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ME6603- FINITE ELEMENT ANALYSIS
UNIT – 1
PART - A
1. State the methods of engineering analysis.
There are three methods of engineering analysis. They are:
1. Experimental methods.
2. Analytical methods.
3. Numerical methods or approximate methods.
2. What is meant by finite element?
A small unit having definite shape of geometry and nodes is called finite element.
3. 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.
4. Give examplesfor the finite element.
1. One dimensional elements: (a) Truss elements.
(b) Bar, Beam elements.
2. Two dimensional elements: (a) Triangular elements.
(b) Rectangular elements.
3. Three dimensional elements: (a) Tetrahedral elements.
(b) Hexahedral elements.
5. What is meant by node or joint? (NOV/DEC2015)
Each kind of finite element has a specific structural shape and is interconnected with
theadjacent elements by nodal points or nodes. At the nodes, degrees of freedom are located. The
forces will act only at nodes and not at any other place in the element.
6. 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 smaller components is known as Discretization.
7. What are the types ofboundary conditions?
There are two types of boundary conditions. They are:
1. Primary boundary condition.
2. Secondary boundary condition.
8. State the three phases offinite element method.
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The three phases are: 1.Preprocessing.
2. Analysis.
3. Post processing.
9. What are structural and non-structural problems?
Structural problems: In structural problems, displacement at each nodal point isobtained.
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 point is obtained. By using these values, properties such as heat flow, fluid flow, etc.,
for each element can be calculated.
10. What are the methods are generally associated with the finite element analysis?
The following two methods are generally associated with the finite element analysis.
They are:
(i) Force method.
(ii) Displacement or stiffness method.
11. Explain force method and stiffness method?
In force method, internal forces are considered as the unknowns of the problem. In
displacement or stiffness method, displacements of the nodes are considered as theunknowns of
the problem. Among them two approaches, displacement method isdesirable.
12. Why polynomial types of interpolation functions are mostly used in FEM?
The polynomial type of interpolation functions are mostly used due to the following
reasons:
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 the
Polynomial.
13. Name the variational methods.
1. Ritz method.
2. Rayleigh-Ritz method.
14. Name the weighted residual methods. (Nov/Dec-2014)(May/June 2016)
1. Point collocation method.
2. Sub domain collocation method.
3. Least squares method.

4. Galerkin's method.
15. What is meant by post processing? [M.Y., April 2001]
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 form.
16. What is Rayleigh-Ritz method? (NOV/DEC 2015)
Rayleigh-Ritz method is a 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.
17. What is meant by Discretization and assemblage?(Nov/Dec 2015)
The art of subdividing a structure into a 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.
18. What is meant by degrees offreedom?
When the force or reactions act at nodal point, node is subjected to deformation. Thedeformation
includes displacement, rotations, and/or strains. These are collectivelyknown as degrees of
freedom.
19. What is "Aspect ratio"?
Aspect ratio is defined as the ratio of the largest dimension ofthe element to the smallest
dimension. 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.
20. What is truss element?
The truss elements are the part of a truss structure linked together by point joints, whichtransmit
only axial force to the element,
21. List the two advantages ofpost-processing.
1. Required result can be obtained in graphical form.
2. Contour diagrams can be used to understand the solution easily and quickly.
22. If a displacement field in x direction is given by u = 2 xl+4 yl+ 6 xy. Determine the
strain in x direction.
23. What are 'It' and 'p' versions offinite element method?

'h' versions 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 numbers of elements are increased.
In 'p' version, the numbers of elements are maintained constant and the order
ofpolynomial approximation of element is increased.
24. During Discretization, mention the places where it is necessary to place a node?
The following places are necessary to place a node during Discretization process.
(i) Concentrated load acting point.
(ii) Cross-section changing point.
(iii) Different material interjunction point.
(iv) Sudden change in load point.
25. 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.
Example: Vibration analysis problems.
26. Name any four FEA software’s.
1. NASTRAN, 2. NISA, 3.ANSYS, 4.COSMOS.
27. 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.
28. Distinguish between potential energyfunction and potential energy functional.
If a system has finite number of degrees of freedom (ql q2 and q3)' then the potential
energy is expressed as,
It is known as function.
If a system has infinite degrees of freedom, then the potential energy is expressed as,
It is known as functional.

29. What doyou mean by constitutive law?
For a finite element, the stress-strain relations are expressed as follows:
This equation is known as constitutive law.
30. Compare the Ritz technique with nodal approximation method (Nov/Dec-2014)
•Rayleigh Ritz method is an approximate method of finding displacements that is based
on the theorem of minimum potential energy.
•The method is restricted to conservative systems that may be linear or non-linear.
31. State the principle of potential energy theorem. (May/June2016)
The deformed state a structure attains upon the application of forces is the equilibrium state of a
structural system. The Potential energy (PE) of a structural system is defined as the sum of the strain
energy (SE) and the work potential (WP). The strain energy is the elastic energy stored in deformed
structure.
32. What are the methods generally associated with finite element Analysis?(May/June2016)
 Numerical methods for approximating
 variational formulation for galerkin method
 Weak formulation
PART - B
1. List and briefly describe the general steps ofthe finite element method. (May/June 2014)
2. Explain the following: (i) Natural Discretization.
(ii) Artificial Discretization.
3. Explain the Discretization process.
4. Explain the following: (i) Variational approach.
(ii) Weighted residual methods.
5. List the advantages, disadvantages and applications ofFEM.
6. Use the Rayleigh-Ritz method to find the displacement of the midpoint of the rod shown in
Fig.

7. A cantilever beam is shown in Fig. It is subjected to an uniformly distributed load w,
Concentrated load Wand moment Moat the free end as shown.
Solve the problem by using Rayleigh-Ritz method.
8. A bar fixed at one end and free at the other end is loaded as shown in Fig. Calculate the 8.
Displacement and stresses using Rayleigh-Ritz procedure.
Compare the solution with exact results using one, two and three terms in the polynomials.
9. Solve the following equation using a. Two parameter trial solution by:
(a) Point collocation method.
(b) Galerkin's method.
(May/June 2014)
Compare the two solutions with the exact solution.
10. Determine a two parameter solution of the following using the Galerkin's method and
compare it with the exact solution.

11. A uniform rod subjected to a uniform axial load is as illustrated. The deformation of the bar
is governed by the differential equation given below. Determine the displacement using weighted
residual method.
AEd2u/dx2 + qo = 0 with boundary conditions u(0) = 0, du/dx at x = L = 0.
(2011)
12. Solve the differential equation for a physical problem expressed as d2
y/dx2
+ 100 = 0 for 0 <
x < 10 with the boundary condition as y(0) = 0 and y(10) = 0 using a) Point Collocation
Method b) Sub Domain Collocation method c) Least squares method and d) Galerkin’s Method.
(2013)
13Derive the characteristic equation for the one dimensional bar element by using piece vise
defined interpolations and weak form of the weighted residual method.
(2012)
14.i) Derive the element level equation for one dimensional bar element based on the stationary
potential of a functional.
ii) List out the general procedure for FEA problems. (2012)
15.
(Nov/Dec 2014)
16
(Nov/Dec 2014)

17. The following differential equations is available for a physical phenomenon
𝑑2𝑦
𝑑𝑥2
-10x2
= 5; 0 ≤ x ≤ 1
The boundary conditions are : y(0) = 0
y(1) = 0
Find an approximate solution of the above differential equation by using Galerkin’s method of
weighted residuals and also compare with exact solutions. (May/June 2016)
18.A beam AB of span ‘l’ simply supported at ends and carrying a concentrated load W at the
centre C as shown in fig. Determine the deflection at midspan by using Rayleigh-ritz method and
compare with exact solution.(May/June 2016)
19. Solve the following differential equations using Galerkin’s method of weighted residuals.
𝑑2𝑦
𝑑𝑥2
+y = 4x; 0 ≤ x ≤ 1
The boundary conditions are : y(0) = 0
y(1) = 0 (May/June 2016)
20. A concentrated load P=50KN is applied at the centre of a fixed beam of length 3m, depth
200mm and width 120mm. calculate the deflection and slope at the midpoint. Assume E= 2×105
N/mm2
.(May/June 2016)
21. Write briefly about Weighted Residual Methods. (Nov/Dec 2015)

22. A simply supported beam is subjected to uniformly distributed load over entire span as
shown in fig. Determine the bending moment and deflection at midspan by using Rayleigh ritz
method.(Nov/Dec 2015)
FEA –UNIT2
PART - A
1. What are the types of loading acting on the structure?
There are three types of loading acting on the body. They are:
(i) Body force (f).
(ii) Traction force (T).
(iii) Point load (P).
2. Define body force (f).
A body force is distributed force acting on every elemental volume ofthe body;
Unit: Force per unit volume.
Example: Self-weight due to gravity.
3. Define Traction force (T).
Traction force is defined as distributed force acting on the surface ofthe body.
Unit: Force per unit area.
Examples: Frictional resistance, viscous drag, surface shear etc.
4. What is Point Load (P).
Point load is force acting at a particular pointwhich causes displacement.
5. What are the basic steps involved in the finite element modelling.
Finite element modelling consists ofthe following:
(i) Discretization of structure.
(ii) Numbering of nodes.
6. What is Discretization?(NOV/DEC 2015)
The art of subdividing a structure into a convenient number of smaller components is
known as Discretization.
5. What are the classifications ofco-ordinates?

The co-ordinates are generally classified as follows:
(i) Global co-ordinates.
(ii) Local co-ordinates.
(iii) Natural co-ordinates.
8. What is Global co-ordinates? [Anna University, June 2005], [Nov 2013]
The points in the entire structure are defined using co-ordinate system is known as global
co-ordinate system.
Example:
9. What is natural co-ordinates? [Nov 2013]
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.
10. Define shapefunction. [Anna University, Dec 2007]
In finite element method, field variables within an element are generally expressed by the
following approximate relation:
11. What are the characteristics ofshape function? (Apr/May 2015)
The characteristics of shape function are as follows:
1.'The shape function has unit value at one nodal point and zero value at other
nodal points.
2, The sum of shape function is equal to one.
12. Why polynomials are generally used as shape/unction?
Polynomials are generally used as shape function due to the following reasons.
1. Differentiation and integration of polynomials are quite easy.
2. The accuracy of the results can be improved by increasing the order of thepolynomial.
3. It is easy to formulate and computerize the finite element equations.
13. How do you calculate the size ofthe global stiffness matrix?

14. Give the general expression for element stiffness matrix.(Nov/Dec 2015)
15. Write down the expression ofstiffness matrixfor one dimensional bar element.
16. State the properties ofa stiffness matrix. [Anna University, Jan 2006]
The properties of a stiffness matrix [ K ] are:
1. It is symmetric matrix.
2. The sum of elements in any column must be equal to zero.
3. It is an unstable element. So, the determinant is equal to zero.
17. Write down the generalfinite element equation.
General finite element equation is,
18. Write down the finite element equation for one dimensional two noded bar element.
The finite element equation for one dimensional two noded bar element is,
19. What is truss?
A truss is defined as a structure, made up of several bars, riveted or welded together.
20. State the assumptions are made while finding the forces in a truss.
The following assumptions are made while finding the forces in a truss.
(i) All the members are pin jointed.

(ii) The truss is loaded only at the joints.
(iiI) The self-weight of the members are neglected unless stated.
21. Write down the expression of stiffness matrix for a truss element.
22. Write down the expression of shape function N and displacement u for one
Dimensional bar element. [Anna University, Jan 2005](Nov/Dec 2015)
For one dimensional bar element,
23. Define total potential energy.
The total potential energy 1t of an elastic body is defined as the sum of total strain energy
U andthe potential energy ofthe external forces, (W).
24. State the principle ofminimum potential energy. [Anna University, Dec 2007]
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 a stable system.
25. What is the stationary property oftotal potential energy? (May/June 2016)

26. State the principles ofvirtual work. [Anna University, Dec 2006]
A body is in equilibrium if the internal virtual work equals the external virtual work for
every kinematically admissible displacement field.
27. Distinguish between essential boundary conditions and natural boundary conditions.
[Anna University, Dec 2006]
There are two types of boundary conditions. They are:
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 boundarycondition or Natural boundary condition:
The boundary conditions which are in the differential form of field variables are known
as secondary boundary condition.
Example: A bar is subjected to axial load. It’s shown in Fig.
In this problem, Displacement u at node 1 =0, that is primary boundary condition.
28. What are/he differences between boundary value problem and initial value
problem.[Anna University, June 2005J
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 aboundary value
problem. The differential equation together with initial conditions subjected to an initial value
problem.
Examples: Boundary value problem.

PART – B
1. For the vertical bar shown in Fig.(i), find the deflection at A and the stress distribution.
Use E = 150 MPa and weight per unit volume =0.05 N'crn",
2. Consider the bar in Fig. (ii). Calculate the nodal displacements, element stresses, andsupport
reactions.

10. Derive an expression of shape functions and the stiffness matrix for one dimensional bar
element based on global co-ordinate approach. (16) (2013)
11. The loading and other parameters for a two bar truss element is shown in fig. Determine
i) Element stiffness matrix for each element
ii) Global stiffness matrix
iii) Nodal displacements
iv) Reaction forces
v) Stresses induced in the elements. Assume E = 200 GPa. (2013, 2014)

12. Determine the shape function for one dimensional quadratic bar element. (2012) (May/June 2016)
13. Find the nodal displacement developed in the plane truss shown in fig. when a vertically downward
load of 1000 N is applied at node 4. The required data are given in the table.
(Nov 2012)
14.
(Nov/Dec 2014)

15.
(Nov/Dec 2014)
16.
(May/June 2015)
16. 17. A steel bar of length 800mm is subjected to an axial load of 3KN as shown in fig.2
Find the nodal displacement of the bar and load vectors.(May/June 2016)

18. Determine the nodal displacement, element stresses and support reactions in the truss
element shown in figure. Assume that points 1 and 3 are fixed. Take E = 70 GPa and A=
200mm2
. (May/June 2016)
19. For a Beam shown in figure. Determine the displacement the displacements and the slopes
at the nodes, the forces in each element and the reactions. E= 200Gpa, I = 1× 10-4
m4
.
(May/June 2016)
20. A steel bar of length 800mm is subjected to an axial load of 3KNas shown in fig. find the
elongation of the bar, neglecting self-weight.(Nov/Dec 2015).

21. Derive the stiffness matrix for 2D truss element. (Nov/Dec 2015).
FEA – UNIT 3
PART - A
1. How do you 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.
2. What is CST element? [Dee. 2007, (Nov/Dec 2015)
Merit:
1. Calculation of stiffness matrix is easier.
Demerit:
1. The strain variation within the element is considered as constant. So, the results will be
poor.
3. What is LST element?
Six noded triangular elements is known as Linear Strain Triangle (LST), which is shown
In Fig. It has twelve unknown displacement degrees of freedom. The displacement
Functions for the element are quadratic instead of linear as in the CST.
4. What is QST element?
Ten noded triangular elements is known as Quadratic Strain Triangle (QST) which is
shown in Fig.3.27. It is also called cubic displacement triangle.

5. What is meant by plane stress analysis?
Plane stress is defined to be a state of stress in which the normal stress (o ) and shear
stress (r) directed perpendicular to the plane are assumed to be zero.
6. Define plane strain analysis.(Nov/Dec 2015).
Plane strain is defined to be a state of strain in which the strain normal to the xyplane and
the shear strains are assumed to be zero.
7. Write a displacementfunction equation for CST element.
8. Write a strain-displacement matrixfor CST element.(May/June 2016)
Strain-Displacement matrix for CST element is,
9. Write down the stress-strain relationship matrixfor plane stress condition.
For plane stress problems.Stress-strain relationship matrix is,
10. Write down the stress-strain relationship matrix for plane strain condition.
For plane strain problems. Stress-strain relationship matrix is.

11. Write down the stiffness matrix equation for two dimensional CST elements.
12. Write down the expressionfor the shape functions for a constant ...uruin triangular
element.
For CST element,
13. What is axisymmetric element? idée. 2007, Anna University]
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 as
axisymmetric element.
14. What are the conditionsfor a problem to be axisymmetric?
1. The problem domain must he symmetric about the axis of revolution.
2. AU boundary conditions must be symmetric about the axis of revolution.
3. All loading conditions must be symmetric about the axis ofrevolution.
15. Write down the displacement equation for an axisymmetric triangular element.

16. Write down the shapefunctions for an axisymmetric triangular element.
17. Give the Strain-Displacement matrix equation for an axisymmetric triangular element.
Strain-Displacement matrix,

22. 18. Write down the Stress-Strain relationship matrix for an axisymmetric triangular
element.(May/June 2016)
19. Give the stiffness matrix equation for an axisymmetric triangular element.
20. 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
21. Calculate the Jacobian of the transformation J for the triangular" element shown
Fig. (i).

22. What is the purpose ofIsoparametric elements? [Jan 2006, Anna University]
It is difficult to represent the curved boundaries by straight edges finite 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 boundaries, a family of elements known as "Isoparametric
elements" are used.
23. Write down tile shape functions for 4 noded rectangular elements using natural
coordinate system.
24.
25.

26.
27.
28. Define superparametric element. [Dec. 2006, Anna University]
If the number of nodes used for defining the "geometry is more than number of nodes
used for defining the displacements, then, it is known as superparametric element.
29. What is meant by subparametric element? [Dec. 2006, Anna University]
If the number of nodes used for defining the geometry is less than number of nodes used
for defining the displacements, then, it is known as subparametric element.
30. What is meant by Isoparametric element?
If the number of nodes used for defining the geometry is same as number of nodes used
for defining the displacements, then, it is known as Isoparametric element.

31. Is beam element an Isoparametric element?
Beam element is not an Isoparametric element since the geometry and displacements are
defined by different order interpolation functions.
32. What is the difference between natural co-ordinate and simple natural co-ordinate?
A natural co-ordinate is one whose value lies between zero and one.
A simple natural co-ordinate is one whose value lies between -1 and +1.
33. Give examples for essential (forced or geometric) and non-essential (natural)
Boundary conditions.
The geometric boundary conditions are displacement, slope, etc. The natural boundary
conditions are bending moment, shear force, etc.
34. What are the types of non-linearity? [Dec. 2007, Anna University]
Types of non-linearity:
(a) Non-linearity in material behavior from point to point.
(b) Non-linearity in loading-deformation relation.
(c) Geometric non-linearity.
(d) Change in boundary condition for different loading.
35.Define path line and stream line.(May/ June 2016)
Streamlines are a family of curves that are instantaneously tangent to the velocity vector of
the flow. These show the direction in which a massless fluid element will travel at any point in
time.
 Path lines are the trajectories that individual fluid particles follow. These can be
thought of as "recording" the path of a fluid element in the flow over a certain period. The
direction the path takes will be determined by the streamlines of the fluid at each moment in
time.
36. Give the application of the plane stress and plane strain problems.(May/ June 2016)

PART – B
(June 2014)
3. Evaluate the stiffness matrix for the element shown in Fig.(iii). The co-ordinates are
shown in units of millimeters. Assume plane stress condition. Let E = 2 x 105 N/mm2,
v = 0.25 and l= 30 mm.

6. The nodal co-ordinates for an axisymmetric triangular element are given below:
r1 = 15 mm, zl = 15 mm; r2= 25 mm,z2= 15 mm; r3=35 mm, z3= 50 mm.
Determine [B] matrix for that element.
7. Determine the stiffness, matrix for the element shown in Fig.

The co-ordinates shown in Fig. are in millimeters. Take E = 2 x 105 N/mm2 and
v = O.25.
8.
9.
10.

11. For a four noded rectangular element shown in Fig. (i), determine the following:
1. Jacobian matrix
2. Strain-Displacement matrix
3. Element stresses
12. For the Isoparametric four noded quadrilateral element shown in Fig.{i), determine the
Cartesian co-ordinates of point P which has local co-ordinates ε= 0.25 and η= 0.25.
13. Calculate the element stresses for the axisymmetric element shown in Fig. The nodal
displacements are:

14. Evaluate the temperature force vector for the axisymmetric triangular element shown in Fig.
The element experiences a 15°C-increase in temperature.
The co-ordinates are in millimeters. Take α = 10 x 10-6
/o
C, E = 2 x 105 N/mm2; v = 0.25.
15. Establish the shape functional of eight node quadrilateral element and represent them
graphically. (June 2011)
16. For the two dimensional loaded plate as shown, determine the nodal displacements and
element stress using plane strain condition considering body force. Take Young’s modulus as
200 GPa, Poisson’s ratio as 0.3 and density as 7800kg/m3
( Nov 2011)
17.
(Nov/Dec 2014)

18.
(Nov/Dec 2014)
19. Calculate the element stiffness matrix and the temperature force vector for the plane stress
element shown in fig . the element experience a 20̊C increase in temperature. Assume coefficient
of thermal expansion is 6 x 10-6
C. take E = 2 x 105
N/mm2
, ѵ = 0.25, t = 5mm.(May/June
2016)
20. Derive the shape function for the constant strain triangular element. (May/June 2016).
21. For the CST element shown in the fig. assemble strain – Displacement matrix. Take t =
20mm, and E = 2 x 105
N/mm2
.(Nov/Dec 2015)

23. For the Isoparametric four noded quadrilateral element shown on fig. determine the Cartesian
coordinates of point P which has local coordinates ɛ = 0.5 and ɳ = 0.5. (Nov/Dec 2015)
24. For the plane strain element shown in fig. the nodal displacements are given as u1 =
0.005mm, v1= 0.002mm, u2 =0.0, v2=0.0, u3 =0.005mm, v3=0.30mm.determine the
element stresses and the principle angle. Take E=70Gpa and poison’s ratio = 0.3 and use
unit thickness for plane strain. All coordinates are in mm.(May/June 2016)
25. Derive the element characteristics of a nine node quadrilateral element. (May/June 2016)
FEA – UNIT 4
PART - A
1. Define frequency ofvibration.
It is the number of cycles described in one second. Unit is Hz.
2. Define Damping ratio.
It is defined as the ratio of actual damping coefficient (C) to the critical damping
coefficient (Cc).

3. 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.
4.What is meant by transverse vibrations?(Nov/Dec2015)
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.
5. Define magnification factor.
The ratio of the maximum displacement of the forced vibration (xmax) to the static
deflection under the static force (X o) is known as magnification factor. .
6.Write down the expression oflongitudinal vibration ofbar element.
Free vibration equation for axial vibration of bar element is,
7. Write down the expression ofgoverning equation for free axial vibration ofrod.
The governing equation for free axial vibration of a rod is given by,
6. Write down the expression ofgoverning equation for transverse vibration of beam.
The governing equation for free transverse vibration of a beam is,

10. What are the types ofEigen value problems?
There are essentially three groups of method of solution,
1. Determinant based methods
2. Transformation based methods
3. Vector iteration methods
11. State the principle ofsuperposition.
It states that for linear systems, the individual responses to several disturbances or driving
functions can be superposed on each other to obtain the total response of thesystem.
12. Define resonance.
When the frequency of external force is equal to the natural frequency of a vibrating,
body, the amplitude of vibration becomes excessively large. This phenomenon is known as
resonance.
13. Define Dynamic Analysis. (Nov/Dec 2015)

When the inertia effect due to the mass of the components is also considered in addition
to the externally applied load, then the analysis is called dynamic analysis.
14. What are methods usedfor solving transient vibrationproblems?
There are two methods for solving transient vibration problems. They are:
1.Mode superposition method
2. Direct integration method.
15. Write down the equation for undamped system of Direct Integration Method in
Central Difference Method.
For an undamped system,
16. State the two differences between direct and iterative methods for solving system of
equations.

PART – B
1. For the one-dimensional bar discredited into three elements as shown in Fig.(i) determine the
lumped-and consistent mass matrix. Let the bar properties be E, p and A
throughout the bar.
2. Consider a cantilever beam as shown in Fig.(ii). Determine the natural frequencies of
Vibration of a cantilever beam of length L, assuming constant values of p, E and A.

3. Using two equal-length finite elements except that the mass matrix are lumped, that is take the
element mass matrix as,
Determine the natural frequencies of the solid circular shaft fixed at one end as shown in Fig.(iii)
and compare with those obtained using consistent mass matrix.
4. Determine all natural frequencies of the simply supported beam as shown in Fig.(iv).
Compare the results obtained using the following,
(i) One-element model.
(ii) Two-element model.
5. Find the natural frequencies of longitudinal vibration of the constrained stepped bar as
Shown in Fig.(v).

6. Consider the system shown in Fig.(vi). Determine the stiffness and mass matrix of the
Structure.
7. It is aimed at determining the largest Eigen value and Eigen vector of the matrix give
below. Perform the necessary calculations following the inverse vector iteration method.
8. Find the Eigen values and Eigen vectors ofthe following matrix by solving the equation,
9.Find the response of the system given below using modal superposition method.
2mq1˝ + 2kq1 – kq2 = 0
Mq2˝ + 2kq2 – kq1 = 0
With the initial conditions q1 = 0, q2 = 1, q1’ = 0, q2’ = 1 (2011)
10. Use the iterative procedures to determine the first and third Eigen values for the structure
shown. Hence determine the second Eigen value and natural frequencies of building. Finally
establish the Eigen vectors and check the rest by applying the orthogonality properties of Eigen
vectors.

(2013)
11. Derive the equation of motion based on weak form for transverse vibration of a beam.
(June 2012, June2014)
12. Consider the uniform cross section bar as shown of length l and made up of material whose
young’s modulus and density is given by E and ρ. Estimate the natural frequencies of axial
vibration of the bar using both consistent and lumped mass matrices. (June 2013)
13.
(Nov/Dec 2015)

14.
(Nov/Dec 2015)
15. Determine the first two natural frequencies of transverse vibration of the cantilever beam
as shown in below figure and plot the mode shapes. (June 2015)
16. For the bar as shown in fig with length 2L, modulus of elasticity E, mass density e, and
cross sectional area A, determine the first two natural frequencies. (Nov/Dec 2015)
17. Determine the Eigen values and natural frequencies of a system whose stiffness and mass
matrices are given below.(Nov/Dec 2015)

18. The nodal co-ordinates for axisymmetric triangular element are given in figure. Evaluate
strain-Displacement matrix for that element.(May/June 2016)
19. Calculate the element stiffness matrix for the axisymmetric triangular element shown in
the fig. the element experiences a 15̊C increase in temperature. The coordinates are in mm.
take α = 10 × 10-6/̊
C , E= 2× 105
N/mm2
, ѵ= 0.25.(May/June 2016)
20. Set up the system of equations governing the free transverse vibrations of a simply
supported beam modeled by two finite elements. Determine the natural frequency of the
system.(May/June 2016)
21. Find the Eigen value and the corresponding Eigen vector of
A=
𝟏 𝟔 𝟏
𝟏 𝟐 𝟎
𝟎 𝟎 𝟑
(May/June 2016)

FEA – UNIT 5
PART - A
1. Define Heat transfer.
Heat transfer can be defined as the transmission of energy from one region to another
region due to temperature difference.
2. Write down the stiffness matrix equation for one dimensional heat conductionelement.
3. Write down the expression ofshape function, N and temperature function, T for one
dimensional heat conduction element.
For one dimensional heat conduction element,
4. Write down the finite element equationfor one dimensional heat conduction with free
end convection.
Finite element equation for one dimensional heat conduction with free end convection is
given by,
5. Write down the governing equation for two-dimensional heat conduction. (May/June
2016)(Nov/Dec 2015)
Governing equation for two-dimensional heat conduction,
6. Write down the shape function for two-dimensional heat transfer. (April/May 2015)
For two-dimensional heat transfer element,

7. Write down the expression for stiffness matrix in two-dimensional heat conduction and
convection.
Stiffness matrix for conduction,
8. Define path line.
A path line is defined as locus of points through which a fluid particle of fixed identity
passes as it moves in space.
9. Define streamline.(Nov/Dec 2015)
A streamline is an imaginary line that connects a series of points in space at a given
instant in such a manner that all particles falling on the line at that instant have velocities whose
vectors are tangent to the line.
10. Define inviscid flow.
An inviscid flow is a frictionless flow characterized by zero viscosity. A viscous flow is
one in which the fluid is assumed to have non-zero viscosity.
11. Write down the expression for governing equation in fluid mechanics, 2-D.
The governing equation for a two-dimensional problem is given by
12. Write down the expression of shape function for 2-D in fluid mechanics.
Shape function for 2D - Fluid mechanics,

13. Write down the expressionfor stiffness matrix in 2Dfluid mechanics.
Stiffness matrix for 2D in fluid mechanics,
14. Write down the expression for velocity gradient in fluid mechanics.
Velocity gradient in 2D fluid mechanics,
15. Name a few FEA packages. (Nov/Dec 2014)
Agros2D, CalculiX, Code Aster, DUNE, FEBio, MaxFEM, Kratos, jFEM
16. What is the difference between natural coordinates and local
coordinates?(May/June2016)
Local Coordinates
“A local coordinates system whose origin is located within the element in order to simplify the
algebraic manipulations in the derivation of the element matrix.”

 The use of natural coordinates in expressing approximate functions is advantageous because
special integration formulas can often be applied to evaluate the integrals in the element
matrix.
 Natural coordinates also play a crucial role in the development of elements with carved
boundaries.
Natural Coordinates
“It is a local coordinate system that permits the specification of a point within the element by a
dimensionless parameter whose absolute magnitude never exceeds unity.”
 It is dimension less.
 They are defined with respect to the element rather than with reference to the global
coordinates.
Part – B

6. Derive a finite element equation for one dimensional heat conduction with free end
convection. (June2013, June2014)
7. i) In the finite element analysis of a two dimensional flow using triangular elements, the
velocity components u and v are assumed to vary linearly within an element e as
ii) Explain the potential function formulation of finite element equations for deal flow problems.
(June2013)
7.
(Nov/Dec 2014)
9.
(Nov/Dec 2014)
10.The figure shows a uniform aluminum fin of diameter 25mm. the root (left end) of the
fin is maintained at a temperature of T∞ = 120°C, Convection takes place from the lateral
(circular) surface and the right (flat) edge of the fin. Assuming k= 200W/m°C, h=
1000W/m°C and T = 20°C, determine the temperature distribution in the fin Using one

dimensional element, Considering two elements. (May/June 2016)
11. For the two dimensional sandy soil region shown in the figure. Determine the potential
distribution. The potential distribution (Fluid head) on the left side is 10m and that on the
right side is 0.0m. The permeability’s are K41=K32= 25×10-5
m/s and K34 = K12 = 0.Assume
unit thickness. (May/June 2016)
12. Derive the shape functions for 4-noded rectangular element by using natural coordinate
system. (May/June 2016)
13. Evaluate the Cartesian coordinate of the point P which has local coordinates ɛ = 0.6 and
ɳ = 0.8 as shown in the fig. (May/June 2016)

14. Derive an expression for temperature function and the shape function for one
dimensional heat conduction element. (Nov/Dec 2015)
15. Derive the stiffness matrix and the load vectors for fluid mechanics in two dimensional
finite elements. (Nov/Dec 2015)

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