V

V.S.B ENGINEERING COLLEGE, KARUR

DEPARTMENT OF AERONAUTICAL ENGINEERING

FINITE ELEMENT ANALYSIS QUESTION BANK

UNIT-I

2 MARK QUESTIONS

1. Mention the various methods of engineering analysis.

The various methods of engineering analysis are

Experimental methods

Analytical methods or Theoretical analysis

Numerical methods or Approximate methods

2. Define experimental methods.

In the experimental methods, prototypes can be used. If we want to change the dimensions of the prototype, we have to disassemble the entire prototype and reassemble it and the testing should be carried out. It needs man power and materials. So, it is time consuming and costly process.

3. What is meant by analytical or Theoretical Analysis?In the analytical methods, problems are expressed by mathematical differential equations. It gives quick and closed form solutions. It is used only for simple geometries and idealized support and loading conditions.4. State numerical methods.

Analytical solutions can be obtained only for certain simplified situations. For problems involving complex material properties and boundary conditions, the engineer prefers numerical methods that give approximate but acceptable solutions.5. Mention the three methods coming under numerical methods.

The three methods coming under numerical methods are Functional approximation

Functional difference method(FDM)

Finite element method(FEM)

6. Define functional approximation.

The classical methods such as Rayleigh-Ritz methods and Galerkin methods are based on functional approximation but vary in their procedure for evaluating the unknown temperatures. Rayleigh-Ritz method is useful for solving complex structural problems, encountered in finite element analysis.

Weighted residual methods or Galerkin methods is useful for solving non-structural problems.

7. State finite difference method.

Finite difference method is useful for solving heat transfer fluid mechanics and structural mechanics problems. It is a general method. It is applicable to any phenomenon for which differential equation along with the boundary conditions is available. It works well for two dimensional regions with boundaries parallel to the co-ordinate axes. 8. What is meant by finite element?A small unit having definite shape or geometry and nodes is called finite elements.9. Mention the advantages or merits of FEA. FEA can handle irregular geometry in a convenient manner.

It handles general lad conditions without difficulty.

Non-homogenous materials can be handled easily.

All the various types of boundary conditions are handled.

Dynamic effects are included.

Higher order elements may be implemented.

Altering the element model with different loads, boundary conditions and other changes in the model can be done easily and cheaply.

Vary the size of the elements to make it possible for using small elements where necessary.10. Mention the disadvantages or demerits of FEA.

The finite element method is a time consuming process.

In order to analyze many numbers of small elements, due to human fatigue, we have to depend on computer package.

The result obtained using FEM will be closer to exact solution only if the system is divided into large number of smaller elements. Otherwise, there may be a considerable variation from the exact solution.

FEM cannot produce exact results as those of analytical methods.11. Mention the applications of FEA.

S.No

Area of StudyAnalyzing Problems

1

Civil engineering structuresAnalysis of trusses, frames, folded plates, shell roofs, bridges and prestressed concrete structures, shear walls.

2

Aircraft structuresAnalysis of aircraft wings, fins, rockets. Spacecraft

3

Mechanical designStress analysis of pressure vessels, pistons, composite materials, linkages and gears.

4Heat conductionTemperature distribution in solids and fluids

5

Hydraulics and Water resources engineeringAnalysis of potential flows, free surface flows. Viscous flows. Analysis of hydraulic structures and dams

6Electrical machines and electromagneticAnalysis of synchronous and induction machines eddy current and core losses in electric machines.

7Nuclear engineeringAnalysis of nuclear pressure vessels and containment structures.

8

GeomechanicsStress analysis in soils, dams, layered piles and machine foundations

9

Biomedical engineeringStress analysis of eyeballs, bones and teeth. Mechanics of heart valves.

12. Mention the general steps in finite element analysis.

The general steps in finite element analysis are

Discretization of structure

Numbering of nodes and elements

Selection of a displacement function or interpolation function

Define the material behavior by using strain-displacement matrix

And stress strain relationship matrix

Derivation of element stiffness matrix and equations

Assemble the element equations to obtain global or total equations

Apply boundary conditions

Solution for the unknown displacements

Computation of the element strains and stresses from the nodal displacements

Interpret the results.13. What is meant by finite element analysis?Finite element method is a numerical method for solving problems of engineering and mathematical physics.

In the finite element method, instead of solving the problem for the 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. Give examples for the finite element.

One dimensional elements: Truss, Beam and bar elements.

Two dimensional elements: Triangular and rectangular elements.

Three dimensional elements: Tetrahedral and Hexahedral element.15. What is meant by node or joint?Each kind of finite element has a specific structural shape and is interconnected with the adjacent elements by nodal points or nodes. At the nodes, degrees of freedom are located. The forces will act at only nodes and not at any other place in the element.

16. What is the basis of finite element method?

Discretization is the basis of finite element method. The art of subdividing a structure into a convenient number of small components is known as Discretization.

17. What are the types of boundary conditions?

1)Primary boundary condition.

2)Secondary boundary condition.

18. List the three phases of finite element method.

The three phases are:

1) Preprocessing

2) Analysis.

3) Post processing

19. What is a structural and non-structural problem?Structural problems: In structural problems, displacement at each node is obtained. By using these displacement solutions, stress and strain in each element can be calculated.

Non-Structural problems: In non-structural problems, temperatures or fluid pressure at each nodal element is obtained. By using these values, properties such as heat flow, fluid flow, etc., for each element can be calculated.

20. What are the methods are generally associated with the finite element analysis?

1) Force method.

2) Displacement or stiffness method.

21. State force method and stiffness method.

In force method, internal forces are considered as unknowns of the problems. In displacement or stiffness method, displacements of nodes are considered as the unknowns of the problems. Among them two approaches, displacement method is desirable.22. Why polynomial type interpolation functions are preferred over trigonometric functions?1) It is easy to formulate and computerize the finite element equations.

2) It is easy to perform differentiation or integration.

3) The accuracy of the results can be improved by increasing the order of polynomial.

23. Name the variational methods.

1) Ritz method

2) Rayleigh-Ritz method.

24. List out the various weighted-residual methods.1) Point collocation method.

2) Sub domain collocation method.

3) Least square method.

4) Galerkins method.

25. Define post processing.

Analysis and evaluation of the solution results is referred to as post processing. Post processor computer programs help the user to interpret the results by displaying them in graphical from.

26. List the advantages of post processing.

1) Required result can be obtained in graphical form.

2) Contour diagrams can be used to understand the solution quickly and easily.

27. Define Rayleigh Ritz method? Rauleigh ritz method is integral approach method which is useful for solving complex structural problems, encountered in finite element analysis. This method is possible only if a suitable functional is available.

28. What is meant by Discretization and assemblage?

The art of subdividing a structure into a convenient number of smaller components is known as discretization. These smaller components are put together. The process of uniting the various elements together is called assemblage.

29. State degrees of freedom.When the force or reaction acts at a nodal point, node is subjected to deformation. The deformation includes displacements, rotations, and/or strains. These are collectively known as degrees of freedom.

30. Define aspect ratio.Aspect ratio is defined as the ratio of the largest dimension of the element to the smallest dimension of the element. In many cases, as the aspect ratio increases, the inaccuracy of the solution increases. The conclusion of many researches is that the aspect ratio should be close to unity as possible.

31. What is truss element?

The truss elements are the part of a truss structure linked together by point joints, which transmit only axial force to the element.

32. What are h and p versions of finite element method?h and p versions are used to improve the accuracy of the finite element method. In h versions, the order of polynomial approximation for all elements is kept constant and the number of elements is increased. In p version, the number of elements is maintained constant and the order of polynomial approximation of element is increased.

33. During Discretization, mention the places where it is necessary to place a node?

The following places are necessary to place a node during dicretization process.

1) Concentrated load acting point.

2) Cross-section changing point.

3) Different material interjunction method.

4) Sudden change in load point.

34. What is the difference between static and dynamic analysis?Static analysis: The solution of problem does not vary with time

Example: Stress analysis of a beam.

Dynamic analysis: The solution of problem varied with the time.

Example: Vibration analysis problems.

35. List out FEM software packages. ANSYS, NASTRAN, COSMOS, NISA.

36. Differentiate between global axes 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 the element to element.

Global axes are defined for the entire system. They are in the same direction for all the elements even though the elements are differently located.

UNIT-II2 MARK QUESTIONS1. Mention the one dimensional elements.

Bar and beam elements are considered as one dimensional elements. These elements are often used to model trusses and frame structures.2. Define bar, beam, truss and frame.

A bar is a member which resist only axial loads, whereas a beam can resist axial, lateral and twisting loads. A truss is an assemblage of bars with pin joints and a frame is an assemblage of beam elements.3. Define numbering of nodes.

In one dimensional problem, each node is allowed to move only in x direction (positive or negative).So, each node has one degrees of freedom.4. Mention the properties of stiffness matrix.

It is a symmetric matrix

The sum of elements in any column must be equal to zero.

It is an unstable element. So, the determinant is equal to zero.

The dimension of the global stiffness matrix is N X N, where N is the number of nodes. This follows from the fact each node has only one degree of freedom

The diagonal coefficients are always positive and relatively large when compared to the off-diagonal values in the same row.5. Define beam.

Beam is a structural member which is supported along the length and subjected to external forces or loads transversely. Beam is sufficiently long when compared to the lateral dimensions.6. Mention the different types of beams.

Cantilever beam

Simply supported beam

Overhanging beam

Fixed beam

Continuous beam7. What are the types of loading acting on the structure?

1) Body forces

2) Traction force

3) Point load

8. Define body force.

A body force is a distributed force acting on every elemental volume of the body.

Example: Self weight due to gravity9. Define traction force.

Traction force is defined as distributed force acting on the surface of the body.

Eg: Frictional resistance, Viscous drag, surface shear etc.,10. Define point load.

Point load is force acting at a particular point which causes displacement.

11. What are the basic steps involved in the finite element modeling?

Finite element modeling consists of the following:

1) Discretization of structure.

2) Numbering of nodes.

12. Define Discretization.The art of subdividing a structure into a convenient number of smaller components is known as discretization.

13. List the types of co-ordinates.

The co-ordinates are generally classified as follows:a) Global co-ordinates

b) Local co-ordinates

c) Natural co-ordinates

14. Define global co-ordinates. The points in the entire structure are defined using co-ordinate system is known as global co-ordinate system.15. What is meant by natural co-ordinates?

A natural co-ordinate system is used to define any point inside the element by a set of dimensionless numbers, whose magnitude never exceeds unity. This system is very useful in assembling of stiffness matrices.

16. Define shape function.

In finite element method, field variables within element are generally expressed by following approximate relation:

Where and are the values of the field variable at the nodes and N1,N2 and N3 are the interpolation functions. N1,N2,N3 are also called shape functions because they are used to express the geometry or shape of the element.17. What are the characteristics of shape function?

The characteristics of shape function are as follows:

1) The shape function has unit value at one nodal point and zero nodal value at other nodal point.

2) The sum of the shape function s equal to one.18. Why polynomials are generally used as shape function?

Polynomials are generally used as the shape function due to the following reasons:

1) Differentiation and integration of polynomials are quite easy.

2) The accuracy of results can be improved by increasing the order of polynomial.

3) It is easy to formulate and computerize the finite element equations.

19. How do you calculate the size of the global stiffness matrix?

Global stiffness matrix size = Number of node x Degrees of freedom per node20. Give the general expression for element stiffness matrix.

Stiffness matrix [K] = [B]T [D][B] dvWhere [B] = strain displacement matrix(Row matrix)

[D] = Stress, Strain relationship matrix.(Row matrix)

21. Write the expression of stiffness matrix for one dimensional bar element.

Stiffness matrix [K] = (AE/L)Where A = Area of the bar element E = Youngs modulus of the bar element

L = Length of the bar element.

22. Write the general finite element equation.

General finite element equation is {F} = [K] {u}

Where {F} = Force vector (column matrix)

[K] = Stiffness matrix (row matrix)

{u} = Degrees of freedom (column matrix)23. Write the general finite element equation for one dimensional bar element two noded bar element.24. Define truss.

A truss is defined as a structure, made up of several bars, riveted or welded together.

25. State the assumptions are made while finding the forces in a truss.

The following assumptions are made while finding the forces in a truss:

1) All the members are pin jointed.

2) The truss is loaded only at the joints.

3) The self weight of the members is neglected unless stated.

26. Write the expression of stiffness matrix for a truss element.

27. Write the expression of shape function N and displacement U for one dimensional bar element.

28. Define total potential energy.

The total potential energy of an elastic body, is defined as the sum of total strain energy U and potential energy of the external forces W,

= U + W

29. State the principle of minimum potential energy.

The principle of minimum potential energy states: Among all the displacement equations that satisfy internal compatibility and the boundary conditions, those that also satisfy the equations of equilibrium make the potential energy a minimum in the stable state.30. What is the stationary property of total potential energy?

If a body is in equilibrium, its total potential energy is stationary. For stable equilibrium, 2 > 0, otherwise is minimum for stable equilibrium.

For neural equilibrium, 2 = 0. In this case, is unchanging.

For unstable equilibrium, 2 < 0, otherwise is maximum.

31. State the principle of virtual work.

A body is in equilibrium if the internal virtual work equals the external virtual work for every kinematically admissible displacement field.

32. Distinguish between essential boundary conditions and natural boundary conditions.

1) Primary boundary condition or Essential boundary condition:

The boundary condition which in terms of field variable is known as primary boundary condition.

2) Secondary boundary condition or Natural boundary condition: The boundary conditions which are in the differential form of filed variables is known as secondary boundary condition.33. What are the differences boundary value problem and initial value problem.

The solution of differential equation is obtained for physical problems which satisfies some specified conditions known as boundary conditions.

The differential equation together with these boundary conditions, subjected to a boundary value problem.

The differential equation together with the initial conditions subjected to an initial value problem.

Example: Boundary value problem

With boundary conditions,

UNIT-III2 MARK QUESTIONS1. Define Discretization.

The art of subdividing a structure into a convenient number of smaller components is known as Discretization. In two dimensional problems, three kinds of finite elements are used. They are

Triangular element

Rectangular element

Quadrilateral element2. Define numbering of nodes.

In two dimensional problems, each node is permitted to move in the two directions (x and y). Hence each node has two degrees of freedom.3. Define heat transfer.

Heat transfer can be defined as the transmission of energy from one region to another region due to temperature difference.4. Mention the three modes of heat transfer.

Conduction

Convection

Radiation5. Define conduction.

Heat conduction is a mechanism of heat transfer from a region of high temperature to a region of low temperature within a medium (solid, liquid or gases) or between different medium in a direct physical contact. In conduction, energy exchange takes place by the kinematic motion or direct impact of molecules. Pure conduction is found only in solids.6. State Convection.

Convection is a process of heat transfer that will occur between a solid surface and a fluid medium when they are at different temperatures. Convection is possible only in the presence of fluid medium.7. What is meant by Radiation?

The heat transfer from one body to another without any transmitting medium is known as radiation. It is electromagnetic wave propagation.8. Define two dimensional elements.

Two dimensional elements are defined by three or more nodes in a two dimensional plane(i.e. x, y plane).The basic element useful for two dimensional analysis is the triangular element.

9. What is CST element?

Three noded triangular element is known as Constant Strain Triangle (CST), which is shown in figure below. It has six unknown displacement degrees of freedom (u1v1, u2v2, u3v2). The element is called CST because it has a constant strain throughout it.Merit:

Calculation of stiffness matrix is easier.

Demerit:

The strain variation within the element is considered as constant.So, the results will be poor.10. What is LST element?Six noded triangular element is known as Linear Strain Triangle (LST), which is shown in figure below. It has twelve unknown displacement degrees of freedom. The displacement functions for the element are quadratic instead of linear as in the CST.

11. What is QST element?

Ten noded triangular element is known as Quadratic Strain Triangle (QST) which is shown in figure below. It is also called cubic displacement triangle.

12. 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.

13. Define plane strain analysis.

Plane strain is defined to be a state of strain in which the strain normal to the xy plane and the shear strains are assumed to bezero.

14. Write a displacement function equation for CST element.

15. Write a strain displacement matrix for CST element.

16. Write the stress strain relationship matrix for plane stress condition.

17. Write the stress strain relationship matrix for plane strain condition.

18. Write the stiffness matrix equation for two dimensional CST elements.

Stiffness matrix [K] = [B]T[D][B]A t

Where [B] = Strain displacement matrix

[D] = Stress strain matrix

A = Area of the element.

t = Thickness of the element

19. Write the expression for the shape function for a CST element.

UNIT-IV2 MARK QUESTIONS1. Define elasticity equations.

Elasticity equations are used for solving structural mechanics problems. These equations must be satisfied if an exact solution to a structural mechanics problem is to be obtained. There are four basic sets of elasticity equations. They are

Strain-displacement relationship equations

Stress-strain relationship equations

Equilibrium equations

Compatibility equations.2. Define Compatibility equations.

Along with the strain displacement equations, we need compatibility equations to ensure that the displacement components U, V and W are single valued continuous functions so that tearing or overlap of elements does not occur.3. What is meant by axi-symmetric element?

Many three dimensional problems in engineering exhibit symmetry about an axis of rotation. Such types of problems are solved by a special two dimensional element called the axisymmetric element.

4. What are the conditions for a problem to be axi-symmetric?

1) The problems domain must be symmetric about the axis of rotation.2) All boundary conditions must be symmetric about the axis of revolution.

3) All loading conditions must be symmetric about the axis of revolution.

5. Write down the shape functions for an axisymmetric triangular element.

6. Write the strain displacement matrix equation for an axi-symmetric triangular element.

7. Write the stress strain relationship matrix for an axi-symmetric triangular element.

8. Write the stiffness matrix equation for an axi-symmetric triangular element.

9. What are the ways in which a three dimensional problem can be reduced to a two dimensional approach.

1. Plane stress: One dimension is too small when compared to other two dimensions.

Example: Gear _ Thickness is small.

2. Plane strain: One dimension is too large when compared to other two dimensions.

Example: Long pipe (Length is long compared to diameter)

3.Axisymmetric: Geometry is symmetric about the axis.

Example: Cooling tower.

10. Calculate the jacobian of the transformation J for the triangular element shown in figure.

UNIT-V2 MARK QUESTIONS1. What is the need of iso-parametric elements?

It is difficult to represent the curved boundaries by straight edge elements. A large number of elements may be used to obtain reasonable resemblance between original body and the assemblage. In order to overcome this drawback, isoparametric elements are used.

2. Define numerical integration or Gaussian quadrature.

The gauss quadrature is one of the numerical integration methods to calculate the definite integrals. In finite element analysis, gauss quadrature method is mostly preferred. 3. Write the shape function for four noded rectangular element using natural co-ordinate system.4. Write the jacobian matrix for four noded quadrilateral elements.

5. Write the stiffness matrix equation for four noded quadrilateral elements.

6. Write the element force vector equation for four noded quadrilateral elements?7. Write the Gaussian quadrature expression for numerical integration.

8. Define super parametric element.

If the number nodes used for defining the geometry is more than number of nodes used for defining the displacements is known as super parametric element.

9. What is meant by sub parametric element?

If the number of nodes used for defining the geometry is less than the number of nodes used for defining the displacements is known as sub parametric element.

10. State iso parametric element.

If the number of nodes used for defining the geometry is same as the number of nodes used for defining the displacements is known as isoparametric element.

11. Is beam element an iso parametric element?

Beam element is not an isoparametric element since the geometry and displacements are defined by different order interpolation functions.

12. What is the difference between natural co-ordinate and simple natural co-ordinate?

13. Give examples for essential and non-essential boundary conditions.

The geometric boundary conditions are displacement, slope, etc. The natural boundary conditions are bending moment, shear force, etc.14. 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.15.Define initial value problem.

An initial value problem is one in which the dependent variable and possibly is derivatives are specified initially.16.Define boundary value problem.

A differential equation is said to describe a boundary value problem if the dependent variable and its derivatives are required to take specified values on the boundary.