What are the classifications of coordinates? Global coordinates Local coordinates Natural coordinates
13.
What is Global coordinates? The points in the entire structure are defined using coordinates system is known as global coordinate system.
14.
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. This system is very useful in assembling of stiffness matrices.
15.
Name any four FEA software’s. ANSYS NASTRAN COSMOS DYNA
16.
During discretization, mention the places where it is necessary to place a node? Concentrated load acting point, Cross-section changing point, different material interjections and sudden change in point load
17.
Why polynomials are generally used as shape function? Differentiation and integration of polynomial are quite easy. The accuracy of the result can be improved by increasing the order of the polynomial. It
is easy to formulate and computerize the finite element equations 18.
How do you calculate the size of the global stiffness matrix? Global stiffness matrix size = Number of nodes X Degrees of freedom per node.
19.
List the two advantages of post processing. Required result can be obtained in graphical form. Contour diagrams can be used to understand the solution easily and quickly.
20.
What is meant by longitudinal vibrations? When the particles of the shaft or disc moves parallel to the axis of the shaft ,then the vibrations are known as longitudinal vibrations.
21.
What is meant by transverse vibrations? When the particles of the shaft or disc move approximately perpendicular to the axis of the shaft, 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. They are Mode superposition method Direct integration method
23.
Write down the expression of shape function, N and temperature function, T for one dimensional heat conduction element.
24.
Write down the finite element equation for one dimensional heat conduction with free end convection
ALPHA COLLEGE OF ENGINEERING Thirumazhisai, Chennai – 600124 DEPARTMENT OF MECHANICAL ENGINEERING (ANNA UNIVERSITY- IMPORTANT QUESTIONS)
UNIT-II ( 2 MARKS) 1.
Why polynomials are generally used as shape function? (Dec-2011,May2011)
2.
State the assumptions are made while finding the forces in a truss. (Dec-2011)
3.
What is shape function? Write the shape functions of 1D linear element. (Dec-2011)
4.
Write the properties of global stiffness matrix. (Dec-2011.May-2011,2010)
5.
Draw the shape function for a 2 nodded line element with one degree of freedom
at each node. (Dec-2010,May-2009) 6.
List the characteristics of shape functions. (May-2010)
7.
Distinguish between 1D bar element and 1D beam element. (Dec-2009)
8.
State the properties of stiffness matrix. (Dec-2009)
9.
Draw the shape function of a 1D line element with 3 nodes. (May-2009)
10.
Write down the lumped mass matrix for the truss element. (May-2009)
11.
Write the stiffness matrix for the simple beam element. (Dec-2008)
12.
What is discretization?
13.
Write down the finite element equation for 1D 2 noded bar element.
14.
What is truss?
15.
Write down the expression of stiffness matrix of a truss element.
16.
What is meant by dynamic analysis? (Dec-2011)
17.
Determine the element mass matrix for 1D dynamic structural problem. Assume 2 noded linear element.
18. What is meant by longitudinal& transverse vibration? 19. Write down the governing equation for longitudinal and transverse vibration analysis. 20. Name a few boundary condition involved in any heat transfer analysis. 21. What are the difference between 2 dimensional scalar variable and vector variable elements? 22. Write down the stiffness matrix equation for one dimensional heat conduction element.
UNIT-II (1-D FINITE ELEMENT ANALYSIS) (PART-B) ( 16 MARKS) 1. Derive the shape function, stiffness matrix and finite element equation for one dimensional two noded bar element. 2. 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, as shown in Fig. Calculate the rotation at point B using FEA.(Dec-2011, May 2010)
3. Find the nodal displacement and element stresses for the bar shown in fig. (May-2011) E=2x105 N/mm2
4. Calculate nodal displacement and element stresses for the truss shown in fig. E=70GPa cross-sectional area A = 2 cm² for all truss members. (May-2011)
5. Consider the bar shown in fig.2 axial force P= 500KN is applied as shown. Determine the nodal displacement, element stresses in each element and reaction forces. (Dec-2011)
6. Determine the force in the members of the truss shown in Fig.3. (May-2011)
7. The stepped bar shown in fig. 1 is subjected to an increase in temperature, ΔT = 80 C. Determine the nodal displacements, element stresses in each element and reactions. Take P1 = 10 KN and P2 = 15 KN (Dec-2009)
8. For the prismatic bar shown in fig. 12(b), generate the stiffness matrix correspond in to the three coordinates indicated. Use the following shape functions.(May-2009) N1 = (-1/2) r (1-r) N2 = (1/2) r (1-r) N3 = 1-r²
9. Derive the shape functions and stiffness matrix for Quadratic bar element.(May-12)
10. Derive the shape functions for a 2D beam element
11. Derive the stiffness matrix for a 2D truss element.
12. A two noded truss element is shown in fig. The nodal displacements are u 1 = 5 mm, u2 = 8 mm. Calculate the displacement at x = l/4, l/3 and l/2.(May-14)
13. For the given two bar truss element as shown in fig. Find the following. (i) displacement at node 1 (ii) stress in the element 1-3. Take E = 70 GPa, A = 200mm2
14. Using 2 equal- length finite elements, determine the natural frequencies of solid circular shaft fixed at one end shown in fig. (Dec-2011)
15. Compute natural frequencies of free longitudinal vibration of a stepped bar shown in fig.
16. Determine the natural frequencies for the truss shown in fig using finite element method. (May-2007)
17. Determine the natural frequencies and mode shapes of transverse vibration for a beam fixed at both ends. The beam may be modeled by 2 elements , each of length L and cross sectional area A. Consider lumped mass matrix approach (May-2011) 18. Consider the simply supported beam whose length 1m, E = 2e11 N/m², A= 300cm², I = 100mm⁴, ρ = 7800 Kg/m³. Determine all natural frequencies using two types of mass matrices.
19. Determine the temperature distribution in one dimensional rectangular cross section as shown in fig. The fin has rectangular cross section and is 8cm long 4cm wide and 1cm thick. Assume that convection heat loss occurs from the end of the fin. Take K = 3W/cm ºC , h = 3W/cm² ºC , T = 20 ºC ( May 2011)
20. A wall of 0.6m thickness having thermal conductivity of 1.2 W/m-k the wall is to be insulated with a material thickness of 0.06 m having an average thermal conductivity of 0.3 W/m-k. 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. Calculate the nodal temperature using FEA. (Dec-2011) 21. For the smooth pipe of variable cross section shown in fig. Determine the potentials at the junctions, the velocities in each pipe. The potentials at the left end is10 m and that at the right end is 2m. The permeability coefficient is 1 m/s.
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.
ALPHA COLLEGE OF ENGINEERING Department of Mechanical Engineering UNIT-III- 2D SCALAR VARIABLE PROBLEMS (Part –A 2Marks) 1.
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. 2.
What is CST element? Three noded triangular elements are known as CST. It has six unknown
displacement degrees of freedom (u1, v1, u2, v2, u3, v3). The element is called CST because it has a constant strain throughout it. 3.
What is LST element? Six noded triangular elements are known as LST. It has twelve unknown
displacement degrees of freedom. The displacement function for the elements are quadratic instead of linear as in the CST. 4.
What is QST element? Ten nodded triangular elements are known as Quadratic strain triangle. It is
also called as cubic displacement triangle. 5.
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. 6.
Define plane strain analysis. Plane strain is defined to be state of strain normal to the xy plane and the
shear strains are assumed to be zero.
7.
Write a strain displacement matrix for CST element?
8.
Write down the stress, strain relationship matrix for plane stress condition
9.
Write down the stress , strain relationship matrix for plane strain condition
10.
Write down the stiffness matrix equation for two dimensional CST elements.
11.
Write down the expression for the shape function for a CST element.
ALPHA COLLEGE OF ENGINEERING Thirumazhisai, Chennai – 600124 DEPARTMENT OF MECHANICAL ENGINEERING UNIT-III- 2D SCALAR VARIABLE PROBLEMS ( Part-A) ( 2 MARKS) 1.
Distinguish between plane stress and plane strain problems. (Dec-2011,2009,2008,May 2011,2008) 2. What is higher order elements? Why are they preferred? (Dec-2011, May 2011,2008) 3. Differentiate CST and LST elements? (Dec-2009) 4. Explain the important properties of CST element? (Dec-2008) 5. What are the advantages of natural coordinates over global coordinates? (Dec-2008) 6. Sketch a 4 node quadrilateral element along with nodal degree of freedom. (Dec-2009) 7. Write the shape functions of 4 node quadrilateral element. (Dec-2011) 8. Write down the shape functions of a CST element. (Dec-2010) 9. Write down the nodal displacement equations for a 2D triangular element (May-2010) 10. Write down the stiffness matrix equation for two dimensional heat conduction and convection. 11. Write down the expression for constitutive matrix for 2D element.
(PART-B) (16 MARKS) 1. If a displacement field is described by u = (-3x² + 2 y² + 6xy) 10⁻⁴ v = (2x + 6y - y²) 10⁻⁴ Determine the direct strains in x and y directions as well as the shear strain at the point x = 1, y = 0. (Dec-2011) 2. 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. (Dec-2011) 3.
Obtain the shape (Dec-2011,May 2011)
functions
of
eight
noded
quadrilateral
elements.
4. Derive the element matrices and vector matrices of 2D heat transfer element.
5.
For the plane strain element whose coordinates are (2,2),(3,4),(4,2) and the nodal displacements are u₁ = 0.005mm, u₂ = 0.002mm, u₃ = 0, u₄ = 0, u₅ = 0.004mm, u₆= 0. Determine the element stresses and element strains. Take E = 200 GPa and v = 0.3.Use unit thickness for plane strain. (May-2010)
6. Determine the element stiffness matrix and the thermal load vector for the plane stress element whose coordinates are (2,2),(3,4),(4,2). The element experiences 20 ºC increases in temperature. Take E = 15e6 N/ cm², v= 0.3, t = 0.5 cm and α = 6e-6/ ºC 7. Derive the stiffness matrix for the equilateral triangular element with plane stress conditions shown in fig 14 (a).Prove that the resulting stiffness matrix is singular. Assume v = 0.3 (Dec-2010)
8. The nodal coordinates of the triangular element are shown in fig 14 (b). At the interior point ‘P’ the x coordinate is 3.3 and N₁ = 0.3. Determine the N₂, N₃ and y coordinate of P. (Dec-2010)
9. Derive the shape functions and strain –displacement matrix of a CST element (May-2011, May-2008) 10. For the configuration shown in fig 5, determine the deflection (displacements) at the point of load application. Use one element model. Assume plane stress condition. (May- 2011).
11. Find the temperature at a point P (10,12.5) inside the triangular element shown with the nodal temperatures given as T1 = 35ºC, T2 = 45ºC, and T3 = 40ºC . Also determine the location of the 42ºC contour line for the triangular element shown in Figure.
12. (i) The (x,y) co-ordinates of nodes, i, j and k of a triangular element are given by (0,0), (3,0) and (1.5, 4) mm respectively. Evaluate the shape functions N1, N₂ & N₃ at an interior point P (2, 2.5) mm for the element. (ii) For the same triangular element, obtain the strain- displacement matrix B. (Dec 2009) 13. Derive the expression for constitutive stress-strain relationship and also reduce it to the problem of plane stress and plane strain. (Dec 2008) 14. Compute the element matrices and vectors for the element shown in fig, when the edge k-j experiences convection heat loss. Take k = 60 W/cm k , Q = 50 W/cm3
ALPHA COLLEGE OF ENGINEERING Department of Mechanical Engineering (ANNA UNIVERSITY- IMPORTANT QUESTIONS) UNIT-IV (TWO DIMENSIONAL VECTOR VARIABLE PROBLEMS) UNIT-IV ( 2 MARKS) 1. Give 4 applications where axisymmetric elements can be used. (May-2011) 2. Write short notes on axisymmetric problems? (Dec-2009,May 2011) 3. Write down the constitutive relationship for the plane stress problem(Dec2010) 4. What do you mean by constitutive law and give constitutive law for axisymmetric problems? (May-2008) 5. Name the 4 basic elasticity equations. 6. Write down the shape functions for an axisymmetric triangular element. 7. What are the conditions for a problem to be axisymmetric. 8. Give the strain-displacement matrix equation for an axisymmetric triangular element 9. Write down the stress-strain relationship matrix for an axisymmetric triangular element. 10. Write down the shape functions for an axisymmetric triangular element. 11. What is meant by thin and thick plate element? 12. Define shell element. 13. What are the assumptions used in thick and thin plate element? 14. What are the assumptions used in shell element? 15. State the types and advantages of shell element PART-B (16 MARKS) 1. Evaluate the temperature force vector for the axisymmetric triangular element whose coordinates are : r1 = 15 mm, z1 = 15 mm ; r2 = 25 mm , z2 = 15 mm ; r3 = 35 mm , z3 = 50 mm. The element experiences a 15°C increases in temperature. Take α = 10x10-6/°C, E = 2x105 N/mm2, v = 0.25. 2. The (r ,z) co-ordinates of nodes i, j and k of an axisymmetric triangular element are given by (3,4), (6,5) and (5, 8) cm respectively, determine the element stresses.
Let E = 210Gpa and v = 0.25. The displacement are u 1 = 0.05 mm; w1 = 0.03 mm, u2 = 0.02 mm; w2 = 0.02 mm, u3 = 0 ; w3 = 0 3. Derive the shape functions, strain – displacement matrix [B] for axisymmetric triangular element 4. Derive the stress-strain relationship matrix [D] for the axisymmetric triangular element. 5. A hollow cylinder of inside diameter 100 mm and outside diameter 140 mm is subjected to an internal pressure of 4 N/mm2. By using 2 elements on the 15 mm length the cylinder, calculate the displacement at inner radius. Take E = 2x105 N/mm2, v = 0.25.
6. Write short notes on plate and shell elements. 7. Define the following terms with suitable examples: (i) Plane stress, Plane strain (ii) Node, Element and Shape functions (iii) Axisymmetric analysis. (May-2010) 8. 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. The (r ,z) co-ordinates of nodes , i, j and k of an axisymmetric triangular element are given by (3,4), (6,5) and (5, 8) cm respectively. The element displacement (in cm) vector is given as u = (0.002, 0.001, 0.001, 0.004,-0.003, 0.007)T. Determine the element strains. (Dec 2009) 10. The nodal co-ordinates for an axisymmetric triangular are given below: r1 = 15 mm, z1 = 15 mm ; r2 = 25 mm , z2 = 15 mm ; r3 = 35 mm , z3 = 50 mm. Determine [B] & [K] matrix for that element. Take E = 2x105 N/mm2, v = 0.25.
ALPHA COLLEGE OF ENGINEERING Department of Mechanical Engineering (ANNA UNIVERSITY- IMPORTANT QUESTIONS) UNIT-IV ( 2 MARKS)
1. What is axisymmetric element? Many three dimensional problems in engineering exhibit symmetry about an axis odf rotation .Such types of problems are solved by a special two dimensional element called as axisymmetric element. 2. What are the conditions for a problem to be axisymmetric? 1. The problem domian must be symmetric about the axis of revolution 2. All boundary conditions must be symmetric about the axis of revolution 3. All loading conditions must be symmetric about the axis of revolution. 3. 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. 4. Define plane strain analysis. Plane strain is defined to be state of strain normal to the xy plane and the shear strains are assumed to be zero.
5. Write down the displacement equation for an axisymmetric triangular element.
6. Write down the shape function for an axisymmetric triangular element.
7. What are the 4 basic elasticity equations? (i) Equlibrium equation (ii) Compatibility equation (iii) Strain-displacement relationship equation (iv) Stress-strain relationship equation 8. Give 4 applications where axisymmetric elements can be used. (i) Pressure vessels (ii) Rocket castings (iii) Cooling towers (iv) Submarine hills (v) Springs. 9. Name different strain available in axisymmetric elements. Radial, Circumferential, Longitudinal and Shear strains. 10. What are the stresses need to be calculated in axisymmetric problems? Radial, Circumferential, Longitudinal and Shear stresses.
ALPHA COLLEGE OF ENGINEERING Department of Mechanical Engineering (ANNA UNIVERSITY- IMPORTANT QUESTIONS) UNIT-V- ISOPARAMETRIC FORMULATION PART-A ( 2 MARKS) 1. Write the Gauss points and weights for 2 point formula of numerical integration. (May-2011,Dec 2011) 2. Write the Lagrangian shape functions for a 1D, 2noded element? (Dec-2008) 3. What is the significance of Jacobian matrix in 2D problems. (Dec-2011) 4. Write down the Jacobian matrix (Dec -2010) 5. What do you meant by isoparametric element? (Dec-2010, June 2009) 6. Define sub-parametric elements 7. Define super-parametric elements 8. What are serendipity elements? 9. What are the stages in FEA? 10. What is the function of processor? 11. What is the function of pre-processor? 12. What is the function of post-processor? 13. State the applications of FEA. PART-B (16 MARKS) 1. Evaluate the integral
using one point and two point Gaussian quadrature to obtain an exact value of the integral. (May-2011) 2. Evaluate the integral
using three point Gaussian quadrature and compare with exact solution.(Dec-2011)
3. Evaluate the integral I = ∫ ∫ (x² + y² + 6xy) dx dy with limits -1 to +1 using three point Gauss Numerical integration. (Dec-2011, May - 2011) 4. (i) The Cartesian (global) co-ordinates of the corner nodes of an isoparametric quadrilateral element are given by (1, 0), (2, 0), (2.5, 1.5) and (1.5, 1).Find its Jacobian matrix. (ii) Distinguish between sub parametric and super parametric elements. 5. Use Gauss quadrature rule (n=2) to numerically ∫ ∫ x2y dx dy with limits -1 to +1 (Dec 2008) 6. A 4 noded rectangular elements whose coordinates are given by (0, 0), (2, 0), (2, 1) and (0, 1). Determine the following (i) Jacobian matrix ,(ii) Strain-displacement matrix and (iii) Element stresses. Take E = 2x105 N/mm2, v = 0.25, u = (0, 0, 0.003, 0.004, 0.006,0.004, 0,0)T and ε =ή=0.Assume plane stress condition. 7. Evaluate the Cartesian coordinates of point P which has the local coordinates of ε =0.6, ή=0.8 for the isoparametric quadrilateral element which has the coordinates (3, 4), (8, 5), (9, 9) and (5, 7). 8. Establish the strain displacement matrix for the linear quadrilateral element whose coordinates are (1, 1), (5,2), (4,5) and (2,4) at ε =0.57735, ή=0.57735 9.
Derive the shape functions of (i) 8 noded rectangular element and (ii) 6 noded triangular element.
10. Derive the shape functions of isoparametric element. 11. Explain in detail about FEA analysis software. 12. Consider the undamped 2 D.O.F system as shown in fig(i).Find the response of the system when the first mass alone is given an initial displacement of unity and released from rest.
ALPHA COLLEGE OF ENGINEERING Department of Mechanical Engineering (ANNA UNIVERSITY- IMPORTANT QUESTIONS) UNIT-V ( 2 MARKS) 1. What is the purpose of Isoparametric Elements? It is difficult to represent the curve boundaries by straight edges elements. A large number of finite elements may be used to obtain reasonable resemblance between original body and the assemblage .In order to overcome this drawback,isoparametric elements are used i.e., for problems involving curved boundries,a family of elements known as “ Isoperimetric elements “ are used. 2. Define super parametric element. 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. 3. 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,then it is known as isoparametric element. 4. Write down the shape function for 4 noded rectangular elements using natural coordinate system.
5. Write down the element force vector equation for four noded quadrilateral elements.
6. Write down the Gaussian quadrature expression for numerical integration.
7. What is the function of pre-processor? It read geometry, material data, boundary conditions and initial conditions of the problem. 8. What is the function of processor? (i) Generate finite element mesh (ii) Calculate element matrices (iii) Assemble element equations (iv) Solve the equations. 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. 10. State the applications of FEA. FEA involves solution of wide variety of problems in Solid mechanics, fluid mechanics, heat transfer, electrical and magnetic fields. 11. Write down the shape function for 4 noded rectangular elements using natural coordinate system.
ALPHA COLLEGE OF ENGINEERING THIRUMAZHISAI, CHENNAI – 600124 DEPARTMENT OF MECHANICAL ENGINEERING
ME 6603 - FINITE ELEMENT ANALYSIS
UNIT WISE IMPORTANT FORMULAE UNIT – I (INTRODUCTION) 1. Initial and Boundary value Problems: (i) When two roots (m1, m2) are real and un-equal The Complementary function y(x) = C1em1x+ C2em2x (ii) When two roots (m1, 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. 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
3. Rayleigh-Ritz Method Total potential energy = Strain Energy – Work done by external forces
π=U–H
(i)
For beam problem, 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.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
2 U = ½ K �� (Where, ��= u -u ) 2
1
H = Fu
UNIT – II ONE DIMENSIONAL FINITE ELEMENT ANALYSIS 1-D BAR ELEMENT, 1. Linear Polynomial Equation, u = ao+a1x 2. Shape functions N1 = N= 2
3. Stiffness matrix,
Where, [B]
Strain displacement relationship matrix.
[D]
Elasticity matrix or Stress-strain relationship matrix.
[D] = [E] = E = Young’s modulus. =A 3
Stiffness matrix 4. General FEA equation is, {F} = [K] {u} Where, {F} is an element force vector [Column matrix]. [K] is a stiffness matrix [Row matrix]. {u} is a nodal displacement [Column matrix].
5. 1D Displacement equation,
6. Force vector due to self weight,
7. Reaction force,
8. Stress,
Where, E = Young’s modulus 4
= 9. Temperature effect, Force,
Stress,
Where, A = Area of cross section of bar element. ∆T = Temperature difference. α
= Coefficient of thermal expansion.
TRUSS ELEMENTS, 1. Stiffness matrix,
Where, A = Area of the truss element E = Young’s modulus of element le = Equivalent length
2. Strain energy,
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3. Finite element general equation,
Where, [K] = stiffness matrix {U} = nodal displacement matrix 4. Stress,
SPRINGS 1. Stiffness matrix,
2. Tensile force,
Where, k = spring constant = change in deformation
BEAMS 1. Shape functions,
6
2. Stiffness matrix,
3. Finite element equation,
Where, L = length of the beam element E = Young’s modulus I = Moment of inertia LONGITUDINAL & TRANSVERSE VIBRATION PROBLEMS
7
ONE DIMENSIONAL HEAT TRANSFER PROBLEMS 1. Finite Element Equation For 1D Heat Conduction with free end Convection
2. Finite Element Equation For 1D Heat Conduction, Convection and Internal Heat Generation 8
UNIT III (2D SCALAR VARIABLE PROBLEMS CONSTANT TRIANGULAR ELEMENT (CST),
1. Shape functions,
Where, 9
2. Displacement functions,
3. Stiffness matrix,
[K] = [B]T [D] [B] A t
4. Strain displacement matrix,
5. Stress - strain matrix in general 2D form,
10
Plane stress condition,
Plane strain condition,
6. Element stress,
11
Maximum stress,
Minimum stress,
7. Principle angle,
8. Element strain,
TEMPETATURE EFFECT OF CST ELEMENT, 1. Initial strain, Plane stress,
Plane strain,
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2. Element temperature force,
Where, t = thickness A= area of the element. 2D HEAT TRANSFER PROBLEMS Stiffness Matrix for both Conduction and Convection
UNIT IV (2D VECTOR VARIABLE PROBLEMS
AXI-SYMMETRIC ELEMENT, 1. Shape function, 13
Where,
2. Strain displacement matrix,
3. Stress strain relation 14
4. Stiffness matrix,
TEMPETATURE EFFECT OF Axisymmetric Element, 3. Initial strain,
4. Element temperature force,
Where, t = thickness A= area of the element.
15
UNIT V-ISOPARAMETRIC FORMULATION Iso Parametric Quadrilateral Element 1. Shape functions,
2. Strain displacement matrix,
16
3. Displacement function, Rectangular element,
17
4. Jacobian matrix,
Where:
5. Force vector,
Where, = 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. Element stress,
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Gaussian Quadrature (Or) Numerical Integration (i)
For 2 point Quadrature
Where, w1 = w2 =1 and x1= √1/3, x2 = -√1/3
(ii)
For 3 point Quadrature
Where, w1 = w3 = 5/9, w2 = 8/9 and x1= √3/5, x2 = 0, x3 = -√3/5
(iii)
For double Integration,
w12f(x1,y1) + w1w2f(x1,y2) + w2w1f(x2,y1) + w22f(x2,y2)
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