finite element method in engineering s.s. rao

PREFACE The finite element method is a numerical method that can be used for the accurate solution of complex engineering problems.. The objective of this book is to introduce the variou

• ISBN: 0750678283

• Publisher: Elsevier Science & Technology Books

• Pub Date: December 2004

PREFACE

The finite element method is a numerical method that 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 potentiali- ties of the method for the solution of different types of applied science and engineering problems were recognized Over the years, the finite element technique 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 a t t e m p t e d 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 student 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 source codes of all the Fortran computer programs can be found at the Web site for the book, www.books.elsevier.com Note that the computer programs are intended for use by students in solving simple problems Although the programs have been tested, no warranty of any kind is implied as to their accuracy

After studying the material presented in the book, a reader will not only be able to understand the current literature of the finite element method but also be in a position to develop short computer programs for the solution of engineering problems In addition, the reader will be in a position to use the commercial software, such as ABAQUS, NASTRAN, and ANSYS, more intelligently

The book is divided into 22 chapters and an appendix Chapter 1 gives an introduction and overview of the finite element method The basic approach and the generality of the method are illustrated through simple examples Chapters 2 through 7 describe the basic finite element procedure and the solution of the resulting equations The finite element discretization and modeling, including considerations in selecting the number and types of elements, is discussed in Chapter 2 The interpolation models in terms of Cartesian and natural coordinate systems are given in Chapter 3 Chapter 4 describes the higher order and isoparametric elements The use of Lagrange and Hermite polynomials

is also discussed in this chapter The derivation of element characteristic matrices and vectors using direct, variational, and weighted residual approaches is given in Chapter 5

o o

X l l l

xiv PREFACE

The assembly of element characteristic matrices and vectors and the derivation of system equations, including the various methods of incorporating the boundary conditions, are indicated in Chapter 6 The solutions of finite element equations arising in equilibrium, eigenvalue, and propagation (transient or unsteady) problems, along with their computer implementation, are briefly outlined in Chapter 7

The application of the finite element method to solid and structural mechan- ics problems is considered in Chapters 8 through 12 The basic equations of solid mechanics namely, the internal and external equilibrium equations, stress-strain rela- tions, strain-displacement relations and compatibility c o n d i t i o n s - - a r e summarized in Chapter 8 The analysis of trusses, beams, and frames is the topic of Chapter 9 The development of inplane and bending plate elements is discussed in Chapter 10 The anal- ysis of axisymmetric and three-dimensional solid bodies is considered in Chapter 11 The dynamic analysis, including the free and forced vibration, of solid and structural mechanics problems is outlined in Chapter 12

Chapters 13 through 16 are devoted to heat transfer applications The basic equations

of conduction, convection, and radiation heat transfer are summarized and the finite element equations are formulated in Chapter 13 The solutions of one- two-, and three- dimensional heat transfer problems are discussed in Chapters 14-16 respectively Both the steady state and transient problems are considered The application of the finite element method to fluid mechanics problems is discussed in Chapters 17-19 Chapter 17 gives a brief outline of the basic equations of fluid mechanics The analysis of inviscid incompressible flows is considered in Chapter 18 The solution of incompressible viscous flows as well as non-Newtonian fluid flows is considered in Chapter 19 Chapters 20-22 present additional applications of the finite element method In particular, Chapters 20-22 discuss the solution of quasi-harmonic (Poisson), Helmholtz, and Reynolds equations, respectively Finally, Green-Gauss theorem, which deals with integration by parts in two and three dimensions, is given in Appendix A

This book is based on the author's experience in teaching the course to engineering students during the past several 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, such as solid mechanics, heat transfer, and fluid mechanics The student can be assigned a term project in which he/she 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/her choice The material of the book is also useful for self study by practicing engineers who would like to learn the method a n d / o r use the computer programs given for solving practical problems

I express my appreciation to the students who took my courses on the finite element method and helped me improve the presentation of the material Finally, I thank my wife Kamala for her tolerance and understanding while preparing the manuscript

c o m p o n e n t s of a c c e l e r a t i o n along x, y, z directions of a fluid

a r e a of cross section of a o n e - d i m e n s i o n a l element; a r e a of a t r i a n g u l a r (plate) e l e m e n t

cross-sectional a r e a of o n e - d i m e n s i o n a l e l e m e n t e cross-sectional a r e a of a t a p e r e d o n e - d i m e n s i o n a l e l e m e n t at n o d e i ( j )

w i d t h of a r e c t a n g u l a r e l e m e n t

b o d y force vector in a fluid = {Bx, By, B~ }T

specific h e a t specific h e a t at c o n s t a n t v o l u m e

c o n s t a n t s

c o m p l i a n c e m a t r i x ; d a m p i n g m a t r i x flexural rigidity of a p l a t e

elasticity m a t r i x ( m a t r i x r e l a t i n g stresses a n d strains)

xvi PRINCIPAL NOTATION

lox , mox , nox

lij , rnij , nij

t h e r m a l conductivities along x g z axes

t h e r m a l conductivities along r O, z axes unit vector parallel to z ( Z ) axis

stiffness m a t r i x of element e in local coordinate system stiffness m a t r i x of element e in global coordinate system stiffness (characteristic) m a t r i x of complete b o d y after incorporation

of b o u n d a r y conditions stiffness (characteristic) m a t r i x of complete body before incorporation of b o u n d a r y conditions

length of one-dimensional element length of the one-dimensional element e direction cosines of a line

direction cosines of x axis direction cosines of a bar element with nodes i and j total length of a bar or fin: Lagrangian

n a t u r a l coordinates of a line element

n a t u r a l coordinates of a triangular element

n a t u r a l coordinates of a t e t r a h e d r o n element distance between two nodes

mass of b e a m per unit length bending m o m e n t in a beam" total n u m b e r of degrees of freedom

in a body, bending m o m e n t s in a plate torque acting about z axis on a prismatic shaft mass m a t r i x of element e in local coordinate system mass m a t r i x of element e in global coordinate system mass m a t r i x of complete b o d y after incorporation of

b o u n d a r v conditions mass m a t r i x of complete b o d y before incorporation of

b o u n d a r y conditions normal direction interpolation function associated with the ith nodal degree of freedom

m a t r i x of shape (nodal interpolation) functions

d i s t r i b u t e d load on a b e a m or plate; fluid pressure

p e r i m e t e r of a fin vector of c o n c e n t r a t e d nodal forces

p e r i m e t e r of a t a p e r e d fin at node i ( j )

external c o n c e n t r a t e d loads parallel to x y, z axes load vector of element e in local coordinate svstem load vector due to b o d y forces of element e in local (global) coordinate system

b o u n d a r y conditions rate of heat flow

r a t e of heat generation per unit volume

r a t e of heat flow in x direction mass flow rate of fluid across section i vertical shear forces in a plate external c o n c e n t r a t e d m o m e n t s parallel to x, y, z axes vector of nodal displacements (field variables) of element e in local (global) c o o r d i n a t e system

vector of nodal displacements of b o d y before incorporation of

b o u n d a r y conditions

m o d e shape corresponding to the frequency czj

n a t u r a l coordinates of a q u a d r i l a t e r a l element

n a t u r a l coordinates of a h e x a h e d r o n element radial, tangential, and axial directions values of (r, s, t) at node i

radius of c u r v a t u r e of a deflected beam;

residual;

region of integration;

dissipation function surface of a b o d y

p a r t of surface of a b o d y surface of element e

p a r t of surface of element e time; thickness of a plate element

c o m p o n e n t s of displacement parallel to x, y, z axes: c o m p o n e n t s of velocity along x, g, z directions in a fluid ( C h a p t e r 17)

vector of displacements = {~ v, w } r volume of a b o d y

velocity vector = {u, t,, {L'} T ( C h a p t e r 17) transverse deflection of a b e a m

a m p l i t u d e of vibration of a b e a m value of W at node i

work done by external forces vector of nodal displacements of e l e m e n t e

x coordinate:

axial direction coordinates of the centroid of a t r i a n g u l a r element (x, y z) coordinates of node i

global coordinates (X Y Z) of node i coefficient of t h e r m a l expansion ith generalized c o o r d i n a t e variation o p e r a t o r

n o r m a l strain parallel to i t h axis shear strain in ij plane

strain in element e strain v e c t o r - {Exx Cyy.ezz,~xy,eyz,ezx} r for a

t h r e e - d i m e n s i o n a l body"

= {e,.~.eoo.ezz,s,.z} T for an a x i s y m m e t r i c b o d y initial strain vector

torsional displacement or twist

c o o r d i n a t e t r a n s f o r m a t i o n m a t r i x of element e

j t h generalized coordinate

d y n a m i c viscosity Poisson's ratio Poisson's ratio in plane ij

p o t e n t i a l energy of a beam:

strain energy of a solid b o d y

c o m p l e m e n t a r y energy of an elastic b o d y

p o t e n t i a l energy of an elastic b o d y Reissner energy of an elastic b o d y strain energy of element e density of a solid or fluid

n o r m a l stress parallel to i t h axis

PRINCIPAL NOTATION xix

p o t e n t i a l function in fluid flow

b o d y force per unit volume parallel to x, g, z axes vector valued field variable with c o m p o n e n t s u, v, and w vector of prescribed b o d y forces

dissipation function for a fluid surface (distributed) forces parallel to x, y, z axes ith field variable

prescribed value of r value of the field variable 0 at node i of element e vector of nodal values of the field variable of element e vector of nodal values of the field variables of complete b o d y after incorporation of b o u n d a r y conditions

vector of nodal values of the field variables of complete b o d y before

i n c o r p o r a t i o n of b o u n d a r y conditions

s t r e a m function in fluid flow frequency of vibration

j t h n a t u r a l frequency of a b o d y rate of r o t a t i o n of fluid a b o u t x axis

a p p r o x i m a t e value of ith n a t u r a l frequency

b o d y force p o t e n t i a l in fluid flow element e

column vector 3~ = X2

t r a n s p o s e of X([ ]) derivative with respect to time x =

Table of Contents

6

Assembly of element matrices and vectors and derivation

of system equations

209

18 Inviscid and incompressible flows 575

T h e basic idea in the finite element m e t h o d is to find the solution of a complicated problem

by replacing it by a simpler one Since the actual problem is replaced by a simpler one

in finding the solution, we will be able to find only an a p p r o x i m a t e solution r a t h e r t h a n the exact solution T h e existing m a t h e m a t i c a l tools will not be sufficient to find the exact solution (and sometimes, even an a p p r o x i m a t e solution) of most of the practical problems Thus, in the absence of any other convenient m e t h o d to find even the a p p r o x i m a t e solution

of a given problem, we have to prefer the finite element method Moreover, in the finite element m e t h o d , it will often be possible to improve or refine the a p p r o x i m a t e solution by spending more c o m p u t a t i o n a l effort

In the finite element m e t h o d , the solution region is considered as built up of m a n y small, interconnected subregions called finite elements As an example of how a finite element model might be used to represent a complex geometrical shape, consider the milling machine s t r u c t u r e shown in Figure 1.1(a) Since it is very difficult to find the exact response (like stresses and displacements) of the machine under any specified c u t t i n g (loading) condition, this s t r u c t u r e is a p p r o x i m a t e d as composed of several pieces as shown

in Figure 1.1(b) in the finite element method In each piece or element, a convenient

a p p r o x i m a t e solution is assumed and the conditions of overall equilibrium of the s t r u c t u r e are derived T h e satisfaction of these conditions will yield an a p p r o x i m a t e solution for the displacements and stresses Figure 1.2 shows the finite element idealization of a fighter aircraft

1.2 HISTORICAL BACKGROUND

A l t h o u g h the n a m e of the finite element m e t h o d was given recently, the concept dates back for several centuries For example, ancient m a t h e m a t i c i a n s found the circumference

of a circle by a p p r o x i m a t i n g it by the p e r i m e t e r of a polygon as shown in Figure 1.3

In t e r m s of the present-day notation, each side of the polygon can be called a

"finite element." By considering the a p p r o x i m a t i n g polygon inscribed or circumscribed, one can obtain a lower b o u n d S (z) or an u p p e r b o u n d S (~) for the true circumference S

F u r t h e r m o r e , as the n u m b e r of sides of the polygon is increased, the a p p r o x i m a t e values

OVERVIEW OF FINITE ELEMENT METHOD

(b) Finite element idealization

Figure 1.1 Representation of a Milling Machine Structure by Finite Elements

Figure 1.2 Finite Element Mesh of a Fighter Aircraft (Reprinted with Permission from Anamet Laboratories, Inc.)

HISTORICAL BACKGROUND

Figure 1.3 Lower and Upper Bounds to the Circumference of a Circle

converge to the true value These characteristics, as will be seen later, will hold true in any general finite element application In recent times, an approach similar to the finite element method, involving the use of piecewise continuous functions defined over trian- gular regions, was first suggested by Courant [1.1] in 1943 in the literature of applied mathematics

The basic ideas of the finite element method as known today were presented in the papers of Turner, Clough, Martin, and Topp [1.2] and Argyris and Kelsey [1.3] The name

finite element was coined by Clough [1.4] Reference [1.2] presents the application of simple finite elements (pin-jointed bar and triangular plate with inplane loads) for the analysis of aircraft structure and is considered as one of the key contributions in the development of the finite element method The digital computer provided a rapid means of performing the many calculations involved in the finite element analysis and made the method practically viable Along with the development of high-speed digital computers, the application of the finite element method also progressed at a very impressive rate The book by Przemieniecki [1.33] presents the finite element method as applied to the solution of stress analysis problems Zienkiewicz and Cheung [1.5] presented the broad interpretation of the method and its applicability to any general field problem With this broad interpretation of the finite element method, it has been found that the finite element equations can also be derived by using a weighted residual method such as Galerkin method or the least squares approach This led to widespread interest among applied mathematicians in applying the finite element method for the solution of linear and nonlinear differential equations Over the years, several papers, conference proceedings, and books have been published on this method

A brief history of the beginning of the finite element method was presented by

G u p t a and Meek [1.6] Books that deal with the basic theory, mathematical foundations, mechanical design, structural, fluid flow, heat transfer, electromagnetics and manufac- turing applications, and computer programming aspects are given at the end of the chapter [1.10-1.32] With all the progress, today the finite element method is consid- ered one of the well-established and convenient analysis tools by engineers and applied scientists

OVERVIEW OF FINITE ELEMENT METHOD

Figure 1.4

S

E x a m p l e 1.1 T h e circumference of a circle (S) is a p p r o x i m a t e d by t h e p e r i m e t e r s of inscribed a n d c i r c u m s c r i b e d n-sided p o l y g o n s as shown in F i g u r e 1.3 Prove t h e following:

lim S ill - - S and lim S ( ~ ) - S r~ - - - , 3 c r l - - - ~

where S (Z) and S (~) d e n o t e t h e p e r i m e t e r s of t h e inscribed and c i r c u m s c r i b e d polygons, respectively

S o l u t i o n If t h e radius of t h e circle is R, each side of t h e inscribed and the c i r c u m s c r i b e d

p o l y g o n can be expressed as (Figure 1.4)

r = 2R sin 7r - , s _ 2 R t a n 7r - (p, , - 1 ,

Thus, t h e p e r i m e t e r s of t h e inscribed and c i r c u m s c r i b e d polygons are given by

S (z) - n r = 2 n R s i n ~ S (~) - n s - 2 n R t a n 7r (E2)

GENERAL APPLICABILITY OF THE METHOD

1.3 GENERAL APPLICABILITY OF THE METHOD

A l t h o u g h the m e t h o d has been extensively used in the field of s t r u c t u r a l mechanics, it has been successfully applied to solve several other types of engineering problems, such

as heat conduction, fluid dynamics, seepage flow, and electric and magnetic fields These applications p r o m p t e d m a t h e m a t i c i a n s to use this technique for the solution of compli- cated b o u n d a r y value and other problems In fact, it has been established t h a t the m e t h o d can be used for the numerical solution of o r d i n a r y and partial differential equations The general applicability of the finite element m e t h o d can be seen by observing the strong similarities t h a t exist between various types of engineering problems For illustration, let

us consider the following phenomena

1.3.1 One-Dimensional Heat Transfer

Consider the t h e r m a l equilibrium of an element of a heated one-dimensional b o d y as shown

in Figure 1.5(a) T h e r a t e at which heat enters the left face can be w r i t t e n as [1.7]

Ox

where k is the t h e r m a l conductivity of the material, A is the area of cross section t h r o u g h

which heat flows (measured p e r p e n d i c u l a r to the direction of heat flow), and O T / O x is the

rate of change of t e m p e r a t u r e T with respect to the axial direction

T h e rate at which heat leaves the right face can be expressed as (by retaining only two t e r m s in the Taylor's series expansion)

Oqx OT 0 ( _ k A O T )

qz + dz qz + ~ z d x - - k A -~z + -~x -~z d x (1.2)

T h e energy balance for the element for a small time dt is given by

Heat inflow + Heat g e n e r a t e d by = Heat outflow + Change in internal

in time dt internal sources in time dt energy during

T h a t is,

OT

qx dt + OA d x dt = qx+dz d t + c p - ~ d x dt (~.3)

GENERAL APPLICABILITY OF THE METHOD

If t h e h e a t source is zero and t h e s y s t e m is in s t e a d y state, we get t h e L a p l a c e e q u a t i o n

1.3.2 One-Dimensional Fluid Flow

In t h e case of o n e - d i m e n s i o n a l fluid flow ( F i g u r e 1.5(b)), we have t h e net mass flow t h e

s a m e at every cross section; t h a t is,

p A u - c o n s t a n t (1.9) where p is t h e density, A is t h e cross-sectional area, a n d u is t h e flow velocity

1.3.3 Solid Bar under Axial Load

For t h e solid rod shown in F i g u r e 1.5(c), we have at any section z,

R e a c t i o n force - (area) (stress) - ( a r e a ) ( E ) ( s t r a i n )

Ou

= A E - - z - = applied force

u x

(1.13) where E is t h e Y o u n g ' s m o d u l u s , u is t h e axial d i s p l a c e m e n t , and A is t h e cross-sectional area If t h e applied load is c o n s t a n t , we can write Eq (1.13) as

0(

A c o m p a r i s o n of Eqs (1.7), (1.12), a n d (1.14) indicates t h a t a solution p r o c e d u r e applica- ble to a n y one of t h e p r o b l e m s can be used to solve the o t h e r s also We shall see how t h e finite e l e m e n t m e t h o d can be used to solve Eqs (1.7), (1.12), a n d (1.14) w i t h a p p r o p r i a t e

b o u n d a r y c o n d i t i o n s in Section 1.5 and also in s u b s e q u e n t chapters

10 OVERVIEW OF FINITE ELEMENT METHOD

1.4 ENGINEERING A P P L I C A T I O N S OF T H E FINITE E L E M E N T M E T H O D

As s t a t e d earlier, the finite element m e t h o d was developed originally for the analysis of aircraft structures However, the general n a t u r e of its t h e o r y makes it applicable to a wide variety of b o u n d a r y value problems in engineering A b o u n d a r y value p r o b l e m is one in which a solution is sought in the d o m a i n (or region) of a b o d y subject to the satisfaction of prescribed b o u n d a r y (edge) conditions on the d e p e n d e n t variables or their derivatives Table 1.1 gives specific applications of the finite element m e t h o d in the three

m a j o r categories of b o u n d a r y value problems, namely, (i) equilibrium or s t e a d y - s t a t e or

t i m e - i n d e p e n d e n t problems, (ii) eigenvalue problems, and (iii) p r o p a g a t i o n or transient problems

In an equilibrium problem, we need to find the s t e a d y - s t a t e displacement or stress distribution if it is a solid mechanics problem, t e m p e r a t u r e or heat flux distribution if it

is a heat transfer problem, and pressure or velocity distribution if it is a fluid mechanics problem

In eigenvalue problems also time will not a p p e a r explicitly T h e y may be considered

as extensions of equilibrium problems in which critical values of certain p a r a m e t e r s are

to be d e t e r m i n e d in addition to the corresponding s t e a d y - s t a t e configurations In these problems, we need to find the n a t u r a l frequencies or buckling loads and mode shapes if it is

a solid mechanics or s t r u c t u r e s problem, stability of laminar flows if it is a fluid mechanics problem, and resonance characteristics if it is an electrical circuit problem

T h e p r o p a g a t i o n or transient problems are t i m e - d e p e n d e n t problems This t y p e of problem arises, for example, whenever we are interested in finding the response of a b o d y under time-varying force in the area of solid mechanics and under sudden heating or cooling in the field of heat transfer

1.5 GENERAL DESCRIPTION OF T H E FINITE E L E M E N T M E T H O D

In the finite element m e t h o d , the actual c o n t i n u u m or b o d y of m a t t e r , such as a solid, liquid, or gas, is represented as an assemblage of subdivisions called finite elements These elements are considered to be interconnected at specified joints called nodes or nodal points T h e nodes usually lie on the element boundaries where adjacent elements are con- sidered to be connected Since the actual variation of the field variable (e.g., displacement, stress, t e m p e r a t u r e , pressure, or velocity) inside the c o n t i n u u m is not known, we assume

t h a t the variation of the field variable inside a finite element can be a p p r o x i m a t e d by

a simple function These a p p r o x i m a t i n g functions (also called i n t e r p o l a t i o n models) are defined in t e r m s of the values of the field variables at the nodes W h e n field equations (like equilibrium equations) for the whole c o n t i n u u m are written, the new unknowns will

be the nodal values of the field variable By solving the field equations, which are gener- ally in the form of m a t r i x equations, the nodal values of the field variable will be known Once these are known, the a p p r o x i m a t i n g functions define the field variable t h r o u g h o u t the assemblage of elements

T h e solution of a general c o n t i n u u m problem by the finite element m e t h o d always follows an orderly step-by-step process W i t h reference to static s t r u c t u r a l problems, the step-by-step procedure can be s t a t e d as follows:

Step (i): Discretization of the s t r u c t u r e

T h e first step in the finite element m e t h o d is to divide the s t r u c t u r e or solution region into subdivisions or elements Hence, the s t r u c t u r e is to be modeled with suitable finite elements T h e number, type, size, and a r r a n g e m e n t of the elements are to be decided

GENERAL DESCRIPTION OF THE FINITE ELEMENT METHOD 13

Step (ii): Selection of a p r o p e r i n t e r p o l a t i o n or displacement model

Since the displacement solution of a complex s t r u c t u r e under any specified load condi- tions cannot be predicted exactly, we assume some suitable solution within an element to

a p p r o x i m a t e t h e unknown solution T h e assumed solution must be simple from a com-

p u t a t i o n a l s t a n d p o i n t , but it should satisfy certain convergence requirements In general, the solution or the i n t e r p o l a t i o n model is taken in the form of a polynomial

Step (iii): Derivation of element stiffness matrices and load vectors

From the assumed displacement model, the stiffness m a t r i x [K (~)] and the load vector /3(r of element e are to be derived by using either equilibrium conditions or a suitable variational principle

Step ( i v ) : Assemblage of element equations to obtain the overall equilibrium equations Since the s t r u c t u r e is composed of several finite elements, the individual element stiff- ness matrices and load vectors are to be assembled in a suitable m a n n e r and the overall equilibrium equations have to be formulated as

where [K] is the assembled stiffness matrix, ~ is the vector of nodal displacements, and

P is the vector of nodal forces for the complete structure

Step ( v ) : Solution for the unknown nodal displacements

T h e overall equilibrium equations have to be modified to account for the b o u n d a r y condi- tions of the problem After the incorporation of the b o u n d a r y conditions, the equilibrium equations can be expressed as

For linear problems, the vector 0 can be solved very easily However, for nonlinear prob- lems, the solution has to be o b t a i n e d in a sequence of steps, with each step involving the modification of the stiffness m a t r i x [K] a n d / o r the load vector P

Step ( v i ) : C o m p u t a t i o n of element strains and stresses

From the known nodal displacements (I), if required, the element strains and stresses can be c o m p u t e d by using the necessary equations of solid or s t r u c t u r a l mechanics

T h e terminology used in the previous six steps has to be modified if we want to e x t e n d the concept to other fields For example, we have to use the t e r m c o n t i n u u m or d o m a i n

in place of structure, field variable in place of displacement, characteristic m a t r i x in place

of stiffness matrix, and element r e s u l t a n t s in place of element strains T h e application

of the six steps of the finite element analysis is illustrated with the help of the following examples

E x a m p l e 1.2 (Stress analysis of a stepped bar) Find the stresses induced in the axially loaded s t e p p e d bar shown in Figure 1.6(a) T h e bar has cross-sectional areas of A (1) and

A (2) over the lengths/(1) and l(2), respectively Assume the following data: A (1) = 2 cm 2

A (2) = 1 cm2;/(1) = 1(2) = 10 cm; E (1) - E (2) - E - 2 • 10 7 N / c m 2" P3 - 1 N

14 OVERVIEW OF FINITE ELEMENT METHOD

S o l u t i o n

(i) Idealization

L e t t h e b a r be c o n s i d e r e d as an a s s e m b l a g e of two e l e m e n t s as s h o w n in F i g u r e 1.6(b) By

a s s u m i n g t h e b a r to be a o n e - d i m e n s i o n a l s t r u c t u r e , we have onlv axial d i s p l a c e m e n t at

a n y p o i n t in t h e e l e m e n t As t h e r e are t h r e e nodes, t h e axial d i s p l a c e m e n t s of t h e nodes,

n a m e l y , (I)1, (I)2, a n d (I)3, will be t a k e n as u n k n o w n s

(ii) Displacement model

In each of t h e e l e m e n t s , we a s s u m e a linear v a r i a t i o n of axial d i s p l a c e m e n t O so t h a t ( F i g u r e 1.6(c))

w h e r e a a n d b are c o n s t a n t s If we c o n s i d e r t h e e n d d i s p l a c e m e n t s q)~:)(6 at x - 0) a n d (I)~ r (~b at x - 1 (r as u n k n o w n s , we o b t a i n

a - (I)(1 r a n d b - ((I)~ ~) - (I)(1~))//(e'

w h e r e t h e s u p e r s c r i p t e d e n o t e s t h e e l e m e n t n u m b e r T h u s

(iii) Element stiffness matrix

T h e e l e m e n t stiffness m a t r i c e s can be d e r i v e d f r o m t h e principle of m i n i m u m p o t e n t i a l energy For this, we w r i t e t h e p o t e n t i a l e n e r g y of t h e b a r ( I ) u n d e r axial d e f o r m a t i o n as

G E N E R A L D E S C R I P T I O N OF T H E FINITE E L E M E N T M E T H O D 15

A(1), E (1) A(2), E (2) , / , /

16 OVERVIEW OF FINITE ELEMENT METHOD

T h i s expression for rr (~) can be w r i t t e n in m a t r i x form as

where q)~) } is the vector of nodal d i s p l a c e m e n t s of e l e m e n t e

- = { q ) l } f ~ a n d ( {q)2} f ~ I~3 ) 2

[K(e)] A(e)E(e) [ I -ii] 1(e) 1 is called the stiffness m a t r i x of e l e m e n t e

Since t h e r e are only c o n c e n t r a t e d loads a c t i n g at t h e nodes of t h e bar (and no d i s t r i b u t e d load acts on t h e bar), t h e work done by e x t e r n a l forces can be expressed as

where Pi d e n o t e s t h e force applied in t h e direction of t h e d i s p l a c e m e n t ~i (i - 1, 2, 3) In this e x a m p l e , P1 - r e a c t i o n at fixed node P2 - 0 and P3 - 1.0

If e x t e r n a l d i s t r i b u t e d loads act on t h e elements, t h e c o r r e s p o n d i n g element load vectors, /3(a~) , will be g e n e r a t e d for each e l e m e n t and t h e individual load vectors will be assembled to g e n e r a t e t h e global load vector of the s v s t e m due to d i s t r i b u t e d load,/3d This load vector is to be a d d e d to t h e global load vector due to c o n c e n t r a t e d loads./5c, to gener- ate the t o t a l global n o d a l load vector of t h e system ~ - Pd + Pc In t h e present e x a m p l e ,

t h e r e are no d i s t r i b u t e d loads on the element: e x t e r n a l load acts only at one node and hence t h e global load vector t5 is t a k e n to be s a m e as t h e vector of c o n c e n t r a t e d loads acting at t h e nodes of t h e system

If t h e bar as a whole is in e q u i l i b r i u m u n d e r t h e loads i 6 = P2 9

where t h e s u m m a t i o n sign indicates tile a s s e m b l y of vectors (not tile a d d i t i o n of vectors)

in which only t h e e l e m e n t s c o r r e s p o n d i n g to a p a r t i c u l a r degree of freedom in different vectors are added

GENERAL DESCRIPTION OF THE FINITE ELEMENT METHOD 17

(iv) Assembly of element stiffness matrices and element load vectors

This step includes the assembly of element stiffness matrices [K (c)] and element load vectors/~(~) to obtain the overall or global equilibrium equations Equation (Ell) can be rewritten as

{Ol}

where [K]~ is the assembled or global stiffness matrix _ }-~2e=l [K (c)], and ~ - 02 is the

03 vector of global displacements For the data given, the element matrices would be

The overall stiffness matrix of the bar can be obtained by assembling the two element stiffness matrices Since there are three nodal displacement unknowns (~1, 02, and 03), the global stiffness matrix, [K], will be of order three To obtain [h'], the elements of [K (1)] and [K (2)] corresponding to the unknowns 01, 02, and 03 are added as shown below:

Note t h a t a s y s t e m a t i c s t e p - b y - s t e p finite element p r o c e d u r e has been used to derive

Eq (E16) If a s t e p - b y - s t e p p r o c e d u r e is not followed, Eq (E16) can be derived in a much simpler way, in this example, as follows:

T h e p o t e n t i a l energy of the s t e p p e d bar Eq (E3) can be expressed using Eqs (E6) and (E8) as

For the given data, Eqs (E18)-(E20) can be seen to reduce to Eq (E~6)

(v) Solution for displacements

If we t r y to solve Eq (El6) for the u n k n o w n s (I)1 (I)2, and (I)3, we will not be able to do

w r i t t e n as

[ K ] ~ - P

o r

GENERAL DESCRIPTION OF THE FINITE E L E M E N T M E T H O D 19

20 OVERVIEW OF FINITE ELEMENT METHOD

where h is t h e convection heat transfer coefficient, p is the p e r i m e t e r , k is t h e t h e r m a l conductivity, A is t h e cross-sectional area, T ~ is t h e s u r r o u n d i n g t e m p e r a t u r e , and To is

t h e t e m p e r a t u r e at the root of t h e fin T h e derivation of Eq ( E l ) is similar to t h a t of

Eq (1.4) except t h a t convection t e r m is also included in t h e derivation of Eq ( E l ) along with the a s s u m p t i o n of 0 = cgT/Ot = 0 T h e p r o b l e m s t a t e d in Eq ( E l ) is equivalent

(E~)

A s s u m e t h e following data: h - 10 W / c m 2 - ~ k - 70 W / c m - ~ T~ = 40~ To = 140~ and L = 5 cm, and the cross section of fin is circular with a radius of 1 cm

S o l u t i o n

Note: Since t h e present p r o b l e m is a heat transfer problem, the t e r m s used in t h e case of solid mechanics problems, such as solid body displacement, strain, stiffness m a t r i x , load vector, and equilibrium equations, have to be replaced by t e r m s such as body, t e m p e r a - ture, gradient of t e m p e r a t u r e , characteristic matrix, characteristic vector, and governing equations, respectively

(i) Idealization

Let t h e fin be idealized into two finite element.s as shown in Figure 1.7(b) If t h e tem-

p e r a t u r e s of t h e nodes are taken as t h e unknowns, there will be three nodal t e m p e r a t u r e unknowns, n a m e l y T1, 7'2, and T3, in the problem

(ii) Interpolation (temperature distribution) model

In each element e (e = 1.2) the t e m p e r a t u r e (T) is a s s u m e d to vary linearly as

where a and b are constants If the nodal t e m p e r a t u r e s T~ ~) (T at x - 0) and T2 (e) (T at

z = l (~)) of element e are taken as unknowns, the c o n s t a n t s a and b can be expressed as

a = T1 (~) and b- (T(2 ~)- T~ ~))/l (e), where l (e) is the length of element e Thus,

(iii) Element characteristic matrices and vectors

T h e element characteristic matrices and vectors can be identified by expressing the func- tional I in m a t r i x form W h e n t h e integral in I is e v a l u a t e d over the length of element e,

we obtain

i ( e ) _ 1 d T + ~ - d ( T ~ - 2 T ~ T ) ] dx (E~)

GENERAL DESCRIPTION OF THE FINITE ELEMENT METHOD 21

Substitution of Eq (E4) into (Es) leads to

(iv) Assembly of element matrices and vectors and derivation of governing equations

As stated in Eq (E2), the nodal temperatures can be determined by minimizing the functional I The conditions for the minimum of I are given by

22 OVERVIEW OF FINITE ELEMENT METHOD

where [K],,~ X2=I[K (e)] is the assembled characteristic matrix, /5 E~=2 1/5(~) is the assembled characteristic vector, and T is the assembled or overall nodal t e m p e r a t u r e

GENERAL DESCRIPTION OF THE FINITE ELEMENT METHOD 23

(v) Solution for nodal temperatures

E q u a t i o n (E19) has to be solved after applying the b o u n d a r y condition, namely,

T (at node 1) = T1 = To = 140~ For this, the first e q u a t i o n of (El9) is replaced by T1 = To = 140 and the remaining two equations are written in scalar form as

governing the velocity distribution u ( x ) is given by Eq (1 12) with the b o u n d a r y condition

u ( x = O) = uo This problem is equivalent to

24 OVERVIEW OF FINITE ELEMENT METHOD

(i) Idealization

Divide t h e c o n t i n u u m into two finite e l e m e n t s as shown in Figure 1.8(b) If t h e values of

t h e p o t e n t i a l function at t h e various nodes are t a k e n as t h e unknowns, t h e r e will be t h r e e quantities, n a m e l y ~1, (I)2, and ~3 to be d e t e r m i n e d in the problem

(ii) Interpolation (potential function) model

T h e p o t e n t i a l function, O(x) is a s s u m e d to vary linearly within an e l e m e n t e (e = 1,2) as

A (~) is the cross-sectional a r e a of e l e m e n t e (which can be t a k e n as

(A1 + A 2 ) / 2 for e = 1 and (,42 + ,43)/2 for e = 2 for simplicity),

~(e) is the vector of nodal u n k n o w n s of e l e m e n t e

GENERAL DESCRIPTION OF THE FINITE ELEMENT METHOD 25 (iv) Governing e q u a t i o n s

PA(2) ) PA(2) + 1(2) i(2)

where Qi is t h e mass flow r a t e across section i (i = 1 , 2 , 3 ) a n d is n o n z e r o w h e n fluid

is e i t h e r a d d e d to or s u b t r a c t e d from t h e t u b e w i t h Q1 = - p A l t t l (negative since u l is

o p p o s i t e to t h e o u t w a r d n o r m a l to section 1), Q2 = 0, a n d Q3 = pA3u3 Since u l = uo is given, Q1 is known, whereas Q3 is u n k n o w n

(v) Solution of governing equations

In t h e t h i r d e q u a t i o n of (E6), b o t h 493 a n d Q3 are u n k n o w n s a n d t h u s t h e given s y s t e m

of e q u a t i o n s c a n n o t be solved Hence, we set 493 = 0 as a reference value a n d t r y to find

t h e values of 491 a n d 492 w i t h r e s p e c t to this value T h e first two e q u a t i o n s of (EB) can be expressed in scalar form as

(vi) C o m p u t a t i o n of velocities of t h e fluid

T h e velocities of t h e fluid in e l e m e n t s 1 a n d 2 can be found as

u in e l e m e n t 1 = u (1) = d0 (element 1)

d z (I) 2 491 l(1) = 1.246uo

26 OVERVIEW OF FINITE ELEMENT METHOD

and u in e l e m e n t 2 - u (2) = dO (element 2)

dx (I) 3 - - (I)2

I(2)

T h e s e velocities will be c o n s t a n t along t h e e l e m e n t s in view of t h e linear r e l a t i o n s h i p

a s s u m e d for O(x) within each element T h e velocity of the fluid at n o d e 2 can be

a p p r o x i m a t e d as

u2 (u (1/ + u ( 2 1 ) / 2 - 1.660u0

T h e t h i r d e q u a t i o n of (E6) can be w r i t t e n as

pA (2) pA~2) (I)2 + ~3 Q3

(e.g., s e p a r a t i o n m e t h o d s (e.g., solution of

of variables and R a y l e i g h - R i t z differential

F i n i t e or discrete

COMPARISON OF FINITE ELEMENT METHOD 27

Figure 1.9 Free Body Diagram of an Element of Beam

1.6.1 Derivation of the Equation of Motion for the Vibration of a Beam [1.9]

By considering the dynamic equilibrium of an element of the beam shown in Figure 1.9,

M - ( M + d M ) + F d x - p d x - ~ - 0 (moment equilibrium about point A)

which can be written, after neglecting the term involving (dx) 2, as

28 OVERVIEW OF FINITE ELEMENT METHOD

For small deflections, Eq (1.20) can be a p p r o x i m a t e d as

where rn is the mass of b e a m per unit length If the cross section of the b e a m is constant

t h r o u g h o u t its length, we have the final b e a m vibration equation

E q u a t i o n (1.26) has to be solved for any given beam by satisfying the associated b o u n d a r y conditions For example, if we consider a fixed-fixed beam the b o u n d a r y conditions to be satisfied are

Ow

Oz = 0

where L is the length of the beam

1.6.2 Exact Analytical Solution (Separation of Variables Technique)

For free vibrations, we assume harmonic motion and hence

w(~ t) - ~ ( ~ ) ~'"' (1.28)

COMPARISON OF FINITE ELEMENT METHOD 29

where W ( x ) is purely a function of x, and aa is the circular natural frequency of vibration

Substituting Eq (1.28) into Eq (1.26) we get

The general solution of Eq (1.29) can be written as

W ( x ) - C1 sin fix + C2 cos/3x + Ca sinh 3x + C4 cosh 3 x (1.31)

where C1-C4 are constants to be determined from the b o u n d a r y conditions In view of

Eq (1.28), the b o u n d a r y conditions for a fixed-fixed beam can be written as

w ( ~ - 0) = w ( ~ = L) - O}

d W ( x _ O ) = dlV

dx ~ ( : r = L) = 0

(1.32)

If we substitute Eq (1.31) into Eqs (1.32), we get four linear homogeneous equations in

the unknowns C1-C4 For a nontrivial solution, the d e t e r m i n a n t of the coefficient matrix

must be zero This gives the condition [1.9]

Equation (1.33) is called the frequency equation and there will be infinite number of solutions to it Let us call the n t h solution of Eq (1.aa) as & L If we use this solution in Eqs (1.30) and (1.31) we obtain the natural frequencies as

30 OVERVIEW OF FINITE ELEMENT METHOD

1.6.3 Approximate Analytical Solution (Rayleigh's Method)

To obtain an approximate solution to the first natural frequency and the mode shape, the Rayleigh's m e t h o d can be used In this m e t h o d we equate the m a x i m u m kinetic energy during motion to the m a x i m u m potential energy For a beam the potential energy rr is given by [1.9]

To find the value of w 2 from Eq (1.42) we assume a certain deflection or mode shape W(x)

known as admissible function t h a t satisfies the geometric boundary conditions but not necessarily the governing equilibrium equation, Eq (1.29), and substitute it in Eq (1.42) Let us take

2rrx)

This satisfies the boundary conditions stated in Eq (1.32) and not the equation of motion,

Eq (1.29) By substituting Eq (1.43) into Eq (1.42), we obtain the approximate value of

COMPARISON OF FINITE ELEMENT METHOD 31

the first natural frequency (5:1) as

L 2 v rn which can be seen to be 1.87% greater t h a n the exact value of

021 - 22"3729 v / E I L2

1.6.4 Approximate Analytical Solution (Rayleigh-Ritz Method)

If we want many natural frequencies, we have to substitute a solution, made up of a series

of admissible functions t h a t satisfy the forced b o u n d a r y conditions, in Eq (1.42) For example, if we want n frequencies, we take

W ( z ) = Clfl (x) + C2f2(z) + + Cnf,~ (x) (1.45) where C1, C2, , Cn are constants and fl, f 2 , , f~ are admissible functions

If we substitute Eq (1.45) into Eq (1.42), we obtain 5:2 as a function of C~, C2, , C,~

Since the actual frequency w will be smaller t h a n 5: [1.9], we want to choose C1, C 2 , , C~ such t h a t they make 5: a minimum For this, we set

32 OVERVIEW OF FINITE ELEMENT METHOD

lead to the following algebraic eigenvalue problem:

1.6.5 Approximate Analytical Solution (Galerkin Method)

To find the a p p r o x i m a t e solution of Eq (1.29) with the b o u n d a r y conditions s t a t e d in

Eq (1.32), using the Galerkin method, we assume the solution to be of the form

I I'(x) - C t f , ( x ) -4- C2f2(.r) + " " + C n f n ( x ) (1.52) where C1, C2 Cn are constants and f l f2 f , are functions t h a t satisfy all the specified b o u n d a r y conditions Since the solution assumed Eq (1.52), is not an exact one,

it will not satisfy Eq (1.29) and we will obtain, upon s u b s t i t u t i o n into the left-hand side

of the equation, a q u a n t i t y different from zero (known as the residual R) The values of

the constants C1, C2 C,, are o b t a i n e d by setting the integral of the residual multiplied

by each of the functions f~ over the length of the b e a m equal to zero: t h a t is,

L

fi R d x = O,

x 0

i = 1.2 n (1.53)

E q u a t i o n (1.53) represents a system of linear homogeneous equations (since the problem

is an eigenvalue problem) in the unknowns C~, C2 C,, These equations can be solved

to find the n a t u r a l frequencies and mode shapes of the problem For illustration, let us consider a t w o - t e r m solution as