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ELEMENTARY FINITE

ELEMENT

METHOD C.S.

DESAI

B&l

D45e 2035801 Desai element finite Lementary 15.624

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ALLEN COUNTY PUBLIC LIBRARY

4JHB|||a|i|_.

ELEMENTARY FINITE ELEMENT METHOD

CHANDRAKANT S. DESAI Professor, Virginia Polytechnic Institute

and State

University, Blacksburg, Virginia

PRENTICE-HALL,

INC., Englewood

Cliffs,

New Jersey 07632

Library of Congress Cataloging

Desai,

Chandrakant

Elementary

finite

(Civil engineering

Bibliography;

in

Publication Data

S.

element method.

and engineering mechanics

series)

p.

Includes index. 1.

Finite element method.

TA347.F5D47

I.

Title.

78-10389

624'. 171

ISBN 0-13-256636-2

Civil Engineering

N. M.

and Engineering Mechanics

Newmark and W.

V»

J.

Series

Hall, Editors

w.

©1979 by Prentice-Hall, Inc., Englewood Cliffs, N.J. 07632

All rights reserved.

may

No

part of this

book

be reproduced in any form or

by any means without permission from the publisher.

in writing

Printed in the United States of America

10

987654321

Prentice-Hall Prentice-Hall Prentice-Hall Prentice-Hall Prentice-Hall Prentice-Hall

International, Inc., London of Australia Pty. Limited, Sydney of Canada, Ltd., Toronto of India Private Limited, New Delhi of Japan, Inc., Tokyo of Southeast Asia Pte. Ltd., Singapore

Whitehall Books Limited,

Wellington,

New Zealand

2G3S8Q1 To

My

Parents,

Maya and

Sanjay

Digitized by the Internet Archive in

2012

http://archive.org/details/elenrientaryfiniteOOdesa

CONTENTS

PREFACE

1.

XI

INTRODUCTION Basic Concept

l

Process of Discretization

and Laws 13 Cause and Effect 14 Review Assignments 14

Principles

References

2.

16

STEPS IN THE FINITE ELEMENT Introduction

17

General Idea References

34

17

METHOD

17

Contents

vi

3.

ONE-DIMENSIONAL STRESS DEFORMATION Introduction

35

35

Explanation of Global and Local Coordinates

36

Local and Global Coordinate System for the

One-Dimensional Problem Interpolation Functions

38

41

Relation Between Local and Global Coordinates

Requirements for Approximation Functions Stress-Strain Relation

Principle of

Minimum

Expansion of Terms Integration

46 Potential Energy

47

52

53

Approach (for assembly) Method 58

Potential Energy

Direct Stiffness

Boundary Conditions 59 Strains and Stresses 64 Formulation by Galerkin's Method Computer Implementation 76 Other Procedures for Formulation

76

Complementary Energy Approach Mixed Approach 79 Bounds 83

76

References

55

67

Advantages of the Finite Element Method Problems 85

4.

85

91

ONE-DIMENSIONAL FLOW Problems

93

102

Bibliography

102

ONE-DIMENSIONAL TIME-DEPENDENT FLOW (Introduction to Uncoupled and Coupled Problems)

Uncoupled Case 103 Time-Dependent Problems 108 One-Dimensional Consolidation Computer Code 124 Problems 130 References

42

43

132

[8, 9]

121

103

Contents

6.

COMBINED COMPUTER CODE FOR ONEDIMENSIONAL DEFORMATION, FLOW, AND TEMPERATURE/CONSOLIDATION Philosophy of Codes Stages

Listing

134

144

and Samples of Input/Output

Beam-Column

177 189

Other Procedures for Formulation Problems 197 References

192

200

ONE-DIMENSIONAL MASS TRANSPORT Introduction

References

Bibliography

202

202

Finite Element Formulation

202

209 209

ONE-DIMENSIONAL OVERLAND FLOW Introduction

References

211

211

Approximation for Overland and Channel Flows Finite Element Formulation 213

10.

172

172

Physical Models

9.

145

BEAM BENDING AND BEAM-COLUMN Introduction

8.

133

134

Problems

7.

vii

213

222

ONE-DIMENSIONAL STRESS WAVE PROPAGATION Introduction

224

Element Formulation Problems 234 Finite

References

Bibliography

235 235

225

224

Contents

viii

11.

TORSION

237

Introduction

237

Triangular Finite Elements

238

Finite Element Formulation

240

Comparisons of Numerical Predictions and Closed

Form

Solutions

252

Approach 254 Review and Comments Hybrid Approach 269 Mixed Approach 284 Stress

Static Condensation Problems 293

References

12.

268

291

297

OTHER FIELD PROBLEMS: POTENTIAL, THERMAL, AND FLUID FLOW Introduction Potential

299

299

Flow

300

Element Formulation 302 Stream Function Formulation 314 Thermal or Heat Flow Problem 321 Seepage 323 Finite

Electromagnetic Problems

327

Computer Code FIELD-2DFE Problems References

332

Bibliography

13.

328

328

332

TWO-DIMENSIONAL STRESS-DEFORMATION ANALYSIS Introduction

333 Plane Deformation Finite

333

Element Formulation

Computer Code Problems References

368 371

354

338

333

Contents

14.

MULTICOMPONENT SYSTEMS: BUILDING FRAME AND FOUNDATION Introduction

Computer Code

373

382

Transformation of Coordinates Problems 392

15.

389

392

PRELUDE TO ADVANCED STUDY AND APPLICATIONS Theoretical Aspects

Bibliography

394

395

396

VARIOUS NUMERICAL PROCEDURES; SOLUTION TO BEAM BENDING PROBLEM

399

SOLUTION OF SIMULTANEOUS EQUATIONS

400

3.

PHYSICAL MODELS

419

4.

COMPUTER CODES

424

Appendix

1.

Appendix

2.

Appendix

Appendix

INDEX

372

372

Various Components

References

jx

429

PREFACE

The finite element method has gained tremendous attention and popularThe method is now taught at most universities and colleges, is researched extensively, and is used by the practicing engineer, industry, and government agencies. The teaching of the method has essentially been concentrated at the postgraduate level. In view of the growth and wide use of the method, however, it becomes highly desirable and necessary to teach it at the underity.

graduate

level.

There are a number of books and publications available on the finite element method. It appears that almost all of them are suitable for the advanced student and require a number of prerequisites such as theories of constitutive or stress-strain laws, mechanics,

and variational

calculus.

Some

of the introductory treatments have presented the method as an extension of matrix methods of structural analysis. This viewpoint necessary, since the finite element

maturity and generality.

It

method has reached a

may no

longer be

significant level

of

has acquired a sound theoretical basis, and in

has been established as a general procedure relevant to engineering and mathematical physics. These developments permit its teaching and use as a general technique from which applications to topics such as mechanics, itself

structures, geomechanics, hydraulics,

as special cases. It

is

and environmental engineering method be treated

therefore essential that the

arise

as a

general procedure and taught as such.

This book sufficiently

is

intended mainly for the undergraduate.

elementary so that

it

Its

approach

is

can be introduced with the background of xi

Preface

xii

essentially

undergraduate subjects. At the same time, the treatment

enough so

that the reader or the teacher interested in various topics such as

broad

is

stress-deformation analysis, fluid and heat flow, overland flow, potential flow,

time-dependent problems, diffusion, torsion, and wave propagation can use

and teach from

it.

of the method and provides a distinct and element method at an elementary

The book brings out the

intrinsic nature

that permits confluence of various disciplines

rather novel approach for teaching the finite level.

Although the book

is

intended mainly for the undergraduate,

it

can

be used for the fresh graduate and the beginner with no prior exposure to the finite element method.

The

prerequisites for understanding the material

be undergraduate mathematics, strength of materials and undergraduate

will

courses in structures, hydraulics, geomechanics and matrix algebra. Intro-

ductory knowledge of computer programming

The

text

written in such a

is

way

that

is

desirable but not necessary.

no prior knowledge of variational

The derivations are presented through the use of Over a period of the last five years or so, the author has taught, based on these prerequisites, an undergraduate course and a course for user groups composed of beginners. This experience has shown that principles

is

necessary.

differential calculus.

undergraduates or beginners equipped with these prerequisites, available to

them

in the

undergraduate curricula at most academic institutions, can under-

stand and use the material presented in this book.

The

first

chapter presents a rather philosophical discussion of the

finite

element method and often defines various terms on the basis of eastern and western concepts from antiquity. The second chapter gives a description of the eight basic steps. Chapters 3-5 cover one-dimensional problems in stress-

deformation analysis and steady and time-dependent flow of heat and

The fundamental

mon

method

fluids.

by showing the comcharacteristics of the formulation for these topics and by indicating the

fact that their

generality of the

is

illustrated

governing equations are essentially similar. The generality

further established by including a

computer code

in

is

chapter 6 that can solve

the three problems in chapters 3-5.

Understanding and using the the use of the computer.

teaching of the method

It is

may

finite

element method

is

closely linked with

the belief of the author that strictly theoretical

not give the student an idea of the details and

the ranges of applicability of the technique. Consequently this text endeavors

and simultaneously with the theoretical and understanding of computer codes. The code in chapter 6 is thoroughly documented and detailed so that it can be used and understood without difficulty. Moreover, a number of rather simple codes are introduced in the later chapters and their applications are included. Details of these codes, designed for the beginner, are given in appendix 4. It is recommended that these or other available codes be used by the student while to introduce the student, gradually

teaching, to the use

learning various topics in chapters 7-14.

Preface

xiii

Chapter 7 introduces the idea of higher-order approximation for the problem of beam bending and beam-column. One-dimensional problems in mass transport (diffusion-convection), overland flow due to rainfall, and wave propagation are covered in chapters 8, 9 and 10, respectively. These problems illustrate, by following the general procedure, formulations for different categories of time-dependent problems.

Chapters 11-14 enter into the realm of two-dimensional problems. The chapter on Torsion (chapter 11) and Other Flow Problems (chapter 12) have

been chosen because they involve only one degree-of-freedom

at a point.

Chapters 13 and 14 cover two-dimensional stress-deformation problems involving two and higher degrees-of-freedom at a point.

The text presents the finite element method by using simple problems. It must be understood, however, that it is for the sake of easy introduction that we have used relatively simple problems. The main thrust of the method, on the other hand, is for solving complex problems that cannot be easily solved by the conventional procedures. In order to emphasize this and to show the reader what kind of complex factors can be handled, chapter 1 5 includes a rather qualitative description of the advanced study and applications of the method. Here, a number of factors and aspects that are not covered in chapters 1-14 are stated and references are given for a detailed study. For a thorough understanding of the finite element method, it is essential that the student perform hand calculations. With this in mind, most chapters include a number of problems to be solved by hand calculations. They also include problems for home assignments and self-study. The formulations have been presented by using both the energy and residual procedures. In the former, the potential, complementary, hybrid, and mixed procedures have been discussed. In the residual procedures, main attention has been given to Galerkin's method. A number of other residual methods are also becoming popular. They are described, therefore, in appendix I, which gives descriptions, solutions and comparisons for a problem by using a number of methods: Closed form, Galerkin, collocation, subdomain, least squares, Ritz, finite difference, and finite element. Formulations by the finite element method usually result in algebraic simultaneous equations. Detailed description of these methods is beyond the scope of this book. Included in appendix to the

commonly used

direct

and

iterative

2,

however, are brief introductions

procedures for solution of algebraic

simultaneous equations. Physical models can help significantly in the understanding of various concepts of the method. Appendix 3 gives the descriptions of some physical

models. Appendix 4 presents details of a number of computer codes relevant to various topics in the text.

one or two undergraduate courses. The second course may overlap with or be an introductory graduate course. Although a

The book can be used

for

Preface

xiv

number of topics have been covered

in the

book, a semester or quarter course

could include a selected number of topics. For instance, a quarter course can

and then one or two topics from the remaining chapters. mechanics and stress-deformation analyses, the topics can be Beam Bending and Beam-Column (chapter 7), and Two-Dimensional Stress Deformation (chapter 13). If time is available (in the case of a semester course), chapter 10 on One-Dimensional Wave Propagation, chapter 11 on Torsion and/or chapter 14 on Multicomponent Systems can be added. A class oriented toward field problems and hydraulics can choose one or more of chapters 8, 9, 11, and 12 in addition to chapters 1-6. Thanks are due to Y. Yamada, University of Tokyo; Peter Hoadley, Vanderbilt University; William J. Hall, University of Illinois at Urbana; E. L. Wilson, University of California at Berkeley; John F. Abel, Cornell University and my colleagues S. Sture, T. Kuppusamy and D. N. Contractor for reading the manuscript and for offering useful comments and suggestions. A number of my students helped in solutions of some of the problems I would like to express special appreciation to John Lightner for his assistance in implementing some of the computer solutions. I realize that it is not easy to write at an elementary level for the finite element method with so many auxiliary disciplines. The judgment of this book cover chapters

For a

1-6,

class interested in

;

;

is

better left to the reader.

All natural systems are essentially continuous or interconnected,

influenced by a large

number of parameters.

we must understand all we make approximations, by

and are

In order to understand

such a system,

the parameters. Since this

possible

selecting only the significant of

is

not

them and neglecting the others. Such a procedure allows understanding of the entire system by comprehending its components taken one at a time. These approximations or models obviously involve errors, and we strive continuously to improve the models and reduce the errors.

Chandrakant

S.

Desai

INTRODUCTION

BASIC CONCEPT current form the finite element (FE)

method was formalized by civil The method was proposed and formulated previously in different manifestations by mathematicians and physicists. The basic concept underlying the finite element method is not new The principle of discretization is used in most forms of human endeavor. Perhaps In

its

engineers.

:

the necessity of discretizing, or dividing a thing into smaller manageable

from a fundamental limitation of human beings in that they cannot see or perceive things surrounding them in the universe in their entirety things, arises

or totality. Even to see things immediately surrounding us, several turns to obtain

we

a.

we must make

jointed mental picture of our surroundings. In other

around us into small segments, and the final one that simulates the real continuous surroundings. Usually such jointed views contain an element of error. In perhaps the first act toward a rational process of discretization, man

words,

discretize the space

assemblage that we visualize

is

divided the matter of the universe into five interconnected basic essences

(Panchmahabhutd), namely, sky or vacuum,

air,

added to them perhaps the most important of all,

water, earth, and time,

by singing

Time created beings, sky, earth, Time burns the sun and time will bring What is to come. Time is the master of everything

^he number

[l].

1

within brackets indicates references at the end of the chapter.

fire,

and

Chapter 1

Introduction

i + N y = [N]{y„], t

2

where

T {y„]

=

[y

t

y2 ]

is

(3-9a)

2

(3-9b)

2

(3-9c)

the vector of nodal coordinates.

In fact an explanation of the concept of isoparametric elements, which

is

can be given at this stage. A comparison of Eq. (3-8) and (3-9) shows that both the displacement v and the coordinate y at a point in the element are expressed by using the same (iso) the most

common

procedure

interpolation functions.

An

now

in use,

element formulation where

similar) functions for describing the deformations in

(or geometry) of an element

This

is

is

we use the same (or and the coordinates

called the isoparametric element concept

rather an elementary example;

we

and more general isoparametric elements.

shall subsequently

[1].

look at other

VARIATION OF ELEMENT PROPERTIES We often tacitly assume that the material properties such as cross-sectional area A and the elastic modulus E are constant within the element. It is not necessary to assume that they are constant. We can introduce required variation, linear or higher order, for these quantities.

For instance, they can

be expressed as linear functions:

E = N,E + N E = [NtfEJ, A = N A + N A = [N]{A„}, E and {A„} r = [A A are the x

T

where [E n } of E and A

= at

[E

x

X

2

2

(3-10)

t

2

2

(3-11)

2]

nodes

1

2]

x

and

vectors of nodal values

2, respectively.

REQUIREMENTS FOR APPROXIMATION FUNCTIONS As we have

stated before, the choice of an approximation function is guided by laws and principles governing a given problem. Thus an approximation function should satisfy certain requirements in order to be acceptable. For general use these requirements are expressed in mathematical language. However, in this introductory treatment, we shall discuss them in rather

simple words.

An

approximation function should be continuous within an element. The

linear function for v [Eqs. (3-7)

and (3-8)]

is

indeed continuous. In other words,

it

does not yield a discontinuous value of v but rather a smooth variation of

v,

and the variation does not involve openings, overlaps, or jumps.

up

The approximation function should provide interelement compatibility by the problem. For instance, for the column prob-

to a degree required

lem involving

axial deformations,

it

is

necessary to ensure interelement

compatibility at least for displacements of adjacent nodes. That

is,

the

approximation function should be such that the nodal displacements between adjacent nodes are the same. This is shown in Fig. 3-3(e). Note that for this case, the higher derivatives such as the first derivatives may not be compatible.

The displacement

at node 2 of element 1 should be equal to the displacement node 1 of element 2. For the case of the one-dimensional element, the linear approximation function satisfies this condition automatically. As indicated in Chapter 2 (Fig. 2-8), satisfaction of displacement compatibility by the linear function does not necessarily fulfill compatibility of first derivative of displacement, that is, slope. For axial deformations, however, if we provide for the compatibility up to only the displacement, we can still expect to obtain reliable and convergent solutions. Often, this condition is at

tied in with the highest order of derivative in the energy function such as the

potential energy. derivative dv/dy

For example,

= €,

is

1

;

in

Eq. (3-21) below, the highest order of

hence, the interelement compatibility should 43

:

One- Dimensional Stress Deformation

44

include order of v at least

up

to

(zero), that

is,

Chapter 3

displacement

v.

In general,

should provide interelement compatibility up to order

the formulation

—

n 1, where n is the highest order of derivative in the energy function. Approximation functions that satisfy the condition of compatibility can be called conformable.

The

other and important requirement

is

that the approximation function

should be complete; fulfillment of this requirement will assure monotonic convergence. Monotonic convergence can be explained in simple terms as a process in which the successive approximate solutions approach the exact solution consistently without changing sign or direction.

For

instance, in

approximate areas approach the exact area in such a way that each successive value of the area is smaller or greater than the previous value of area for upper and lower bound solutions, respectively. Completeness can be defined in a number of ways. One of the ways is to Fig. l-8(c) the

relate

it

to the characteristics of the chosen approximation function. If the

function for displacement approximation allows for rigid body displacements

(motions) and constant states of strains (gradients), then the function can be

considered to be complete.

mode For

A

rigid

body motion represents a displacement

that the element can experience without development of stresses in

it.

instance, consider the general polynomial for v as

v

=a + 2

cc 2

y

In Eqs. (3-3a) and (3-6)

|

+

a 3y 2

+

a 4 j> 3

we have chosen

the general polynomial as

+

•

•

•

+

n cc n+

(3-3d)

iy

a linear polynomial by truncating

shown by the

vertical

dashed

line.

The

linear

approximation contains the constant term a, which allows for the rigid body displacement mode. In other words, during this mode, the element remains

and does not experience any strain or stress, that is, a 2 y = 0. The requirement of constant state of strain (e y) for the one-dimensional column deformation is fulfilled by the linear model, Eq. (3-3a) because of the existence of term a 2 y. This condition implies that as the mesh is refined, that in each element is, the elements become smaller and smaller, the strain e y

rigid

approaches a constant value. In the case of one-dimensional plane deformations in the column, the

condition of constant state includes only e y

may

— the

first

derivative or gradient

and more general problems such as beam and plate bending. In such cases, it will be necessary to satisfy the constant strain state requirement for all such generalized strain or gradients of the unknowns involved, e.g., see Chapters 7, 11-14. In addition to monotonic convergence, we may be interested in the rate of

v.

Additional constant strain states

exist in other

of convergence. This aspect is often tied in with the completeness of the polynomial expansion used for the problem. For instance, for the onedimensional column problem, completeness of the approximation function requires that a linear function, that

is,

a polynomial of order («) equal to one

Chapter 3

is

One- Dimensional Stress Deformation

needed. Completeness of the polynomial expansion requires that

including and up to the satisfied for the linear

from Eq.

first

order should be included. This

model, Eq. (3-3a) since

(3-3d). In the case of

cubic approximation function is

45

all

terms up to n

beam bending, Chapter is

6,

to

and includes the order

It

may happen

not include satisfy the

all

we

all

it

terms

automatically

=

1

are chosen

shall see that a

Then

required to satisfy completeness.

necessary to choose a polynomial expansion such that

up

is

includes

all

it

terms

3.

that an approximation function of a certain order (n)

may

terms from the polynomial expansion, Eq. (3-3d), and

still

requirements of rigid body motion and constant states of

As an example of a complete approximation

satisfying rigid

strain.

body motion and

constant state of strain in a two-dimensional problem, but not complete

of the polynomial expansion, see Chapter 12, Eq. (12-9). For two-dimensional problems, the requirement of completeness of polynomial expansion can be explained through the polynomial expansion represented by using Pascal's triangle, see Chaps. 11-13. Here we have given rather an elementary explanation of the requirements for approximation functions. The subject is wide in scope and the reader interested in advanced analysis of finite element method will encounter the subject quite often. For example, the completeness requirement can be further explained by using the so-called "patch test" developed by Irons [2]; it is discussed in refs. [3] and [4]. Moreover, the approximation model should satisfy the requirements of isotropy or geometric invariance [5]. These topics are considered beyond the scope of this text. in the sense

Step

3.

Define Strain-Displacement and Stress-Strain Relations

For stress-deformation problems, the actions or causes (Chapter 2) are and the effects or responses become strains, deformations, and stresses. The basic parameter is the strain or rate of change of deformation. The link connecting the action and response is the stress-strain or constitutive law of the material. It is necessary to define relations between strains and displaceforces,

ments and stresses and strains for the derivation of element equations in Step 4. Hence, in this step we consider these two relations. We note at this stage that although we use familiar laws from strength of materials and elasticity for the stress-deformation

problem,

the relations relevant to specific topics.

flow problem (Chapter

4),

the relation

in later chapters

we

shall use

For instance, in the case of the fluid between gradient and fluid head and

Darcy's law will be used.

Returning to the axial deformation of the column element, the displacement relation, assuming small strains, can be expressed as ey

= ^> ay

strain-

(3-12a)

One- Dimensional Stress Deformation

46

Chapter 3

where e y = axial strain. Since we have chosen to use the local coordinate L and since our aim is to find dv/dy in the global system, we can use the chain rule of differentiation as

€

Now, from Eq.

(3-2)

dv

,~ 101 v (3 " 12b)

= d_(l^M)=\ dy\

dy

dL

dL dv -ayTL'

we have d_L

and from Eq.

dv = Ty =

>

(3-12c)

1/2

(3-8)

= ^[±(1 -

L)v t

+ i(l +

=

L)v 2 ]

1]N.

±[-1

(3-12d)

Substitution of Eqs. (3-12b)-(3-12d) into Eq. (3-12a) leads to

= -}-[-

e,

1

1]H

(3-13a)

or

=

{€,} (1

where

[B]

=

(1//)

x

x [— 1

1)

x

{q} 2) (2

1)

is

a one-dimensional problem, the strain vector

contains only one term, and the matrix [B]

{e y}

(3-13b)

,

x

can be called the strain-displacement transforma-

1]

tion matrix. Because this

[B] (1

is

only a row vector.

retain this terminology for multidimensional problems

;

We

shall

however, with multi-

dimensional problems these matrices will have higher orders.

The student can

easily see that Eq. (3-1 2d) indicates constant value

strain within the element; this

the displacement.

is

because

we have chosen

of

linear variation for

We can then call this element a constant-strain-line element.

STRESS-STRAIN RELATION For

simplicity,

we assume

elastic (Fig. 3-4).

that the material of the

column element

is

linearly

This assumption permits use of the well-known Hooke's

law,

ay

— Ey € y

(3- 14a)

,

or in matrix notation, {«,} (1

where

[C]

is

X

1)

=

[C] (1

X

{€,}

1)(1

X

,

(3-14b)

1)

the stress-strain matrix. Here, for the one-dimensional case,

matrices in Eq. (3- 14b) consist of simply one scalar term.

One- Dimensional Stress Deformation

Chapter 3

47

i\

A^ = U cl u

-

III

Figure 3-4 Linear elastic constitutive or stress-strain (Hooke's) law.

Substitution of Eq. (3-13) into Eq. (3-14b)

now

allows us to express {a y}

in terms of {q} as

{«y } Step

A

4.

(3-15)

Derive Element Equations

number of procedures

Among

= [C][B]{qJ.

are available for deriving element equations.

and residual methods. Principles based on and complementary energies and hybrid and mixed methods are used within the framework of variational methods. As described in Chapter 2 and in Appendix 1, a number of schemes such as Galerkin, collocation, and least squares fall under the category of residual methods. We shall use some of these methods in this chapter and subsequently in other chapters. these are the variational

potential

PRINCIPLE OF In simple words,

MINIMUM POTENTIAL ENERGY if

a loaded elastic body

is

in equilibrium

under given geo-

metric constraints or boundary conditions, the potential energy of the

deformed body assumes a stationary value. In the case of linear elastic bodies is a minimum; since most problems we consider involve this specialization, for convenience we shall use the term minimum. Figure 3-5 shows a simple axial member represented by a linear spring in equilibrium, the value

with spring constant k(F/L). Under a load P, the spring experiences a displacement equal to v. The potential or the potential energy n p of the spring is composed of two load (see Chapter 2) parts, strain energy U and potential p of the external

n,

The

strain energy

U

W u+ W

:

B.

(3-16)

can be interpreted as the area under the stress-strain when we minimize II,, we differentiate or

curve (Fig. 3-4). Mathematically,

:

One- Dimensional Stress Deformation

48

Chapter 3

IP

I

n

w^% Figure 3-5 Idealized linear spring.

take variations of Tl p with respect to the displacement v. While doing this we assume that the force remains constant, and we can relate variation of work

W by the load and the potential of the load as

done

SW

6Wr

(3-17)

where 8 denotes arbitrary change, variation, or perturbation. For our purpose

we can consider

it

to imply a series of partial differentiations.

sign in Eq. (3-17) occurs because the potential

into is

work by

these loads,

W

p

The negative

of external loads

is

lost

W. Then the principle of minimum potential energy

expressed as

sn B = su + 8wD =

su-sw =

(3-18)

o.

There are two ways that we can determine the minimum of II,: manual and mathematical. Both involve essentially an examination of the function represented by lip until we find a minimum point. For simple understanding, we first consider the manual procedure and write the potential energy for the spring (Fig. 3-5), assuming undeformed state of the spring as the datum for potentials, as

n

-Pv Ikv - Pv, 2

(3-19a)

where kv = force in the spring and %(kv)v denotes strain energy as the area under the load-displacement curve (Fig. 3-5). Since the load in the spring goes from to kv, we have to use average strain energy. The term Pv denotes the potential of load P; since we have assumed P to be constant, this term does not include £. We further assume that P = 10 units and k = 10 units per unit deformation. Then

U p = \\0v 2

Now we

search for the

;

a positive v

is

=

5v 2

-

minimum by examining

various values of deformations table

lOv

assumed

v.

The

(3- 19b)

10v.

values of the potential Il p for

results are

shown

in the following

to act in the direction of the applied load

One- Dimensional Stress Deformation

Chapter 3

V

n,

-2.000 -1.000

+40.000

49

+ 15.000

0.000

0.000

0.125

-1.1719 -2.1875 -3.7500 -5.0000

0.250 0.500 1.000

2.000

0.0000

3.000

15.0000

4.000

40.0000

5.000

75.0000

etc.

Figure 3-6 shows a plot of lip versus value at v will

=

1.

deform by

On

Hence, under 1

known

is

can be seen that lip has a

when

minimum

in equilibrium,

we can perform the procedure of "going to and minimum by using mathematics. It is

11, to find its

that a function assumes a

derivative

v. It

10 units, the spring,

unit.

the other hand,

on the function

P=

zero.

Applying

minimum

\kvbv

Pdv

well

value at a point(s) where

this principle to II, in

SU,

fro"

Eq.

(3- 19a),

we

its

obtain

=0

or (kv

- P)Sv = 0.

(3-20a)

Figure 3-6 Variation of potential energy.

/

i

40

Is** 30 20

\

\ \ 10

\

\ -3

-1

^^" '2

3

Minimum

point

1

-10 -20 -30

4

5

v

One- Dimensional Stress Deformation

50

Since Sv

is

arbitrary, the term in parentheses

Chapter 3

must vanish. Therefore,

kv-P = or

=P,

kv

(3-20b)

which is the equation of equilibrium for the spring. STl p = in Eq. (3-20) is analogous to equating d\\ p \dv = 0, which will result in the same equilibrium equation (3-20b). Substitution of the numerical values gives \0v

Therefore, v

We method

=

1.0 unit, the

=

10.

same answer

as before.

note here that in most problems solved by using the

Hp

is

a function of a large

ments. Consequently,

it

is

number of parameters

most economical and

matical methods because the manual procedure

finite

element

or nodal displace-

direct to use the is

mathe-

cumbersome and

often

impossible.

The mathematical procedure involved minimization of lip. For simplicity, we may view the process as simply taking derivatives of Tl p In general, how.

ever, the minimization will involve calculus of variations. In

treatment in this book

we

Now we return to the column element (Fig.

n> =

most of the

shall use the simple differentiation concept.

3-7)

and write FL

as

[5]

M

c e dv iff ^ ' >

~

iff

fvdv

~

if

T'vds

S P„v„

Figure 3-7 Generic column element with loads.

jp„

© U\®

©n

% \

^

(3-2 la)

One- Dimensional Stress Deformation

Chapter 3

=

where \o y e y

strain energy per unit

Ty =

(weight) per unit volume,

S

area,

= part

x

M — number

maximum

A

value of

vt

level,

M=

Y = body

=

Pa =

applied

displacement corresponding to

the joint forces

P a is now appropriate. Pu can be treated as

Pih

Since a

applied to the total structure,

is

force

2.

contribution of the joint force

we

volume,

of points at which joint forces are applied; here the

comment concerning

joint force

=

surface loading or traction per unit surface

of the surface on which surface loading acts,

nodal (joint) forces at local

and

V

volume,

51

Pig

applied at point

i

a local

in the global sense. Later

more convenient to add contributions of concentrated when we consider potential energy of the entire body; becomes clearer when the total structure is considered.

shall see that

it is

joint or nodal forces their relevance

The terms

Eq. (3-21 a) are essentially similar to those in Eq.

in

now we

except that

assign

volume

(3- 19a),

to the element instead of treating

it

as a

spring.

For the present, we assume that the cross-sectional area A of the element is

constant; then Eq. (3-2 la) reduces to

- A\

Tyvdy -

n,

=

^

Ty

is

the (surface) loading per unit length along the centerline of the

Here

idealized line

o y e y dy

column

Yvdy-

£ Pav

t

(3-21b)

.

(Fig. 3-7).

Equation (3-2 lb) can

now

be expressed

in

terms of the local coordinate

system by using the transformation of Eq. (3-2) as

dy

=

(3-22)

-L-dL.

Therefore, 1

11,

=

^

Next we

ay€ y

dL-^C

YvdL-l-j TyvdL-tP»Vi'

(«lc)

J"'

and a y from Eqs.

substitute for v, €y9

(3-8), (3-13),

and

(3-14),

respectively, in Eq. (3-21c) to obtain, in matrix notation,

n* = T

j\ _1

F

(1

x

1)(1

-yj "'

^WL "t£

[C]

{€'

x

1)(1

[N] (1

x

x

2) (2

x

(1

1)

dL-^

T,

{q} 1)(1

x

[N]

1)

x 2)(2 x

Pu (1

x

f dL

{q} 1)(1

x

1)

(3-21d)

v,

l)(l

x

1)

One- Dimensional Stress Deformation

52

Chapter 3

or

=^

n

{qf

[W

x

x

(1

Al

2) (2

x

-yj

[B]

x 1)(1 x

1)(1

[N] (1

[C]

Y

{q}

x

2) (2

[N]

x

1)(1

x

2) (2

1)

M r

x

x

2) (2

x

1)(1

(3-2 le)

T

1

(1

1)

dL

fy dL-

{q}

dL

{q}

(1

1)

x

1)(1

x

1)

where Y and Ty are assumed to be uniform. Equation (3-21) represents a quadratic function expressed in terms of v and v 2 In matrix notation, transposing in Eq. (3-2 Id) is necessary to make .

x

the matrix multiplication in {e } r [C]{€ >,} consistent so as to yield the scalar 3;

a y e y = Eel m Eq. (3-2 lc). The need for transposing will become when we expand the terms in Eq. (3-2 lc). The last term denotes sum-

(energy) term clear

mation,

P uv

l

+P

M=

2 iV 2 , if

2.

EXPANSION OF TERMS

We now them

consider the

first

three terms in Eq. (3-2 le) one by one

and expand

as follows:

First term

"-*!>

Eqs. (3-23a), (3-23b), and (3-23c) and the last term in

One- Dimensional Stress Deformation

Chapter 3

¥l',[i Notice that

2) to find its

+

invoke the principle of

minimum

+ L>

2

| + yd + L)»4 ]rfi

fir

3

3

J'

~

¥

-

%^ J'

/',

- P!,«, -

[t°

~~

L)v '

+ T (I + LKl rfL

[yd " iK + yd +

(PL

+ P*u)v - (P|, + 2

£)»,]

= R

1

>




kzi

k 33

Here kt4 x 8

(i

=

1, 2,

3) is

(3-37b)

R

v3 Ij

-0