Tài liệu finite element methods for structural engineers ppt

General Approach—Using Displacement or Shape Functions 29 44 Transformation of Reference Coordinate Systems 33 Chapter 5 DERIVATION OF STIFFNESS PROPERTIES FOR SL Pin-Jointed Plane Bar E

Finite Element Method

for Structural Engineers (A Basic Approach)

“The book presents the basic ideas of the finite element method so that it can be Used as a

textbook in the curriculum for undergraduate and graduate engineering courses In the

presentation of fundamentals and derivations care had been taken not to use an advanced

‘mathematical approach, rather the use of matrix algebra and calculus is made Further effort

isbeing made o include the intricacies ofthe computer programming aspect, rather the material

is presented in a mannersso that the readers can understand the basic principles using hand

alculations However, a list of computer codes is given, Several ilustralive examples are

presented in a detailed stepwise manner to explain the various steps in the application ofthe

method A fairly comprehensive references listat the end of each chaplet is given foradditional

information and further study,

Wall N AL-Rifale is Professor of Civil Engineering at the University of Technology, Baghdad, Iraq He obtained his Ph.D ftom the University College, Cardiff, U.K in 1978 Dr Wail

established the Civil Engineering Department at the Engineering College in Baghdad and was:

the Head for nearly seven years He received the Telford Premium Prize from the Inslitution of

Civil Engineering (London) in 1976 His main areas of research are: Box girder bridges, folded

plate siructures, frames and shear walls including dynamic analysis, He is the author of three

books on structural analysisinArabic:

‘Ashok K Govil is Professor in the Department of Applied Mechanies, Motilal Nehru Regional

Engineering College, Allahabad, India and was also Head of the same department for over five

years He oblained B.E degree in Civil Engineering (1963) from BITS, Pilani, India, and M.S

(4969) and Ph.D., (1977) from the University oflawa, Iowa City, U.S.A, Dr Govil's main areas of

fesearch are: Optimal design of structures, fal-safe design of structures, and finite element

method He has writen several research papers and technical reports, and developed many

‘computer programmes for optimal design of structures including dynamic analysis end

‘ulnerabilly reduction,

(SEN: 978412242410

[Ill

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Finite Element

Method for Structural Engineers

(A Basic Approach)

Wail N Al-Rifaie

Ashok K Govil

Copyright © 2008, New Age International (P) Ltd., Publishers

Published by New Age International (P) Lid,, Publishers

First Edition: 2008

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PREFACE

‘The finite clement method is one of the most popular numerieal tech- niques used for obtaining an approximate solution of complex problems in various fields of engineering In the beginning, the method was developed

fs an extension of matrix methods for the analysis of structural engineering problems However later it has also been recognized as a most powerful

‘method for analysing problems in other fields of engincering, suck as duid mechanies, soil mechanics, rock mechanics, heat low, etc The generality of its application coupled with the availabilty of high speed electronic digital computers, has put finite clement method in wide use It has also been in- cluded in the curriculum of engineering colleges

The aim of this book is to present the basic ideas of the finite element method so that it ean be used as a text book in the eurrievlum for under- graduate and graduate engineering courses Tn the presentation of funda- mentals and derivations, care has been taken to use matrix algebra and calculus only, rather than an advanced approach It may be mentioned further that the computer programming aspect has not been included but the material is presented in such a manner that the readers ean understand the basic principles using hand calculations However, a list of computer codes is given in Appendix G

The book is divided into eight chapters The first Chapter intreduees the basic concepts of the finite clement method and in Chapter 2, basic equa- tions of elasticity are presented Chapter 3 discusses the structural idealiza- tion and also describes the commonly used elements in structural aualysis, In Chapter 4, methods for determining stlfiwess characteristics und transform= ation of matrices are given In Chapters Sand 6, element stiffness properties are derived In Chapter 7, general formulation of the finite clement method

is presented with the help of suitable examples Finally in Chapter 8,

‘examples are presented to illustrate the various basie steps in the appl tion of the method The references listed at the end of cach cltupter are those in which readers can find additional information or detailed developments,

Tn the end, six appendices ave provided which give details of matrix algebra, reactive forces, section properties, solution of simultancous equations, SI units and bibliography of some additional references

22 Stress and Strain Components _9

2.3 Equations of Static Equilibrium 11

2.4 Strain—Displacement Equations 13

2.5 Compatibility Equations 13

.6 Generalized Hooke's Law (Constitutive Equations) 14

2⁄1 Plane Strain and Plane Stress 15

References 18

3.1 Inbodueion 19

ization or Diseretization 19 3.3 Types of Structural Elements 22

References 26

Chapter 4 METHODS OF DETERMINING STIFFNESS

PROPERTIES AND ITS TRANSFORMATION 21

41 Introduelion 27

4/2 Biementay Approach Using Basic Definition 28

vi Contents

4.3 General Approach—Using Displacement or Shape Functions 29

44 Transformation of Reference Coordinate Systems 33

Chapter 5 DERIVATION OF STIFFNESS PROPERTIES FOR

SL Pin-Jointed Plane Bar Elements 40

5.2 Plane Ream Elements 49

53 Pin-Jointed Space Bar Elements 68

54 Space Beam Elements 71

References 75

Chapter 6_DERIVATION OF STIFFNESS PROPERTIES FOR

6.1 Triangular Plate Elements (In-Plane Forces) 76

62_Rectangular Plate Elements (In-Plane'Forces) 84

63 Triangular Plate Elements (In-Bending) 92

64 Rectangular Plate Elements ([n-Bending) 99

References 108

Chapter 7_ FORMULATION OF FINITE ELEMENT METHOD 109

74 Assembly of the Overall Stiffness Matrix 116

15 Elimination of Restrained Degrees of Freedom 122

7.6 Determination of Equivalent Applied Nodal Forces 123

3.1 Calculation of Nodal Displacements, Forces and Stresses 130

References l3}

Chapter §_ APPLICATION OF FINITE ELEMENT METHOD 133 81_Analvsis of PinJointed Structure 133

8.2 Analysis of Beam _148

‘Contents 8.3 Analysis of Deep Beam 157

Problems 162

References 167

Appendix B_ REACTIVE FORCES FOR RESTRAINED BEAM ELEMENTS 18

Appendix D_ SOLUTION OF SIMULTANEOUS EQUATION — 186 Appendix E_ SI UNITS FOR STRUCTURAL ENGINEERS 192

CHAPTER 1

INTRODUCTION TO FINITE ELEMENT METHOD

1.1 INTRODUCTION AND GENERAL DESCRIPTION

‘The finite clement method represents an extension of matrix methods for the analysis of framed structures to the analysis of the continuum struc- tures The basic philosophy of this method is to replace the structure or the continuum having an unlimited or infinite number of unknowns by a

‘mathematical model which has a limited or finite number of unknowns at certain chosen discrete points The method isextremely poworful as it helps

to accurately analyse structures with complex geometrical properties and loading conditions In finite element method, a structure or a continuum, as shown in Fig 1.1(@) and (b), is discretized and idealized by using a mathematical model which is an assembly of subdivisions or discrete elements These discrete elements, known as finite clements, arc assumed to be interconnec- ted only at the joints called nodes Simple functions, such as polynomials, are chosen in terms of unknown displacements (endjor theit derivatives) at the nodes to approximate the variation of the actual displacements over each finite element The external loading is also transformed into equiva~ Tent forces applied at the nodes Next, the behaviour of each element independently and later as an assembly of these elements is obtained by relating their response to that of the nodes in such a way that the following basic conditions are satisfied at each node:

1, The equations of equil

2 The compat 3 The material constitutive relationship

‘The equations, which are obtained using the above conditions, are in the form of force-displacement relationship Finally, the force-displacement

‘equations are modified to incorporate the given boundary conditions and then these modified equations are solved to obtain displacements tthe nodes which are the basic unknowns in the finite element method An example illustrating the basic procedure of the method described above is presented in the subsequent section.

2 Finite Element Method for Structural Engineers

Introduction to Finite Element Method 3

2A simple linear spring eystem,

This system has two linear springs connected in series with spring sifnestes

%¿ and kạ The left-hand end is rigidly fixed while the right-hand ends free to move Further, it is assumed that under the action of applied load, these springs can have displacements in x-coordinate direction Thus, forces, displacements and spring stiffnesses are the only parameters in this system, Also fora linear spring, the applied force F is proportional to the resulting displacement A and may be expressed as

where k is the constant of proportionality which defines the stiffness of the spring, Thus, knowing the value of k and F in Eq (1.1), the displacement will be given by

2)

which is sufficient to describe fully the deformed state of an elastic spring

‘Now considering the system in Fig 1.2, the first step in the finite ele- ment method is to subdivide the system into diserete elements Defining each spring to be an element, the idealized linear spring system consists of

‘two elements and three nodes as shown in Fig 1.3 The forces (F,, Fy Fs) and the associated displacements (A,, Ay, 4s) at the nodes are also shown

in the figure For convenience the direction of the forces and the displace ments shown in Fig, 1.3 are taken as positive

J ttementcen Element (ep?

Fig 1.3 Discretization of the system

Next, in order to obtain the response of each clement, free body diagrams of isolated spring elements are considered, as shown lá,

Tt may be noted that for these elements, interpolation functions to describe the displacements over the clement are not needed and may be obtained directly using Eq (1 2) Now, assume that node (1) is displaced by A, due

to force Fy with node (2) fixed in position as shown in Fig 1.5(a) Using

Eq, (I.1), the force-displacement relation for the node (1) may be expressed

as

4 Finite Element Method for Structural Bagincers

where Fy is the reactive force at node (2) due to A, at node (1)

Similarly, now assume that node (1) is fixed and node (2) is displaced

by 4, due to force Fay applied at node (2), as shown in Fig 1.5 (b) Then,

‘the force-displacement equation at node (2) may be expressed as

Introduction to Finite Element Method 5 faa

where Fig is the reactive force at node (1) due to A, at node (2)

Now using the principle of superposition, one obtains the case shown

in Fig 1.5 (©) i-c., adding up algebraically the two cases shown in Fig

1.5(a) and (b) Thus, we obtain

and

where F, and F, are the forces at node (1) and (2), respectively Substitut- ing the Values of Fy, Fay Fx and Fy in Eqs (1.9) and (1.10), we obtain the

forces in terms of displacements as

{Fy = fayeoyaye (4)

Equation (1.14) defines the clement force-displacement or equilibrium equation for element (e,) In this equation, square matrix [A}*? is known as clement stiffness matrix; the column vectors {4} and (F)( are defined as nodal displacement and nodal force vectors, respectively ‘The superscript defines the element number

Following a similar approach as for element (c,), the force-displace- ment relationship for element (e,) may be expressed as

Fy of fe =®] (Ai

condensed form

‘Once the equilibrium or force-displacement equation for cach element

is known, the equation for the whole system may then be obtained by following the principle of superposition

6 Finite Element Method for Structural Engineers

‘Although, the matrices J? and (Ff? are of the same order, they may not be added directly as they relate to different sets of displacements For this simple example, the equilibrium equations may easily be expanded by inserting rows and columns of zeros in such a way that both sets of equa~ tions are related to all the possible displacements (ic As, Ay and 1) of the system Thus, Eqs (1.13) and (1.15) after expansion, may be expres- sed as:

For element (e)

Ry [ke —h 07 (ay at=|—k & 0] 4a,t, and 19)

ml Lo oo od bs For clememt (2)

A) fo 0 0] gai a†=|l0 ky —| {Ai di

xi lo —k, kel Uy

Using the principle of superposition and applying the rule of matrix addi- tion (see Appendix A) we obtain

A) [kh 0) [An E=|—k k+th =H t d19)

RJ Lo TH ky) HA,

It may be pointed out that the procedure of expanding element equilibrium

‘equations is lengthy and hence direct superposition is used as explained in Chapter 7

In condensed form, Eq (1.19) may be expressed as

“The matrix equation (1.20) represents the equilibrium or force-isplace-

‘ment equation for the linear spring system shown in Fig 1.2 In this equa- tion, square matrix [K] is known as overall stiffness matrix and is elements are usually subseripted as ky to denote their location in the ith row and Rh column; the column vectors {A} and {F) are known as nodal displace- ‘ment and nodal force vectors, respectively

It may be noted that the form of equilibrium or force-dsplacement equation (1,20) remains the same, irespectve of the type of problem and idealization.

Introduction to Finite Element Method 7

Now to obtain the solution of the problem, we use Eq (1.19) or its

‘condensed form Eq, (1.20), It may be noted that so far no limitations has been placed on any of the displacements 44 and 3 Hence, application

‘of aay external loading will result in moving the system as & rigid body

‘Thus, before solving for unknown displacements, Eq (1.19) needs to be modified to incorporate boundary conditions so as to prevent the rigid body ‘motion of the structure For the linear spring system shown in Fig 1.2, node (I) is fixed, ic A, ~ 0; hence rewriting Eq (1 19) in partitioned form

Pea = Ky (Oa — a) (128) where Ay =0; By and by are given by Eqs (1.24) and (1.25), respectively.

8 Finite Element Method for Structural Engineers

‘This completes the solution using finite element method It may be men- tioned that the finite element method involves extensive computations,

‘mostly repetitive in nature Hence the method is suited for computer pro-

‘gramming and solutions of the problems can be obtained easily using pro- gramming on electronic digital computers However, in this book, without Boing into the intricacies of computer programming, the basic concepts and the development of the method are presented in a simple manner Further for easy understanding of the various steps in the method, illustrative examples with hand catculations are given in Chapter 8 These examples ate taken from the field of structural engineering, nevertheless the method is general and can also be applied equally well to other fields of engineering

CHAPTER 2

BASIC EQUATIONS FROM LINEAR

ELASTICITY THEORY

24 INTRODUCTION

To provide a ready reference for the development of the general theory

of the finite clement method applied to the problems of structural analysis, some of the concepts and basic equations of linear elasticity theory are summarized in this chapter The equations are given without derivation or proof and apply to homogeneous isotropic materials 2.2 STRESS AND STRAIN COMPONENTS

10 Finite Element Method for Structural Engincers

they act over the surface of the body, they are called surface forcesand are expressed in corms of force per unit area; if they are distributed throughout the volume of the body, they are called body forces and are expressed in terms of force per unit volume For example, a force such a5 the hydro- static pressure which is distributed over the surface of a body, is called a surface force; while gravitational and centrifugal forces, which are distri- buted over the volume of the body are called body forees

The state of stress, which exists in a body acted upon by external forces, is completely defined in terms of six components of stress as shown

where on 6y, % = components of normal stresses,

‘ay Tyee Tx = Components of shear stresses

It may be pointed out here that we have designated only three components

of shear stress because only these are independent In the next section by considering the equilibrium of an elemental volume, it is shown that

Stresses acting on a positive face of the elemental volume in a positive coordinate direction are positive; those acting on 2 negative face in a nega- tive direction are positive; and al the others are negative A positive face

is one on which a normal vector directed outward from the element, points in a positive direction

Basic Equations from Linene Elasticity Theory 11 22.2 Strain Components

Corresponding to the six stress components, the state of strain ata point can be divided into six strain components, In a vector form, the state of strain is expressed as

where x, €, aud ¢¢ = components of normal strains

‘Yer Yrs and Yor = components of shear strains

The notation and sign convention used for the strain components are the same as those for the stress components,

23 EQUATIONS OF STATIC EQUILIBRIUM

‘The equilibrium of an elastic body in a state of stress is governed by three partial differential equations for the nine stress components These equations are derived by considering the equilibrium of forces and moments

‘acting on an elemental volume of a body Consider a small rectangular Parallelopiped of a body shown in Fig 2.2 It is subjected to a general system of positive three-dimensional stresses as well as to Positive body force-components X, ¥, Z in x, y, 2 directions, respectively

‘Summing all forces in the x-direction, and using the condition 3 F, = 0, wwe get

‘Thus, the condition 8 F, = 0 gives

Be Boe 4 Btn tet Get x0 3a)

12 Finite Element Method for Structural Engineers

as (dedydz) is not necessarily equal to zeto Similarly, summing forces in and z directions, we obtain

=o, Sry Sey Bey

Fig 2.2 Stress and body force components on en elemental volume of a boty

On Face OAA'0": 9p tus te

On Face BCC: oy + 2% dy, tye + PE dy, oye + HE dy On Face OACB ï cụ ta tợ 1

VAC: a0, One Đy

On Face O'ACB': on 8 do sự + ĐT đờ sy + Ty de

Likewise, a balance of moments about the three coordinate directions

shows that in the absence of body moments

Sey Tim Tự = tor, 8d Ton = Tae G30 Equations (2.3, b, «) must be satisfied at all points of the body The stessts vary throughout the body, and at the surface of boundary they rust be in equilibrium with the forces applied on the surface Let the

‘components of the surface forees per unit area on the boundary of the body be denoted by Ÿ, Ÿ, Z, then equilibrium at the surface is as follows

Veal opm tay tM te

Basic Equations from Linear Elasticity Theory 13 where J, m, and ø represent the direction cosines of the outward normal to the surface at the point of interest

24 STRAIN—DISPLACEMENT EQUATIONS

‘The deformed shape of an clastic structure under a given system of

loads can be described completely by three displacements v, v and w in x,

‘y and z directions respectively The positive directions of the displacements

correspond to the positive directions of the coordinate axes The relations

between the components of strain and the displacement components are

au, dw, buaU , aor | wow

5z lấy Í Scar + eax t oe ox

For small deformations, the strain-displacement reli

‘the expressions for the strain components, given by Eq (2.5), can be sim-

plified by retaining only the first order or linear terms and neglecting the

second order terms, that is

25 COMPATIBILITY EQUATIONS

In order that steain-displacement Eq (2.6) give a set of single-valted continuous solution for the three displacement components, some condi- tions must be imposed on the components of strain, These conditions,

| 14 Finite Element Method for Structural Engineers

known as compatiblity conditions, are obtained by eliminating displace-

‘ment components in Eq (2.6) and can be expressed as follows

‘This is known as generalized Hooke’s law and can be expressed as

Basie Equations from Linesr Elesiely ‘Theaty 15

“The matrix [C]is termed as the material stiffness matrix, while its in verse [D] is the material flexibility matrix Equation (2.9) or (2.10) repre- sents the constitutive law for a linear, elastic, anisotropic, and homogeneous

‘material For homogencous isotropic clastic materials, only two physical constants are required to express all the clastic constants in Hooke's Law Hence, in terms of Young's modulus (E) and Poisson’s rato (s), the matrices [C] and [DỊ can be expressed as

27 PLANE STRAIN AND PLANE STRESS

In many practical situations the three dimensional problem can be redeed to « two dimensions! problem There are two important types of two-dimensional problems ia linear elasticity theory which are Known as plane strain and phine stress problems These two cases are discussed separately and the foregoing equations of elasticity theory are

16 Finite Element Method for Structural Engineers

presented ina simplified form In both the cases, itis assumed that the body foree Z is zero, and ¥ and ¥ are functions of x and y only

2.7.1 Plane strain

It i state of strain in which eg = rye = Yer = 0 and only strains s„ cy and yx, exist, The body suffers displacements in one plane only, ie., Z component of body force is taken as zero

Stress Components:

alee % SP (2.150)

strain Components: O=le 6 tw 16)

Static Equilibrium Equations:

Basic Equations from Linear Elasticity Theory 17 2.7.2 Plane stress

In plane stress problem, the components of stress normal to the x-» plane are 2070 it., 6, = ter = Ty: =0-

2.26 Constittive Equations:

where

G29

It should be pointed out that in a state of plane stress, ¢ is not equal

to zero, but is given by

X6 +)

18 Finite Element Method for Structural Engineers

REFERENCES

2⁄1 Timoshenko, $., and J.N Goodivr, Theory of Elasticity 3rd ed

‘McGraw-Hill Book Company, New York, 1951

2.2 Wang, CT., Applied Elasticity, McGraw-Hill Book Company, New ‘York, 1953

23 Love, AE, A Treatise on Mathematical Theory of Elasticity, Dover ions, New York, 1944

Mathematical Theory of Elasticity 2nd ed., MeGraw- Hill Book Company, New York, 1956

2.5 Seckler, EE, Elasticity in Engineering, John Wiley 1952 & Sons, New York 2.6 Southwell, RV., Án Introduction 10 the Theory of Elasticity, Oxford University Press, Oxford, 1936

24

‘a mathematical model which should be as near as possible equivalent

to the actual continuum Such a formulation of a model is referred to as structural idealization of discretization In this chapter, the underlying principles of structural idealization are discussed followed by the most commonly used structural elements, which are employed in the subsequent chapters, are described

3.2 STRUCTURAL IDEALIZATION OR DISCRETIZATION

The continuum is physical body, structure or a solid which needs to

be analysed The subdivision or discretization process of the continuum is essentially an exercise of engineering judgement These subdivisions are called elements, and are connected to the adjacent elements only at limi- ted number of points called nodes (Fig 3.1) ‘Thus, inthe idealization ofthe continuum, we have to decide the number, shape, size and configuration

of the clements in such a way that the original body is represented by it as closely as possible Hence, the general objective of such an idealizai to discretize the body into finite number of elements sufficiently small so that the simple displacement models can adequately approximate the true solution, At the same time, it may be pointed out that too many small subdivisions will lead to extra computation effort No effort here is being made to discuss as to how many elementsshould be employed in any particular problem, rather it is suggested that wo or three cases with điferent numberof elements or fineness of meshes should be considered The results thus obtained can be used in establishing the rate of con vergence and enhance confidenes in the idealization employed The structures, in general, may be divided into two categories

1 Skeletal structures—Trusses, beams, Frames, etc

2, Continuous structures—Folded plates, box girders, deep beams, ete

‘The structural idealization and analysis of skeletal structures

20 Finite Element Method for Structural Engineers

pose no problem and can be done accurately as the assumed mathe

‘matical model is similar to the actual structure, The elements in the model formulation of these structures may easily be defined by the lengths between the two nedes as shown in Fig 3.1 In the second type of struc-

Typical nodes Typical elements,

fa) Plone truss JWeical neges Typical elements

(1 Beam Wicol elertentz flea ‘nodes

⁄

(e) Frome Fig 3.1 Skeletal structures,

Structural Tdealizaton 2L

(0) Detoils of folded piate mese!

(6) Delals of finite elomont ideatization (8.1)

Fig, 3.2 Continuum structure

where abrupt changes in geometry, loading or material properties occur Examples of these natural subdivisions are shown in Fig 3.3

Novos “2 Nodes (a) Concentrated loads and supports

Node (©) Abrupt change in toaging

22 Flaite Elemeot Method for Structural Engineers

Node (c) Abrupt change in section properties

Steel copper

Nooe (6) Abrupt ehange in moterial properties Fig, 3,3 Natural subdivision of structure or continuum

It is evident from the above discusssion that the structural idealization

is simply a process whereby a complex continuum is modelled asan assembly

of discrete structural elements satisfying the conditions stated in Chapter

1, Now in the next sub-section, various types of structural elements are described

33 TYPES OF STRUCTURAL ELEMENTS

The structural elements are of various types The shape or configura- tion of these discrete elements depends upon the geometry of the continuum and upon the number of independent space coordinates (e.g x,y, of 2) necessary to describe the problem Thus, based on the space coordinates, a finite element can be classified as a simple one-, twor, or three-dimensional element Here in this section, only the most commonly used one-, and two- dimensional elements with straight boundaries are described and in the, next chapter, stiffness characteristics for these elements are derived

3.3.1 One-Dimensional Elements

A one-dimensional clement may be represented by a straight line whose ends, such as 1 and 2, are nodal points (Fig 3.4 a) These elements are referenced in a coordinate system Fnown as local or element coordinate system In this sytem, x-axis is defined by element axis which is a line joining the two nodes of the elements One-dimensional elements are used ‘when the geometry, material properties and dependent variables such as displacements can all be expressed in terms of one independent space coor- dinate which is measured along the element axis Examples of structures using this type of element are skeletal structures such as trusses, frames, etc (Fig 31), It may be pointed out that for skeletal structures, element and global coordinate systems generally do not coincide

Structural Técalzation 23 33.11 Pin-Jointed Bar Element

‘The pin-jointed bar or truss clement shown in Fig 3.4 (b), is the simplest structural element and is assumed to be pin connected at both the ends The bar element is also assumed to have constant cross-sectional area (4) and modulus of elasticity (E) over its length (L), external loads are applied at the nodes and the effect of self-weight is neglected Thus for a pplane structure, this element has four degrees of frecdom, two at each node;

‘whereas for a space truss itis six, three at each node, This element carries &

‘one-dimensional stress distribution as it is assumed to resist only axial force,

ores {b) Sir jainted plane bor or truss element

beam or fame clement Fig, 3.4 One dimensional elements,

©

24 Flaite Slement Method for Structural Engineers

33.2 Two-Dimensional Elements

‘The elements shown in Figs 3.5 and 3.6 are two-diinensional elements

‘These elements are of constant thickness and with straight boundaries ‘Many problems in solid mechani:s such as plane strain, plane stress, plato bending, eto., could be idealized using these elements It may be noted that for small deflections, the in-plane and transverse deformations can be un- coupled [3.5] Thus, these elements for in-plane forces and in bending are

a (#)_ Rectangular plato eloment (In-plano forcos)

Fig 3.5 Two dimensional elements with in-plane forces, 33.211 Trlangular Plate Element (In-Plane Forces)

Structural Tdealization 25 3.3.2.2 Rectangular Plate Element (In-Plane Forces)

‘Rectangular plate element can be obtained by combining two triangular plate elements This element has four nodes with two degrees of freedom

at cach node, Thus each rectangular element with in-plane forces has eight degrees of freedom as shown in Fig 3.5 (b)

33:23 Triangular Plate Element (In-Bending) Ta thịs case, elements are subjected to only bending, ic., out-of-plane forces The clement has nine degrees of freedom, three at each node Le, transverse displacement and rotations about x and y axes (Fig 3.6 a)

(b) Rectangula: plate etement in bending

Fig 3.6 Two dimensional elements in bending

26 Finle Element Method for Structural Engineers

3.3.24 Rectangular Plate Element (In-Bending)

Like in the previous case, only bending is considered, The element has twelve degrees of freedom, three at each node_as shown in Fig 3.6 (b) Ta the next Chapter, methods for deriving element stiffaess properties are presented and in Chapters 5 and 6 these properties are derived for the clements discussed in this chapter

REFERENCES

3.1 Rockey, K.C., and H.R Evans, “An Experimental and Finite Ele- ment Study of the Behaviour of Folded Plate Roofs Containing Large Openings”, International Asso for Bridge and Structural Engi- neering, Vol 36-I of the “Publications”, Zurich 1976

3.2 Desai, S Chandrakant and Join F., Abel, Introduction to the Finite Element Method—A Numerical Method for Engineering Analysis Van Nostrand Reinhold Company, New York, 1972 3.3, Gaylord, Jr Edwin H., and Charles N., Gaylord (Edited by), Strue- ‘Company New York, 1979 tural Engineering Hand -Book, Second Bdition, McGraw-Hill Book 3.4 Zienkiewicz, O.C., BM., Irons, J Ergatoudis.S Ahmad and F.O., Scott, “Isoparametric and Associated Element Families for Two- and three-Dimensional Analysis” FEM Tapir The Technical University of Norway, Trondheim 1969 3.5 Timoshenko, SP and S., Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill Book Company, New York, 1959

CHAPTER 4

METHODS OF DETERMINING STIFFNESS

PROPERTIES AND ITS TRANSFORMATION

4.1 INTRODUCTION

It has been mentioned earlier (Chapter 1) that the basis of finite ele- ment method is the representation of @ continuum or a structure by an assemblage of discrete elements called finite elements, These elements are assumed fo be interconnected only at node points In order to determine

isplacement relationship or stiffness characteristics of the whole is required in this analysis, we must obtain first the stiff-

28 Finle Hlement Method for Structural Engineers

‘ness properties of individual elements These stiffness properties are deter-

‘mined, herein, using the following two approaches 1 ‘Elementary Approach—using basic definition

2 “General Approach—using displaceinent or shape functions

In this chapter, the two approaches for obtaining clement stiffness properties are discussed; and later in subsequent Chapters 5 and 6, these methods are applied to pin-jolated bar elements, beam elements, trian- gular and rectangular plate clements The general principles discussed may ‘also be used for deriving the required stiffness properties of other types of lements

It should be mentioned that it is convenient to develop stiffness charact- evistes particularly for one-dimensional elements in reference to local or clement coordinate system (Fig 4.1) Analysis using finite element method requires that these characteristics ‘ot orient in the direction of local coordinate system However, itis always be referenced in global or overall coordinate system which in general does for the entire structure or contiauum should possible to transform element stiffness characteristics from one coordinate system to another coordinate system The general procedure for achieving this transformation is also presented herein

‘ments The single-headed arrows denote translations, whereas the double- headed arrow denote rotations Thus, at node j the translations are num- dered 1, 2, and 3 and the rotations are numbered 4, 5, and 6 Similarly,

at the other node k, numbers 7, 8, and 9 denote translations and 10, 11, and 12 denote rotations

Now, in order to obtain the element stiffness characteristics by this approach, unit nodal displacements are imposed one ata time while all other nodal displacements are retained at zero The resulting forces exerted

‘on the element are obtained by determining the values of the restraint for-

es required to hold the distorted member in equilibrium These restraint forces due to unit displacements define the element stifnesses Finally using the principle of superposition, the desired element stiffness character- istics are more commonly known as element stiffness matrix is obtained, ‘The elementary approach discussed above uses the basic definition of stiffness It enables the reader to have a clear picture of the force-displace-

‘ment concept and to obtain the element stiffaess matrix without much

‘mathematical manipulations This method is applied to obtain the stiffness

Methods of Determining Stiffess Properties and its Transformation 29

Fig 4.2 Restrained element, matrices of pin-jointed truss and beam elements For other types such a5 triangular or rectangular plate elements which are more complex, this elementary approach becomes cumbersome Hence a general approach is considered

43 GENERAL APPROACH —USING DISPLACFMENT OR

‘SHAPE FUNCTIONS

‘The general derivation approach was first used by Turner at cl [4.1] 0 derive the stiffness matrix for e triangular plate clement for plane stress problems (sce Chapter 2) This approach is general and can be used for any other type of element as well There are four essential steps in this

derivation, which can be expressed as follows

STEP I: For each element, choose a set of functions that defines displace-

ments uniquely within the element These functions are called shape functions, displacement functions, or displacement fields Express these displacement functions in terms of the nodal displacements

STEP 2 : Introduce the strain-displacement equations and thereby deter-

ine the state of element strain corresponding to the assumed displacement field

STEP 3: Write the constitutive equations relating stress to strain These

‘equations introduce the influence of the material properties of the element

STEP 4: Write force-displacement relationship and identify the element

stiffness matrix

30 Finita Element Method for Structural Engineers

Now using the steps presented above, stiffness matrix for a general finite

us, 3,2) {fle ¥,2)) =} vs, 9» 2), for 3D elements (42)

W635 2) where 1, v and w are displacements within the element at (x, y,2) Hereafter the arguments (x, ¥, z) shall not be used for ease in writing

‘Now to obtain displacement function in terms of nodal displacements,

it can be expressed in matrix form as

where [N] = matrix of shape functions

{A} = column vector of nodal displacements

‘These quantities depend upon the type and dimensions of element In

‘expanded form, Eq (4.3) can be expressed as

{Ag}, {8;} = subvectors of nodal displacement vector (4)

STEP 2: With displacements within the element known, the strains at any point can be determined In matrix form, strain-displacement

equations ean be expressed as

Methods of Determining Stiffess Properties and Ís Trandarmaion 3L

fe) = [8] {8} (4.5)

In the above equation, matrix [8] i given by

where [L] matrix of a suitable linear operator

It may be pointed out here that in case [Z] is not linear operator, strain vector cannot be expressed by Eq (4.5)

‘Asan example of linear operator, consider a plane stress element For this element, the strain-displacement equations are (see Chapter 2):

STEP 3: Assuming general linear elastic behaviour of the material of clement (¢), the relationship between stresses and strains will be

linear and of the form

32 Finite Element Method for Structural Engineers

In the above equation, the matrix [9] is defined as

STEP 4: For writing the force-displacement relationship, we consider the virtual work approach It states that for an elastic body to be in

equilibrium, the total virtual work is equal to zero Mathemati cally, it can be expressed as

where 817, = Virtual work due to extemal forces

317, = Virtual work of internal forces per unit volume Now in order to obtain the element stiffness characteristics, 17, and

BB are expressed in terms of force and displacement components, and then substituted in Eq, (4.12) Finally, the clement stiffness matrix is id fied The derivation procedure is straight forward and is given below with~

‘out much detailed explanation

Let (F} represent a vector of nodal forces corresponding to nodal

displacement vector (A); the victual work 8¥7, may be given by

3% = BAIT (FD 3)

where (8A) = vector of virtual nodal displacements

‘The virtual work of internal forces per unit volume is given by

(16), respectively, we obtain

Fivally, substituting in Eq (4.12) the values of #, and ðW from Eqs (4.13) and (4.17), respectively, we obtain

‘Methods of Determining Stiffaess Properties and lis Transformation 33,

that the nodal displacement vector is indepeudent of x, y and z inates, we obtain after rearranging the above equation,

where the stiffness matrix denoted by [K] is identified as

i= Ệ (ar (al ta) av (21)

‘Thus, the force-displacement relationship and the stiffness matrix for

an element is given by Eqs (4.20) and (4.21), respectively ‘The dimension

‘of clement stiffness matrix [X] will depend upon the degrees of freedom associated with the clement

44 TRANSFORMATION OF REFERENCE (COORDINATE SYSTEMS

In the preceding chapter, various types of elements were described and

it was seen that for one-dimensional elements, local and global reference coordinates are different, whereas for two-dimensional elements, these

wu (OXV2:Glopal oordmate ayers OV oat coordinate system

Fig 43 Coordinate systems (orthogona)