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PROGRESS IN AEROSPACE SCIENCES
THE FINITE
ELEMENT METHOD
IN ENGINEERING
By
S. S. RAO
Professor of Mechanical Engineering
San Diego State University, San Diego, USA
and Indian Institute of Technology, Kanpur, India
PERGAMON PRESS
OXFORD · NEW YORK · TORONTO · SYDNEY · PARIS · FRANKFURT
U.K. Pergamon Press Ltd., Headington Hill Hall,
Oxford OX3 OBW, England
U.S.A. Pergamon Press Inc., Maxwell House, Fairview Park,
Elmsford, New York 10523, U.S.A.
CANADA Pergamon Press Canada Ltd., Suite 104,
150 Consumers Rd., Willowdale, Ontario M2J 1P9, Canada
AUSTRALIA Pergamon Press (Aust.) Pty. Ltd., P.O. Box 544,
Potts Point, N.S.W. 2011, Australia
FRANCE Pergamon Press SARL, 24 rue des Ecoles,
75240 Paris, Cedex 05, France
FEDERAL REPUBLIC Pergamon Press GmbH, 6242 Kronberg-Taunus,
OF GERMANY Hammerweg 6, Federal Republic of Germany
Copyright ©1982 S. S. Rao
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system or transmitted
in any form or by any means: electronic, electrostatic,
magnetic tape, mechanical, photocopying, recording or
otherwise, without permission in writing from the
publishers.
First edition 1982
British Library Cataloguing in Publication Data
Rao, S S
The finite element method in engineering -
(Pergamon international library)
1. Finite element method 2. Engineering
mathematics
I. Title
620\001'515353 TA347.F5
ISBN 0-08-025467-5 (Hardcover)
025466 7 (Flexicover)
Library of Congress Catalog Card no: 80-40817
In order to make this volume available as economically
and as rapidly as possible the typescript has been
reproduced in its original form. This method unfor
tunately has its typographical limitations but it is hoped
that they in no way distract the reader.
Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter
TO
LORD SRI VIGHNESWARA
PREFACE
The finite element method is a numerical method which can be used for the accurate
solution of complex engineering problems. The method was first developed in 1956
for the analysis of aircraft structural problems. Thereafter, within a decade, the
potentialities of the method for the solution of different types of applied science
and engineering problems were recognized. Over the years, the finite element tech
nique has been so well established that today it is considered to be one of the
best methods for solving a wide variety of practical problems efficiently. In fact
the method has become one of the active research areas for applied mathematicians.
One of the main reasons for the popularity of the method in different fields of
engineering is that once a general computer program is written, it can be used for
the solution of any problem simply by changing the input data.
The objective of this book is to introduce the various aspects of finite element
method as applied to engineering problems in a systematic manner. It is attempted
to give details of development of each of the techniques and ideas from basic
principles. New concepts are illustrated with simple examples wherever possible.
Several Fortran computer programs are given with example applications to serve the
following purposes:
— to enable the reader to understand the computer implementation of the theory
developed;
— to solve specific problems;
— to indicate procedure for the development of computer programs for solving any
other problem in the same area.
The book is divided into eight chapters and an appendix. Chapter 1 gives an intro
duction and overview to the finite element method. The numerical solution of var
ious types of finite element equations is considered in chapter 2. Chapter 3
presents the general procedure of finite element analysis. The development of
higher order and isoparametric elements is discussed in chapter 4. The application
of finite element method for static and dynamic solid and structural mechanics
problems like frames, plates and solid bodies is considered in chapter 5. Chapter
6 deals with the solution of one-, two- and three-dimensional steady state and
transient heat transfer problems. In chapter 7 the finite element solution of
fluid mechanics problems is considered. Additional applications and generalization
of the finite element method are presented in chapter 8. Finally, Green-Gauss
theorem, which deals with integration by parts in two and three dimensions, is
given in appendix A.
vu
vili Preface
This book is an outgrowth of the author's experience in teaching the course to
engineering students during the past eight years. A basic knowledge of matrix
theory is required in understanding the various topics presented in the book. More
than enough material is included for a first course at the senior or graduate level.
Different parts of the book can be covered depending on the background of students
and also on the emphasis to be given on specific areas like solid mechanics, heat
transfer and fluid mechanics. The student can be given a term project in which he
is required to either modify some of the established elements or develop new finite
elements, and use them for the solution of a problem of his own choice. The
material of this book is also expected to be useful for practising engineers who
would like to learn the method and/or use the computer programs given for solving
practical problems.
I would like to express my appreciation to the students who took my course on the
finite element method at I.I.T. Kanpur and helped me in improving the earlier
versions of the manuscript. I am especially grateful to my graduate students
Kultar Singh, Srinivasa Rao, Ramana and Bhoomaiah and my brother Sudarsana Rao for
their help in verifying the computer programs. I wish to acknowledge the partial
financial support given by the Ministry of Education, Government of India, under
the Quality Improvement Program. Finally I wish to thank my wife Kamala and
daughters Sridevi and Shobha for their tolerance and understanding while preparing
this manuscript.
Kanpur S. S. Rao
1 January, 1980
PPRRIINNCCIIPPAALL NNOOTTAATTIIOONN
a length of a rectangular element
a , a , a components of acceleration along x, y, z directions of a fluid
x y z
A area of cross-section of a one-dimensional element; area of a
triangular (plate) element
(e) cross-sectional area of one-dimensional element e
A
cross-sectional area of a tapered one-dimensional element at node
A *.{ A 0. )
i(j)
b width of a rectangular element
B body force vector in a fluid = {B B ,B 1T
x' y z
specific heat
o
specific heat at constant volume
Q
constants
Cj, C2y ...
compliance matrix; damping matrix
[C]
flexural rigidity of a plate
D
elasticity matrix (matrix relating stresses and strains)
[D]
Young's modulus; total number of elements
E
Young's modulus of element e
E
Young's modulus in a plane defined by axis i
E..
functions of s
f1M y fz(x)> ··
shear force in a beam
F
acceleration due to gravity
g
shear modulus
G
shear modulus in plane i j
G. .
convection heat transfer coefficient
h
HQÌ (χ) Lagrange polynomial associated with node i
jth order Hedte polynomial
HU)J
xxxxiiii Principal Notation
i (-1 ) 1/2
I functional to be extremized;
potential energy;
area moment of inertia of a beam
i, (I) unit vector parallel to x (X) axis
jvö) contribution of element e to the functional I
I area moment of inertia of a cross-section about z-axis
J polar wment of inertia of a cross-section
j (j) unit vector parallel to y (Y) axis
[J] Jacobian matrix
k thermal conductivity
fc«o K,> k„ thermal conductivities along x, 9, z axes
j, y z thermal conductivities along r, 8, z axes
kr, kg, kz unit vector parallel to z (Z) axis
fc (Ä)
stiffness matrix of element e in local coordinate system
[ k^e']
stiffness matrix of element e in global coordinate system
stiffness (characteristic) matrix of complete body after
[X] = [K^j] incorporation of boundary conditions
stiffness (characteristic) matrix of complete body before
[ K] = [KjA
incorporation of boundary conditions
~ ~
length of one-dimensional element
Z
length of the one-dimensional element e
l^e'
direction cosines of a line
I , I , i„
direction cosines of x-axis
x9 y z
Zox> mox9 nox direction cosines of a bar element with nodes i and j
Av-,·» rn·-, n-j total length of a bar or fin;
<Ί) 1,Q TsQ Lagrangian
L
natural coordinates of a line element
Lj, L2 natural coordinates of a triangular element
L\ L9 Li natural coordinates of a tetrahedron element
9 2
L\ L'i* L3, Lit distance between two nodes
9
£ mass of beam per unit length
m bending moment in a beam;
total number of degrees of freedom in a body
M
bending moments in a plate
M , M , M torque acting about z-axis on a prismatic shaft
x y xy mass matrix of element e in local coordinate system
M
mass matrix of element e in global coordinate system
[m(e']
mass matrix of complete body after incorporation of boundary
[^/(e)] conditions
[M] mass matrix of complete body before incorporation of boundary
conditions
[M]
conditions
Principal Notation xxiii
normal direction
N.^ interpolation function associated with the -£th nodal degree of
freedom
[N] matrix of shape (nodal interpolation) functions
P distributed load on a beam or plate;
fluid pressure
P perimeter of a fin
K : vector of concentrated nodal forces
v
pi ' perimeter of a tapered fin at node £(«/)
external concentrated loads parallel to x, y, z axes
load vector of element e in local coordinate system
-»■(el 1 (?<*>) load vector due to body forces of element e in local (global)
coordinate system
W1 <?<«>> load vector due to initial strains of element e in local (global)
coordinate system
^
» (ÊW)
V load vector due to surface forces of element e in local (global)
coordinate system
$W 1 - ipfh
vector of nodal forces (characteristic vector) of element e in
global coordinate system
P = <V vector of nodal forces of body after incorporation of boundary
conditions
1-
{£*> vector of nodal forces of body before incorporation of boundary
conditions
a rate of heat flow
k rate of heat generation per unit volume
rate of heat flow in jc-direction
*x
e* mass flow rate of fluid across section i
«*· % vertical shear forces in a plate
V β3 external concentrated moments parallel to x9 y, z axes
vector of nodal displacements (field variables) of element e in
local (global) coordinate system
Q vector of nodal displacements of body before incorporation of
boundary conditions
Q. mode shape corresponding to the frequency ω .
ΰ υ
r, s natural coordinates of a quadrilateral element
r, s, t natural coordinates of a hexahedron element
r, Θ, z radial, tangential and axial directions
(r., 8·, t.) values of (r, s, t) at node i
R radius of curvature of a deflected beam;
residual;
region of integration;
dissipation function
S surface of a body
xxiv Principal Notation
Si y £2 part of surface of a body
S surface of element e
^Ι » ^2 part of surface of element e
t time;
thickness of a plate element
T temperature;
temperature change;
kinetic energy of an elastic body
T. temperature at node i
TQ temperature at the root of fin
Tœ surrounding temperature
τΨ' temperature at node i of element e
Τκ ' vector of nodal temperatures of element e
T vector of nodal temperatures of the body before incorporation
of boundary conditions
u flow velocity along x-direction;
axial displacement
u, V, W components of displacement parallel to x, y, z axes;
components of velocity along x, y, z directions in a fluid
(Chapter 7)
·> T
U vector of displacements = {u, v, w}
V volume of a body
-*■ T
velocity vector = {u, v, w) (Chapter 7)
V
transverse deflection of a beam
w
amplitude of vibration of a beam
W
value of W at node i
W.
work done by external forces
Wp
W^e' vector of nodal displacements of element e
x 3c- coord inate ;
axial direction
coordinates of the centroid of a triangular element
((x^Q,i yQ)yv 2ΐ> (x, y, z) coordinates of node i
zi> global coordinates (X, Y, 2) of node i
a coefficient of thermal expansion
a ^. ith generalized coordinate
6
variation operator
ε . .
normal strain parallel to ith axis
shear strain in ij plane
e(e)
strain in element e
T
strain vector = {E xx' €yyI EZZ. €Xg* € yz, E ~ ~for) a three-
dimensional body;
T
= E ~ E~ ~,Er ~z 3 ,for an axisymmetric body
