ME6603 - FINITE ELEMENT ANALYSIS UNIT - II NOTES AND QUESTION BANK

R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / JAN – MAY 2017
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 21
UNIT – II – ONE-DIMENSIONAL PROBLEMS
PART – A
2.1) Write a note on node numbering scheme.
2.2) What do you mean by node and element?
2.3) What are the types of problems treated as one dimensional problem?
[AU, May / June – 2013]
2.4) Highlight at least two rules to guide the placement of the nodes when obtaining
approximate solution to a differential equation. [AU, April / May – 2010]
2.5) Define shape function.
[AU, Nov / Dec – 2007, April / May – 2009, May / June – 2014]
2.6) What is a shape function? [AU, Nov / Dec – 2009]
2.7) Give the properties of shape function. [AU, Nov / Dec – 2014]
2.8) Differentiate shape function from displacement model.
2.9) Draw the shape function of a two noded line element. [AU, April / May – 2009]
2.10) Draw the shape function of a two noded line element with one degree of freedom at
each node. [AU, Nov / Dec – 2010]
2.11) Draw the shape function for one dimensional line element with three nodes.
[AU, April / May – 2009]
2.12) State the properties of stiffness matrix. [AU, Nov / Dec – 2009, 2010, 2011]
2.13) Write the properties of the stiffness matrix? [AU, April / May – 2015]
2.14) List out the stiffness matrix properties. [AU, May / June – 2012]
2.15) State the characteristics of shape function. [AU, May / June – 2011]
2.16) List the characteristics of shape functions. [AU, April / May – 2010]
2.17) When does the stiffness matrix of a structure become singular?
[AU, Nov / Dec – 2012]
2.18) State the significance of shape function.

2.19) Write the element stiffness matrix for a two noded linear element subjected to axial
loading.
2.20) Write the stiffness matrix for the simple beam element given below.
[AU, Nov / Dec – 2008]
2.21) What are the properties of global stiffness matrix? [AU, April / May – 2011]
2.22) Write the properties of Global Stiffness Matrix of a one dimensional element.
[AU, May / June – 2012]
2.23) Differentiate global stiffness matrix from elemental stiffness matrix.
2.24) What do you mean by banded matrix?
2.25) How will you find the width of a band?
2.26) How do you calculate the size of the global stiffness matrix?
2.27) List the properties of the global stiffness matrix. [AU, April / May – 2010]
2.28) What is the effect of node numbering on assembled stiffness matrix?
[AU, Nov / Dec – 2013]
2.29) Give a brief note on the following
(a) elimination approach (b) penalty approach.
2.30) Name the factors which affect the number element in the given domain.
2.31) State the requirements to be fulfilled by the approximate solution for its
convergence towards the actual solution.
2.32) What do you mean by continuity weakening?
2.33) Compare the linear polynomial approximation and quadratic polynomial
approximation.
2.34) Why polynomials are generally used as shape function?
[AU, Nov / Dec – 2011, 2016]

2.35) Why are polynomial terms preferred for shape functions in finite element method?
2.36) What do you mean by error in FEA solution?
2.37) What are the types of load acting on the structure?
2.38) Define traction force (T).
2.39) What is a truss? [AU, May / June – 2014]
2.40) State the assumptions are made while finding the forces in a truss.
[AU, Nov / Dec – 2011]
2.41) How are thermal loads input in finite element analysis?
[AU, Nov / Dec – 2007, April / May – 2009]
2.42) What is an interpolation function? [AU, May / June – 2012]
2.43) Why are polynomial types of interpolation functions preferred over trigonometric
functions? [AU, Nov / Dec – 2007, April / May – 2009]
2.44) What is an equivalent nodal force? [AU, April / May – 2008]
2.45) What are called higher order elements?
[AU, April / May – 2008, Nov / Dec – 2010, 2011]
2.46) What is higher order element? [AU, Nov / Dec – 2011]
2.47) What do you mean by higher order elements? [AU, Nov / Dec – 2008]
2.48) Why higher order elements are required for FE analysis? [AU, Nov / Dec – 2012]
2.49) What are higher order elements and why are they preferred?
2.50) What are the characteristics of shape functions?
2.51) Derive the shape functions for a 1D quadratic bar element.
2.52) Plot the variations of shape function for 1 – D beam element.
[AU, Nov / Dec – 2010]

2.53) Illustrate element connectivity information considering beam elements.
[AU, Nov / Dec – 2013]
2.54) Write about the effective global nodal forces of beam element.
[AU, May / June, Nov / Dec – 2016]
2.55) When do we resort to 1 D quadratic spar elements? [AU, April / May – 2011]
2.56) Obtain any one shape function for a quadratic cubic spar element.
[AU, Nov / Dec – 2013]
2.57) Mention two advantages of quadratic spar element over linear spar element.
[AU, Nov / Dec – 2013]
2.58) Give the Lagrangen equation of motion and obtain the shape functions for quadratic
coordinate transformation. [AU, May / June, Nov / Dec – 2016]
2.59) Give a brief note on the sources of error in FEA.
2.60) State the significance of post processing the solution in FEA.
2.61) What is meant by Post Processing? [AU, Nov / Dec – 2016]
2.62) Define dynamic analysis. [AU, May / June – 2014, Nov / Dec – 2015]
2.63) What is dynamic analysis? Give examples [AU, Nov / Dec – 2010, 2016]
2.64) What is mean by dynamic analysis? [AU, Nov / Dec – 2011]
2.65) List the types of dynamic analysis problems. [AU, May / June – 2012]
2.66) Define normal modes. [AU, May / June – 2013]
2.67) What is meant by transverse vibrations?
[AU, May / June – 2014, Nov / Dec – 2015]
2.68) What is the principle of mode superposition technique?
[AU, Nov / Dec – 2013, 2016]
2.69) Sketch two 3D elements exhibiting linear strain behavior. [AU, April / May – 2011]
2.70) What is the influence of element distortion on the analysis results?

2.71) Determine the element mass matrix for one-dimensional, dynamic structural
analysis problems. Assume the two-node, linear element. [AU, Nov / Dec – 2011]
2.72) What are consistent and lumped mass techniques? [AU, Nov / Dec – 2013]
2.73) Write down the expression of governing equation for free axial vibration of rod and
transverse vibration of beam. [AU, May / June – 2016]
2.74) Specify the consistent mass matrix for a beam element.
[AU, Nov / Dec – 2013, 2016]
2.75) Derive the mass matrix for 1D linear bar element. [AU, April / May – 2015]
2.76) Consistent mass matrix gives accurate results than lumped mass matrix in dynamic
analysis of bar element- Justify. [AU, May / June – 2016]
2.77) Comment on the accuracy of the values of natural frequencies obtained by using
lumped mass matrices and consistent mass matrices. [AU, Nov / Dec – 2012]
2.78) Write down the governing equation for 1D longitudinal vibration of a bar fixed at
one end and give the boundary conditions. [AU, April / May – 2015]
2.79) Explain consistent load vector. [AU, Nov / Dec – 2010]
2.80) What do you mean by Lumped mass matrix? [AU, May / June – 2011]
2.81) Write down the lumped mass matrix for the truss element.[AU, April / May – 2009]
2.82) What type of analysis preferred in FEA when the structural member subjected to
transient vibrations? [AU, May / June – 2016]
2.83) What are the types of Eigen value problems? [AU, May / June – 2012]
2.84) What is meant by mode superposition technique? [AU, May / June – 2013]
2.85) What do you know about radially symmetric problem?
2.86) Write the boundary condition for a cantilever beam subjected to point load at its
free end.
2.87) For a one dimensional fin problem, what are all the boundary conditions that can be
specified at the free end?

2.88) Write the analogies between structural, heat transfer and fluid mechanics.
[AU, Nov / Dec – 2014]
2.89) What are the boundary conditions in FEA heat transfer problem?
[AU, May / June – 2016]
2.90) Write down the one dimensional heat conduction equation.
2.91) Distinguish between homogenous and non – homogenous boundary conditions.
[AU, Nov / Dec – 2013]
2.92) Mention two natural boundary conditions as applied to thermal problems.
[AU, Nov / Dec – 2015]
2.93) Write down the expression of shape function and temperature function for one
dimensional heat conduction. [AU, May / June – 2011]
2.94) Write down the governing differential equation for the steady state one dimensional
conduction heat transfer. [AU, Nov / Dec – 2010, 2012]
2.95) Derive the convection matrix for a 1D linear bar element.
[AU, April / May – 2015, Nov / Dec – 2016]
2.96) Write down the stiffness matrix equation for one dimensional heat conduction
element. [AU, Nov / Dec – 2011]
2.97) Define static condensation. [AU, Nov / Dec – 2010]
2.98) Determine the load vector for the beam element shown in Figure
[AU, Nov / Dec – 2012]
2.99) Write the element stiffness matrix of a truss element. [AU, May / June – 2012]

2.100) Write down the expression of stiffness matrix for a truss element.
[AU, Nov / Dec – 2015]
2.101) Sketch a typical truss element showing local global transformation.
2.102) Differentiate global and local coordinates. [AU, May / June – 2013]
2.103) State the differences between a bar element and a truss element.
PART – B
2.104) What are the different types of elements? Explain the significance of each.
[AU, Nov / Dec – 2010]
2.105) Derive and sketch the quadratic shape function for the bar element.
[AU, May / June – 2011]
2.106) Derive the shape function of a quadratic 1 – D element. [AU, Nov / Dec – 2011]
2.107) Derive the shape functions for one dimensional linear element using direct method.
[AU, May / June – 2013]
2.108) Determine the shape function and element matrices for quadratic bar element.
[AU, May / June – 2012]
2.109) Derive the shape functions for One-Dimensional Quadratic Bar element.
[AU, May / June – 2016]
2.110) Derive the Lagrangian Polynomials the shape functions for one dimensional three
noded bar element. Plot the variations of the same. Hence derive the stiffness matrix
and load vector. [AU, April / May – 2015]
2.111) Derive the stiffness matrix and finite element equation for one dimensional bar.
[AU, Nov / Dec – 2011]
2.112) Derive the stiffness matrix and body force vector for a quadratic spar element.
[AU, Nov / Dec – 2013]
2.113) Obtain an expression for the shape function of a linear bar element.

2.114) Write the stiffness matrix for a 1D two noded linear element.
[AU, Nov / Dec – 2014]
2.115) Derive shape functions and stiffness matrix for a 2D rectangular element.
[AU, Nov / Dec – 2012]
2.116) Derive the equation of motion based on weak form for transverse vibration of a
beam. [AU, May / June – 2012, 2014]
2.117) Explain the direct integration method using central difference scheme for predicting
the transient dynamic response of a structure. [AU, Nov / Dec – 2013]
2.118) Derive the consistent mass matrix for a truss element in its local coordinate system.
[AU, Nov / Dec – 2012]
2.119) Derive the finite equations for the time – dependent stress analysis of one
dimensional bar. [AU, May / June – 2011]
2.120) 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. [AU, May / June – 2016]
2.121) Derive the shape function for a 2 noded beam element and a 3 noded bar element.
[AU, Nov / Dec – 2008]
2.122) Why is higher order elements needed? Determine the shape functions of an eight
noded rectangular element. [AU, Nov / Dec – 2007, April / May – 2009]
2.123) Derive the shape functions for a 2D beam element.
[AU, Nov / Dec – 2007, April / May – 2008, 2009]
2.124) Derive the shape functions for a 2D truss element.
[AU, Nov / Dec – 2007, April / May – 2008, 2009]
2.125) Derive the stiffness matrix for 2D truss element. [AU, Nov / Dec – 2015]
2.126) A two noded truss element as shown in figure. The nodal displacements are u1 = 5
mm and u2 = 8mm. Calculate the displacement at 𝑥 =
𝑙
4
,
𝑙
3
𝑎𝑛𝑑
𝑙
2
[AU, May / June – 2014]

2.127) Consider the rod (a robot arm) as shown below, which is rotating at constant
angular velocity  = 30 rad/sec. Determine the axial stress distribution in the rod,
using two quadratic elements. Consider only the centrifugal force. Ignore bending
of the rod.
2.128) A link of 2m, pin – jointed at one end, is rotating at angular velocity 5 rad / sec. the
cross – sectional area of link is 2 * 10-3
m2
. Determine the nodal displacements
using two linear spar elements. Take E = 200GPa and ρ = 7850 kg/m3
.
[AU, Nov / Dec – 2013]
2.129) A steel rod of length 1m is subjected to an axial load of 5 kN as shown in figure.
Area of cross section of the rod is 250 mm2
. Using 1 – D element equation solve for
the deflection of the bar, E = 2*105
N/mm2
. Use four elements.
[AU, Nov / Dec – 2010]

2.130) A steel bar of length 800mm is subjected to an axial load of 3kN as shown in figure.
Find the elongation of the bar, neglecting self-weight. [AU, Nov / Dec – 2015]
2.131) A steel bar of length 800 mm is subjected to an axial load of 3 kN as shown in fig.
Find the nodal displacements of the bar, and load vectors. [AU, May / June – 2016]
2.132) A column of length 500mm is loaded axially as shown in figure. Analyze the
column and evaluate the stress and strain at salient points. The Young’s modulus
can be taken as E. Take A1 = 62.5mm2
and A2 = 125mm2

2.133) Determine the nodal displacements, stress and strain for the bar shown Fig.
[AU, Nov / Dec – 2014]
2.134) Consider a bar as shown in figure. Young’s Modulus E = 2*105
N/mm2
. A1 = 2 cm2
,
A2 = 1 cm2
and force of 100 N. Determine the nodal displacement.
[AU, Nov / Dec – 2010]

2.135) Consider the bar shown in Figure Axial force P = 30 kN is applied as shown.
Determine the nodal displacement, stresses in each element and reaction forces
[AU, May / June – 2012]
2.136) Consider the bar as shown in figure. Axial force P1 = 20 kN and P2 = 15 kN is
applied as shown in figure. Determine the nodal displacements, stresses in each
element and reaction forces. [AU, April / May – 2011]
2.137) Find the nodal displacement and elemental stresses for the bar shown in Figure.

2.138) An axial load P = 300 x 103
N is applied at 200
C to the rod as shown below. The
temperature is then raised to 600
C
a) Assemble the stiffness (K) and load (F) matrices.
b) Determine the nodal displacements and element stresses.
2.139) The stepped bar shown in fig is subjected to an increase in temperature, T=80o
C.
Determine the displacements, element stresses and support reactions.
[AU, Nov / Dec – 2009]

2.140) Axial load of 500N is applied to a stepped shaft, at the interface of two bars. The
ends are fixed. Obtain the nodal displacements and stresses when the element is
subjected to all in temperature of 100˚C. Take E1 = 70*103
N/mm2
, E2 = 200*103
N/mm2
, A1 = 900mm2
, A2 = 1200mm2
, α1 = 23*10-6
/ ˚C, α2 = 11.7*10-6
/ ˚C, L1 =
200mm, L2 = 300mm. [AU, Nov / Dec – 2011]
2.141) Consider a bar as shown below having a cross sectional area Ae = 1.2 in2
and
Young’s modulus E = 30 x 106
psi If q1 = 0.02 in and q2 = 0.025 in, determine the
following:
a) The displacement at the point P b) The strain  and stress 
c) The element stiffness matrix and d) The strain energy in the element.
A finite element solution using one – dimensional, two – noded elements has been
obtained for a rod as shown below.
Displacement are as follows
T
mm
Q 0.1]-0,0.6,[-0.2, , E = 1N/mm2
, area of each
element = 1 mm2
, L1-2 = 50 mm, L2-3 = 80 mm, L3-4 = 100 mm.

i) According to the finite element theory, plot the displacement u(x) versus x.
ii) According to the finite element theory, plot the strain (x) versus x.
iii) Determine the B matrix for element 2-3.
iv) Determine the strain energy in the element 1-2 using .
2
1
kqqU T

2.142) Consider the bar, loaded as shown below. Determine the nodal displacements,
element stresses and support reactions. Solve this problem by adopting elimination
method for handling boundary conditions. (value of E = 200 x 109
N/m2
).
2.143) For the bar element as shown in fig. Calculate the nodal displacements and
elemental stresses. [AU, Nov / Dec – 2016]

2.144) In the figure shown below load P = 60kN is applied. Determine the displacement
field, stress and support reactions in the body. Take E = 20 kN/mm2
[AU, May / June – 2011]
2.145) Consider the bar as shown below. Determine the nodal displacements, element
stresses and support reactions. (E = 200 x 109
N/m2
) [AU, Nov / Dec – 2014]
2.146) An axial load P = 385 KN is applied to the composite block as shown below.
Determine the stress in each material.

2.147) For a vertical rod as shown below, find the deflection at A and the stress
distribution. E = 100 MPa and weight per unit volume = 0.06 N/cm3
. Comment on
the stress distribution.
2.148) Consider a brick wall of thickness 0.3 m, k = 0.7 W/m˚C. The inner surface is at
28˚C and the outer surface is exposed to cold air at -15˚C. The heat transfer
coefficient associated with the outside surface is 40 W/m2
˚C. Determine the steady
state temperature distribution within the wall and also the heat flux through the
wall. Use two 1D elements and obtain the solution. [AU, Nov / Dec – 2013]
2.149) Consider a brick wall as shown in figure of thickness L = 30cm, K = 0.7 W/m˚C.
The inner surface is at 28˚C and the outer surface is exposed to cold air at -15˚C.
The heat transfer coefficient associated with the outside surface is h = 40 W/m2
˚C.
Determine the steady state temperature distribution within the wall and also the heat

flux through the wall. Use a two element model. Assume one dimensional flow.
2.150) A composite wall consists of three materials as shown in figure. The outer
temperature is T0 = 20˚C. Convection heat transfer takes place on the inner surface
of the wall with T∞ = 800˚C and h = 25W/m2
˚C. Determine the temperature
distribution in the wall. [AU, May / June – 2011]
2.151) A composite wall consists of three materials as shown in figure. The inside wall
temperature is 200º C and the outside air temperature is 50ºC with a convection

coefficient of 10W/cm2
ºC. Determine the temperature along the composite wall.
[AU, April / May – 2015, Nov / Dec – 2016]
2.152) A composite wall is made of three different materials. The thermal conductivity of
the various sections are k1 = 2 W/cm ˚C, k2 = 1 W/cm ˚ C, k3 = 0.2 = W/cm ˚C. The
thickness of the wall for the section is 1cm, 5cm and 4cm respectively. Determine
the temperature values of nodal points within the wall. Assume the surface area to
unity. The left edge of the wall is subjected to a temperature of 30˚C and the right
side of the wall is at 10˚C. [AU, Nov / Dec – 2011]
2.153) Figure shows a sandwiched composite wall. Convection heat loss occurs on the left
surface and the temperature on the right surface is constant. Considering a unit area
and with the parameters given, use three linear elements (one for each layer) and
(i) Determine the temperature distribution through the composite wall and
(ii) Calculate the flux on the right surface of the wall. [AU, Nov / Dec – 2012]

2.154) A wall of 0.6m thickness having thermal conductivity of 1.2 W/m-K the wall is to
be insulated with a material of thickness 0.06 m having an average thermal
conductivity of 0.3 W/m-K. The inner surface temp is 1000˚C and outside of the
insulation is exposed to atmospheric air at 30˚C with heat transfer co-efficient of 35
N/m2
K. Calculate the nodal temperature using FEA. [AU, Nov / Dec – 2011]
2.155) A long bar of rectangular cross section having thermal conductivity of 1.5 W/m˚
C is
subjected to the boundary condition as shown below. Two opposite sides are
maintained at uniform temperature of 180 0
C. One side is insulated and the
remaining side is subjected to a convection process with T = 85˚
C and h = 50
W/m2˚
C. Determine the temperature distribution in the bar.

2.156) Compute the steady state temperature distributions in the plate shown in Fig. by
discretizing the domain of interest using triangular elements. Assume Thermal
Conductivity k = 1.5W/m°C. [AU, Nov / Dec – 2014]
2.157) A steel rod of diameter d = 2 cm, length l =5 cm and thermal conductivity K = 50
W/m˚C is exposed at one end to a constant temperature of 320˚C. The other end is
in ambient air of temperature 20˚C with a convection co-efficient of h = 100
W/m2
˚C. Determine the temperature at the midpoint of the rod using FEA.
[AU, Nov / Dec – 2011]
2.158) Determine the temperature distribution in one dimensional rectangular cross-section
as shown in Figure. The fin has rectangular cross-section and is 8cm long 4cm wide
and 1cm thick. Assume that convection heat loss occurs from the end of the fin.
Take h = 3W / cm˚C, h = 0.1 W / cm2
˚ C,T ∞ = 20˚C. [AU, April / May – 2011]

2.159) Determine the temperature distribution along a circular fin of length 5 cm and
radius 1 cm. The fin is attached to a boiler whose wall temperature 140°C and the
free end is open to the atmosphere. Assume Tα = 40°C, h = 10W/cm2
°C, K =
70W/cm°C. [AU, Nov / Dec – 2014]
2.160) Calculate the temperature distribution in stainless steel fin shown in figure. The
region can be discretized into five elements and six nodes.[AU, April / May – 2009]
2.161) Calculate the temperature distribution in stainless steel fin shown in figure. The
region can be discretized into three elements of equal size. [AU, Nov / Dec – 2016]
2.162) The figure shows a uniform Aluminium fin of diameter 25 mm. 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 =
200 W /m °C, h = 1000 W /m2
°C and T = 20 °C, determine the temperature
distribution in the fin using one dimensional element, considering two elements.
[AU, May / June – 2016]

2.163) Using two equal-length finite elements, determine the natural frequencies of the
solid circular shaft fixed at one end shown in figure. [AU, Nov / Dec – 2011]
2.164) For the bar as shown in figure with length 2L, modulus of elasticity E, Mass density
ρ and cross sectional area A, determine the first two natural frequencies.
[AU, Nov / Dec – 2015]
2.165) Obtain the natural frequencies of vibration for a stepped steel bar of area 625mm2
for the length of 250mm and 312.5mm2
for the length of 125mm. The element is
fixed at larger end [AU, Nov / Dec – 2007]

2.166) Determine the Eigen values and frequencies for the stepped bar shown in the figure.
Take E = 20 * 1010
N/m2
and self-weight = 8500 kg.m3
[AU, May / June – 2005]
2.167) Determine the Eigen values for the stepped bar shown in fig.
[AU, Nov / Dec – 2016]

2.168) Find the natural frequency of longitudinal vibration of the unconstrained stepped
bar as shown in figure. [AU, Nov / Dec – 2006]
2.169) Determine the first two natural frequencies of longitudinal vibration of the stepped
steel bar shown in Fig and plot the mode shapes. All dimensions are in mm. E = 200
GPa and density 0.78 kg/cc. [AU, Nov / Dec – 2014]
2.170) Determine the first two natural frequencies of longitudinal vibration of bar as shown
in figure assuming that the bar is discretised into two elements as shown in figure. E
and ρ represents the Young’s modulus and mass density of material of the bar.

2.171) A vertical plate of thickness 40 mm is tapered with widths of 0.15m and 0.075m at
top and bottom ends respectively. The plate is fixed at the top end. The length of the
plate is 0.8m. Take Young's modulus as 200 GPa and density as 7800 kg/m3
. Model
the plate with two spar elements. Determine the natural frequencies of longitudinal
vibration and the mode shapes. [AU, Nov / Dec – 2013]
2.172) Determine the natural frequencies and mode shapes of a system whose stiffness and
mass matrices are given below [AU, May / June – 2008, 2014]
[ 𝑘] =
2AE
L
[
3 −1
−1 1
] [ 𝑚] =
𝜌𝐴𝐿
12
[
6 1
1 12
]
2.173) Determine the eigen values and natural frequencies of a system whose stiffness and
mass matrices are given below. [AU, Nov / Dec – 2015]
[ 𝑘] =
2AE
L
[
3 −1
−1 1
] [ 𝑚] =
𝜌𝐴𝐿
12
[
6 1
1 12
]
2.174) Consider a two-bar supported by a spring shown in figure. Both bars have E = 210
GPa and A=5.0 x10-4
m2
. Bar one has a length of 5m 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. [AU, Nov / Dec – 2009]

2.175) Each of the three bars of the pin – jointed frame shown in figure has a cross
sectional area of 1000mm2
with E = 200GPa. Solve for displacements.
[AU, Nov / Dec – 2013]
2.176) Find the deflection at the free end under its own weight, using divisions of
a) 1 element b) 2 elements c) 4 elements d) 8 elements and e) 16 elements
Then plot the number of elements versus deflection.

2.177) For the discretization of beam elements as shown below, number the nodes so as to
minimize the bandwidth of the assembled stiffness matrix (K)
2.178) The elements of a row or column of the stiffness matrix of a bar element sum up to
zero, but not so for a beam element. Explain why this is so.
2.179) For the beam problem shown below, determine the tip deflection and the slope at
the roller support.

2.180) For the beam and loading as shown in figure. Determine the slopes at the two ends
of the distributed load and the vertical deflection at the mid-point of the distributed
load. Take E = 200GPa and I = 4*106
mm4
[AU, May / June – 2011]
2.181) Analyze the beam shown in figure using finite element technique. Determine the
rotations at the supports. Give E = 200GPa and I = 4 * 106
mm4
[AU, Nov / Dec – 2013]
2.182) Determine the maximum deflection and slope in the beam, loaded as shown in
figure. Determine also the reactions at the supports. E = 200GPa, I = 20*10-6
m4
, q
= 5kN/m and L =1 m. [AU, April / May – 2015]

2.183) For the beam shown in figure, determine the displacements and the slopes at the
nodes, the forces in each element and the reactions. E = 200 GPa, I = 1 X 10-4
m4
[AU, May / June, Nov / Dec – 2016]
2.184) Find the deflection and slope for the following beam section at which point load is
applied.
2.185) Solve the following beam as shown below, clamped at one end and spring support
at other end. A linearly varying transverse load of maximum magnitude of 100
N/cm applied over the span of 4 cm to 10 cm. Take EI = 2 x 107
N/cm2
,
2
10

EI
K
.

2.186) Obtain the deflection at the midpoint of the beam shown below and determine the
reaction.
2.187) The simply supported beam shown in figure is subjected to a uniform transverse
load, as shown. Using two equal-length elements and work-equivalent nodal loads
obtain a finite element solution for the deflection at mid-span and compare it to the
solution given by elementary beam theory. [AU, April / May - 2010]
2.188) Determine the displacements and slopes at the nodes for the beam shown in figure.
Take k=200kN / m, E=70GPa and I=2x10-4
m4
. [AU, Nov / Dec – 2012]

2.189) Determine the nodal displacements and slopes for the beam shown in Figure. Find
the moment at the midpoint of element 1. [AU, Nov / Dec – 2012]
2.190) Determine the first two natural frequencies of transverse vibration of the cantilever
beam as shown in figure and plot the mode of shapes. [AU, April / May – 2015]
2.191) Determine the natural frequencies and mode shapes of transverse vibration for a
beam fixed at both ends. The beam may be modelled by two elements, each of
length L and cross-sectional area A. Consider lumped mass matrix approach.
2.192) Find the natural frequencies of transverse vibrations of the cantilever beam shown
in Figure using one beam element. [AU, Nov / Dec – 2012]

2.193) Consider a uniform cross – section bar as shown in figure of length “L” made up of
a material whose Young’s modulus and density are given by E and ρ. Estimate the
natural frequencies of axial vibration of the bar using both lumped and consistent
mass matrix. [AU, May / June – 2005, Nov / Dec – 2016]
2.194) Compute material frequencies of free transverse vibration of a stepped beam shown
in figure. [AU, May / June – 2003]
2.195) Determine the natural frequencies of transverse vibration for a beam fixed at both
ends. The beam may be modelled by two elements each of length L and cross –
sectional area A. The use of symmetry boundary condition is optional.
[AU, May / June – 2008]

2.196) Determine the displacement of node 1 and the stress in element 3, for the three-bar
truss as shown below. Take A = 250 mm2
, E = 200 GPa for all elements.
2.197) 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 = 200 mm2
[AU, May / June , Nov / Dec – 2016]

2.198) Determine the force in the members of the truss as shown in figure.
Take E = 200 GPa [AU, May / June – 2012]
2.199) The loading and other parameters for a two bar truss element is shown in figure
Determine [AU, May / June – 2013]
(i) The element stiffness matrix for each element
(ii) Global stiffness matrix
(iii) Nodal displacements
(iv) Reaction forces
(v) The stresses induced in the elements. Assume E = 200 GPa.

2.200) Calculate nodal displacement and elemental stresses for the truss shown in Figure.
E= 70Gpa.cross-sectional area A = 2cm2
for all truss members.
2.201) Find the horizontal and vertical displacements of node 1 for the truss shown below.
Take A = 300 mm2
, E = 2 x105
N/mm2
for each element.
2.202) Each of the five bars of the pin jointed truss shown in figure below has a cross
sectional area 20 sq. cm. and E = 200 GPa.

(i) Form the equation F = KU where K is the assembled stiffness matrix of the
structure.
(ii) Find the forces in all the five members. [AU, April / May – 2008]
2.203) Analyze the truss shown in figure and evaluate the stress resultants in member (2).
Assume area of cross section of all the members in same. E = 2 * 105
N/mm2
[AU, Nov / Dec – 2010]
2.204) Determine the joint displacements, the joint reactions, element forces and element
stresses of the given truss elements. [AU, April / May - 2010]

Elements A
cm2
E
N/m2
L
m
Global
Node
connection
 Degree
1 32.2 6.9e 10 2.54 2 to 3 90
2 38.7 20.7e10 2.54 2 to 1 0
3 25.8 20.7e10 3.59 1 to 3 135
2.205) Determine the force in the members of the truss shown in figure.
2.206) For the two bar truss shown in figure, determine the displacements of node 1 and
stress in element 1 – 3. [AU, May / June – 2014]

2.207) Find the nodal displacement developed in the planer truss shown in Figure when a
vertically downward load of 1000 N is applied at node 4. The required data are
given in the Table. [AU, May / June – 2012]
Element No.
‘e’
Cross – Sectional area A
cm2
Length l (e)
cm
Young’s Modulus
E(e)
N/mm2
1 2 √2 50 2 * 106
2 2 √2 50 2 * 106
3 1 √2.5 100 2 * 106
4 1 √2 100 2 * 106

2.208) Determine the natural frequencies for the truss shown in figure using finite element
method. [AU, May / June – 2007]

2.209) Derive the interpolation function for the one dimensional linear element with a
length “L” and two nodes, one at each end, designated as “i” and ” j”. Assume the
origin of the coordinate system is to the left of node “i”.
[AU, April / May - 2010]
Figure shows the one-dimensional linear element

ME6603 - FINITE ELEMENT ANALYSIS UNIT - II NOTES AND QUESTION BANK

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ME6603 - FINITE ELEMENT ANALYSIS UNIT - II NOTES AND QUESTION BANK