ME6603 - FINITE ELEMENT ANALYSIS

R.M.K. COLLEGE OF ENGINEERING
AND TECHNOLOGY
R.S.M. NAGAR, PUDUVOYAL-601206
DEPARTMENT OF MECHANICAL
ENGINEERING
ME6603 – FINITE ELEMENT ANALYSIS
VI SEM MECHANICAL ENGINEERING
Regulation 2013
QUESTION BANK
PREPARED BY
R.ASHOK KUMAR M.E (Ph.D)

R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ ME6603 / VI / MECH / DEC 2017 – MAY 2018
FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 2
ME6603 FINITE ELEMENT ANALYSIS
OBJECTIVES:
 To introduce the concepts of Mathematical Modeling of Engineering Problems.
 To appreciate the use of FEM to a range of Engineering Problems.
UNIT I INTRODUCTION 9
Historical Background – Mathematical Modeling of field problems in Engineering – Governing Equations
– Discrete and continuous models – Boundary, Initial and Eigen Value problems– Weighted Residual
Methods –Variational Formulation of Boundary Value Problems – Ritz Technique – Basic concepts of the
Finite Element Method.
UNIT II ONE-DIMENSIONAL PROBLEMS 9
One Dimensional Second Order Equations – Discretization – Element types- Linear and Higher order
Elements – Derivation of Shape functions and Stiffness matrices and force vectors- Assembly of Matrices -
Solution of problems from solid mechanics and heat transfer. Longitudinal vibration frequencies and mode
shapes. Fourth Order Beam Equation –Transverse deflections and Natural frequencies of beams.
UNIT III TWO DIMENSIONAL SCALAR VARIABLE PROBLEMS 9
Second Order 2D Equations involving Scalar Variable Functions – Variational formulation –Finite Element
formulation – Triangular elements – Shape functions and element matrices and vectors. Application to
Field Problems - Thermal problems – Torsion of Non circular shafts –Quadrilateral elements – Higher
Order Elements.
UNIT IV TWO DIMENSIONAL VECTOR VARIABLE PROBLEMS 9
Equations of elasticity – Plane stress, plane strain and axisymmetric problems – Body forces and
temperature effects – Stress calculations - Plate and shell elements.
UNIT V ISOPARAMETRIC FORMULATION 9
Natural co-ordinate systems – Isoparametric elements – Shape functions for iso parametric elements – One
and two dimensions – Serendipity elements – Numerical integration and application to plane stress
problems - Matrix solution techniques – Solutions Techniques to Dynamic problems – Introduction to
Analysis Software. TOTAL: 45 PERIODS
OUTCOMES:
 Upon completion of this course, the students can able to understand different mathematical
Techniques used in FEM analysis and use of them in Structural and thermal problem
TEXT BOOK:
 Reddy. J.N., “An Introduction to the Finite Element Method”, 3rd Edition, Tata McGraw-Hill, 2005
 Seshu, P, “Text Book of Finite Element Analysis”, Prentice-Hall of India Pvt. Ltd., New Delhi,
2007.

COURSE OUTCOMES:
After successful completion of the course, the students should be able to
MAPPING OF COURSE OUTCOMES WITH PROGRAM OUTCOMES:
CO No. Course Outcomes
Highest
Cognitive
Level
C313.1 Calculate the solution for BVP using numerical techniques. K3
C313.2
Compute structural and thermal problems utilizing 1D problem
formulation.
K3
C313.3 Use 2D scalar formulation for solving thermal and torsion problems K3
C313.4
Use 2D vector formulation for solving plane stress, plane strain and
axisymmetric problems
K3
C313.5 Use iso-parametric formulation for complex contour domain K3
C313.6
Compute the real time structural and thermal problems using finite
element techniques.
K3
Course
Out
Comes
Level
of CO
Program Outcomes
Program Specific
Outcomes
K3 K4 K5 K5
K3,K5
,K6
A3 A2 A3 A3 A3 A3 A2
PSO-
1
PSO-
2
PSO-
3PO-
1
PO-
2
PO-
3
PO-
4
PO-5
PO-
6
PO-
7
PO-
8
PO-
9
PO-
10
PO-
11
PO-
12
C313.1 K3 3 2
C313.2 K3 3 2 2
C313.3 K3 3 2 2 2
C313.4 K3 3 2 2
C313.5 K3 3 2 2
C313.6 K3 3 1 2 3 2

UNIT – I – INTRODUCTION
PART – A
1.1) What is the finite element method?
1.2) How does the finite element method work?
1.3) What are the main steps involved in FEA. [AU, April / May – 2011]
1.4) Write the steps involved in developing finite element model.
1.5) What are the basic approaches to improve a finite element model?
[AU, Nov / Dec – 2010]
1.6) What are the methods generally associated with the finite element analysis?
[AU, May / June – 2016]
1.7) Write any two advantages of FEM Analysis. [AU, Nov / Dec – 2012]
1.8) Why polynomial type interpolation functions are mostly used in FEM?
[AU, April / May – 2017]
1.9) What are the methods generally associated with finite element analysis?
1.10) List any four advantages of finite element method. [AU, April / May – 2008]
1.11) What are the applications of FEA? [AU, April / May – 2011]
1.12) Define finite difference method.
1.13) What is the limitation of using a finite difference method? [AU, April / May – 2010]
1.14) Define finite volume method.
1.15) Differentiate finite element method from finite difference method.
1.16) Differentiate finite element method from finite volume method.
1.17) What do you mean by discretization in finite element method?
1.18) What is discretization? [AU, Nov / Dec – 2010, 2015]
1.19) What is meant by node or joint? [AU, May / June – 2014]
1.20) What is meant by node? [AU, Nov / Dec – 2015, 2016]
1.21) List the types of nodes. [AU, May / June – 2012]

1.22) Define degree of freedom.
1.23) What is meant by degrees of freedom? [AU, Nov / Dec – 2012]
1.24) State the advantage of finite element method over other numerical analysis
methods.
1.25) State the fields to which FEA solving procedure is applicable.
1.26) What is a structural and non-structural problem?
1.27) Distinguish between 1D bar element and 1D beam element.
[AU, Nov / Dec – 2009, May / June – 2011]
1.28) Write the equilibrium equation for an elemental volume in 3D including the body
force.
1.29) How to write the equilibrium equation for a finite element? [AU, Nov / Dec – 2012]
1.30) Classify boundary conditions. [AU, Nov / Dec – 2011]
1.31) What are the types of boundary conditions?
1.32) What do you mean by boundary condition and boundary value problem?
1.33) Write the difference between initial value problem and boundary value problem.
1.34) What are the difference between boundary value problem and initial value problem?
1.35) What are the different types of boundary conditions? Give examples.
[AU, May / June – 2012]
1.36) List the various methods of solving boundary value problems.
[AU, April / May – 2010, Nov / Dec – 2016]
1.37) Write down the boundary conditions of a cantilever beam AB of span L fixed at A
and free at B subjected to a uniformly distributed load of P throughout the span.
[AU, May / June – 2009, 2011]
1.38) Briefly explain force method and stiffness method.
1.39) What is aspect ratio?

1.40) Write a short note on stress – strain relation.
1.41) Write down the stress strain relationship for a three dimensional stress field.
1.42) If a displacement field in x direction is given by 𝑢 = 2𝑥2
+ 4𝑦2
+ 6𝑥𝑦 Determine
the strain in x direction. [AU, May / June – 2016]
1.43) State the effect of Poisson’s ratio.
1.44) Define total potential energy of an elastic body.
1.45) What is the stationary property of total potential energy? [AU, May / June – 2016]
1.46) Write the potential energy for beam of span L simply supported at ends, subjected
to a concentrated load P at mid span. Assume EI constant.
[AU, April / May, Nov / Dec – 2008]
1.47) State the principle of minimum potential energy.
[AU, Nov / Dec – 2007, 2013, April / May – 2009]
1.48) State the principle of minimum potential energy theorem. [AU, May / June – 2016]
1.49) How will you obtain total potential energy of a structural system?
[AU, April / May – 2011, May / June – 2012]
1.50) Write down the potential energy function for a three dimensional deformable body
in terms of strain and displacements. [AU, May / June – 2009]
1.51) What should be considered during piecewise trial functions?
1.52) What do you understand by the term “piecewise continuous function”?
[AU, Nov / Dec – 2013]
1.53) Write about weighted residual method. [AU, May / June – 2016]
1.54) Distinguish between the error in solution and Residual. [AU, April / May – 2015]
1.55) Name the weighted residual methods. [AU, Nov / Dec – 2011]
1.56) List the various weighted residual methods. [AU, Nov / Dec – 2014, 2017]

1.57) What is the use of Ritz method? [AU, Nov / Dec – 2011]
1.58) What is Rayleigh – Ritz method?
[AU, May / June – 2014, Nov / Dec – 2015, 2016]
1.59) Mention the basic steps of Rayleigh-Ritz method. [AU, April / May – 2011]
1.60) Highlight the equivalence and the difference between Rayleigh Ritz method and the
finite element method. [AU, Nov / Dec – 2012]
1.61) Distinguish between Rayleigh Ritz method and finite element method.
[AU, Nov / Dec – 2013]
1.62) Distinguish between Rayleigh Ritz method and finite element method with regard
to choosing displacement function. [AU, Nov / Dec – 2010]
1.63) Compare the Ritz technique with the nodal approximation method.
[AU, Nov / Dec – 2014]
1.64) Why are polynomial types of interpolation functions preferred over trigonometric
functions? [AU, April / May – 2009, 2017, May / June – 2013]
1.65) Why are polynomial types of interpolation functions mostly preferred?
[AU, Nov / Dec – 2017]
1.66) What is meant by weak formulation? [AU, May / June – 2013, April / May – 2017]
1.67) What are the advantage of weak formulation? [AU, April / May – 2015]
1.68) Define the principle of virtual work.
1.69) Differentiate Von Mises stress and principle stress.
1.70) What do you mean by constitutive law?
[AU, Nov / Dec – 2007, April / May – 2009, 2017]
1.71) What are h and p versions of finite element method?
1.72) What is the difference between static and dynamic analysis?
1.73) Mention two situations where Galerkin’s method is preferable to Rayleigh – Ritz
method. [AU, Nov / Dec – 2013]

1.74) What is Galerkin method of approximation? [AU, Nov / Dec – 2009]
1.75) What is a weighted resuidal method? [AU, Nov / Dec – 2010]
1.76) Distinguish between potential energy and potential energy functional.
1.77) What are the types of Eigen value problems? [AU, May / June – 2012]
1.78) Name a few FEA packages. [AU, Nov / Dec – 2014]
1.79) Name any four FEA software
1.80) List any four finite element analysis software packages. [AU, Nov / Dec – 2017]
PART – B
1.81) Explain the step by step procedure of FEA. [AU, Nov / Dec – 2010]
1.82) Explain the general procedure of finite element analysis. [AU, Nov / Dec – 2011]
1.83) List and briefly describe the general steps of the finite element method.
[AU, May / June – 2014]
1.84) Briefly explain the stages involved in FEA.
1.85) Explain the step by step procedure of FEM. [AU, Nov / Dec – 2011]
1.86) List out the general procedure for FEA problems. [AU, May / June – 2012]
1.87) Compare FEM with other methods of analysis. [AU, Nov / Dec – 2010]
1.88) Define discretization. Explain mesh refinement. [AU, Nov / Dec – 2010]
1.89) Explain the various aspects pertaining to discretization, process in finite element
modeling analysis. [AU, Nov / Dec – 2013]
1.90) Explain the process of discretization of a structure in finite element method in
detail, with suitable illustrations for each aspect being & discussed.
[AU, Nov / Dec – 2012]
1.91) Discuss procedure using the commercial package (P.C. Programs) available today
for solving problems of FEM. Take a structural problem to explain the same.
[AU, Nov / Dec – 2011]

1.92) State the importance of locating nodes in finite element model.
[AU, Nov / Dec – 2011]
1.93) Write briefly about weighted residual methods. [AU, Nov / Dec – 2015]
1.94) Write a brief note on the following.
(a) isotropic material
(b) orthotropic material
(c) anisotropic material
1.95) What are initial and final boundary value problems? Explain.
[AU, Nov / Dec – 2010]
1.96) Explain the Potential Energy Approach [AU, Nov / Dec – 2010]
1.97) Explain the principle of minimization of potential energy. [AU, Nov / Dec – 2011]
1.98) Explain the four weighted residual methods. [AU, Nov / Dec – 2011]
1.99) Explain Ritz method with an example. [AU, April / May – 2011]
1.100) Explain Rayleigh Ritz and Galerkin formulation with example.
[AU, May / June – 2012]
1.101) Write short notes on Galerkin method? [AU, April / May – 2009]
1.102) Discuss stresses and equilibrium of a three dimensional body.
[AU, May / June – 2012]
1.103) Derive the element level equation for one dimensional bar element based on the
station- of a functional. [AU, May / June – 2012]
1.104) Derive the characteristic equations for the one dimensional bar element by using
piece-wise defined interpolations and weak form of the weighted residual method?
[AU, May / June – 2012]
1.105) Develop the weak form and determine the displacement field for a cantilever beam
subjected to a uniformly distributed load and a point load acting at the free end.
[AU, Nov / Dec – 2013]

1.106) Explain Gaussian elimination method of solving equations.
1.107) Write briefly about Gaussian elimination? [AU, April / May – 2009]
1.108) The following differential equation is available for a physical phenomenon.
𝑑
𝑑𝑥
(𝑥
𝑑𝑢
𝑑𝑥
) −
2
𝑥2
= 0, 1 ≤ 𝑥 ≤ 2
Boundary conditions are, x = 1 u = 2
x = 2 𝑥
𝑑𝑢
𝑑𝑥
= −
1
2
Find the value of the parameter a, by the following methods.
(i) Collocation (ii) Sub – Domain (iii) Least Square
(iv) Galerkin [AU, Nov / Dec – 2017]
𝑑2
𝑦
𝑑𝑥2
+ 50 = 0, 0 ≤ 𝑥 ≤ 10
Trial function is 𝑦 = 𝑎1 𝑥(10 − 𝑥)
Boundary conditions are, y (0) = 0 y (10) = 0
Find the value of the parameter a1, by the following methods.
(i) Collocation (ii) Sub – Domain (iii) Least Square
(iv) Galerkin
𝑑2
𝑦
𝑑𝑥2
+ 50 = 0, 0 ≤ 𝑥 ≤ 10
Trial function is 𝑦 = 𝑎1 𝑥(10 − 𝑥)
Boundary conditions are, y (0) = 0 y (10) = 0
Find the value of the parameter a1, by the following methods.

(i) Least Square (ii) Galerkin [AU, April / May – 2017]
1.111) Discuss the following methods to solve the given differential equation :
𝐸𝐼
𝑑
2
𝑦
𝑑𝑥2
𝑀( 𝑥) = 0
with the boundary condition y(0) = 0 and y(H) = 0
(i) Variant method (ii) Collocation method. [AU, April / May – 2010]
1.112) The differential equation of a physical phenomenon is given by
𝑑2
𝑦
𝑑𝑥2
+ 𝑦 = 4𝑥, 0 ≤ 𝑥 ≤ 1
The boundary conditions are: y(0)=0; y(1)=1; Obtain one term approximate solution
by using Galerkin's method of weighted residuals.
[AU, May / June – 2014, 2016, Nov / Dec – 2016]
1.113) Find the approximate deflection of a simply supported beam under a uniformly
distributed load ‘P‘ throughout its span. Using Galerkin and Least square residual
method. [AU, May / June – 2011]
1.114) A simply supported beam of length Land uniform section is subjected to a linearly
varying load of intensity zero at the left end and intensity q0 at the right end.
Compute the deflection and bending moment at the midpoint by using basic
Galerkin method. [AU, Nov / Dec – 2017]
1.115) Solve the differential equation for a physical problem expressed as
𝑑2 𝑦
𝑑𝑥2 + 100 =
0, 0 ≤ 𝑥 ≤ 10 with boundary conditions as y (0) = 0 and y (10) = 0 using
(i) Point collocation method
(ii) Sub domain collocation method
(iii) Least squares method and
(iv) Galerkin method. [AU, May / June – 2013, April / May – 2017]

1.116) Solve the differential equation for a physical problem expressed as
𝑑2 𝑦
𝑑𝑥2 + 50 =
0, 0 ≤ 𝑥 ≤ 10 with boundary conditions as y (0) = 0 and y (10) = 0 using the
trail function 𝑦 = 𝑎1 𝑥 (10 − 𝑥) Find the value of the parameters a1 by the
following methods.
(i) Point collocation method
(ii) Sub domain collocation method
(iii) Least squares method and
(iv) Galerkin method. [AU, Nov / Dec – 2011]
1.117) Using a collocation method, find the solution of given governing equation
𝑑2 𝜑
𝑑𝑥2 +
𝜑 + 𝑥 = 0, 0 ≤ 𝑥 ≤ 1 subject to the boundary conditions 𝜑 (0) = 𝜑(1) =
0. Use 𝑥 =
1
4
and
1
1
as the collocation points. [AU, Nov / Dec – 2017]
1.118) Solve the following equation using a two – parameter trial solution by the
(a) Collocation method (𝑅 𝑑 = 0 𝑎𝑡 𝑥 =
1
3
𝑎𝑛𝑑 𝑥 =
2
3
)
(b) Galerkin method.
Then, compare the two solutions with the exact solution
𝑑𝑦
𝑑𝑥
+ 𝑦 = 0, 0 ≤ 𝑥 ≤ 1
y (0) = 1
1.119) Determine the Galerkin approximation solution of the differential equation
𝐴
𝑑2 𝑢
𝑑𝑥2 + 𝐵
𝑑𝑢
𝑑𝑥
+ 𝐶 = 0, 𝑢(0) = 𝑢( 𝑙) = 0
1.120) Solve the following differential equation using Galerkin’s method.
D
𝑑2 𝜑
𝑑𝑥2 + 𝑄 = 0, 0 ≤ 𝑥 ≤ 𝐿
subjected to 𝜑(0) = 𝜑0 𝑎𝑛𝑑 𝜑( 𝐿) = 𝜑1 [AU, April / May – 2011]

1.121) A physical phenomenon is governed by the differential equation
𝑑2 𝑤
𝑑𝑥2 − 10𝑥2
= 5 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 1 The boundary conditions are given by
𝑤(0) = 𝑤(1) = 0. By taking two-term trial solution as 𝑤( 𝑥) = 𝑐1 𝑓1( 𝑥) +
𝑐2 𝑓2(𝑥) with, 𝑓1( 𝑥) = 𝑥 ( 𝑥 − 1) 𝑎𝑛𝑑 𝑓2 = 𝑥2
(𝑥 − 1) find the solution of
the problem using the Galerkin method. [AU, Nov / Dec – 2009]
1.122) Determine the two parameter solution of the following using Galerkin method.
𝑑2 𝑦
𝑑𝑥2 = − cos 𝜋𝑥 , 0 ≤ 𝑥 ≤ 1, 𝑢(0) = 𝑢(1) = 0 [AU, Nov / Dec – 2012]
𝑑2
𝑦
𝑑𝑥2
− 10𝑥2
= 5, 0 ≤ 𝑥 ≤ 1
With boundary conditions y (0) = 0, y (l) = 1. Find an approximate solution of the
above differential equation by using Galerkin's method of weighted residuals and
also compare with exact solution [AU, May / June – 2016]
1.124) A physical phenomenon is governed by the differential equation
𝑑2 𝑤
𝑑𝑥2 − 10𝑥2
=
5 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 1. The boundary conditions are given by w (0) = w (1) = 0.
Assuming a trail function 𝑤(𝑥) = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥2
+ 𝑎3 𝑥3
. Determine using
Garlerkin method the variation of “w” with respect to “x”. [AU, Nov / Dec – 2016]
1.125) Using Collocation method, find the maximum displacement of the tapered rod as
shown in Fig. E = 2 *107
N/cm2
ρ = 0.075N/cm3
[AU, Nov / Dec – 2014]

1.126) Find the deflection at the centre of the simply supported beam of span length 'l'
subjected to uniformly distributed load throughout its length as shown in Figure
using (i) point collocation method (ii) sub-domain method.
1.127) A cantilever beam of length L is loaded with a point load at the free end. Find the
maximum deflection and maximum bending moment using Rayleigh-Ritz method
using the function 𝑦 = 𝐴{ 1 − cos (
𝜋𝑥
2𝐿
)} Given: EI is constant.
1.128) Compute the slope deflection and reaction forces for the cantilever beam of length
L carrying uniformly distributed load of intensity 'fo'. [AU, Nov / Dec – 2014]
1.129) A simply supported beam carries uniformly distributed load over the entire span.
Calculate the bending moment and deflection. Assume EI is constant and compare
the results with other solution. [AU, Nov / Dec – 2012]
1.130) Determine the expression for deflection and bending moment in a simply supported
beam subjected to uniformly distributed load over entire span. Find the deflection
and moment at midspan and compare with exact solution using Rayleigh-Ritz
method. Use 𝑦 = 𝑎1 sin (
𝜋𝑥
𝑙
) + 𝑎2 sin (
3𝜋𝑥
𝑙
) [AU, Nov / Dec – 2008]
1.131) Compute the value of central deflection in the figure below by assuming
𝑦 =
𝑎𝑠𝑖𝑛𝜋𝑥
𝐿
The beam is uniform throughout and carries a central point load P.
[AU, Nov / Dec – 2007, April / May – 2009]

1.132) A concentrated load P = 50 kN is applied at the centre of a fixed beam of length 3
m, depth 200 mm and width 120 mm. Calculate the deflection and slope at the
midpoint. Assume E = 2 x 105
N/mm2
[AU, May / June – 2016]
1.133) A beam AB of span '1' 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
[AU, May / June – 2016, Nov / Dec – 2016]
1.134) If a displacement field is described by
𝑢 = (−𝑥2
+ 2𝑦2
+ 6𝑥𝑦)10−4
𝑣 = (3𝑥 + 6𝑦 − 𝑦2)10−4
Determine the direct strains in x and y directions as well the shear strain at the point
x = 1, y =0. [AU, April / May – 2011]
1.135) In a solid body, the six components of the stress at a point are given by x= 40
MPa, y = 20 MPa, z = 30 MPa, yz = -30 MPa, xz = 15 MPa and xy = 10 MPa.

Determine the normal stress at the point, on a plane for which the normal is (nx, ny,
nz) = ( ½, ½, 2
1 )
1.136) In a plane strain problem, we have
x = 20,000 psi y = - 10,000 psi E = 30 x 10 6
psi,  = 0.3.
Determine the value of the stress z.
1.137) For the spring system shown in figure, calculate the global stiffness matrix,
displacements of nodes 2 and 3, the reaction forces at node 1 and 4. Also calculate
the forces in the spring 2. Assume, k1 = k3 = 100 N/m, k2 = 200 N/m, u1 = u4= 0 and
P=500 N. [AU, April / May – 2010]
1.138) Use the Rayleigh – Ritz method to find the displacement of the midpoint of the rod
shown in figure. [AU, April / May – 2011]

1.139) Consider the differential equation
𝑑2 𝑦
𝑑𝑥2 + 400𝑥2
= 0 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 1 subject
to boundary conditions 𝑦(0) = 0; 𝑦(1) = 0 The functional corresponding to this
problem, to be extremized is given by 𝐼 = ∫ {−0.5 (
𝑑𝑦
𝑑𝑥
)
2
+ 400𝑥2
𝑦
1
0
1.140) 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) [AU, Nov / Dec – 2009]
1.141) A bar of uniform cross section is clamped at one end and left free at the other end. It
is subjected to a uniform load axial load P as shown in figure. Calculate the
displacement and stress in the bar using three terms polynomial following Ritz
method. Compare the results with exact solutions. [AU, May / June – 2011]
1.142) A simply Supported beam subjected to uniformly distributed load over entire span
and it is subjected to a point load at the centre of the span. Calculate the deflection
using Rayleigh-Ritz method and compare with exact solutions.
[AU, May / June – 2013, April / May – 2017]
1.143) A simply Supported beam subjected to uniformly distributed load over entire span
as shown in figure. Calculate the bending moment and deflection at midspan using
Rayleigh-Ritz method. [AU, Nov / Dec – 2015, 2016]

1.144) A simply supported beam (span L and flexural rigidity EI) carries two equal
concentrated loads at each of the quarter span points. Using Raleigh – Ritz method
determine the deflections under the two loads and the two end slopes.
1.145) Analyze a simply supported beam subjected to a uniformly distributed load
throughout using Rayleigh Ritz method. Adopt one parameter trigonometric
function. Evaluate the maximum deflection and bending moment and compare with
exact solution. [AU, Nov / Dec – 2010]
1.146) Solve for the displacement field for a simply supported beam, subjected to a
uniformly distributed load using Rayleigh – Ritz method. [AU, Nov / Dec – 2013]
1.147) Derive the governing equation for a tapered rod fixed at one end and subjected to its
own self weight and a force P at the other end as shown in fig. Let the length of the
bar be l and let the cross section vary linearly from A1 at the top fixed end to A2 at
the free end. E and γ represents the Young’s modulus and specific weight of the
material of the bar. Convert this equation into weak form and hence determine the
matrices for solving using Ritz technique. [AU, April / May – 2015]

1.148) Use the Rayleigh – Ritz method to find the displacement field u(x) of the rod as
shown below. Element 1 is made of aluminum and element 2 is made of steel. The
properties are
Eal = 70 GPa A1 = 900 mm2
L1 = 200 mm
Est = 200 GPa A2 = 1200 mm2
L2 = 300 mm
Load = P = 10,000 N. Assume a piecewise linear displacement.
Field u = a1 + a2x for 0  x  200 mm, and u = a3 + a4 x for 200  x  500 mm.
1.149) A fixed beam length of 2L m carries a uniformly distributed load of a w(in N / m)
which run over a length of ‘L’ m from the fixed end, as shown in Figure. Calculate
the rotation at point B using FEA. [AU, Nov / Dec – 2011]
1.150) A rod fixed at its ends is subjected to a varying body force as shown in Figure. Use
the Rayleigh-Ritz method with an assumed displacement field 𝑢( 𝑥) = 𝑎0 +
𝑎1 𝑥 + 𝑎2 𝑥2
to find the displacement u(x) and stress σ(x). Plot the variation of the
stress in the rod. [AU, Nov / Dec – 2012]

1.151) A uniform rod subjected to a uniform axial load is illustrated in Figure. The
deformation of the bar is governed by the differential equation given below.
Determine the displacement using weighted residual method.
1.152) A steel rod is attached to rigid walls at each end and is subjected to a distributed
load T(x) as shown below.
a) Write the expression for potential energy.
b) Determine the displacement u(x) using the Rayleigh – Ritz method.
Assume a displacement field u(x) = a0 + a1 x + a2 x2
.
1.153) Derive the stress – strain relation and strain – displacement relation for an element
in space.
1.154) Derive the equation of equilibrium in case of a three dimensional stress system.
[AU, Nov / Dec – 2008]
1.155) What is constitutive relationship? Express the constitutive relations for a linear
elastic isotropic material including initial stress and strain. [AU, Nov / Dec – 2009]
1.156) Give a detailed note on the following:

(a) Rayleigh Ritz method (b) Galerkin method
(c) Least square method and (d) Collocation method
1.157) Determine using any Weighted Residual technique the temperature distribution
along the circular fin of length 6cm and 1cm. the fin is attached to a boiler whose
wall temperature is 140ºC and the free end is insulated. Assume the convection
coefficient h = 10 W/cm2
ºC. Conduction coefficient K = 70 W/ cm ºC and T∞ =
40ºC. The governing equation for the heat transfer through the fin is given by
−
𝑑
𝑑𝑥
[𝐾𝐴(𝑥)
𝑑𝑇
𝑑𝑥
] + ℎ𝑝(𝑥)(𝑇 − 𝑇∞) = 0 Assume appropriate boundary conditions
and calculate the temperatures at every 1cm from the left end.
1.158) Give a one – parameter Galerkin solution of the following equation, for the two
domain’s shown below. .1
2
2
2
2














y
u
x
u
1.159) Find the Eigen values and Eigen vectors of the matrix.
[
𝟑 −𝟏 𝟎
−𝟏 𝟐 −𝟏
𝟎 −𝟏 𝟑
]
1.160) Find the Eigen values and Eigen vectors of the matrix.
[
𝟑 𝟏𝟎 𝟓
−𝟐 −𝟑 −𝟒
𝟑 𝟓 𝟕
]

1.161) Find the Eigen value and the corresponding Eigen vector of 𝐴 = [
1 6 1
1 2 0
0 0 3
]
[AU, May / June – 2016]
1.162) Describe the Gaussian elimination method of solving equations.
1.163) Explain the Gaussian elimination method for the solving of simultaneous linear
algebraic equations with an example. [AU, April / May – 2008]
1.164) Solve the following system of equations using Gauss elimination method.
[AU, Nov / Dec – 2010]
x1 – x2 + x3 = 1
-3x1 + 2x2 – 3x3 = -6
2x1 – 5x2 + 4x3 = 5
1.165) Solve the following system of equations by Gauss Elimination method.
2x1 – 2x2 – x4 = 1
2x2 + x3 + 2x4 = 2
x1 – 2x2 + 3x3 – 2x4 = 3 [AU, May / June – 2012]
x2 + 2x3 + 2x4 = 4
1.166) Solve the following equations by Gauss elimination method.
28r1 + 6r2 = 1
6r1 + 24r2 + 6r3 = 0
6r2 + 28r3 + 8r4 = -1
8r3 + 16r4 = 10 [AU, Nov / Dec – 2010, 2012]
1.167) Use the Gaussian elimination method to solve the following simultaneous
equations:
4x1 + 2x2 – 2x3 – 8x4 = 4
x1 + 2x2 + x3 = 2

0.5x1 – x2 + 4x3 + 4x4 = 10
–4x1 – 2x2 – x4 = 0 [AU, April / May – 2009]
1.168) Solve the following system of equations using Gauss elimination method.
x1 + 3x2 + 2x3 = 13
– 2x1 + x2 – x3 = –3
- 5x1 + x2 + 3x3 = 6 [AU, Nov / Dec – 2009]

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.
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. [AU, Nov / Dec – 2017]

2.19) Write the element stiffness matrix for a two noded linear element subjected to axial
loading.
2.20) Write the stiffness matrix for a one dimensional 2 noded linear element
[AU, Nov / Dec – 2017]
2.21) Write the stiffness matrix for the simple beam element given below.
[AU, Nov / Dec – 2008]
2.22) What are the properties of global stiffness matrix? [AU, April / May – 2011]
2.23) Write the properties of Global Stiffness Matrix of a one dimensional element.
[AU, May / June – 2012]
2.24) Differentiate global stiffness matrix from elemental stiffness matrix.
2.25) What do you mean by banded matrix?
2.26) How will you find the width of a band?
2.27) How do you calculate the size of the global stiffness matrix?
2.28) List the properties of the global stiffness matrix. [AU, April / May – 2010]
2.29) What is the effect of node numbering on assembled stiffness matrix?
[AU, Nov / Dec – 2013]
2.30) Give a brief note on the following
(a) elimination approach (b) penalty approach.
2.31) Name the factors which affect the number element in the given domain.
2.32) State the requirements to be fulfilled by the approximate solution for its
convergence towards the actual solution.
2.33) What do you mean by continuity weakening?
2.34) Compare the linear polynomial approximation and quadratic polynomial
approximation.

2.35) Why polynomials are generally used as shape function?
[AU, Nov / Dec – 2011, 2016]
2.36) Why are polynomial terms preferred for shape functions in finite element method?
2.37) What do you mean by error in FEA solution?
2.38) What are the types of load acting on the structure?
2.39) Define traction force (T).
2.40) How are thermal loads input in finite element analysis?
2.41) What is an interpolation function? [AU, May / June – 2012]
2.42) Why are polynomial types of interpolation functions preferred over trigonometric
functions? [AU, Nov / Dec – 2007, April / May – 2009]
2.43) What is an equivalent nodal force? [AU, April / May – 2008]
2.44) What are called higher order elements?
[AU, April / May – 2008, Nov / Dec – 2010, 2011]
2.45) What is higher order element? [AU, Nov / Dec – 2011]
2.46) What do you mean by higher order elements? [AU, Nov / Dec – 2008]
2.47) Why higher order elements are required for FE analysis? [AU, Nov / Dec – 2012]
2.48) What are higher order elements and why are they preferred?
2.49) What are the characteristics of shape functions?
2.50) Derive the shape functions for a 1D quadratic bar element.
2.51) Give the Governing equation and the primary and secondary variables associated
with the one dimensional beam element. [AU, Nov / Dec – 2017]

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, April / May – 2017]
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, April / May – 2017]
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, April / May – 2017]
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.
2.66) Define normal modes. [AU, May / June – 2013]
2.67) What is meant by transverse vibrations?
[AU, May / June – 2014, Nov / Dec – 2015, April / May – 2017]
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) Derive the consistent mass matrix lD-2 noded bar element. [AU, Nov / Dec – 2017]
2.77) Consistent mass matrix gives accurate results than lumped mass matrix in dynamic
analysis of bar element- Justify. [AU, May / June – 2016]
2.78) Comment on the accuracy of the values of natural frequencies obtained by using
lumped mass matrices and consistent mass matrices. [AU, Nov / Dec – 2012]
2.79) 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]
2.80) Write the natural frequency of bar of length 'L', Young's modulus 'E" and cross
section 'A' fixed at one end and carrying lumped mass 'M' at the other end.
[AU, Nov / Dec – 2017]
2.81) Write down the expression of longitudinal vibration of bar element.
2.82) Explain consistent load vector. [AU, Nov / Dec – 2010]
2.83) What do you mean by Lumped mass matrix? [AU, May / June – 2011]
2.84) Write down the lumped mass matrix for the truss element.[AU, April / May – 2009]

2.85) What type of analysis preferred in FEA when the structural member subjected to
transient vibrations? [AU, May / June – 2016]
2.86) What are the types of Eigen value problems? [AU, May / June – 2012]
2.87) What is meant by mode superposition technique? [AU, May / June – 2013]
2.88) What do you know about radially symmetric problem?
2.89) Write the boundary condition for a cantilever beam subjected to point load at its
free end.
2.90) For a one dimensional fin problem, what are all the boundary conditions that can be
specified at the free end?
2.91) Write the analogies between structural, heat transfer and fluid mechanics.
[AU, Nov / Dec – 2014]
2.92) What are the boundary conditions in FEA heat transfer problem?
2.93) Write down the one dimensional heat conduction equation.
2.94) Distinguish between homogenous and non – homogenous boundary conditions.
[AU, Nov / Dec – 2013]
2.95) Mention two natural boundary conditions as applied to thermal problems.
[AU, Nov / Dec – 2015]
2.96) Write down the expression of shape function and temperature function for one
dimensional heat conduction. [AU, May / June – 2011]
2.97) Write down the governing differential equation for the steady state one dimensional
conduction heat transfer. [AU, Nov / Dec – 2010, 2012]
2.98) Derive the convection matrix for a 1D linear bar element.
2.99) Write down the stiffness matrix equation for one dimensional heat conduction
element. [AU, Nov / Dec – 2011]

2.100) Define static condensation. [AU, Nov / Dec – 2010]
2.101) Determine the load vector for the beam element shown in Figure
[AU, Nov / Dec – 2012]
2.102) What is a truss? [AU, May / June – 2014]
2.103) State the assumptions are made while finding the forces in a truss.
[AU, Nov / Dec – 2011]
2.104) What is the need for co-ordinate transformation for solving truss problem?
[AU, Nov / Dec – 2017]
2.105) Write the element stiffness matrix of a truss element. [AU, May / June – 2012]
2.106) Write down the expression of stiffness matrix for a truss element.
[AU, Nov / Dec – 2015]
2.107) Sketch a typical truss element showing local global transformation.
2.108) Differentiate global and local coordinates. [AU, May / June – 2013]
2.109) State the differences between a bar element and a truss element.
PART – B
2.110) What are the different types of elements? Explain the significance of each.
[AU, Nov / Dec – 2010]
2.111) Derive and sketch the quadratic shape function for the bar element.
[AU, May / June – 2011]
2.112) Derive the shape function of a quadratic 1 – D element. [AU, Nov / Dec – 2011]
2.113) Derive the shape functions for one dimensional linear element using direct method.
[AU, May / June – 2013]

2.114) Determine the shape function and element matrices for quadratic bar element.
[AU, May / June – 2012]
2.115) Derive the shape functions for One-Dimensional Quadratic Bar element.
[AU, May / June – 2016]
2.116) 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]
2.117) Derive the stiffness matrix and finite element equation for one dimensional bar.
[AU, Nov / Dec – 2011]
2.118) Derive the stiffness matrix and body force vector for a quadratic spar element.
[AU, Nov / Dec – 2013]
2.119) Obtain an expression for the shape function of a linear bar element.
2.120) Write the stiffness matrix for a 1D two noded linear element.
[AU, Nov / Dec – 2014]
2.121) Derive shape functions and stiffness matrix for a 2D rectangular element.
[AU, Nov / Dec – 2012]
2.122) Derive the equation of motion based on weak form for transverse vibration of a
beam. [AU, May / June – 2012, 2014]
2.123) Explain the direct integration method using central difference scheme for predicting
the transient dynamic response of a structure. [AU, Nov / Dec – 2013]
2.124) Derive the consistent mass matrix for a truss element in its local coordinate system.
[AU, Nov / Dec – 2012]
2.125) Derive the finite equations for the time – dependent stress analysis of one
dimensional bar. [AU, May / June – 2011]

2.126) 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]
2.127) Derive the shape function for a 2 noded beam element and a 3 noded bar element.
[AU, Nov / Dec – 2008]
2.128) Why is higher order elements needed? Determine the shape functions of an eight
noded rectangular element. [AU, Nov / Dec – 2007, April / May – 2009]
2.129) Derive the shape functions for a 2D beam element.
2.130) Derive the shape functions for a 2D truss element.
2.131) Derive the stiffness matrix for 2D truss element. [AU, Nov / Dec – 2015]
2.132) 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]
2.133) 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.

2.134) 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]
2.135) 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]
2.136) 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]

2.137) 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]
2.138) 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
2.139) Determine the nodal displacements, stress and strain for the bar shown Fig.
[AU, Nov / Dec – 2014]

2.140) 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]
2.141) 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]

2.142) Consider a bar as shown in Figure. An axial load of 200 kN is applied at point p.
Take A1 = 2400 mm2
, E1 = 70 * 109
N/mm2
, A2 = 600 mm2
, E2 = 200 * 109
N/mm2
.
Calculate the following, (i) The nodal displacement at point p, (ii) Stress in each
element (iii) Reaction force. [AU, April / May – 2017]
2.143) 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]
2.144) Find the nodal displacement and elemental stresses for the bar shown in Figure.

2.145) A bar of length 1000 mm is made of brass and aluminium and is subjected to loads
as shown in Figure. AC is made of brass. Its length is 500 mm with area 1000 mm2
and modulus elasticity is 105 GPa. CE is made of aluminium. Its length is 500 mm,
area 2000 mm2 and modulus of elasticity is 70 GPa. The loads are applied at the
mid points of AC and CE. Compute stress developed in two materials.
[AU, Nov / Dec – 2017]
2.146) 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.

2.147) 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]
2.148) 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]
2.149) 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

2.150) 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
).

2.151) For the bar element as shown in fig. Calculate the nodal displacements and
elemental stresses. [AU, Nov / Dec – 2016]
2.152) 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]
2.153) 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]

2.154) An axial load P = 385 KN is applied to the composite block as shown below.
Determine the stress in each material.
2.155) 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.

2.156) 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]
2.157) 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.

2.158) 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]
2.159) 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.

2.160) 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]
2.161) 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]
2.162) 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]
2.163) 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.
2.164) 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]
2.165) 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]
2.166) 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]
2.167) 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]
2.168) 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]
2.169) 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]

2.170) 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.
2.171) A metallic fin 20 mm wide and 4 mm thick is attached to a furnace whose wall
temperature is 180°C. The length of the fin is 120 mm. If the thermal conductivity
of the material of the fin is 350 W/m°C and convection coefficient is 9 W/m2
°C,
determine the temperature distribution assuming that the tip of the fin is open to the
atmosphere and that the ambient temperature is 25°C. [AU, Nov / Dec – 2017]
2.172) 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]

2.173) 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]
2.174) Evaluate the natural frequencies of axial vibration of the bar shown in Figure for the
following cases (i) Bar is considered as a single element (ii) Bar consists of two
elements. [AU, Nov / Dec – 2017]
2.175) 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]

2.176) 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]
2.177) Determine the Eigen values for the stepped bar shown in fig.
[AU, Nov / Dec – 2016]

2.178) Find the natural frequency of longitudinal vibration of the unconstrained stepped
bar as shown in figure. [AU, Nov / Dec – 2006]
2.179) 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]
2.180) 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.
[AU, April / May – 2015, 2017]

2.181) 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]
2.182) 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
]
2.183) 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
]
2.184) 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]

2.185) 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]
2.186) 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.

2.187) For the discretization of beam elements as shown below, number the nodes so as to
minimize the bandwidth of the assembled stiffness matrix (K)
2.188) 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.
2.189) For the beam problem shown below, determine the tip deflection and the slope at
the roller support.

2.190) 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]
2.191) 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]
2.192) 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]

2.193) 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, April / May – 2017]
2.194) Find the deflection and slope for the following beam section at which point load is
applied.
2.195) 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
.

2.196) Obtain the deflection at the midpoint of the beam shown below and determine the
reaction.
2.197) 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]
2.198) 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]

2.199) 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]
2.200) A cantilever beam of length 4 m is subjected to a linearly varying load of intensity
60kN/m at the left end (fixed end) and zero intensity at the free end as shown in
Figure. Calculate the displacement and rotation at the free end. The cross section of
the beam is a rectangle of width 300 mm and depth 200 mm. E = 200 GPa
[AU, Nov / Dec – 2017]

2.201) Determine the maximum deflection for the beam loaded as shown in Figure,
Young’s modulus 200 GPa and density 0. 78 x 106
kg/m3
. The beam is of “T” cross
section shown in Figure. [AU, Nov / Dec – 2017]
2.202) 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, 2017]
2.203) 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.
2.204) Find the natural frequencies of transverse vibrations of the cantilever beam shown
in Figure using one beam element. [AU, Nov / Dec – 2012]
2.205) 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]
2.206) Compute material frequencies of free transverse vibration of a stepped beam shown
in figure. [AU, May / June – 2003]
2.207) 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]
2.208) Obtain the natural frequencies and mode shapes of the transverse vibration of
cantilever beam shown in Figure E is the modulus of elasticity, ρ is the density and

A is the cross section of the beam. Neglect rotation of the beam.
[AU, Nov / Dec – 2017]
2.209) 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.
2.210) 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, April / May – 2017]
2.211) Determine the force in the members of the truss as shown in figure.

Take E = 200 GPa [AU, May / June – 2012]
2.212) 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.
2.213) Calculate nodal displacement and elemental stresses for the truss shown in Figure.
E= 70Gpa.cross-sectional area A = 2cm2
for all truss members.

2.214) Consider a three bar truss as shown in figure. Take E = 2 * 105
N/mm2
. Calculate
the nodal displacement. Take A1 = 2000 mm2
, A2 = 2500mm2
and A3 = 2500mm2
2.215) 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.

2.216) 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]
2.217) 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]

2.218) 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

2.219) Determine the force in the members of the truss shown in figure.
2.220) For the two bar truss shown in figure, determine the displacements of node 1 and
stress in element 1 – 3. [AU, May / June – 2014]
2.221) 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
2.222) Determine the natural frequencies for the truss shown in figure using finite element
method. [AU, May / June – 2007]

2.223) Calculate the stress developed in the members of the truss as shown in Figure. E =
200 GPa. Area of the member AB is 20cm2
and its length is 5m. Members BC and
AC have the same cross sectional area and is equal to 25 cm2
.
[AU, Nov / Dec – 2017]
2.224) 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

UNIT – III – TWO DIMENSIONAL SCALAR VARIABLE PROBLEMS
PART – A
3.1) Name few 2-D elements along with a neat sketch.
3.2) State the differences between 2D element and 1D element.
3.3) How do you define two dimensional elements? [AU, May / June – 2014]
3.4) Define Lagrange’s interpolation.
3.5) What is geometric Isotropy? [AU, May / June – 2013, Nov / Dec – 2016]
3.6) Write the Lagrangean shape functions for a 1D, 2 noded elements.
[AU, Nov / Dec – 2008]
3.7) Write the relation to obtain the size of the stiffness matrix for a linear quadrilateral
element having Ux and Uy as dof.
3.8) Why is the 3 noded triangular element called as a CST element?
[AU, Nov / Dec – 2010]
3.9) Write down the interpolation function of a field variable for three-node triangular
element. - [AU, April / May – 2010]
3.10) What is a CST element? [AU, April / May – 2011, Nov / Dec – 2015]
3.11) Why a CST element so called? [AU, Nov / Dec – 2014, 2017]
3.12) Write down the shape functions associated with the three noded linear triangular
element and plot the variation of the same. [AU, April / May – 2015, 2017]
3.13) Draw the shape functions of a CST element. [AU, Nov / Dec – 2010]
3.14) Explain the important properties of CST elements. [AU, Nov / Dec – 2008]
3.15) Write a note on CST element. [AU, May / June – 2011]
3.16) What is QST (Quadratic strain Triangle) element? [AU, May / June – 2014]
3.17) What is QST Element? [AU, April / May – 2017]
3.18) Write briefly about LST and QST elements.
3.19) What are CST and LST elements? [AU, Nov / Dec – 2009]

3.20) What is an LST element? [AU, Nov / Dec – 2016]
3.21) Define LST element. [AU, Nov / Dec – 2012]
3.22) Write the displacement function equation for CST element.
3.23) Write the strain – displacement matrix for CST element. [AU, Nov / Dec – 2016]
3.24) Differentiate CST and LST elements. [AU, Nov / Dec – 2007, April / May – 2009]
3.25) Give the Jacobian matrix for a CST element and state its significance.
[AU, Nov / Dec – 2013]
3.26) Evaluate the following area integrals for the three node triangular element
∫ 𝑁𝑖 𝑁𝑗
2
𝑁𝑘
3
𝑑𝐴 [AU, May / June – 2012]
3.27) A triangular element is shown in Figure and the nodal coordinates are expressed in
mm. Compute the strain displacement matrix. [AU, Nov / Dec – 2012]
3.28) Write down the shape functions for a 4 noded bi-linear rectangular element.
[AU, Nov / Dec – 2017]
3.29) What do you mean by the terms : c0
,c1
and cn
continuity?
3.30) Distinguish between C0, C1 and C2 continuity elements.
3.31) What are the different problems governed by 2D scalar field variables?

3.32) Use various number of triangular elements to mesh the given domain in the order of
increasing solution refinement.
3.33) Define Pascal triangle.
3.34) Write the significance of Pascal triangle in developing triangular elements.
3.35) Plot the variation of shape function with respect node of a 3 noded triangular
element.
3.36) Write down the nodal displacement equations for a two dimensional triangular
elasticity element. [AU, April / May – 2010]
3.37) Define a plane stress condition. [AU, Nov / Dec – 2011]
3.38) What is meant by plane stress analysis? [AU, Nov / Dec – 2016]
3.39) State the condition for plane stress problem.
3.40) Give one example each for plane stress and plane strain problems.
[AU, Nov / Dec – 2008]
3.41) Distinguish between plane stress and plane strain problems. [AU, Nov / Dec – 2009]
3.42) Distinguish plane stress and plane strain conditions. [AU, Nov / Dec – 2010]
3.43) Differentiate between plane stress and plane strain. [AU, Nov / Dec – 2017]
3.44) Define plane strain with suitable example. [AU, Nov / Dec – 2012]
3.45) Define plane strain analysis. [AU, Nov / Dec – 2011, 2015]

3.46) Define a plane stress problem with a suitable example.
3.47) Explain plane stress problem with an example. [AU, April / May – 2011]
3.48) Explain plane stress conditions with example. [AU, May / June – 2011]
3.49) Give at least one example each for plane stress and plain strain analysis.
[AU, April / May – 2015, 2017]
3.50) Give the application of plane stress and plane strain problems.
[AU, May / June – 2016]
3.51) Write down the stress-strain relationship matrix for plane strain condition.
3.52) Write down the strain displacement relation. [AU, April / May – 2011]
3.53) State whether plane stress or plane strain elements can be used to model the
following structures. Justify your answer. [AU, Nov / Dec – 2012]
(a) A wall subjected to wind load
(b) A wrench subjected to a force in the plane of the wrench.
3.54) Define path line and streamline. [AU, May / June – 2016]
3.55) Write the assumptions used to define the given problem as plane stress problem.
3.56) Write the assumptions used to define the given problem as plane strain problem.
3.57) Using general stress - strain relation, obtain plane stress equation.
3.58) Beginning with general elastic stress-strain relation, derive the plane strain
condition.
3.59) What are the differences between 2 Dimensional scalar variable and vector variable
elements? [AU, Nov / Dec – 2009]
3.60) What are the ways which a three dimensional problems can be reduced to a two
dimensional approach? [AU, April / May – 2017]
3.61) What are the ways by which a 3D problem can be reduced to a 2D problem?
[AU, Nov / Dec – 2014, 2017]

3.62) How to reduce a 3D problem into a 2D problem? [AU, Nov / Dec – 2012]
3.63) Write down the governing equation for two-dimensional steady state heat
conduction. [AU, May / June – 2014]
3.64) Write down the governing differential equation for a two dimensional steady-state
heat transfer problem. [AU, Nov / Dec – 2009]
3.65) Write down the conduction matrix for a three noded linear triangular element.
3.66) Sketch a two dimensional differential control element for heat transfer and obtain
the heat diffusion equation. [AU, Nov / Dec – 2012]
3.67) Define element capacitance matrix for unsteady state heat transfer problems.
[AU, May / June – 2013]
3.68) Name a few boundary conditions involved in any heat transfer analysis.
3.69) Mention two natural boundary conditions as applied to thermal problems.
3.70) Consider a wall of a tank containing a hot liquid at a temperature T0 with an air
stream of temperature Tx passed on the outside, maintaining a wall temperature of
TL at the boundary. Specify the boundary conditions. [AU, April / May – 2009]
3.71) State the assumptions in the theory of pure torsion. [AU, Nov / Dec – 2016]
3.72) Give the governing equation of torsion problem. [AU, May / June – 2012]
3.73) Write the governing equation for the torsion of non-circular sections and give the
associated boundary conditions. [AU, Nov / Dec – 2017]
3.74) Write the step by step procedure of solving a torsion problem by finite element
method. [AU, April / May – 2011]
3.75) Outline the step by step procedure of handling torsion problem using the finite
element method. [AU, May / June – 2012]

PART – B
3.76) Determine the shape functions for a constant strain triangular (CST) element in
terms of natural coordinate system. [AU, Nov / Dec – 2008, May / June – 2014]
3.77) Derive the shape function for the constant strain triangular element.
[AU, May / June – 2016]
3.78) What are shape functions? Derive the shape function for the three noded triangular
elements. [AU, Nov / Dec – 2011]
3.79) Derive the expression of shape function for heat transfer in 2D element.
3.80) A two noded line element with one translational degree of freedom is subjected to a
uniformly varying load of intensity P1 at node 1 and P2 at node 2. Evaluate the
nodal load vector using numerical integration. [AU, Nov / Dec – 2012]
3.81) The temperature at the four corners of a four – noded rectangle are T1, T2 T3 and T4.
Determine the consistent load vector for a 2-D analysis, aimed to determine the
thermal stresses. [AU, Nov / Dec – 2007, April / May – 2009]
3.82) Establish the finite element equations including force matrices for the analysis of
two dimensional steady – state fluid flows through a porous medium using
triangular element. [AU, Nov / Dec – 2013]
3.83) Explain the potential function formulation of finite element equations for ideal flow
problems. [AU, May / June – 2013]
3.84) Develop stiffness coefficients due to torsion for a three dimensional beam element.
3.85) Determine the shape functions N1, N2 and N3 at the interior point p for the
triangular element shown in figure [AU, May / June – 2014]

3.86) The nodal co-ordinates of the triangular element is as shown below. At the interior
point P, the x- co-ordinate is 3.3 and N1 = 0.3. Determine N2, N3 and the y – co-
ordinate at point P.
3.87) The (x, y) co-ordinates of nodes i, j and k of a triangular element are given by (0, 0),
(3, 0) and (1.5, 4) mm respectively. Evaluate the shape functions N1, N2 and N3 at an
interior point P (2, 2.5) mm for the element. For the same triangular element, obtain
the strain-displacement relation matrix B. [AU, Nov / Dec – 2009]
3.88) The (x, y) co-ordinates of nodes i, j and k of a triangular element are given by (0, 0),
(3, 0) and (1.5, 4) mm respectively. Evaluate the shape functions N1, N2 and N3 at
an interior point P (2, 2.5) mm for the element. For the same triangular element,
obtain the strain-displacement relation matrix B for the above same triangular and
explain how stiffness matrix is obtained assuming scalar variable problem
[AU, Nov / Dec – 2016]
3.89) Derive the interpolation function 14 for the quadratic triangular element as shown
below.

3.90) Derive the interpolation function of a corner node in a cubic serendipity element.
3.91) Find the temperature at a point P(1,1.5) inside the triangular element shown with
the nodal temperatures given as T1 = 400
C, TJ = 340
C, and TK = 460
C. Also
determine the location of the 420
C contour line for the triangular element shown in
figure below. [AU, April / May - 2008]
3.92) Calculate the temperature at the point for a three noded triangular element as shown
in figure. The nodal values are T1 = 40˚C, T2 = 34˚C and T3 = 46˚C. Point A is
located at (2, 1.5). Assume the temperature is linearly varying in the element. Also
determine the location of 42˚C contour line. [AU, May / June – 2011]
3.93) Nodal values of the triangular element is shown in Figure. Evaluate element shape
functions and calculate the value of temperature at a point whose coordinates are
given (5, 7). [AU, Nov / Dec – 2017]

3.94) Calculate the value of pressure at the point A which is inside the 3 noded triangular
elements as shown in figure. The nodal values are φ1 = 40 MPa, φ2 = 34 MPa and φ3
= 46 MPa, Point A is located at (2, 1.5) Assume pressure is linearly varying in the
element. Also determine the location of 42 MPa contour line.
[AU, May / June – 2013]
3.95) A bilinear rectangular element has coordinates as shown in figure and the nodal
temperatures are T1 = 100ºC, T2= 60ºC, T3 = 50ºC and T4 = 90ºC. Compute the
temperature at the point whose coordinates are (2.5, 2.5). Also determine the 80ºC
isotherm. [AU, April / May – 2015]

3.96) Determine the pressure at the location (7, 4) in a rectangular plate with the data
shown in Figure and also draw 50 MPa contour line. [AU, Nov / Dec – 2017]
3.97) A CST element has nodal coordinates 1 (0,0), 2 (5,0), 3 (0,4). The element is
subjected to a body force f = x3 N/m3. Determine the nodal force vector. Take the
element thickness as 0.3m. [AU, Nov / Dec – 2013]
3.98) Compute the elemental stress vectors for the following element, assuming plane
stress conditions. The nodal displacements in ‘mm’ [q] = [0 1 1 0 1
1]T
. The temperature increase in the element is 5˚C. Take E = 200 GPa and µ = 0.3.
The thermal coefficient of expansion is 11 * 10-6
/˚C. The thickness of the material
is 1 mm. [AU, April / May - 2011]

3.99) Calculate the element stiffness matrix and thermal force vector for the plane stress
element shown in figure below. The element experiences a rise of 100
C.
3.100) Calculate the element stiffness matrix and the temperature force vector for the plane
stress element shown in fig. The element experiences a 20°C increase in
temperature. Assume coefficient of thermal expansion is 6*10-6
/ °C. Take E = 2 x
105
N/mm2
, v = 0.25, t = 5 mm. [AU, May / June – 2016]

3.101) Determine the element stiffness matrix and the thermal load vector for the plane
stress element shown in figure. The element experiences 20o
C increase in
temperature [AU, April / May - 2010]
3.102) Obtain the finite element equations for the following element. The thermal
conductivity (k) of the material of the element is 2 W/ mK. The convective heat
transfer coefficient (h) is 3 W/m2
K. The ambient temperature (Tf) is 25˚ C. The
thickness (t) of the material is 1mm. Assume convection along the edge ‘jk’ alone.
3.103) Compute the elemental stress vectors for the following element, assuming plane
stress conditions. The nodal displacements in ‘mm’ [q] = [0 1 1 0 1
1]T
. The temperature increase in the element is 5˚C. Take E = 200 GPa and µ =

0.3. The thermal coefficient of expansion is 11 * 10-6
/˚C. The thickness of the
material is 1 mm. [AU, April / May - 2011]
3.104) Determine the temperature and heat fluxes at a location (2, 1) in a square plate as
shown in figure. Draw the isothermal for 125°C. T1 = 100°C, T2 = 150°C, T3 =
200°C, T4 = 50°C [AU, Nov / Dec – 2010]
3.105) 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.

3.106) 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]
3.107) Determine the temperature distribution in the rectangular fin shown in Figure. The upper half
can be meshed taking into account symmetry using triangular elements.
[AU, Nov / Dec – 2017]

3.108) The plane wall shown below is 0.5 m thick. The left surface of the wall is
maintained at a constant temperature of 2000
C and the right surface is insulated.
The thermal conductivity K = 25 W/Mo
C and there is a uniform heat generation
inside the wall of Q = 400 W/m3
. Determine the temperature distribution through
the wall thickness using linear elements.
3.109) Compute the steady state temperature distribution for the plate shown in the figure
below. A constant temperature of T0 = 1500
C is maintained along the edge y = w
and all other edges have zero temperature. The thermal conductivities are Kx = Ky
= 1. Assume w = L = 1 and thickness t = 1

3.110) A two dimensional fin is subjected to a heat transfer by conduction and convention.
It is discretized as shown in figure into two elements using linear triangular
elements. Derive the conduction, and thermal load vector. How is convection
accounted for solving the problem using finite element method?
3.111) The heat generated by a slab of thickness 150 mm is 1.5 MW/m3
and its thermal
conductivity coefficient is 150 W/m°C. The temperature on its surfaces (left and
right) are 150°C. Calculate the temperature distribution across the thickness of the
slab by diving into four elements. Assume that the area for conduction is 1 m2
[AU, Nov / Dec – 2017]
3.112) Compute the element stiffness matrix and vectors for the element shown in figure
when the edge 2 – 3 and 3 – 1 experience heat loss. [AU, May / June – 2012]

e
3.113) Compute the element matrices and vectors for the element shown below, when the
edges jk and ik experience convection heat loss.
3.114) Compute element matrices and vectors for the elements shown in figure when the
edge kj experiences convection heat loss. [AU, Nov / Dec – 2009]

3.115) The triangular element shown in figure is subjected to a constant pressure 10
N/mm2
along the edge ij. Assume E = 200 Gpa, Poisson’s ratio  = 0.3 and
thickness of the element = 2 mm. The coefficient of thermal expansion of the
material  = 2 x10-6
/ o
C and T = 50o
C. Determine the constitutive matrix (stress-
strain relationship matrix D) and the nodal force vector for the element.
[AU, Nov / Dec - 2009]
3.116) Derive element force vector when linearly varying pressure acts on the side joining
nodes jk of a triangular element shown in Figure and body force of 25N/mm2
acts
downwards. Thickness = 5mm. [AU, April / May – 2011]
3.117) Find the expression for nodal vector in a CST element shown in figure subjected to
pressures Px1 on side 1. [AU, Nov / Dec – 2008]

3.118) Derive the expression for nodal vector in a CST element subjected to pressures Px1,
Py1 on side 1, Px2, Py2, on side 2 and Px3, Py3 on side 3 as shown in figure.
[AU, Nov / Dec – 2013, 2016]
3.119) For the square shaft of cross section 1 cm x 1 cm as shown in Figure. It was decided
to determine the stress distribution using FEM by solving for the stress function
values. Considering geometric and boundary condition symmetry 1/8th
of the cross
section was modeled using two triangular elements and one bilinear rectangular
element as shown. The element matrices are given below. Carry out the assembly
and solve for the unknown stress function values. [AU, Nov / Dec – 2017]
𝐹𝑜𝑟 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝐾 =
1
2
[
1 −1 0
−1 2 −1
0 −1 1
] 𝑟 = [
29.1
29.1
29.1
]

𝐹𝑜𝑟 𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 𝐾 =
1
6
[
4 −1 −2 −1
−1 4 −1 −2
−2 −1 4 −1
−1 −2 −1 4
] 𝑟 = [
43.6
43.6
43.6
43.6
]

UNIT – IV – TWO DIMENSIONAL VECTOR VARIABLE PROBLEMS
PART – A
4.1) Write a displacement function equation for CST element. [AU, May / June – 2016]
4.2) Give the stiffness matrix equation for an axisymmetric triangular element.
4.3) What is axisymmetric element?
4.4) Give examples of axisymmetric problems. [AU, May / June – 2012]
4.5) What is an axisymmetric problem? [AU, April / May – 2011]
4.6) Write short notes on axisymmetric problems.
4.7) What is meant by axi-symetric field problem? Given an example.
[AU, Nov / Dec – 2009]
4.8) When are axisymmetric elements preferred? [AU, Nov / Dec – 2013]
4.9) List the conditions for the problem to be assumed axisymmetric.
[AU, Nov / Dec – 2017]
4.10) State the situations where the axisymmetric formulation can be applied.
4.11) Give four applications where axisymmetric elements can be used.
4.12) List the applications of axisymmetric elements. [AU, May / June – 2016]
4.13) State the applications of axisymmetric elements. [AU, Nov / Dec – 2010]
4.14) Write down the stress-strain relationship matrix for an axisymmetric triangular
element. [AU, May / June – 2016]
4.15) Write down the constitutive relationship for axisymmetric problem.
4.16) Write down the constitutive relationship for the plane stress problem.
[AU, Nov / Dec – 2010]

ME6603 - FINITE ELEMENT ANALYSIS

ME6603 - FINITE ELEMENT ANALYSIS

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