FEA unit 1 to 5 qb

ME6603 FEA : Question Bank
Department of Mechanical Engineering, Sri Eshwar College of Engineering
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Sri Eshwar College of Engineering
Department of Mechanical Engineering
III Year / VI Semester
[2013 – 2017 Batch]
ME 6603 – Finite Element Analysis
Question Bank
UNIT 1 – Introduction
Part A
Q.No Questions CO# BT AU Month/Year
1. What is the basis of finite element method? 1 R Apr/May 2009
2.
What are the methods are generally associated with the
finite element analysis?
1 R
3.
What is meant by post processing?
List out two advantages of post-processing.
1 R Apr/May 2013
4.
Give two sketches of structures that have both discrete
elements and continuum
1 R Apr/May 2012
5. Name the weighted residual methods. 1 R
6.
Write about the Galerkin’s residual method (or) what are the
basic steps of Gallerkin method?
1 R Apr/May 2012
Nov/Dec 2013
7. On what basis, collocation of points are selected 1 U Nov/Dec 2012
8. What is Rayleigh-Ritz method? 1 R
9.
Compare Rayleigh-Ritz method with Nodal approximation
method
1 U Nov/Dec 2012
10.
What is the difference between Ritz technique and Galerkin
technique?
1 U Nov/Dec 2011
11. State the principle of minimum potential energy. 1 R Nov/Dec 2011
Apr/May 2012
12. What is meant by discretization and assemblage? 1 U
13.
What is meant by degrees of freedom? State with example
the degrees of freedom at a node in Finite Element analysis.
1 R
14. Define total potential energy. 1 R
15.
What are the difference between boundary value problem
and initial value problem?
1 R

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16. What is "Aspect ratio"? 1 R
17. What is interpolation function? 1 R Apr/May 2012
18.
What are 'h' and 'p' versions / refinements of finite element
method?
1 R Nov/Dec 2012
19.
Derive the Governing equation for a bar fixed at one end and
subjected to self weight and a point load at the free end and
give the associated boundary conditions.
1 U Apr/May 2014
20.
List out various weighted residual methods and differentiate
between residue and error in solution.
1 R Apr/May 2015
21. What is the limitation of using a finite difference method 1 U Apr/May 2010
22. List the various method of solving boundary value problems 1 R Apr/May 2010
23. State the advantages of Gaussian Elimination technique 1 R Apr/May 2013
24.
What is the principle of skyline solution based on Gaussian
elimination?
1 R Nov/Dec 2013
25. What are the advantages of week formulation? 1 U Apr/May 2015
26. What should be considered during piecewise trial function? 1 U Apr/May 2011
27. Mention the basic steps of Rayleigh Ritz method. 1 R Apr/May 2011
28. What is Gallerkin method of approximation? 1 R Nov/Dec 2009
29.
What are Eigen value problems? Give at least two examples
for the same.
1 R Apr/May 2014
30.
Differentiate between primary and secondary variables with
suitable examples
1 R Nov/Dec 2013
31. State the advantages of Guass elimination technique 1 U Apr/May 2013
32. List the types of nodes 1 R Apr/May 2012
33. What is meant by node or joint? 1 R Apr/May 2014
34. Define p-refinement. 1 R Nov/Dec 2012
Part B
1.
a) Discuss the factors to be considered in discretization of a
domain (10)
b) Solve the following equation using the gauss elimination
method (6)
1 U Apr/May 2013

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2x1 + 3x2 + x3 = 9
x1 + 2x2 + 3x3 = 6
3x1 + x2 + 2x3 = 0
2.
a) The following differential equation is available for a
physical phenomenon. (d²w/dx²)-10x2 = 5 for 0 ≤ x ≤ 10,
The Boundary conditions are, w(0) = 0, w(1) = 0. By taking
two-term trial function as w(x) = C1 f1(X) + C2 f2(X) with f1
(X) = x (x-1) and f2 (X) = x2 (x-1) Find the value of the
parameter by using Galerkin’s Method. (10)
b) Solve the following equation using the gauss elimination
method (6)
x1 + 3x2 + 2x3 = 13
-2x1 + x2 - x3 = -3
-5x1 + x2 + 3x3 = 6
1 U Nov/Dec 2009
3.
A beam AB of span ‘l’ simply supported at ends and carrying
a point load W at the centre ‘C’. Determine the deflection at
mid span by using Rayleigh-Ritz method and compare with
exact result.
1 U Nov/Dec 2009
4.
A SSB subjected to UDL over entire span and it is subjected
to a point load at centre of the span. Calculate the bending
moment and deflection at midspan by using Rayleigh- Ritz
method and compare with exact solution.
1 U Apr/May 2013
5.
The differential equation of a physical phenomenon is given
by d²y/d²x + 500x² = 0; 0≤ x ≤ 1. By using the trial function,
y = a1 (x – x3) + a2 (x –x5), calculate the value of the
parameters a1 and a2 by the following (i) Point collocation
Method; (ii) Subdomain collocation Method; (iii) Least
square Method; (iv) Galerkin’s Method. The boundary
conditions are: y (0) = 0, y(1) = 0.
1 U
6. List and briefly describes the general steps involved in FEA 1 U Nov/Dec 2012
Apr/May 2014
7.
Consider the differential equation
d²y
d²x
+ 400x² = 0; 0≤ x ≤ 1.
Subjected to the boundary conditions y (0) = 0 and y (1) = 0.
The functional corresponding to this problem to be
1 U Nov/Dec 2009

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extremized is given by
I = {−0.5(dy/dx)2
+ 400 x2
y}
1
0
Find the solution of the problem using Rayleigh-Ritz method
by considering a two term solution as
y(x) = c1 x (1 - x) + c2 x2 ( 1 - x)
8.
Discuss the following methods to solve the given differential
equation. EI
d²y
d²x
– M(x) = 0
With the boundary conditions y (0) = 0 and y(l) = 0 ; using
a) Variational Method b) Collocation Method
1 U Apr/May 2010
9.
i) Describe the historical background of FEM
ii) Explain with relevance of FEA for solving design problems
with the aid of examples.
1 U Nov/Dec 2013
10.
A rod fixed at its ends is subjected to a varying body force as
shown. Use the Rayleigh Ritz technique with an assumed
displacement field u = a0 + a1x + a2x2 to determine
displacement u(x) and stress σ (x)
1 U Nov/Dec 2013
11.
Determine using any weighted residual technique the
temperature distribution along a circular fin of length 6 cm
and radius 1 cm. The fin is attached to a boiler whose wall
temperature is 140oC and the free end is insulated. Assume
convection coefficient h = 10 W/cm2 0C. Conduction
coefficient K=70 W/cm2 0C and T∞ = 400C. The governing
equation for heat transfer through the fin is given by
Assume appropriate boundary conditions and calculate the
temperatures at every 1 cm from the left end.
1 U Apr/May 2015
12.
i) Discuss the importance of FEA in assisting design process.
ii) Solve the ordinary differential equation
1 U Apr/May 2013

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Subject to the boundary conditions y(0)=y(1)= 0 using the
Gallerkin method with the trial functions No(x)=0;
N1(x)=x(1-x2) [10 marks]
13.
The differential equation for a phenomenon is given by
The boundary conditions are y(0)=0; y(5)=0
Find the approximate solution using any classical technique.
Start with minimum possible approximate solution
1 U Apr/May 2012
14.
i) list and briefly describe the general steps of finite element
method
ii) Derive an equation to find the displacement at node 2 of
fixed – fixed beam subjected to axial load P at node 2 using
Rayleigh Ritz method. Find out stress at nodal points.
Draw the displacement and stress variation diagram.
1 U Apr/May 2012
15.
The differential equation for a phenomenon is given by
The boundary conditions are y(0)=0; y(1)=0
Find the one term approximate solution using Gallerkin’s
method of weighted residuals.
1 U Apr/May 2014
16.
Determine using any numerical technique, the temperature
distribution along a circular fin of length 8cm and radius
1cm. The fin is attached to a boiler whose wall temperature
is 120 °C and the free end is insulated. Assume convection
coefficient h=10 W/cm2 °C, Conduction coefficient K= 70
W/cm2 °C and T∞ = 40°C. Calculate the temperatures at
every 1cm from the left end
1 U Nov/Dec 2013

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UNIT II – ONE-DIMENSIONAL PROBLEMS
Part A
1. Define body force (f). 2 R
2. What are the factors which govern the selection of nodes? 2 R Nov/Dec 2012
3. Define shape function. 2 R Nov/Dec 2011
4. List the characteristics of shape function 2 R Apr/May 2010
5.
Why polynomial term preferred for shape functions in finite
element method?
2 U Apr/May 2011
6.
What is the need for coordinate transformation in solving truss
problems?
2 R Nov/Dec 2013
7. What are the types of Non-linearity 2 R Apr/May 2012
8. State the properties of a stiffness matrix 2 R Nov/Dec 2013
Nov/Dec 2011
9. What is natural co-ordinates? 2 R
10.
Write down the expression of stiffness matrix for one dimensional
bar element.
2 R
11.
Write down the expression of shape function N and displacement
u for one dimensional bar element.
2 R
12. Distinguish between 1 D bar element and 1D beam element 2 R Nov/Dec 2009
13. Write down the expression of stiffness matrix for a truss element. 2 R
14. State the assumptions made while finding the forces in a truss. 2 R
15. Derive the convection matrix for a 1D linear bar element 2 U Apr/May 2015
16.
Write down the conduction matrix for a three noded linear
triangular element
2 R Apr/May 2015
17. Derive the mass matrix for a 1 D linear bar element. 2 U Apr/May 2015
18.
Write down the governing equation and for 1 D longitudinal
vibration of a bar fixed at one end and give the boundary
conditions
2 R Apr/May 2015
19. Define Damping ratio. 2 R
20. Define magnification factor. 2 R
21.
Write down the expression of longitudinal vibration of bar
element.
2 R
22.
Write down the expression of governing equation for free axial
vibration of rod.
2 R

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23.
What are the methods used for solving transient vibration
problems?
2 R
24. Define dynamic analysis. 2 R Apr/May 2014
25. Define lumped and consistent mass matrix. 2 R
Part B
1.
For a tapered plate of uniform thickness t=10mm as shown in fig.
Find the displacements at the nodes by forming into two element
model. The bar has mass density ρ = 7800kg/m3, young’s
modulus, E=2x105MN/m2. In addition to self-weight, the plate is
subjected to a point load p=10kN at its centre. Also determine the
reaction force at the support.
2 U
2.
Consider a bar shown in shown fig. An axial load of 200kN is
applied at point p. Take A1=2400 mm², E1=70x109N/m2, A2=600
mm², E2=200x109N/m2. Calculate the following: (a) The nodal
displacement as point p. (b) Stress in each material. (c) Reaction
force. 2 U
3.
Determine the nodal displacements, element stresses and support
reactions for the stepped bar loaded as shown in Fig., P1= 100 kN
and P2=75 kN. The details of each section of the bar are tabulated
below:
Portion Material E (GPa) Area (mm2)
A Steel 200 1200
B Aluminium 70 800
2 U Nov/Dec 2013

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4.
Determine the nodal displacements, stress and strain for the bar
shown in Fig.,
2 U Nov/Dec 2014
5.
The structure shown in fig (i). is subjected to an increase in
temperature of 80°C. Determine the displacements, stresses and
support reactions. Assume the following data:
2 U
6.
For the beam and loading shown in fig. Calculate the rotations at
B and C.E=210GPa, I=6x106mm4.
2 U
7.
Determine the maximum deflection and the slope in the beam,
loaded as shown in Fig., Determine also the reactions at the
2 U Apr/May 2015

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supports. E = 200 GPa, I = 20 x 10-6 m4, q = 5 kN/m and L = 1m
8.
Derive the stiffness matrix for a beam element from the
fundaments. (AU Nov 2011)
2 U Nov/Dec 2011
9.
Derive using Lagrangian Polynomials the shape functions for a
one dimensional three noded bar element. Plot the variation of the
same. Hence derive the stiffness matrix and load vector
2 U Apr/May 2015
10.
Fig shows the pin-jointed configuration. Determine the nodal
displacements and stresses in each elements
]
2 U Apr/May 2013
11.
Determine the deflection and slope in the beam, loaded as shown
in Fig., at the mid-span and at the tip. Determine also the reactions
at the fixed end. E1 = 200 GPa. E2 = 85GPa, I = 20 x 10-6 m4
2 U
12.
For the plane truss shown in fig. determine the horizontal and
vertical displacements of nodal and the stresses in each element.
All elements have E=201GPa and A=4x10-4 m2
2 U Nov/Dec 2013

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13.
Consider a two-bar truss supported by a spring shown in Fig.,
Both bars having E = 210 GPa and A = 5.0 x 10-4 m2. Bar one has a
length of 5 m and bar two has a length of 10 m. The spring
stiffness is k = 2 kN/m. Determine the horizontal and vertical
displacements at the joint 1 and stresses in each bar
2 U Nov/Dec 2009
14.
Find the natural frequency of longitudinal vibration of the
unconstrained stepped bar as shown in fig.
2 U Nov/Dec 2006
15.
For the one-dimensional bar shown in fig. determine the natural
frequencies of longitudinal vibration using two elements of equal
length. Take E = 2x105 N/mm2, ρ = 0.8x10-4N/mm3, and
L=400mm. 2 U
16.
Determine the eigen values and natural frequencies of a system
whose stiffness and mass matrices are given below.
2 U
Apr/May 2014
Apr/May 2008
17.
Derive the equation of motion based on weak formulation for a
transverse vibration of a beam
2 U Apr/May 2014
18.
Derive a finite element equation for one dimensional heat
conduction with free end convection
2 U Apr/May 2014
19.
Derive the stiffness matrix and load vectors for fluid mechanics in
two dimensional finite element
2 U Apr/May 2014
20. A Composite wall consists of three materials as shown in fig., The 2 U Apr/May 2015

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inside wall temperature is 200°C and the outside air temperature
is 50°C with a convection coefficient of 10 W/cm2 °C. Determine
the temperature along the composite wall
21.
A circular fin of 40 mm diameter is fixed to a base maintained at
50˚C as shown in figure. The fin is insulated on the surface
expected the end face which is exposed to air at 25 ˚C. The length
of the pin is 1000 mm, the fin is made of metal with thermal
conductivity of 37 W/m K. If the convection heat coefficient with
air is 15 W/m2 K. Find the temperature distribution at 250, 500,
750 and 1000 mm from the base
2 U Apr/May 2012
22.
i) Derive the shape function for the one dimensional quadratic
element in Natural coordinates [8]
ii) Derive the stiffness matrix for heat transfer using shape
functions for a four noded quadrilateral element
2 U Nov/Dec 2012
23.
Determine the temperature distribution in one dimensional
rectangular cross section as shown in fig. The fin has rectangular
cross section and is 8 cm long wide and 1cm thick. Assume that
convection heat loss occurs from the end of the fin. Take K = 3
W/cm˚ C, h=3 W/cm2˚C, T=20˚C
2 U Apr/May 2011

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UNIT III - TWO DIMENSIONAL SCALAR VARIABLE PROBLEMS
Part A
1. How do you define a two dimensional elements? 3 R Apr/May 2014
2.
Write down the shape function for 4 noded rectangular element
using natural coordinates system
3 R
3. What is QST (Quadratic strain Triangular) element? 3 R Apr/May 2014
4.
Write down the shape functions associated with the three noded
linear triangular element and plot the variation of the same
3 R Apr/May 2015
5.
Write down the nodal displacement equations for a two
dimensional triangular elasticity element
3 R Apr/May 2010
6. What is meant by plane stress analysis? 3 R
7.
Give at least one example each for plane stress and plane strain
analysis
3 U Apr/May 2015
8.
List the importance of two dimensional plane stress and plane
strain analysis
3 U Apr/May 2012
9. Distinguish between plane stress and plane strain problems 3 R Apr/May 2009
Apr/May 2013
10. Why a CST element So called? 3 U Nov/Dec 2014
11. Distinguish between CST and LST elements 3 U Apr/May 2013
12.
Specify the strain displacement matrix of CST element and
comment on it
3 R Nov/Dec 2013
13. What are called higher order element? 3 R Apr/May 2008
Nov/Dec 2008
14.
Write down the governing equation for Two dimensional steady
state heat conduction
3 R Apr/May 2014
Part B
1.
Determine three points on the 50°C contour line for the
rectangular element shown in figure. The nodal values are Φi
=42° C, Φj = 54°C, Φk=56°C and Φm = 46°C
3 U Apr/May 2010

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2.
Find the temperature at a point P (2,1.5) inside the
triangular elements with the nodal temperatures given as
Ti=40°C, Tj=44°C and Tk=46°C. The nodal coordinates are i (0,0),
j(4,0.5) and k(3,6)
3 U Nov/Dec 2012
3.
A bilinear rectangular element has the coordinates as shown in
Fig., and the nodal temperatures are T1 = 100° C, T2= 60°C,
T3=50°C and T4=90°C. Compute the temperature at the point
whose coordinates are (2.5, 2.5). Also determine the 80°C
isotherm.
3 U Apr/May 2015
4.
Derive the strain-displacement matrix for constant strain
triangular element.
3 U
5.
Determine the shape functions for a constant triangular (CST)
element in terms of natural co-ordinate system
3 U Apr/May 2014
6.
Determine the shape functions N1,N2 and N3 at the interior point
p for the triangular element show in the figure
3 U Apr/May 2014
7.
Evaluate the element stiffness matrix for the triangular element
shown in fig., under plane stress conditions. Take E = 2e5
N/mm², µ = 0.3 and t = 10mm.
3 U Nov/Dec 2010
8.
Calculate the element stress σx, σy, τxy σ1 and σ2 and the
principle angle θp for the element shown in figure. The nodal
3 U Apr/May 2009

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displacement are u1 = 2mm, u2 = 0.5mm, u3 = 3mm and v1 =
1mm, v2 = 0, v3 = 1mm. Let E = 210 GPa, µ = 0.25. Assume
plane stress condition.
9.
For the plane stress element shown in fig, the nodal
displacements are given as; u1=0.005mm, u2=0.002mm,
u3=0.0mm, u4=0.0mm and u5=0.004mm, u6=0.0mm. Determine
the element stresses. Take E = 200 Gpa and γ=0.3.Use unit
thickness for the plane strain
3 U Apr/May 2010
10.
Calculate the displacement and stress in the given triangular
plate, fixed along one edge and subjected to concentrated load at
its free end, Take E=70Gpa, thickness of the plate = 10 mm and
Poisson’s ration = 0.3
3 U Apr/May 2012
11.
For the plane strain elements shown in fig, the nodal
displacements are given as u1=0.005mm, v1=0.002mm,
u2=0.0,v2=0.0,u3=0.005mm and v3=0.30mm. Determine the
element stresses and the principle angle. Take E = 70 Gpa and
poisson’s ratio =0.3 and use unit thickness for plane strain. All
the coordinates are in mm.
3 U Apr/May 2012

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12.
Determine the element stiffness matrix and the thermal load
vector for the plane stress element shown in fig. The element
experiences 20°C increase in temperature. Take E = 15e6 N/cm2,
γ=0.25, t=0.5 cm and α=6 e-6/°C
3 U Apr/May 2010
13.
Triangular elements are used for the stress analysis of plate
subjected to inplane loads. The (x,y) coordinates of nodes i,j and
k of an element are given by (2,3), (4,1) and (4,5) mm
respectively. The nodal displacements are given as
u1=2.0mm, u2=0.5mm, u3=3.0mm
v1=1.0mm, v2=0.0mm, v3=0.5mm
Determine element stresses, Let E = 160Gpa, Poisson’s ratio =
0.25 and thickness of the element t=10mm.
3 U Apr/May 2013
14.
a) The (x,y) co-ordinates 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 and N3 at an interior point P
(2,2.5)mm for the element [4]
b) For the same triangular element, obtain the strain-
displacement relation matrix B [12]
3 U Apr/May 2009
15.
For the two-dimensional loaded plate shown in fig., determine
the nodal displacements and element stress using plane strain
condition considering the body force. Take E as 200 Gpa,
poisson’s ratio as 0.3 and density as 7800 kg/m3(AU April 2011)
3 U Apr/May 2011

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16.
Consider the triangular element shown in fig. The element is
extracted from a thin plate of thickness 0.5 cm. The material is
hot rolled low carbon stell. The nodal co-ordinates are xi = 0; yi =
0, xj = 0, yj = -1, xk = 2 and yk= -1 cm. Determine the element
stiffness matrix. Assuming plane stress analysis. Take 0.3 and E =
2.1 x 107 N/cm2 (AU May 2012)
3 U Apr/May 2012
17.
Derive the characteristic matrix for a two dimensional heat
conduction problem using triangular element by galerkin’s
approach
3 U Nov/Dec 2013
18.
Consider a rectangular plate of length 3500 mm and width 2500
mm having a thickness of 300 mm. It is subjected to a uniform
heat source of 200 W/m3 acting over the whole body. The
temperature of the top side of the body is maintained at 130˚C.
The body is insulated on the other edges. Take the thermal
conductivity of the materials as 35 W/m˚C. Determine the
temperature distribution using triangular elements.
3 U Nov/Dec 2013
19.
A two dimensional fin is subjected to heat transfer by conduction
and convection. It is discretised as shown in fig., in to two
elements using linear triangular elements. Derive the conduction
and thermal load vector. How is convection accounted for in
solving the problem using finite element method? (AU Apr 2015)
3 U Apr/May 2015

Page17
20.
Determine the element matrices and vectors for the LST element
shown in fig. The nodal coordinates are i (1,1), j(5,2) and k(3,5).
Convection takes place along the edge jk. (AU May 2013)
3 U Apr/May 2013
UNIT IV – TWO DIMENSIONAL VECTOR VARIABLE PROBLEMS
Part A
1.
What are the differences between 2 dimensional scalar
variable and vector variable elements?
4 R Apr/May 2009
2.
Write down the displacement equation for an axisymmetric
triangular element.
4 R
3.
Give the strain displacement matrix equation for an
axisymmetric triangular element.
4 R
4.
Write the stress-strain relationship matrix equation for an
axisymmetric triangular element.
4 R
5.
List the required conditions for a problem assumed to be
axisymmetric.
4 U Apr/May 2015
6.
Give four examples of practical applications of axisymmetric
elements
4 U
Apr/May 2012
Apr/May 2013
Apr/May 2011
7. Sketch a finite element model for a long cylinder subjected to 4 U Nov/Dec 2013

Page18
an internal pressure using axisymmetric elements
8. What is meant by axi-symmetric element? 4 R Nov/Dec 2007
9.
List out the various 2-dimesional Finite elements that are
generally used for 2-Dimensional vector variable problems.
4 U Nov/Dec 2012
10. Name any two problems related to axisymmetric case. 4 U Nov/Dec 2012
11.
Write the material stiffness matrix for (i) plane stress (ii)
plane strain 2D structural problems.
4 R Nov/Dec 2011
12.
With suitable examples and the governing equations
distinguish between vector and scalar variable problems.
4 U Nov/Dec 2013
Part B
1.
For the axisymmetric elements shown in fig., determine the
element stresses and stiffness matrix. Let E =210 GPa and µ
= 0.25. The co-ordinates all are in millimetres. The nodal
displacement are u1 = 0.05mm, u2 = 0.02mm, u3 = 0mm and
w1 = 0.03mm, w2 = 0.02, w3 = 0mm.
4 U Apr/May 2009
2.
a) What are the non-zero strain and stress components of
axisymmetric element? Explain [4]
b) Derive the stiffness matrix of an axisymmeric element
using potential approach [12]
4 U Apr/May 2013
3. Derive constitutive matrix for axisymmetric analysis 4 U Nov/Dec 2012
4.
Establish the traction force vector and estimate the nodal
forces corresponding to a uniform radial pressure of 7 bar
acting on an axisymmetric element as shown in figure. Take
E = 200 GPa and Poisson’s ratio = 0.25
4 U Apr/May 2012

Page19
5.
The (x,y) 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 q= [0.002, 0.001, 0.004, -0.003, 0.007]T. Determine
the element strains.
4 U Apr/May 2009
6.
i) For the member shown, typical element has coordinates as
shown in Fig. E = 210 GPa and µ=0.3
1) If the member is subjected to internal pressure how
will you analyse the problem? (3marks)
2) Determine the strain displacement relations and the
strain displacement matrix. (5marks)
3) How will you solve for stresses in this member?
(3 marks)
ii) Explain how you will go about analyzing the stresses in a
solid flywheel. How will you model the same and what
elements would you choose? (5 marks)
4 U Apr/May 2014
7.
A circular aluminium rod is having a length of 700 cm. The
area of cross section is 60 cm2. The bar is subjected to an
axial compressive load of 50 kN at the fixed end. Calculate
the maximum displacement and stress developed in the bar.
Solve using two dimensional coordinates
4 U Nov/Dec 2012

Page20
8.
Establish the shape functions and derive the strain
displacement matrix for an axisymmetric triangular element.
4 U Nov/Dec 2012
9.
Calculate the element stiffness matrix and the thermal force
vector for the axisymmetric triangular element.
r1 = 6, z1=7, r2 = 8, z2=7, r3 = 9, z3=10
The element experiences a 150C increase in temperature.
The coordinates are in mm.
Take α = 10 x 10-6/oC ; E = 210 GPa; µ=0.25
4 U
10.
The nodal coordinate for an axisymmetric triangular
element are given below
r1 = 20, z1=40, r2 = 40, z2=40, r3 = 30, z3=60
Evaluate [B] matrix and [K] Matrix for that element.
4 U
UNIT V - ISOPARAMETRIC FORMULATION
Part A
1. What do you mean by isoparamertic formulation? 5 R Nov/Dec 2007
2. Define Isoparametric elements. 5 R Nov/Dec 2008
3.
State the basic laws on which isoparametric concept is
developed.
5 R Apr/May 2008
4.
Write the natural co-ordinates for the point P of the
triangular element. The point P is the C.G of the triangle.
5 R Nov/Dec 2008
5. Define superparmetric element. Give an example. 5 R &
U
Apr/May 2009
Nov/Dec 2013
6.
Write down the gauss Intergration formula for triangular
domains.
5 R Apr/May 2009
7. When are isoparametric elements used? 5 U Apr/May 2013
8. Name a few FEA Packages. 6 U Nov/Dec 2014
9.
Write the shape function for the 1D quadratic isoparametric
element
5 R Nov/Dec 2014
10.
Write the shape function for the 4 noded and 8 noded
isoparametric element
5 R Nov/Dec 2013
11. What are force vectors? Give an example 5 R Apr/May 2013
12.
What is post processing? Give an example
List the two advantages of post processing.
6 R Apr/May 2013
13. What is the salient feature of the isoparametric element? 5 U Apr/May 2012
14. When do we resort 1D quadratic spar element? 5 U Apr/May 2012

Page21
15.
Write down Jacobian matrix for 4 noded quadrilateral
elements.
5 R
16.
Write down stiffness matrix equation for 4 noded
isoparametric quadrilateral elements.
5 R
17. Define sub parametric element. 5 R
18. Is beam element an Isoparametric element? 5 U
19.
What is the difference between natural coordinate and
simple natural coordinate?
5 R
20.
With suitable examples explain what are serendipity
elements are?
5 U Nov/Dec 2013
21.
Derive the shape functions for linear isoparametric
triangular element and plot the variation of the same.
5 R Nov/Dec 2013
22.
Name the important load boundary conditions to be defined
in any commercial FEA preprocessor for the case of
structural problems.
6 U Nov/Dec 2012
23.
Obtain the value of integral using Gauss two point
integration scheme u1,2 = ±0.57735, W 1, 2 =1
5 U Nov/Dec 2011
Part B
1.
Establish the strain displacement matrix for the linear
quadrilateral element as shown in figure at gauss point r =
0.57735 and s = - 0.57735
5 U Nov/Dec 2007
2.
Derive the element stiffness matrix for a linear
isoparametric quadrilateral element.
5 U Nov/Dec 2007
Apr/May 2008
3.
Integrate
F(x)=10+(20x)-(3x2/10)+(4x3/100)-(-5x4/1000)+(6x5/10000)
5 U Apr/May 2008

Page22
between 8 and 12. Use Gauss quadrature rule.
4.
Write short notes on
a) Uniqueness of mapping of isoparametric elements [6]
b) Jacobian Matrix [5]
c) Gauss Quadrature integration technique[5]
5 U Nov/Dec 2008
5.
i)Use gauss quadrature rule (n=2) to numerically integrate
[10]
ii) Using natural coordinate derive the shape function for a
linear quadrilateral element [6]
5 U Nov/Dec 2008
6.
For a triangular element shown in figure, compute the
stiffness matrix by using isoparamteric formulation and
numerical integration with one point quadrature rule.
E = 2x103 kN/cm2, µ=0.3
5 U Apr/May 2009
7.
Evaluate the following integral using two point gauss
quadrature
5 U Apr/May 2009
8.
The integral f(x) = cos x / (1-x2) dx between the limits -1 and
+1 by using 3 point Gaussian quadrature.
5 U Apr/May 2010
9.
i) The Cartesian (global) coordinates of the corner nodes of a
quadrilateral element are given by (0,-1), (-2,3), (2,4) and
(5,3). Find the coordinate transformation between the global
and local (natural) coordinates. Using this, determine the
Cartesian coordinates of the point defined by (r,s) = (0.5,0.5)
in the global coordinate system. [8]
ii) Evaluate the integral and compare with exact solution [8]
5 U Nov/Dec 2008

Page23
10.
i) The Cartesian (global) coordinates of the corner nodes of a
quadrilateral element are given by (1,0), (2,0), (2.5,1.5) and
(1.5,1).Find its Jacobian matrix [12]
ii) Distinguish between subparametric and superparametric
elements [4]
5 U Nov/Dec 2009
11.
Use Gaussian quadrature to obtain an exact value of the
intergral
5 U Apr/May 2010
12.
For the four noded element shown in figure. determine the
Jacobian and evaluate its value at the point (1/2,1/3)
5 U Apr/May 2015
13.
i) What are natural coordinates? What are the advantages of
the same? (3 marks)
ii) Explain with suitable examples why we resort to
isoparametric transformation. Differentiate between
isoparametric, sub parametric and super parametric
elements. (5 marks)
iii) For the four noded element shown in figure determine
the Jacobian and evaluate its value at the point (1/2, 1/3).
(8marks)
5 U Nov/Dec 2013

Page24
14.
i) Using Gauss Quadrature evaluate the following integral
using 1 2 and 3 point Integration. Compare with exact
solution.(8 marks)
ii) Evaluate the shape functions for a corner node and mid
side node of a quadratic triangular serendipity element and
plot its variation. (8 marks)
5 U Nov/Dec 2013
15.
Using Gauss Quadrature evaluate the following integral
using 1 2 and 3 point Integration. Compare with exact
solution.(8 marks)
5 U Nov/Dec 2013
16.
i) Derive the jacobian matrix for triangular element with the
(x,y) coordinates of the nodes are (1.5,2), (7,3.5) and (4,7) at
nodes i, j, k.
ii) Find the jacobian transformation for four noded
quadrilateral element with the (x,y) coordinates of the nodes
are (0,0), (2,0), (2,1), (0,1) at nodes i, j, k, l. Also find the
jacobian at point whose natural coordinates are (0,0)
5 U Nov/Dec 2014
17.
i) Consider the four noded isoparametric quadrilateral
element with the (x,y) coordinates of the nodes are (5,5),
(11,7), (12,15), (4,10). Compute the jacobian matrix and its
determinant at the element centroid. (10)
ii) Using Gauss Quadrature evaluate the following integral
using 2 point Integration. Compare with exact solution.
(6 marks)
5 U Apr/May 2013
18.
The nodal displacements of a rectangular element having
nodal coordinates (0,0), (4,0), (4,2), (0,2) are u1 = 0mm, v1 =
0mm, u2 = 0.1mm, v2 = 0.05 mm, u3 = 0.05 mm, v3 = -0.05
5 U Apr/May 2013

Page25
mm, u4 = v4=0 mm. Determine the stress matrix at r = 0, s = 0
using the isoparametric formulation. Take E = 210 GPa,
poisson’s ratio = 0.25
19.
Evaluate the integrals using appropriate Guassian
quadrature.
5 U Apr/May 2012
20.
Derive the element characteristics of a four noded
quadrilateral element.
5 U Apr/May 2012
21.
Define the following terms with suitable examples.
a) Plane stress and plane strain
b) Node, element and shape function
c) Iso-parametric element
d) Axisymmetric analysis
5 U Apr/May 2010
22.
Explain the concept of iso-parametric element formation in
finite element analysis and hence derive the stiffness matrix
of a 1D iso-parametric bar element.
5 U Nov/Dec 2012
23.
Write short notes on:
(i) Serendipity elements
(ii) FE Solution methodology for dynamic problems
(iii) Material stiffness (constitutive) matrix for plane stress
and axisymmetric problems
(iv) Numerical integration by Gauss Quadrature
5 U Nov/Dec 2012
24.
i) Using Gauss Quadrature evaluate the following integral
(8 marks)
ii) Evaluate the shape functions for a corner node and mid
side node of a quadratic quadrilateral serendipity element
and plot its variation.
5 U Apr/May 2014
25. i) Why do we resort to natural coordinate, transformation? 5 U Apr/May 2014

Page26
Differentiate between subparametric, isoparametric and
superparametric elements. State when you would use' each
of these elements. (8 marks)
ii). For the four noded element shown in fig. Determine the
Jacobian and evaluate its value at the point
(-1/1.732, -1/1.732). Comment on the value that you obtain.
(8marks)
26.
Establish the shape functions of a eight noded quadrilateral
element and represent them graphically.
5 U Apr/May 2011
27.
A four node quadrilateral element is defined by nodal
coordinates in mm as 1(3,8), 2(10,5), 3(12,18) and 4( 5,16).
The nodal displacement vector is given by
q=[0,0,2,2,1.6,1.2,0,0.6]T. Evaluate the stress at the point
P(7,12) of the element, assuming plane stress condition.
Take youngs modulus and poisson’s ratio as 30 x 106 N/m2
and 0.3
5 U Nov/Dec 2013
28.
A four noded rectangular element is shown in fig. Determine
the following:
i) Jacobian matrix
ii) Strain displacement matrix
iii) Element Stresses
Take E = 2 x 105 N/mm2, µ = 0.25
u=[0,0,0.003,0.004,0.006,0.004,0,0]T Ɛ=0, ƞ=0.
Assuming Plane stress condition
5 U Apr/May 2012

FEA unit 1 to 5 qb

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