introduction to finite elements in engineering

\FORTRAN Programs in Fortran Language No Programs in C WE Programs in Visual Basic \EXCELVB Programs in Excel Visual Basic \EXAMPLES Example Data Files .inp extension displacements a

\FORTRAN Programs in Fortran Language

No Programs in C

WE Programs in Visual Basic

\EXCELVB Programs in Excel Visual Basic

\EXAMPLES Example Data Files (.inp extension)

displacements along coordinate directions at point (X; y, Z)

components of body force per unit volume, at point (x, y, z) components of traction force per unit area, at point (.x, y,z) om the surface

strain components, ¢ are normal strains and y are engineering shear strains

‘stress components, o are normal stresses and 7 are engineering shear stresses

Potential energy = U + WP, where U = strain energy, WP = work potential

vector of displacements of the nodes (degrees of freedom or DOF) of an element, dimension (NDN*NEN, 1)—see next Table for explanation of NDN and NEN

vector of displacements of ALL the nodes of an element, dimension (NN*NDN, 1)—see next Table for explanation of NN and NDN

element stiffness matrix; strain energy in element, U, = }q'kq

global stiffness matrix for entire structure: 1 = !Q?KQ - Q'F

body force in element e distributed to the nodes of the element traction force in element ¢ distributed to the nodes of the element virtual displacement variable; counterpart of the real displacement u(x y,z)

vector of virtual displacements of the nodes in an element; counterpart of q

shape functions in En{ coordinates, material matrix, strain-displacoment matrix respectively uw = Nq,¢ = Bq and = DBq

Structure of Input Files‘

TITLE (*)

PROBLEM DESCRIPTION (*}

NN NE NM NDIM NEN NON (*)

422 2 3 2 - 1 Line of data, 6 entries per Tine

ND NL NMPC (*}

s 2 0 - 1 Line of data, 3 entries

Node# Coordinate#l Coordinate#NDIM (*)

2 3 2

3 0 2 ~ NN Lines of data, (NDIM+i)entries

4 0 0

Elem# Node#l Node#fNEN Mat# Element Characteristics't (*)

1 4 1 2 1 0.5 0 Ì -NE Lines of data,

2 3 4 2 2 0.5 0 | (NEN+2+ #of Char.)entrfes

bore Specified Displacement (*)

MAT# Material Properties*? (*)

1 30e6 0.25 12e-6 ~-NM Lines of data, (1+ # of prop entries

2 20e6 0.3 9,

B1 i B2 j B3 (Multipoint constraint: 81*Qi+B2*Qj=B3) (Ψ)

} -NMPC Lines of data, 5 entries

THEAT1D and HEAT2D Programs need extra boundary data about flux and convection (See Chapter 10.)

(*) = DUMMY LINE - necessary

Note: No Blank Lines must be present in the input file

See below for description of element characteristics and material properties

Main Program Variables

NN = Number of Nodes; NE = Number of Elements; NM = Number of Different Materials

NDIM = Number of Coordinates per Node (e.g NDIM = 2 for 2-D,or = 3 for 3-D): NEN = Number of Nodes per

Element (og,NEN = 3 for 3-noded triangular element, or = 4 for a 4-noded quadrilateral) NDN = Number of Degrees of Freedom per Node (eg.,NDN = 2 for a CST element, or = 6 for 3-D beam clement)

ND = Number of Degrees of Freedom along which Displacement is Specified = No of Boundary Conditions

NL = Number of Applied Component Loads (along Degrees of Freedom)

NMPC = Number of Multipoint Constraints; NO = Total Number of Degrees of Freedom = NN * NDN

Program Element Characteristics Material Properties

FRAME2D Area, Inertia, Distributed Load E ị

HEAT2D Element Heat Source Thermal Conductivity, &

BEAMKM Inertia, Area Ep |

Introduction to Finite

Elements in Engineering

Includes bibliographical references and index

ISBN 0-13-061591-9

1 Finite element method.2 Engineering mathematics, I Belugundu, Ashok D., U.Tide

TA347.F5 C463 2001 6207.001'51535 dc21

Vice President and Editoriat Director, ECS: Marcia J, Honon Acquisitions Editor: Laura Fischer

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‘The author and publisher ofthis book have used their best efforts in preparing this book These efforts include the development, re- search, and testing of the theories and programs to determine their effectiveness The author and publisher make no warranty of an kind, expressed or implied, with regard to these programs or the documentation contained in this book The author and publisher shall

‘ot be lable in any event for incidental or consequential damages ia connection with, or arising out of, the furnishing, performance, ot use of these programs

“Visual Basic” and “Excel” are registered trademarks of the Microsoft Corporation, Redmond, WA

“MATLAB isa registered trademark of The MathWorks, Inc.,3 Apple Hill Drive, Natick, MA 01760-2098, Printed in the United States of America

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ISBN 0-13-061891-9

Pearson Education Lid,, London Pearson Education Australia Pty, Lid., Sydney Pearson Education Singapore, Ple Ltd

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To our parents

15 1.6

17 1.8 1.9

1.10

111 1.12 1.13 1,14

21

pe

Introduction 1 Historical Background 1 Outline of Presentation 2 Stresses and Equilibrium 2 Boundary Conditions 4 Strain~Displacement Relations 4 Stress-Strain Relations 6

Special Cases, 7

Temperature Effects 8

Potential Energy and Equilibrium;

The Rayleigh-Ritz Method 9

Potential Energy Tl, 9 Rayleigh-Ritz Method, 11

Galerkin’s Method 13 Saint Venant’s Principle 16

Von Mises Stress 17

Computer Programs 17 Conclusion 18

te

viii Contents

22

Matrix Multiplication, 23 Transposition, 24

Differentiation and Integration, 24

Square Matrix, 25

Diagonal Matrix, 25

Identity Matrix, 25 Symmetric Matrix, 25

Upper Triangular Matrix, 26

Determinant of a Matrix, 26

Matrix Inversion, 26 Eigenvalues and Eigenvectors, 27

Positive Definite Matrix, 28 Cholesky Decomposition, 29

Gaussian Elimination 29

General Algorithm for Gaussian Elimination, 30 Symmetric Matrix, 33

Symmetric Banded Matrices, 33

Solution with Multiple Right Sides, 35 Gaussian Elimination with Column Reduction, 36

Skyline Solution, 38

Frontal Solution, 39

2.3 Conjugate Gradient Method for Equation Solving 39

Conjugate Gradient Algorithm, 40

3.3 Coordinates and Shape Functions 48

3.4 The Potential-Energy Approach 52

Element Stiffness Matrix, 53 Force Terms, 54

3.5 The Galerkin Approach 56

Element Stiffness, 56 Force Terms, 57

3.6 Assembly of the Global Stiffness Matrix and Load Vector 58

37 Propertiesof K 61 3.8 The Finite Element Equations: Treatment of Boundary Conditions 62

Types of Boundary Conditions, 62

Elimination Approach, 63

Penalty Approach, 69 Multipoint Constraints, 74

4.3 Three-Dimensional Trusses 114 4.4 Assembly of Global Stiffness Matrix for the Banded and Skyline Solutions 116

Assembly for Banded Solution, 116 Input Data File, 119

Isoparametric Representation, 135

Potential-Energy Approach, 139

Element Stiffness, 140 Force Terms, 141 Galerkin Approach, 146 Stress Calculations, 148

Temperature Effects, 150

5.4 Problem Modeling and Boundary Conditions 152

Some General Comments on Dividing into Elements, 154

Rotating Flywheel, 185 Surface Traction, 185

Stress Calculations, 190

Temperature Effects, 191

6.4 Problem Modeling and Boundary Conditions 191

Cylinder Subjected to Internal Pressure, 191 Infinite Cylinder, 192

Press Fit on a Rigid Shaft, 192 Press Fit on an Elastic Shaft, 193

Belleville Spring, 194

Thermal Stress Problem, 195

Input Data File, 197

73 Numerical Integration 214

Two-Dimensional Integrals, 217 Stiffness Integration, 217

Stress Calculations, 218

74 Higher Order Elements 220

Nine-Node Quadrilateral, 220 Eight-Node Quadrilateral, 222 Six-Node Triangle, 223

7.5 Four-Node Quadrilateral for Axisymmetric Problems 225 7.6 Conjugate Gradient Implementation

of the Quadrilateral Element 226

8.5 Shear Force and Bending Moment 245 8.6 Beamson Elastic Supports 247

9 THREE-DIMENSIONAL PROBLEMS IN STRESS ANALYSIS 275

91 Introduction 275 9.2 Finite Element Formulation 276

Element Stiffness, 279 Force Terms, 280

93 Stress Calculations 280 9.4 Mesh Preparation 281 9.5 Hexahedral Elements and Higher Order Elements 285 9.6 Problem Modeling 287

97 Frontal Method for Finite Element Matrices 289

Connectivity and Prefront Routine, 290

Element Assembly and Consideration of Specified dof, 290

Elimination of Completed dof 291

Backsubstitution, 291

Consideration of Multipoint Constraints, 291

Input Data File, 292

One-Dimensional Heat Conduction, 309

One-Dimensional Heat Transfer in Thin Fins, 316 Two-Dimensional Steady-State Heat Conduction, 320

Two-Dimensional Fins, 329

Preprocessing for Program Heat2D, 330

xii Contents

10.3 10.4

Acoustics, 343

Boundary Conditions, 344

One-Dimensional Acoustics, 344 1-D Axial Vibrations, 345 Two-Dimensional Acoustics, 348

11.3 11.4

11.5

11.6 11.7 11.8

Introduction 367

Formulation 367

Solid Body with Distributed Mass, 368

Element Mass Matrices 370

Evaluation of Eigenvalues and Eigenvectors 375

Properties of Eigenvectors, 376 Eigenvalue-Eigenvector Evaluation, 376 Generalized Jacobi Method, 382 Tridiagonalization and Implicit Shift Approach, 386 Bringing Generalized Problem to Standard Form, 386

Rigid Body Modes 394 Conclusion 396

Input Data File, 397

Problems 399

Program Listings, 404

nt

Region and Block Representation, 411

Block Corner Nodes, Sides, and Subdivisions, 412

Preface

The first edition of this book appeared over 10 years ago and the second edition fol- lowed a few years later We received positive feedback from professors who taught from the book and from students and practicing engineers who used the book We also benefited from the feedback received from the students in our courses for the past 20 years We have incorporated several suggestions in this edition The underly- ing philosophy of the book is to provide a clear presentation of theory, modeling, and implementation into computer programs The pedagogy of earlier editions has been retained in this edition

New material has been introduced in several chapters Worked examples and exercise problems have been added to supplement the learning process Exercise prob- lems stress both fundamental understanding and practical considerations Theory and computer programs have been added to cover acoustics, axisymmetric quadrilateral elements, conjugate gradient approach, and eigenvalue evaluation Three additional pro- grams have now been introduced in this edition All the programs have been developed

to work in the Windows environment The programs have a common structure that should enable the users to follow the development easily The programs have been pro- vided in Visual Basic, Microsoft Excel/Visual Basic, MATLAB, together with those pro- vided earlier in QBASIC, FORTRAN and C The Solutions Manual has also been updated

Chapter 1 gives a brief historical background and develops the fundamental con-

cepts, Equations of equilibrium, stress-strain relations, strain—displacement relations, and the principles of potential energy are reviewed The concept of Galerkin’s method

is introduced

Properties of matrices and determinants are reviewed in Chapter 2 The Gaussian

elimination method is presented, and its relationship to the solution of symmetric band-

ed matrix equations and the skyline solution is discussed Cholesky decomposition and

conjugate gradient method are discussed

Chapter 3 develops the key concepts of finite element formulation by consider-

ing one-dimensional problems The steps include development of shape functions,

derivation of element stiffness, formation of global stiffness, treatment of boundary conditions, solution of equations, and stress calculations Both the potential energy approach and Galerkin’s formulations are presented Consideration of temperature effects is included

xv

xvi

Preface Finite element formulation for plane and three-dimensional trusses is developed

in Chapter 4 The assembly of global stiffness in banded and skyline forms is explained Computer programs for both banded and skyline solutions are given

Chapter 5 introduces the finite element formulation for two-dimensional plane stress and plane strain problems using constant strain triangle (CST) elements Problem modeling and treatment of boundary conditions are presented in detail Formulation for orthotropic materials is provided Chapter 6 treats the modeling aspects of axisym- metric solids subjected to axisymmetric loading Formulation using triangular elements

is presented Several real-world problems are included in this chapter

Chapter 7 introduces the concepts of isoparametric quadrilateral and higher order elements and numerical integration using Gaussian quadrature Formulation for axi- symmetric quadrilateral element and implementation of conjugate gradient method for quadrilateral element are given

Beams and application of Hermite shape functions are presented in Chapter 8 The chapter covers two-dimensional and three-dimensional frames

Chapter 9 presents three-dimensional stress analysis Tetrahedral and hexahe-

dral elements are presented The frontal method and its implementation aspects are

discussed

Scalar field problems are treated in detail in Chapter 10 While Galerkin as well

as energy approaches have been used in every chapter, with equal importance, only Galerkin's approach is used in this chapter This approach directly applies to the given differential equation without the need of identifying an equivalent functional to mini-

mize Galerkin formulation for steady-state heat transfer, torsion, potential flow, seep-

age flow, electric and magnetic fields, fluid flow in ducts, and acoustics are presented Chapter 11 introduces dynamic considerations Element mass matrices are given

Techniques for evaluation of eigenvalues (natural frequencies) and eigenvectors (mode

shapes) of the generalized eigenvalue problem are discussed, Methods of inverse itera- tion, Jacobi, tridiagonalization and implicit shift approaches are presented

- Preprocessing and postprocessing concepts are developed in Chapter 12 Theory and implementation aspects of two-dimensional mesh generation, least-squares ap-

Proach to obtain nodal stresses from element values for triangles and quadrilaterals,

and contour plotting are presented

Prepare the data in an efficient manner

số We thank Nels Madsen, Auburn University; Arif Masud, University of Illinois, Chicago, Robert L Rankin, Arizona State University; John §, Strenkowsi, NC State Uni- versity, and Hormoz Zareh, Portland State University, who reviewed our second edi-

iS that helped us improve the book

sity who took his courses, He expresses hi

who gave valuable feedback after teaching a course from the second edition We thank

me

Preface xvii

our production editor Fran Daniele for her meticulous approach in the final produc-

tion of the book

Ashok Belegundu thanks his students at Penn State for their feedback on the

course material and programs He expresses his gratitude to Richard C Benson, chair- man of mechanical and nuclear engineering, for his encouragement and appreciation He also expresses his thanks to Professor Victor W Sparrow in the acoustics department and

to Dongjai Lee, doctoral student, for discussions and help with some of the material in

the book His late father’s encouragement with the first two editions of this book are an

ever present inspiration

We thank our acquisitions editor at Prentice Hall, Laura Fischer, who has made this

a pleasant project for us

TIRUPATHI R CHANDRUPATLA ASHOK D BELEGUNDU

xviii

About the Authors

Tirupathi R Chandrupatla is Professor and Chair of Mechanical Engineering at Rowan University, Glassboro, New Jersey He received the B.S degree from the Regional En-

gineering College, Warangal, which was affiliated with Osmania University, India He re-

ceived the M.S degree in design and manufacturing from the Indian Institute of

Technology, Bombay He started his career as a design engineer with Hindustan Machine Tools, Bangalore He then taught in the Department of Mechanical Engineering at I.L.T,

Bombay He pursued his graduate studies in the Department of Aerospace Engineer- ing and Engineering Mechanics at the University of Texas at Austin and received his Ph.D in 1977 He subsequently taught at the University of Kentucky Prior to joining

Rowan, he was a Professor of Mechanical Engineering and Manufacturing Systems

Engineering at GMI Engineering & Management Institute (formerly General Motors Institute), where he taught for 16 years

Dr Chandrupatla has broad research interests, which include finite element analy-

sis, design, optimization, and manufacturing engineering He has published widely in these areas and serves as a consultant to industry Dr Chandrupatla is a registered Pro- fessional Engineer and also a Certified Manufacturing Engineer He is a member of ASEE, ASME, NSPE, SAE, and SME

Ashok D Belegundu is a Professor of Mechanical Engineering at The Pennsylvania

State University, University Park He was on the faculty at GMI from 1982 through

1986 He received the Ph.D degree in 1982 from the University of Iowa and the B.S de-

gree from the Indian Institute of ‘Technology, Madras He was awarded a fellowship to spend a summer in 1993 at the NASA Lewis Research Center, During 1994-1995, he

obtained a grant from the UK Science and Engineering Research Council to spend his

sabbatical leave at Cranfield University, Cranfield, UK

Dr Belegundu’s teaching and r jects for government and industry He is an associat € editor of Mechani Structures and Machines, He is also a member of ASME and a: in Associate fellow of AIAA is OF Sire

Denice

Introduction to Finite Elements in Engineering

The finite element method has become a powerful tool for the numerical solution of a

wide range of engineering problems Applications range from deformation and stress

analysis of automotive, aircraft, building, and bridge structures to field analysis of heat flux, fluid flow, magnetic flux, seepage, and other flow problems With the advances in computer technology and CAD systems, complex problems can be modeled with rela- tive ease Several alternative configurations can be tested on a computer before the first prototype is built All of this suggests that we need to keep pace with these develop- ments by understanding the basic theory, modeling techniques, and computational as-

pects of the finite element method In this method of analysis, a complex region defining

a continuum is discretized into simple geometric shapes called finite elements The ma- terial properties and the governing relationships are considered over these elements and expressed in terms of unknown values at element corners An assembly process,

duly considering the loading and constraints, results in a set of equations Solution of

these equations gives us the approximate behavior of the continuum

1.2 HISTORICAL BACKGROUND

Basic ideas of the finite element method originated from advances in aircraft structur-

al analysis In 1941, Hrenikoff presented a solution of elasticity problems using the

“frame work method.” Courant’s paper, which used piecewise polynomial interpolation over triangular subregions to model torsion problems, appeared in 1943 Turner, et al

derived stiffness matrices for truss, beam, and other elements and presented their find- ings in 1956 The term finite element was first coined and used by Clough in 1960

In the early 1960s, engineers used the method for approximate solution of prob-

lems in stress analysis, fluid flow, heat transfer, and other areas A book by Argyris in 1955

on energy theorems and matrix methods laid a foundation for further developments in finite element studies The first book on finite elements by Zienkiewicz and Cheung was published in 1967 In the late 1960s and early 1970s, finite element analysis was applied

to nonlinear problems and large deformations Oden’s book on nonlinear continua ap- peared in 1972

2 Chapter 1 Fundamental Concepts

Mathematical foundations were laid in the 1970s New element development, con-

vergence studies, and other related areas fall in this category

Today, the developments in mainframe computers and availability of powerful mi- crocomputers has brought this method within reach of students and engineers working

in small industries

1.3 OUTLINE OF PRESENTATION

In this book, we adopt the potential energy and the Galerkin approaches for the pre-

sentation of the finite clement method The area of solids and structures is where the method originated, and we start our study with these ideas to solidify understanding For this reason, several early chapters deal with rods, beams, and elastic deformation prob-

lems The same steps are used in the development of material throughout the book, so that the similarity of approach is retained in every chapter The finite element ideas are

then extended to field problems in Chapter 10 Every chapter includes a set of problems

and computer programs for interaction

We now recall some fundamental concepts needed in the development of the fi- nite element method

1.4 STRESSES AND EQUILIBRIUM

A three-dimensional body occupying a volume V and having a surface $ is shown in

Fig 1.1 Points in the body are located by x, y, z coordinates, The boundary is con- strained on some region, where displacement is specified On part of the boundary, dis-

FIGURE 1.1 Three-dimensional body

_mm==—=—_

Section 1.4 Stresses and Equilibrium 3

tributed force per unit area T, also called traction, is applied Under the force, the body deforms The deformation of a point x ( = [x, y,z]") is given by the three components

The body force acting on the elemental volume dV is shown in Fig 1.1 The surface trac-

tion T may be given by its component values at points on the surface:

T =[7,.7,„TJ (3)

Examples of traction are distributed contact force and action of pressure A load P act-

ing at a point i is represented by its three components:

P= (PPP IF (14)

‘The stresses acting on the elemental volume dV are shown in Fig 1.2 When the volume

dV shrinks to a point, the stress tensor is represented by placing its components in a

FIGURE 1.2 Equilibrium of elemental volume,

4 Chapter 1 Fundamental Concepts

{3 X 3) symmetric matrix However, we represent stress by the six independent com-

ponents as in

© = [05,0 Op, TTT ry) (L5)

where o,,0y, 0, are normal stresses and 7y,, 7;;, Txy, are shear stresses, Let us consid-

er equilibrium of the elemental volume shown in Fig, 1.2 First we get forces on faces by multiplying the stresses by the corresponding areas Writing LF, = 0, =F, = 0, and

ZF, = Oand recognizing dV = dx dy dz, we get the equilibrium equations

aa, Ủy | aT,

Referring to Fig 1.1, we find that there are displacement boundary conditions and sur-

face-loading conditions If wis specified on part of the boundary denoted by S,, we have

u=donS, (1.7)

‘We can also consider boundary conditions such as u = a, where ais a given displacement

We now consider the equilibrium of an elemental tetrahedron ABCD, shown in Fig 1.3, where DA, DB, and DC are parallel to the x-, y-, and Z-axes, respectively, and area ABC, denoted by dA, lies on the surface Ifn = [m„, ty, n,]T is the unit normal to

aA, then area BDC = n,dA, area ADC = ny@A, and area ADB = n,dA Considera-

tion of equilibrium along the three axes directions gives

On, + TyMy + Tyn, = T,

Sy, where the tractions are applied

as loads distributed over small, but

1.6 STRAIN-DISPLACEMENT RELATIONS

We represent the strains in a vector form that corresponds to the stresses in Eq.15,

© [es 6s 24 Myce Vaes Yay] (L9)

where e,,¢,, and e, are normal strains And y,,,„,„

Figure 1.4 gives the deformation of the dx

we consider here Also considering other faces,

and y,, are the engineering shear strains

¬dy face for small deformations, which

we can write

es

6 Chapter 1 Fundamental Concepts

du av dw av | aw au | aw au | avy" (1.10)

ox’ dy” az az ay’ dz ax’ dy ủx

These strain relations hold for small deformations

1.7 STRESS-STRAIN RELATIONS

For linear elastic materials, the stress-strain relations come from the generalized Hooke’s law For isotropic materials, the two material properties are Young’s modulus (or mod- ulus of elasticity) E and Poisson’s ratio » Considering an elemental cube inside the

body, Hooke’s law gives

Section 1.7 Stress-Strain Relations 7 Special Cases

One dimension In one dimension, we have normal stress ¢ along x and the

corresponding normal strain ¢, Stress-strain relations (Eq 1.14) are simply

o=Ee (1.16) Two dimensions In two dimensions, the problems are modeled as plane stress

and plane strain

Plane Stress A thin planar body subjected to in-plane loading on its edge sur- face is said to be in plane stress, A ring press fitted on a shaft, Fig, 1.5a, is an example Here Stresses o,,7,,, and 7,, are set as zero The Hooke’s law relations (Eq 1.11) then give us

-

t

| 8 Chapter1 Fundamental Concepts

The inverse relations are given by

which is used as o = De

Plane Strain Ya long body of uniform cross section is subjected to transverse loading along its length, a small thickness in the loaded area, as shown in Fig 1.5b,can

be treated as subjected to plane strain Here €,, y.,, y,, ate taken as zero Stress ơ; may not be zero in this case The stress-strain relations can be obtained directly from Egs 1.14 and 1.15:

%7{1xzz0ñ-z)j| 7? 1y 9 5 (119)

D here is a (3 X 3) matrix, which relates three stresses and three strains,

Anisotropic bodies, with uniform orientation, can be considered by using the ap- propriate D matrix for the material

1.8 TEMPERATURE EFFECTS

If the temperature rise AT (x, y, z) with respect to the original state is known, then the associated deformation can be considered easily For isotropic materials, the tempera- ture rise AT results in a uniform strain, which depends on the coefficient of linear ex-

pansion a of the material a, which represents the change in length per unit temperature

rise, is assumed to be a constant within the range of variation of the temperature Also,

this strain does not cause any stresses when the body is free to deform The temperature

strain is represented as an initial strain:

€o = [@AT, @AT, @AT,0,0,0]" (4.20)

The stress-strain relations then become

For plane stress and plane strain, note that o = [

and that D matrices are as given in Eqs 1.18 and 1

T

Ơ.,Øy,T„]Ï and e = [é„,6„YyÌ"

-19, respectively

Section 1.9 Potential Energy and Equilibrium; The Rayleigh-Ritz Method 9 1.9 POTENTIAL ENERGY AND EQUILIBRIUM;

THE RAYLEIGH-RITZ METHOD

In mechanics of solids, our problem is to determine the displacement u of the body shown

in Fig 1.1, satisfying the equilibrium equations 1.6 Note that stresses are related to strains,

which, in turn, are related to displacements This leads to requiring solution of second-

order partial differential equations Solution of this set of equations is generally referred

to as an exact solution Such exact solutions are available for simple geometries and load-

ing conditions, and one may refer to publications in theory of elasticity For problems of

complex geometries and general boundary and loading conditions, obtaining such solutions

is an almost impossible task Approximate solution methods usually employ potential en-

ergy or variational methods, which place less stringent conditions on the functions

Potential Energy, I

The total potential energy IT of an elastic body, is defined as the sum of total strain

energy (U) and the work potential:

TI = Strain energy + Work potential

(U) (WP) (124)

For linear elastic materials, the strain energy per unit volume in the body is 307e For

the elastic body shown in Fig 1.1, the total strain energy U is given by

U= ; | eteav 2w (125)

The work potential WP is given by

WP=- [ota - [sms - XuP, (126)

The total potential for the general elastic body shown in Fig, 1.1 is

1 =3 [oreav- [wea - [was - Sule, (1.27)

V vo ‘Ss +

We consider conservative systems here, where the work potential is independent

of the path taken In other words, if the system is displaced from a given configuration

and brought back to this state, the forces do zero work regardless of the path The po-

tential energy principle is now stated as follows:

Principle of Minimum Potential Energy

For conservative systems, of all the kinematically admissible displacement fields,

those corresponding to equilibrium extremize the total potential energy If the

extremum condition is a minimum, the equilibrium state is stable

10 Chapter 1 Fundamental Concepts

Kinematically admissible displacements are those that satisfy the single-valued nature of displacements (compatibility) and the boundary conditions In problems where

displacements are the unknowns, which is the approach in this book, compatibility is

automatically satisfied

To illustrate the ideas, let us consider an example of a discrete connected system

Example 1.1

Figure E1.1a shows a system of springs The total potential energy is given by

H= ;kiêi + šoổi + 2koốế + 2kBf — Rin — Fay

where ô;, ô;, ô;, and 5, are extensions of the four springs Since 8; = q, — đa, Ô; = 4› 5; = 4, — q, and 8, = —g;, we have

Th = Shula ~ 42)? + 3hoa} + dhslgs — a2)? + $kag3 — Fas — Fags

where q;, @, and g, are the displacements of nodes 1,2, and 3, respectively

For equilibrium of this three degrees of freedom system, we need to minimize I] with

Tespect to đ¡, đ;, and g; The three equations are given by

aq KN &) + kage ~ bsg - ga) = 0

oT kl - ag, BG 7 &) + kegs - B= 0

‘These equilibrium equations can be put in the form of Kg = Fas follows:

ky ky 0 4 F,

Th Ath thy ch ay=4o (1.29)

If, on the other hand, we Proceed to write oon the equilibri i -

sidering the equilibrium of each separate non Sduafions oƒ the system by com

node, as shown in Fig E1.1b, we can write

ec

Section 1.9 Potential Energy and Equilibrium; The Rayleigh-Ritz Method 14

ko, =F ksỗ; ~ kiỗi — kạô; = 0

Rayleigh-Ritz Method For continua, the total potential energy II in Eq 1.27 can be used for finding an ap-

proximate solution The Rayleigh-Ritz method involves the construction of an assumed displacement field, say,

u= Sae{x.y.z) i =1to€

v= Dad,(x, y,z) j=£+1tom (130)

wr Sad, yz) k=m + lton

n>m>€é The functions ¢; are usually taken as polynomials Displacements x, v, w must be kine- matically admissible That is, u, », w must satisfy specified boundary conditions Intro-

ducing stress-strain and strain-displacement relations, and substituting Eq 1.30 into

Eq 1.27 gives

TI = M(a,,42, ,4,) (1.31) where r = number of independent unknowns Now, the extremum with Tespect to a;,

( = 1 tor) yields the set of r equations

iy;

oa, (1.32)

Deen cee

ET

Section 1.10 Galerkin's Method = 13

Then du/dx = 2a,(—1 + x) and

where the coefficients ; are arbitrary, except for requiring that $ satisfy homogeneous

(zero) boundary conditions where i is prescribed The fact that ¢ in Eq, 1.37 is constructed ina similar manner as i in Eq 1.36 leads to simplified derivations in later chapters

Galerkin’s method can be stated as follows:

Usually, in the treatment of Eq 1.38 an integration by parts is involved The order of the

derivatives is reduced and the natural boundary conditions, such as surface-force con- ditions, are introduced

Galerkin’s method in elasticity Let us turn our attention to the equilibrium

equations 1.6 in elasticity Galerkin’s method requires

Oa, Oty | Te Ory 8a, ar

Section 1.10 Galerkin’s Method 15

is an arbitrary displacement consistent with the boundary conditions ofu.Ifn = [1,,7,,#,]"

is a unit normal at a point x on the surface, the integration by parts formula is

[e4 =- [zue+ {me (1.40)

where a and @ are functions of (x, y, z) For multidimensional problems, Eq 1.40 is usu-

ally referred to as the Green~Gauss theorem or the divergence theorem Using this for-

mula, integrating Eq 1.39 by parts, and rearranging terms, we get

_ {z«0a + [ea + [low + nyt + NA) v Vv ý by

cí@) = l# ad, ab, Ad, + a6 ag, + 99; aby + ef (142)

ax" ay’ az’ az ay” az ax? ay ax

is the strain corresponding to the arbitrary displacement field d

On the boundary, from Eq 1.8, we have (n,o, + m,7„y + n;7¿,) — T,, and so on

At point loads (n,0, + n,t,, + 1,7,,) dS is equivalent to P,, and so on These are the

natural boundary conditions in the problem Thus, Eq 1.41 yiclds the Galerkin’s “vari-

ational form” or “weak form” for three-dimensional stress analysis:

where # is an arbitrary displacement consistent with the specified boundary conditions

of u We may now use Eq 1.43 to provide us with an approximate solution

For problems of linear elasticity, Eq 1.43 is precisely the principle of virtual work

¢ is the kinematically admissible virtual displacement The principle of virtual work may

be stated as follows:

Principle of Virtual Work

A body is in equilibrium if the internal virtual work equals the external virtu-

al work for every kinematically admissible displacement field (@, €())

We note that Galerkin’s method and the principle of virtual work result in the same set of equations for problems of elasticity when same basis or coordinate func-

tions are used Galerkin’s method is more general since the variational form of the type

Eq 1.43 can be developed for other governing equations defining boundary-value prob-

lems Galerkin’s method works directly from the differential equation and is preferred

to the Rayleigh-Ritz method for problems where a corresponding function to be min-

imized is not obtainable

d du u=0 atx=0

489 „=0 atx=2

Multiplying this differential equation by ¢, and integrating by parts, we get

2 ~ eat ta (gentt) « (seat) =

1.11 SAINT VENANT’S PRINCIPLE

We often have to make approximations in defining boundary conditions to represent a support-structure interface For instance, consider a cantilever beam, free at one end

and attached to a column with rivets at the other end Questions arise as to whether the riveted joint is totally rigid or partially rigid, and as to whether each point on the cross section at the fixed end is specified to have the same boundary conditions Saint Venant considered the effect of different approximations on the solution to the total problem Saint Venant $ principle states that as long as the different approximations are statical-

ly equivalent, the resulting solutions will be valid Provided we focus on regions suffi-

ciently far away from the support That i ent 3 h - is, the sol i i ifi i

within the immediate vicinity of the support hon may significantly difer ony

1.12

Section 1.13 Computer Programs 17

VON MISES STRESS

Von Mises stress is used as a criterion in determining the onset of failure in ductile ma- terials The failure criterion states that the von Mises stress oy,, should be less than the yield stress oy of the material In the inequality form, the criterion may be put as

Ovm = oy (1.44) The von Mises stress oyy is given by

where J, and J, are the first two invariants of the stress tensor For the general state of

stress given by Eq 1.5, 7, and ï; are given by

l=ơ,+ơy+ơ,

L= a0, + oo, + 0,0, — Ty — The ~ Thy (1.46)

In terms of the principal stresses 71, a, and ơ;, the two invariants can be written as

h =ơi +ơa+ơi

šb = 0iØ¿ + ØđạØy + 030;

It is easy to check that von Mises stress given in Eq 1.45 can be expressed in the form

Loos

vụ = (ơi — ơ¿}” + (a, — 3) + (ơy — ơi} (1.47)

For the state of plane stress, we have

h =ơ,+ gy

L= ow, ty (1.48) and for plane strain

h=ơ,+ơy+ơ,

41.49)

= 2

1b = đ,ữy + 0yơ, + ơ,đy — Tây

where ơ, = (ơ, + ơ,)

1.13 COMPUTER PROGRAMS

Computer use is an essential part of the finite element analysis Well-developed, well- maintained, and well-supported computer programs are necessary in solving engineer- ing problems and interpreting results Many available commercial finite element packages fulfill these needs, It is also the trend in industry that the results are acceptable only when solved using certain standard computer program packages The commercial packages provide user-friendly data-input platforms and elegant and easy to follow dis- play formats However, the packages do not provide an insight into the formulations and solution methods Specially developed computer programs with available source codes enhance the learning process We foliow this philosophy in the development of this

hi

18 Chapter 1 Fundamental Concepts

book Every chapter is provided with computer programs that parallel the theory The curious student should make an effort to see how the steps given in the theoretical de- velopment are implemented in the programs Source codes are provided in QBASIC, FORTRAN, C, VISUALBASIC, Excel Visual Basic, and MATLAB Example input and output files are provided at the end of every chapter We encourage the use of com- mercial packages to supplement the learning process

1.14 CONCLUSION

In this chapter, we have discussed the necessary background for the finite elemeat method We devote the next chapter to discussing matrix algebra and techniques for solving a set of linear algebraic equations

HISTORICAL REFERENCES

1 Hrenikoff, A., “Solution of problems in elasticity by the frame work method,” Journal of Ap- plied Mechanics, Transactions of the ASME 8: 169-175 (1941)

2 Courant, R., “Variational methods for the solution of problems of equilibrium and vibra-

tions.” Bulletin of the American Mathematical Society 49: 1-23 (1943)

3 Turner, M.J.,R W Clough, H C Martin, and L J Topp, “Stiffness and deflection analysis of complex structures,” Journal of Aeronautical Science 23(9); 805-824 (1956)

4 Clough, R W.,“‘The finite element method in plane stress analysis.” Proceedings American So-

ciety of Civil Engineers, 2d Conference on Electronic Computation, Pittsburgh, Pennsylva- nia, 23; 345-378 (1960)

5 Argyris J.H.,“Energy theorems and structural analysis.” Aircraft Engineering, 26: Oct—Nov., 1954; 27: Feb~May, 1955

6 Zienkiewicz, O C., and Y.K Cheung, The Finite Element Method in Structural and Continu-

um Mechanics (London: McGraw-Hill, 1967)

Oden, IT, Finite Elements of Nonlinear Continua (New York: McGraw-Hill, 1972)

PROBLEMS

1.1 Obtain the D matrix given b

(Eq 1.11)

12 Ina plane strain problem, we have

y Eq 1.15 using the generalized Hooke’s law relations

a, = 20.000 psi, o, = ~10000 psi

E = 30 X 10p, p = 03

Determine the value of the stress a

1.3 Ifa displacement field is described by

w= (~x? + 2y? + Gxy)104

v= (3x + 6y — y*)10-4

determine e,, e,, „y at the point x = 1, y=0

1.4 Develop a deformation field u(x, y) › Đ(x, y) that describ i inite

element shown From this determin: es the deformation of the fin

© €x> €v, Yey Interpret your answer

Dene —

FIGURE P1.5

(a) Write down the expressions for ¢,, ¢, and y,,

{b) Plot contours of ¢,, ¢,, and y,, using, say, MATLAB software,

(c) Find where e, is a maximum within the square

6 In a solid body, the six components of the stress at a point are given by 7, = 40 MPa,

o, = 20MPa,c, = 30 MPa,r,, = —30MPa,7,, = 15MPa,andr,, = 10 MPa Determine

the normal stress at the point, on a plane for which the normal is (7,, 1,7.) = ($,4,1/V2)

(Hint: Note that normaal stress ¢, = T,n, + Tyny + Tanz)

1.7 For isotropic materials, the stress-strain relations can also be expressed using Lame’s con-

stants A and 1, as follows:

o, = hey + 2pe,

oy = dey + 2.6,

o, = Ae, + 20,

Tyy = BY ues Tre = MY az Tay = BY xy

Here «, = €, + €, + €, Find expressions for A and yw in terms of E and »

20 Chapter 1 Fundamental Concepts

1⁄8, A long rod is subjected to loading and a temperature increase of 30°C The total strain at

a point is measured to be 1.2 x 10° If E = 200 GPa and a = 12 x 10°/°C, determine

the stress at the point

1.9 Consider the rod shown in Fig P1.9, where the strain at any point x is given by

€, = 1 + 2x7 Find the tip displacement 8

z rm eae

FIGURE P1.11

1.12 A rod fixed at its ends is subjected to a vai method with an assumed displacement ment u(x) and stress a(x)

L13 Use the Rayleigh-Ritz method to find the dis

Element 1 is made of aluminum, and elemen

rying body force as shown Use the Rayleigh-Ritz

field u = a) + a,x + ax? to determine displace- placement field u(x) of the rod in Fig, P1.13

t 2 is made of steel The Properties are

Ey = 70GPa, A, = 900mm?, L, = 200mm

Ex, = 200GPa, A, = 1200mm?, L, = 300mm

1.14 Use Galerkin’s method to find the displacement at the midpoint of the rod (Figure P1.11)

115 Solve Example 1.2 using the potential energy approach with the polynomial u = a, + a2x

(b) Determine the displacement u(x) using the Rayleigh—-Ritz method Assume a dis-

placement field u(x) = ag + a)x + a)x* Plot u versus x

{€) Plot o versus x

1.17 Consider the functionat for minimization given by

“ fdy\2

1= ƒ (2) dx + $h(ay ~ 800)?

with y = 20atx = 60 Givenk = 20,4 = 25, and L = 60, determine ap, a; and a; using

the polynomial approximation y(x) = ay + a,x + a2x in the Rayleigh-Ritz method