Roark's formulas for stress and strain

1.2 State Properties, Units, and Conversions The basic state properties associated with stress analysis include thefollowing: geometrical properties such as length, area, volume,centroid

Roark’s Formulas for Stress and Strain

WARREN C YOUNG RICHARD G BUDYNAS

Seventh Edition

McGraw-Hill

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Contents

List of Tables vii

Preface to the Seventh Edition ix

Preface to the First Edition xi

Terminology State Properties, Units, and Conversions Contents.

Chapter 2 Stress and Strain: Important Relationships 9Stress Strain and the Stress–Strain Relations Stress Transformations.

Strain Transformations Tables References.

Methods of Loading Elasticity; Proportionality of Stress and Strain.

Factors Affecting Elastic Properties Load–Deformation Relation for a Body.

Plasticity Creep and Rupture under Long-Time Loading Criteria of Elastic

Failure and of Rupture Fatigue Brittle Fracture Stress Concentration.

Effect of Form and Scale on Strength; Rupture Factor Prestressing Elastic

Stability References.

Equations of Motion and of Equilibrium Principle of Superposition.

Principle of Reciprocal Deflections Method of Consistent Deformations

(Strain Compatibility) Principles and Methods Involving Strain Energy.

Dimensional Analysis Remarks on the Use of Formulas References.

Chapter 5 Numerical Methods 73The Finite-Difference Method The Finite-Element Method The Boundary-

Element Method References.

Measurement Techniques Electrical Resistance Strain Gages Detection of

Plastic Yielding Analogies Tables References.

Chapter 7 Tension, Compression, Shear, and Combined

Bar under Axial Tension (or Compression); Common Case Bar under Axial

Tension (or Compression); Special Cases Composite Members Trusses.

Body under Pure Shear Stress Cases of Direct Shear Loading Combined

Stress.

Straight Beams (Common Case) Elastically Stressed Composite Beams and

Bimetallic Strips Three-Moment Equation Rigid Frames Beams on

Elastic Foundations Deformation due to the Elasticity of Fixed Supports.

Beams under Simultaneous Axial and Transverse Loading Beams of

Variable Section Slotted Beams Beams of Relatively Great Depth Beams of

Relatively Great Width Beams with Wide Flanges; Shear Lag Beams with

Very Thin Webs Beams Not Loaded in Plane of Symmetry Flexural Center.

Straight Uniform Beams (Common Case) Ultimate Strength Plastic, or

Ultimate Strength Design Tables References.

Bending in the Plane of the Curve Deflection of Curved Beams Circular

Rings and Arches Elliptical Rings Curved Beams Loaded Normal to Plane

of Curvature Tables References.

Straight Bars of Uniform Circular Section under Pure Torsion Bars of

Noncircular Uniform Section under Pure Torsion Effect of End Constraint.

Effect of Longitudinal Stresses Ultimate Strength of Bars in Torsion.

Torsion of Curved Bars Helical Springs Tables References.

Common Case Bending of Uniform-Thickness Plates with Circular

Boundaries Circular-Plate Deflection due to Shear Bimetallic Plates.

Nonuniform Loading of Circular Plates Circular Plates on Elastic

Foundations Circular Plates of Variable Thickness Disk Springs Narrow

Ring under Distributed Torque about Its Axis Bending of

Uniform-Thickness Plates with Straight Boundaries Effect of Large Deflection.

Diaphragm Stresses Plastic Analysis of Plates Ultimate Strength Tables.

References.

iv Contents

Chapter 12 Columns and Other Compression Members 525Columns Common Case Local Buckling Strength of Latticed Columns.

Eccentric Loading: Initial Curvature Columns under Combined

Compression and Bending Thin Plates with Stiffeners Short Prisms under

Eccentric Loading Table References.

Chapter 13 Shells of Revolution; Pressure Vessels; Pipes 553Circumstances and General State of Stress Thin Shells of Revolution under

Distributed Loadings Producing Membrane Stresses Only Thin Shells of

Revolution under Concentrated or Discontinuous Loadings Producing

Bending and Membrane Stresses Thin Multielement Shells of Revolution.

Thin Shells of Revolution under External Pressure Thick Shells of

Revolution Tables References.

Chapter 14 Bodies in Contact Undergoing Direct Bearing

Stress due to Pressure between Elastic Bodies Rivets and Riveted Joints.

Miscellaneous Cases Tables References.

General Considerations Buckling of Bars Buckling of Flat and Curved

Plates Buckling of Shells Tables References.

Dynamic Loading General Conditions Body in a Known State of Motion.

Impact and Sudden Loading Approximate Formulas Remarks on Stress

due to Impact Temperature Stresses Table References.

Static Stress and Strain Concentration Factors Stress Concentration

Reduction Methods Table References.

Table.

Composite Materials Laminated Composite Materials Laminated

Composite Structures.

Contents v

9.4 Formulas for Curved Beams of Compact Cross-Section Loaded Normal to the

13.4 Formulas for Discontinuity Stresses and Deformations at the Junctions of Shells

viii List of Tables

Preface to the Seventh Edition

The tabular format used in the fifth and sixth editions is continued inthis edition This format has been particularly successful when imple-menting problem solutions on a programmable calculator, or espe-cially, a personal computer In addition, though not required inutilizing this book, user-friendly computer software designed toemploy the format of the tabulations contained herein are available.The seventh edition intermixes International System of Units (SI)and United States Customary Units (USCU) in presenting exampleproblems Tabulated coefficients are in dimensionless form for conve-nience in using either system of units Design formulas drawn fromworks published in the past remain in the system of units originallypublished or quoted

Much of the changes of the seventh edition are organizational, suchas:

j Numbering of equations, figures and tables is linked to the cular chapter where they appear In the case of equations, thesection number is also indicated, making it convenient to locatethe equation, since section numbers are indicated at the top of eachodd-numbered page

parti-j In prior editions, tables were interspersed within the text of eachchapter This made it difficult to locate a particular table anddisturbed the flow of the text presentation In this edition, allnumbered tables are listed at the end of each chapter before thereferences

Other changes=additions included in the seventh addition are asfollows:

j Part 1 is an introduction, where Chapter 1 provides terminologysuch as state properties, units and conversions, and a description ofthe contents of the remaining chapters and appendices The defini-

tions incorporated in Part 1 of the previous editions are retained inthe seventh edition, and are found in Appendix B as a glossary.

j Properties of plane areas are located in Appendix A

j Composite material coverage is expanded, where an introductorydiscussion is provided in Appendix C, which presents the nomen-clature associated with composite materials and how availablecomputer software can be employed in conjunction with the tablescontained within this book

j Stress concentrations are presented in Chapter 17

j Part 2, Chapter 2, is completely revised, providing a more hensive and modern presentation of stress and strain transforma-tions

compre-j Experimental Methods Chapter 6, is expanded, presenting morecoverage on electrical strain gages and providing tables of equationsfor commonly used strain gage rosettes

j Correction terms for multielement shells of revolution werepresented in the sixth edition Additional information is provided

in Chapter 13 of this edition to assist users in the application ofthese corrections

The authors wish to acknowledge and convey their appreciation tothose individuals, publishers, institutions, and corporations who havegenerously given permission to use material in this and previouseditions Special recognition goes to Barry J Berenberg and UniversalTechnical Systems, Inc who provided the presentation on compositematerials in Appendix C, and Dr Marietta Scanlon for her review ofthis work

Finally, the authors would especially like to thank the many cated readers and users of Roark’s Formulas for Stress & Strain It is

dedi-an honor dedi-and quite gratifying to correspond with the mdedi-any individualswho call attention to errors and=or convey useful and practicalsuggestions to incorporate in future editions

Warren C YoungRichard G Budynas

x Preface to the Seventh Edition

Preface to the First Edition

This book was written for the purpose of making available a compact,adequate summary of the formulas, facts, and principles pertaining tostrength of materials It is intended primarily as a reference book andrepresents an attempt to meet what is believed to be a present need ofthe designing engineer

This need results from the necessity for more accurate methods ofstress analysis imposed by the trend of engineering practice Thattrend is toward greater speed and complexity of machinery, greatersize and diversity of structures, and greater economy and refinement

of design In consequence of such developments, familiar problems, forwhich approximate solutions were formerly considered adequate, arenow frequently found to require more precise treatment, and manyless familiar problems, once of academic interest only, have become ofgreat practical importance The solutions and data desired are often to

be found only in advanced treatises or scattered through an extensiveliterature, and the results are not always presented in such form as to

be suited to the requirements of the engineer To bring together asmuch of this material as is likely to prove generally useful and topresent it in convenient form has been the author’s aim

The scope and management of the book are indicated by theContents In Part 1 are defined all terms whose exact meaningmight otherwise not be clear In Part 2 certain useful general princi-ples are stated; analytical and experimental methods of stress analysisare briefly described, and information concerning the behavior ofmaterial under stress is given In Part 3 the behavior of structuralelements under various conditions of loading is discussed, and exten-sive tables of formulas for the calculation of stress, strain, andstrength are given

Because they are not believed to serve the purpose of this book,derivations of formulas and detailed explanations, such as are appro-priate in a textbook, are omitted, but a sufficient number of examples

are included to illustrate the application of the various formulas andmethods Numerous references to more detailed discussions are given,but for the most part these are limited to sources that are generallyavailable and no attempt has been made to compile an exhaustivebibliography.

That such a book as this derives almost wholly from the work ofothers is self-evident, and it is the author’s hope that due acknowl-edgment has been made of the immediate sources of all material herepresented To the publishers and others who have generouslypermitted the use of material, he wishes to express his thanks Thehelpful criticisms and suggestions of his colleagues, Professors E R.Maurer, M O Withey, J B Kommers, and K F Wendt, are gratefullyacknowledged A considerable number of the tables of formulas havebeen published from time to time in Product Engineering, and theopportunity thus afforded for criticism and study of arrangement hasbeen of great advantage

Finally, it should be said that, although every care has been taken toavoid errors, it would be oversanguine to hope that none had escapeddetection; for any suggestions that readers may make concerningneeded corrections the author will be grateful

Raymond J Roarkxii Preface to the First Edition

Part 1

Introduction

be analyzed quite effectively independently without the need for anelaborate finite element model In some instances, finite elementmodels or programs are verified by comparing their solutions withthe results given in a book such as this Contained within this book aresimple, accurate, and thorough tabulated formulations that can beapplied to the stress analysis of a comprehensive range of structuralcomponents.

This chapter serves to introduce the reader to the terminology, stateproperty units and conversions, and contents of the book

1.1 Terminology

Definitions of terms used throughout the book can be found in theglossary in Appendix B

1.2 State Properties, Units, and Conversions

The basic state properties associated with stress analysis include thefollowing: geometrical properties such as length, area, volume,centroid, center of gravity, and second-area moment (area moment ofinertia); material properties such as mass density, modulus of elasti-city, Poisson’s ratio, and thermal expansion coefficient; loading proper-ties such as force, moment, and force distributions (e.g., force per unitlength, force per unit area, and force per unit volume); other proper-

ties associated with loading, including energy, work, and power; andstress analysis properties such as deformation, strain, and stress.Two basic systems of units are employed in the field of stressanalysis: SI units and USCU units.y

SI units are mass-based unitsusing the kilogram (kg), meter (m), second (s), and Kelvin (K) ordegree Celsius (C) as the fundamental units of mass, length, time,and temperature, respectively Other SI units, such as that used forforce, the Newton (kg-m=s2), are derived quantities USCU units areforce-based units using the pound force (lbf), inch (in) or foot (ft),second (s), and degree Fahrenheit (F) as the fundamental units offorce, length, time, and temperature, respectively Other USCU units,such as that used for mass, the slug (lbf-s2=ft) or the nameless lbf-

s2=in, are derived quantities Table 1.1 gives a listing of the primary SIand USCU units used for structural analysis Certain prefixes may beappropriate, depending on the size of the quantity Common prefixesare given in Table 1.2 For example, the modulus of elasticity of carbonsteel is approximately 207 GPa ¼ 207  109Pa ¼ 207  109N=m2 Pre-fixes are normally used with SI units However, there are cases whereprefixes are also used with USCU units Some examples are the kpsi(1 kpsi ¼ 103psi ¼ 103lbf =in2), kip (1 kip ¼ 1 kilopound ¼ 1000 lbf ), andMpsi (1 Mpsi ¼ 106psi)

Depending on the application, different units may be specified It isimportant that the analyst be aware of all the implications of the unitsand make consistent use of them For example, if you are building amodel from a CAD file in which the design dimensional units are given

in mm, it is unnecessary to change the system of units or to scale themodel to units of m However, if in this example the input forces are in

(derived units)

USCU unit, y symbol (derived units)

Stress, pressure Pascal, Pa (N=m 2 ) psi (lbf=in 2 )

y

In stress analysis, the unit of length used most often is the inch.

y SI and USCU are abbreviations for the International System of Units (from the French Syste´me International d’Unite´s) and the United States Customary Units, respectively.

Newtons, then the output stresses will be in N=mm2, which is correctlyexpressed as MPa If in this example applied moments are to bespecified, the units should be N-mm For deflections in this example,the modulus of elasticity E should also be specified in MPa and theoutput deflections will be in mm.

Table 1.3 presents the conversions from USCU units to SI unitsfor some common state property units For example, 10 kpsi ¼ð6:895  103Þ  ð10  103Þ ¼68:95  106Pa ¼ 68:95 MPa Obviously, themultiplication factors for conversions from SI to USCU are simply thereciprocals of the given multiplication factors

Prefix, symbol Multiplication factor

USCU units to SI units

To convert from USCU to SI Multiply by

1.3 Contents

The remaining parts of this book are as follows

relationships associated with stress and strain, basic materialbehavior, principles and analytical methods of the mechanics ofstructural elements, and numerical and experimental techniques instress analysis

applica-tions associated with the stress analysis of structural components.Topics include the following: direct tension, compression, shear, andcombined stresses; bending of straight and curved beams; torsion;bending of flat plates; columns and other compression members; shells

of revolution, pressure vessels, and pipes; direct bearing and shearstress; elastic stability; stress concentrations; and dynamic andtemperature stresses Each chapter contains many tables associatedwith most conditions of geometry, loading, and boundary conditions for

a given element type The definition of each term used in a table iscompletely described in the introduction of the table

area The second appendix provides a glossary of the terminologyemployed in the field of stress analysis

The references given in a particular chapter are always referred to

by number, and are listed at the end of each chapter

Part 2

Facts; Principles; Methods

2.1 Stress

Stress is simply a distributed force on an external or internal surface

of a body To obtain a physical feeling of this idea, consider beingsubmerged in water at a particular depth The ‘‘force’’ of the water onefeels at this depth is a pressure, which is a compressive stress, and not

a finite number of ‘‘concentrated’’ forces Other types of force tions (stress) can occur in a liquid or solid Tensile (pulling rather thanpushing) and shear (rubbing or sliding) force distributions can alsoexist

distribu-Consider a general solid body loaded as shown in Fig 2.1(a) Piand

pi are applied concentrated forces and applied surface force tions, respectively; and Ri and ri are possible support reaction forceand surface force distributions, respectively To determine the state ofstress at point Q in the body, it is necessary to expose a surfacecontaining the point Q This is done by making a planar slice, or break,through the body intersecting the point Q The orientation of this slice

distribu-is arbitrary, but it distribu-is generally made in a convenient plane where thestate of stress can be determined easily or where certain geometricrelations can be utilized The first slice, illustrated in Fig 2.1(b), isarbitrarily oriented by the surface normal x This establishes the yzplane The external forces on the remaining body are shown, as well asthe internal force (stress) distribution across the exposed internal

surface containing Q In the general case, this distribution will not beuniform along the surface, and will be neither normal nor tangential

to the surface at Q However, the force distribution at Q will havecomponents in the normal and tangential directions These compo-nents will be tensile or compressive and shear stresses, respectively.Following a right-handed rectangular coordinate system, the y and zaxes are defined perpendicular to x, and tangential to the surface.Examine an infinitesimal area DAx¼DyDz surrounding Q, as shown

in Fig 2.2(a) The equivalent concentrated force due to the forcedistribution across this area is DFx, which in general is neithernormal nor tangential to the surface (the subscript x is used todesignate the normal to the area) The force DFx has components inthe x, y, and z directions, which are labeled DFxx, DFxy, and DFxz,respectively, as shown in Fig 2.2(b) Note that the first subscript

Figure 2.1

denotes the direction normal to the surface and the second gives theactual direction of the force component The average distributed forceper unit area (average stress) in the x direction isy

s

sxx ¼DFxx

DAxRecalling that stress is actually a point function, we obtain the exactstress in the x direction at point Q by allowing DAx to approach zero.Thus,

sxx¼ lim

DA x !0

DFxx

DAxor,

sxx¼dFxx

Stresses arise from the tangential forces DFxy and DFxzas well, andsince these forces are tangential, the stresses are shear stresses.Similar to Eq (2.1-1),

y Standard engineering practice is to use the Greek symbols s and t for normal (tensile

or compressive) and shear stresses, respectively.

Since, by definition, s represents a normal stress acting in the samedirection as the corresponding surface normal, double subscripts areredundant, and standard practice is to drop one of the subscripts andwrite sxx as sx The three stresses existing on the exposed surface atthe point are illustrated together using a single arrow vector for eachstress as shown in Fig 2.3 However, it is important to realize that thestress arrow represents a force distribution (stress, force per unitarea), and not a concentrated force The shear stresses txy and txz arethe components of the net shear stress acting on the surface, where thenet shear stress is given byy

ðtxÞnet¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

t2

xyþt2 xz

q

ð2:1-4Þ

To describe the complete state of stress at point Q completely, itwould be necessary to examine other surfaces by making differentplanar slices Since different planar slices would necessitate differentcoordinates and different free-body diagrams, the stresses on eachplanar surface would be, in general, quite different As a matter offact, in general, an infinite variety of conditions of normal and shearstress exist at a given point within a stressed body So, it would take aninfinitesimal spherical surface surrounding the point Q to understandand describe the complete state of stress at the point Fortunately,through the use of the method of coordinate transformation, it is onlynecessary to know the state of stress on three different surfaces todescribe the state of stress on any surface This method is described inSec 2.3

The three surfaces are generally selected to be mutually cular, and are illustrated in Fig 2.4 using the stress subscript notation

perpendi-y Stresses can only be added as vectors if they exist on a common surface.

as earlier defined This state of stress can be written in matrix form,where the stress matrix ½s is given by

3

Except for extremely rare cases, it can be shown that adjacent shearstresses are equal That is, tyx ¼txy, tzy¼tyz, and txz¼tzx, and thestress matrix is symmetric and written as

3

in one direction are zero This situation is referred to as a case of planestress Arbitrarily selecting the z direction to be stress-free with

sz¼tyz¼tyz¼0, the last row and column of the stress matrix can

be eliminated, and the stress matrix is written as

2.2 Strain and the Stress–Strain Relations

As with stresses, two types of strains exist: normal and shear strains,which are denoted by e and g, respectively Normal strain is the rate ofchange of the length of the stressed element in a particular direction

Shear strain is a measure of the distortion of the stressed element, andhas two definitions: the engineering shear strain and the elasticityshear strain Here, we will use the former, more popular, definition.However, a discussion of the relation of the two definitions will beprovided in Sec 2.4 The engineering shear strain is defined as thechange in the corner angle of the stress cube, in radians.

the element as shown in Fig 2.6 We see that the element increases inlength in the x direction and decreases in length in the y and zdirections The dimensionless rate of increase in length is defined asthe normal strain, where ex, ey, and ezrepresent the normal strains in

the x, y, and z directions respectively Thus, the new length in anydirection is equal to its original length plus the rate of increase(normal strain) times its original length That is,

Dx0¼Dx þ exDx; Dy0¼Dy þ eyDy; Dz0¼Dz þ ezDz ð2:2-1Þ

There is a direct relationship between strain and stress Hooke’s lawfor a linear, homogeneous, isotropic material is simply that the normalstrain is directly proportional to the normal stress, and is given by

If the strains in Eqs (2.2-2) are known, the stresses can be solved forsimultaneously to obtain

stresses can be first illustrated by examining the effect of txy alone asshown in Fig 2.7 The engineering shear strain gxyis a measure of theskewing of the stressed element from a rectangular parallelepiped InFig 2.7(b), the shear strain is defined as the change in the angle BAD.That is,

gxy¼ ff 0A0D0

where gxy is in dimensionless radians

For a linear, homogeneous, isotropic material, the shear strains inthe xy, yz, and zx planes are directly related to the shear stresses by

where the material constant, G, is called the shear modulus

It can be shown that for a linear, homogeneous, isotropic materialthe shear modulus is related to Poisson’s ratio by (Ref 1)

3

Now consider the element, shown in Fig 2.8(b), to correspond to thestate of stress at the same point but defined relative to a different set ofsurfaces with normals in the x0, y0, and z0directions The stress matrixcorresponding to this element is given by

3

To determine ½sx0 y 0 z 0 by coordinate transformation, we need toestablish the relationship between the x0y0z0 and the xyz coordinatesystems This is normally done using directional cosines First, let usconsider the relationship between the x0 axis and the xyz coordinatesystem The orientation of the x0axis can be established by the angles

yx0 x, yx0 y, and yx0 z, as shown in Fig 2.9 The directional cosines for x0aregiven by

lx0¼cos yx0 x; mx0¼cos yx0 y; nx0¼cos yx0 z ð2:3-3ÞSimilarly, the y0 and z0axes can be defined by the angles yy0 x, yy0 y, yy0 z

and yz0 x, yz0 y, yz0 z, respectively, with corresponding directional cosines

ly0¼cos yy0 x; my0¼cos yy0 y; ny0 ¼cos yy0 z ð2:3-4Þ

lz0 ¼cos yz0 x; mz0 ¼cos yz0 y; nz0 ¼cos yz0 z ð2:3-5Þ

It can be shown that the transformation matrix

3

transforms a vector given in xyz coordinates, fVgxyz, to a vector in x0y0z0

coordinates, fVgx0 y 0 z 0, by the matrix multiplication

3

The stress transformation by Eq (2.3-8) can be implemented veryeasily using a computer spreadsheet or mathematical software Defin-ing the directional cosines is another matter One method is to definethe x0y0z0 coordinate system by a series of two-dimensional rotationsfrom the initial xyz coordinate system Table 2.2 at the end of thischapter gives transformation matrices for this

5 MPa

Determine the state of stress on an element that is oriented by first rotating

about the new x axis

transformations for each rotation From Fig 2.10(a), the vector componentsfor the first rotation can be represented by

35

xyz

37xyz

26

37

xyz

Equation (c) is of the form of Eq (2.3-7) Thus, the transformation matrix is

½T ¼

24

6

6

p2ffiffiffi

26

3p

26

6

6

p2ffiffiffi

2

6

37

26

3p

26

37

This matrix multiplication can be performed simply using either a computerspreadsheet or mathematical software, resulting in

3

5 MPa

stress on one particular surface, a complete stress transformationwould be unnecessary Let the directional cosines for the normal ofthe surface be given by l, m, and n It can be shown that the normalstress on the surface is given by

s ¼ sxl2þsym2þszn2þ2txylm þ 2tyzmn þ 2tzxnl ð2:3-10Þand the net shear stress on the surface is

t ¼ ½ðsxl þ txym þ tzxnÞ2þ ðtxyl þ sym þ tyznÞ2

þ ðtzxl þ tyzm þ sznÞ2 s21=2 ð2:3-11Þ

The direction of t is established by the directional cosines

3

5 kpsi

Determine the normal and shear stress on a surface at the point where thesurface is parallel to the plane given by the equation

numbers of the plane and are simply the coefficients of x, y, and z terms ofdirectional cosines of the normal to the surface are simply the normalizedvalues of the directional numbers, which are the directional numbers dividedby

l ¼ 2= ffiffiffiffiffiffi14

pÞ

14

pÞð3= ffiffiffiffiffiffi14

p

14

pÞð2= ffiffiffiffiffiffi14

pÞ2

From Eq (2.3-12), the directional cosines for the direction of t are

ffiffiffiffiffiffi14

sz¼tyz¼tzx¼0 Plane stress transformations are normally formed in the xy plane, as shown in Fig 2.11(b) The angles relatingthe x0y0z0 axes to the xyz axes are

½sxy ¼ sx txy

txy sy

ð2:3-13Þ

sx0¼sxcos2y þ sysin2y þ 2txycos y sin y

sy0¼sxsin2y þ sycos2 xycos y sin y ð2:3-16Þ

tx0 y 0 sx syÞsin y cos y þ txyðcos2 2yÞ

If the state of stress is desired on a single surface with a normalrotated y counterclockwise from the x axis, the first and third equa-tions of Eqs (2.3-16) can be used as given However, using trigono-metric identities, the equations can be written in slightly differentform Letting s and t represent the desired normal and shear stresses

on the surface, the equations are

so it will not be represented here (see Ref 1)

normal stresses occur on surfaces where the shear stresses are zero.These stresses, which are actually the eigenvalues of the stressmatrix, are called the principal stresses Three principal stressesexist, s1, s2, and s3, where they are commonly ordered as s15 s25 s3

Considering the stress state given by the matrix of Eq (2.3-1) to beknown, the principal stresses spare related to the given stresses by

To avoid the zero solution of the directional cosines of Eqs (2.3-18), thedeterminant of the coefficients of lp, mp, and npin the equation is set tozero This makes the solution of the directional cosines indeterminatefrom Eqs (2.3-18) Thus,

y Mathematical software packages can be used quite easily to extract the eigenvalues (sp) and the corresponding eigenvectors (lp, mp, and np) of a stress matrix The reader is urged to explore software such as Mathcad, Matlab, Maple, and Mathematica, etc.

For the following stress matrix, determine the principal stresses and thedirectional cosines associated with the normals to the surfaces of eachprincipal stress

conventional ordering,

The directional cosines associated with each principal stress are determined

This results in

Equations (b), (c), and (d) are no longer independent since they were used to

in this example, any two of the above can be used Consider Eqs (b) and (c),which are independent A third equation comes from Eq (2.3-19), which is

simulta-neously, consider the following approach

solving these simultaneously gives m1¼n1¼12 These values of l1, m1, and n1

do not satisfy Eq (2.3-19) However, all that remains is to normalize theirvalues by dividing by

If two of the principal stresses are equal, there will exist an infinite set

of surfaces containing these principal stresses, where the normals ofthese surfaces are perpendicular to the direction of the third principalstress If all three principal stresses are equal, a hydrostatic state ofstress exists, and regardless of orientation, all surfaces contain thesame principal stress with no shear stress

in Fig 2.11(a), the shear stresses on the surface with a normal in the zdirection are zero Thus, the normal stress sz¼0 is a principal stress.The directions of the remaining two principal stresses will be in the xyplane If tx0 y 0 ¼0 in Fig 2.11(b), then sx0would be a principal stress, spwith lp¼cos y, mp¼sin y, and np¼0 For this case, only the first two

of Eqs (2.3-18) apply, and are

ðsx spÞcos y þ txysin y ¼ 0

txycos y þ ðsy spÞsin y ¼ 0 ð2:3-21Þ

As before, we eliminate the trivial solution of Eqs (2.3-21) by settingthe determinant of the coefficients of the directional cosines to zero.That is,

Equation (2.3-22) is a quadratic equation in sp for which the twosolutions are

Each solution of Eq (2.3-23) can then be substituted into one of Eqs.(2.3-21) to determine the direction of the principal stress Note that if

sx ¼sy and txy ¼0, then sx and sy are principal stresses and Eqs.(2.3-21) are satisfied for all values of y This means that all stresses inthe plane of analysis are equal and the state of stress at the point isisotropic in the plane

Figure 2.12(a) illustrates the initial state of stress, whereas the orientation

of the element containing the in-plane principal stresses is shown in Fig.2.12(b)

general stress state have been determined using the methods justdescribed and are illustrated by Fig 2.13 The 123 axes represent thenormals for the principal surfaces with directional cosines determined

by Eqs (2.3-18) and (2.3-19) Viewing down a principal stress axis(e.g., the 3 axis) and performing a plane stress transformation in theplane normal to that axis (e.g., the 12 plane), one would find that theshear stress is a maximum on surfaces 45 from the two principalstresses in that plane (e.g., s1, s2) On these surfaces, the maximumshear stress would be one-half the difference of the principal stresses[e.g., tmax¼ ðs1 s2Þ=2] and will also have a normal stress equal to theaverage of the principal stresses [e.g., save¼ ðs1þs2Þ=2] Viewingalong the three principal axes would result in three shear stress

maxima, sometimes referred to as the principal shear stresses Thesestresses together with their accompanying normal stresses arePlane 1; 2: ðtmaxÞ1;2¼ ðs1 s2Þ=2; ðsaveÞ1;2¼ ðs1þs2Þ=2Plane 2; 3: ðtmaxÞ2;3¼ ðs2 s3Þ=2; ðsaveÞ2;3¼ ðs2þs3Þ=2Plane 1; 3: ðtmaxÞ1;3¼ ðs1 s3Þ=2; ðsaveÞ1;3¼ ðs1þs3Þ=2

ð2:3-24ÞSince conventional practice is to order the principal stresses by

s15 s25 s3, the largest shear stress of all is given by the third ofEqs (2.3-24) and will be repeated here for emphasis:

containing the principal stresses was shown in Fig 2.12(b), where axis 3 wasthe z axis and normal to the page Determine the maximum shear stress andshow the orientation and complete state of stress of the element containingthis stress

principal stresses are repeated in Fig 2.14(a) and (b), respectively Themaximum shear stress will exist in the 1, 3 plane and is determined by

To establish the orientation of these stresses, view the element along the axis

shown in Fig 2.14(c)

The directional cosines associated with the surfaces are found throughsuccessive rotations Rotating the xyz axes to the 123 axes yields

123

xyz

37

xyz