1. MECHANICALDESIGNHANDBOOK

2. This page intentionally left blank 3. MECHANICALDESIGNHANDBOOKMeasurement, Analysis, and Controlof Dynamic SystemsHarold A. Rothbart EditorDean EmeritusCollege of Science and EngineeringFairleigh Dickinson UniversityTeaneck, N.J.Thomas H. Brown, Jr. EditorFaculty AssociateInstitute for Transportation Research and EducationNorth Carolina State UniversityRaleigh, N.C.Second EditionMcGRAW-HILLNew York Chicago San Francisco Lisbon London MadridMexico City Milan New Delhi San Juan SeoulSingapore Sydney Toronto 4. Copyright 2006, 1996 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured inthe United States of America. 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For more information about this title, click hereCONTENTSvContributors viiForeword ixPreface xiAcknowledgments xiiiPart 1 Mechanical Design FundamentalsChapter 1. Classical Mechanics 1.3Chapter 2. Mechanics of Materials 2.1Chapter 3. Kinematics of Mechanisms 3.1Chapter 4. Mechanical Vibrations 4.1Chapter 5. Static and Fatigue Design 5.1Chapter 6. Properties of Engineering Materials 6.1Chapter 7. Friction, Lubrication, and Wear 7.1Part 2 Mechanical System AnalysisChapter 8. System Dynamics 8.3Chapter 9. Continuous Time Control Systems 9.1Chapter 10. Digital Control Systems 10.1 7. vi CONTENTSChapter 11. Optical Systems 11.1Chapter 12. Machine Systems 12.1Chapter 13. System Reliability 13.1Part 3 Mechanical Subsystem ComponentsChapter 14. Cam Mechanisms 14.3Chapter 15. Rolling-Element Bearings 15.1Chapter 16. Power Screws 16.1Chapter 17. Friction Clutches 17.1Chapter 18. Friction Brakes 18.1Chapter 19. Belts 19.1Chapter 20. Chains 20.1Chapter 21. Gearing 21.1Chapter 22. Springs 22.1Appendix A. Analytical Methods for Engineers A.1Appendix B. Numerical Methods for Engineers B.1Index follows Appendix B 8. CONTRIBUTORSWilliam J. Anderson Vice President, NASTEC Inc., Cleveland, Ohio (Chap. 15, Rolling-Ellement Bearings)William H. Baier Director of Engineering, The Fitzpatrick Co., Elmhurst, Ill. (Chap. 19, Belts)Stephen B. Bennett Manager of Research and Product Development, Delaval TurbineDivision, Imo Industries, Inc., Trenton, N.J. (Chap. 2, Mechanics of Materials)Thomas H. Brown, Jr. Faculty Associate, Institute for Transportation Research and Education,North Carolina State University, Raleigh, N.C. (Co-Editor)John J. Coy Chief of Mechanical Systems Technology Branch, NASA Lewis Research Center,Cleveland, Ohio (Chap. 21, Gearing)Thomas A. Dow Professor of Mechanical and Aerospace Engineering, North Carolina StateUniversity, Raleigh, N.C. (Chap. 17, Friction Clutches, and Chap. 18, Friction Brakes)Saul K. Fenster President Emeritus, New Jersey Institute of Technology, Newark, N.J. (App. A,Analytical Methods for Engineers)Ferdinand Freudenstein Stevens Professor of Mechanical Engineering, Columbia University,New York, N.Y. (Chap. 3, Kinematics of Mechanisms)Theodore Gela Professor Emeritus of Metallurgy, Stevens Institute of Technology, Hoboken, N.J.(Chap. 6, Properties of Engineering Materials)Herbert H. Gould Chief, Crashworthiness Division, Transportation Systems Center, U.S.Department of Transportation, Cambridge, Mass. (App. A, Analytical Methods for Engineers)Bernard J. Hamrock Professor of Mechanical Engineering, Ohio State University, Columbus,Ohio (Chap. 15, Rolling-Element Bearings)John E. Johnson Manager, Mechanical Model Shops, TRW Corp., Redondo Beach, Calif.(Chap. 16, Power Screws)Sheldon Kaminsky Consulting Engineer, Weston, Conn. (Chap. 8, System Dynamics)Kailash C. Kapur Professor and Director of Industrial Engineering, University of Washington,Seattle, Wash. (Chap. 13, System Reliability)Robert P. Kolb Manager of Engineering (Retired), Delaval Turbine Division, Imo Industries,Inc., Trenton, N.J. (Chap. 2, Mechanics of Materials)Leonard R. Lamberson Professor and Dean, College of Engineering and Applied Sciences,West Michigan University, Kalamazoo, Mich. (Chap. 13, System Reliability)Thomas P. Mitchell Professor, Department of Mechanical and Environmental Engineering,University of California, Santa Barbara, Calif. (Chap. 1, Classical Mechanics)Burton Paul Asa Whitney Professor of Dynamical Engineering, Department of MechanicalEngineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pa. (Chap. 12,Machine Systems)viiCopyright 2006, 1996 by The McGraw-Hill Companies, Inc. Click here for terms of use. 9. viii CONTRIBUTORSJ. David Powell Professor of Aeronautics/Astronautics and Mechanical Engineering, StanfordUniversity, Stanford, Calif. (Chap. 10, Digital Control Systems)Abillo A. Relvas ManagerTechical Assistance, Associated Spring, Barnes Group, Inc., Bristol,Conn. (Chap. 22, Springs)Harold A. Rothbart Dean Emeritus, College of Science and Engineering, Fairleigh DickensonUniversity, Teaneck, N.J. (Chap. 14, Cam Mechanisms, and Co-Editor)Andrew R. Sage Associate Vice President for Academic Affairs, George Mason Universtiy,Fairfax, Va. (Chap. 9, Continuous Time Control Systems)Warren J. Smith Vice President, Research and Development, Santa Barbara Applied Optics, asubsidiary of Infrared Industries, Inc., Santa Barbara, Calif. (Chap. 11, Optical Systems)David Tabor Professor Emeritus, Laboratory for the Physics and Chemistry of Solids,Department of Physics, Cambridge University, Cambridge, England (Chap. 7, Friction, Lubrication,and Wear)Steven M. Tipton Associate Professor of Mechanical Engineering, University of Tulsa, Tulsa,Okla. (Chap. 5, Static and Fatigue Design)George V. Tordion Professor of Mechanical Engineering, Universit Laval, Quebec, Canada(Chap. 20, Chains)Dennis P. Townsend Senior Research Engineer, NASA Lewis Research Center, Cleveland,Ohio (Chap. 21, Gearing)Eric E. Ungar Chief Consulting Engineer, Bolt, Beranek, and Newman, Inc., Cambridge, Mass.(Chap. 4, Mechanical Vibrations)C. C. Wang Senior Staff Engineer, Central Engineer Laboratories, FMC Corporation, SantaClara, Calif. (App. B, Numerical Methods for Engineers)Erwin V. Zaretsky Chief Engineer of Structures, NASA Lewis Research Center, Cleveland, Ohio(Chap. 21, Gearing) 10. FOREWORDMechanical design is one of the most rewarding activities because of its incrediblecomplexity. It is complex because a successful design involves any number of individualmechanical elements combined appropriately into what is called a system. The wordsystem came into popular use at the beginning of the space age, but became somewhatoverused and seemed to disappear. However, any modern machine is a system andmust operate as such. The information in this handbook is limited to the mechanicalelements of a system, since encompassing all elements (electrical, electronic, etc.)would be too overwhelming.The purpose of the Mechanical Design Handbook has been from its inception to pro-videthe mechanical designer the most comprehensive and up-to-date information onwhat is available, and how to utilize it effectively and efficiently in a single referencesource. Unique to this edition, is the combination of the fundamentals of mechanicaldesign with a systems approach, incorporating the most important mechanical subsystemcomponents. The original editor and a contributing author, Harold A. Rothbart, is one ofthe most well known and respected individuals in the mechanical engineering community.From the First Edition of the Mechanical Design and Systems Handbook published overforty years ago to this Second Edition of the Mechanical Design Handbook, he has con-tinuedto assemble experts in every field of machine designmechanisms and linkages,cams, every type of gear and gear train, springs, clutches, brakes, belts, chains, all mannerof roller bearings, failure analysis, vibration, engineering materials, and classical mechanics,including stress and deformation analysis. This incredible wealth of information, whichwould otherwise involve searching through dozens of books and hundreds of scientific andprofessional papers, is organized into twenty-two distinct chapters and two appendices. Thisprovides direct access for the designer to a specific area of interest or need.The Mechanical Design Handbook is a unique reference, spanning the breadth anddepth of design information, incorporating the vital information needed for a mechanicaldesign. It is hoped that this collection will create, through a system perspective, the level ofconfidence that will ultimately produce a successful and safe design and a proud designer.ixHarold A. RothbartThomas H. Brown, Jr.Copyright 2006, 1996 by The McGraw-Hill Companies, Inc. Click here for terms of use. 11. This page intentionally left blank 12. PREFACEThis Second Edition of the Mechanical Design Handbook has been completely reorga-nizedfrom its previous edition and includes seven chapters from the Mechanical Designand Systems Handbook, the precursor to the First Edition. The twenty-two chapters con-tainedin this new edition are divided into three main sections: Mechanical DesignFundamentals, Mechanical System Analysis, and Mechanical Subsystem Components. Itis hoped that this new edition will meet the needs of practicing engineers providing thecritical resource of information needed in their mechanical designs.The first section, Part I, Mechanical Design Fundamentals, includes seven chapterscovering the foundational information in mechanical design. Chapter 1, ClassicalMechanics, is one of the seven chapters included from the Second Edition of theMechanical Design and Systems Handbook, and covers the basic laws of dynamics andthe motion of rigid bodies so important in the analysis of machines in three-dimensionalmotion. Comprehensive information on topics such as stress, strain, beam theory, andan extensive table of shear and bending moment diagrams, including deflection equa-tions,is provided in Chap. 2. Also in Chap. 2 are the equations for the design ofcolumns, plates, and shells, as well as a complete discussion of the finite-elementanalysis approach. Chapter 3, Kinematics of Mechanisms, contains an endless numberof ways to achieve desired mechanical motion. Kinematics, or the geometry ofmotion, is probably the most important step in the design process, as it sets the stagefor many of the other decisions that will be made as a successful design evolves.Whether its a particular multi-bar linkage, a complex cam shape, or noncircular gearcombinations, the information for its proper design is provided. Chapter 4, MechanicalVibrations, provides the basic equations governing mechanical vibrations, including anextensive set of tables compiling critical design information such as, mechanicalimpedances, mechanical-electrical analogies, natural frequencies of basic systems, tor-sionalsystems, beams in flexure, plates, shells, and several tables of spring constantsfor a wide variety of mechanical configurations. Design information on both static anddynamic failure theories, for ductile and brittle materials, is given in Chap. 5, Static andFatigue Design, while Chap. 6, Properties of Engineering Materials, covers the issuesand requirements for material selection of machine elements. Extensive tables and chartsprovide the experimental data on heat treatments, hardening, high-temperature and low-temperatureapplications, physical and mechanical properties, including properties forceramics and plastics. Chapter 7, Friction, Lubrication, and Wear, gives a basic overviewof these three very important areas, primarily directed towards the accuracy requirementsof the machining of materials.The second section, Part II, Mechanical System Analysis, contains six chapters, thefirst four of which are from the Second Edition of the Mechanical Design and SystemsHandbook. Chapter 8, Systems Dynamics, presents the fundamentals of how a complexdynamic system can be modeled mathematically. While the solution of such systems willbe accomplished by computer algorithms, it is important to have a solid foundation onxiCopyright 2006, 1996 by The McGraw-Hill Companies, Inc. Click here for terms of use. 13. how all the components interactthis chapter provides that comprehensive analysis.Chapter 9, Continuous Time Control Systems, expands on the material in Chap. 8 byintroducing the necessary elements in the analysis when there is a time-dependentinput to the mechanical system. Response to feedback loops, particularly for nonlineardamped systems, is also presented. Chapter 10, Digital Control Systems, continueswith the system analysis presented in Chaps. 8 and 9 of solving the mathematicalequations for a complex dynamic system on a computer. Regardless of the hardwareused, from personal desktop computers to supercomputers, digitalization of the equa-tionsmust be carefully considered to avoid errors being introduced by the analog todigital conversion. A comprehensive discussion of the basics of optics and the passageof light through common elements of optical systems is provided in Chap. 11, OpticalSystems, and Chap. 12, Machine Systems, presents the dynamics of mechanical sys-temsprimarily from an energy approach, with an extensive discussion of Lagrangesequations for three-dimensional motion. To complete this section, Chap. 13, SystemReliability, provides a system approach rather than addressing single mechanical elements.Reliability testing is discussed along with the Weibull distribution used in the statisticalanalysis of reliability.The third and last section, Part III, Mechanical Subsystem Components, contains ninechapters covering the most important elements of a mechanical system. Cam layout andgeometry, dynamics, loads, and the accuracy of motion are discussed in Chap. 14 whileChap. 15, Rolling-Element Bearings, presents ball and roller bearing, materials of construc-tion,static and dynamic loads, friction and lubrication, bearing life, and dynamic analysis.Types of threads available, forces, friction, and efficiency are covered in Chap. 16, PowerScrews. Chapter 17, Friction Clutches, and Chap. 18, Friction Brakes, both contain anextensive presentation of these two important mechanical subsystems. Included are thetypes of clutches and brakes, materials, thermal considerations, and application to varioustransmission systems. The geometry of belt assemblies, flat and v-belt designs, and beltdynamics is explained in Chap. 19, Belts, while chain arrangements, ratings, and noise aredealt with in Chap. 20, Chains. Chapter 21, Gearing, contains every possible gear type,from basic spur gears and helical gears to complex hypoid bevel gears sets, as well as theintricacies of worm gearing. Included is important design information on processing andmanufacture, stresses and deflection, gear life and power-loss predictions, lubrication, andoptimal design considerations. Important design considerations for helical compression,extension and torsional springs, conical springs, leaf springs, torsion-bar springs, powersprings, constant-force springs, and Belleville washers are presented in Chap. 22, Springs.This second edition of the Mechanical Design Handbook contains two new appen-dicesnot in the first edition: App. A, Analytical Methods for Engineers, and App. B,Numerical Methods for Engineers. They have been provided so that the practicingengineer does not have to search elsewhere for important mathematical informationneeded in mechanical design.It is hoped that this Second Edition continues in the tradition of the First Edition,providing relevant mechanical design information on the critical topics of interest tothe engineer. Suggestions for improvement are welcome and will be appreciated.Harold A. RothbartThomas H. Brown, Jr.xii PREFACE 14. ACKNOWLEDGMENTSOur deepest appreciation and love goes to our families, Florence, Ellen, Dan, and Jane(Rothbart), and Miriam, Sianna, Hunter, and Elliott (Brown). Their encouragement, help,suggestions, and patience are a blessing to both of us.To our Senior Editor Ken McCombs, whose continued confidence and support hasguided us throughout this project, we gratefully thank him. To Gita Raman and herwonderful and competent staff at International Typesetting and Composition (ITC) inNoida, India, it has been a pleasure and honor to collaborate with them to bring thisSecond Edition to reality.And finally, without the many engineers who found the First Edition of theMechanical Design Handbook, as well as the First and Second Editions of theMechanical Design and Systems Handbook, useful in their work, this newest editionwould not have been undertaken. To all of you we wish the best in your career andconsider it a privilege to provide this reference for you.Harold A. RothbartThomas H. Brown, Jr.xiiiCopyright 2006, 1996 by The McGraw-Hill Companies, Inc. Click here for terms of use. 15. This page intentionally left blank 16. MECHANICALDESIGNHANDBOOK 17. This page intentionally left blank 18. PART1MECHANICAL DESIGNFUNDAMENTALSCopyright 2006, 1996 by The McGraw-Hill Companies, Inc. Click here for terms of use. 19. This page intentionally left blank 20. CHAPTER 1CLASSICAL MECHANICSThomas P. Mitchell, Ph.D.ProfessorDepartment of Mechanical and Environmental EngineeringUniversity of CaliforniaSanta Barbara, Calif.1.31.1 INTRODUCTION 1.31.2 THE BASIC LAWS OF DYNAMICS 1.31.3 THE DYNAMICS OF A SYSTEM OFMASSES 1.51.3.1 The Motion of the Center of Mass 1.61.3.2 The Kinetic Energy of a System 1.71.3.3 Angular Momentum of a System(Moment of Momentum) 1.81.4 THE MOTION OF A RIGID BODY 1.91.5 ANALYTICAL DYNAMICS 1.121.5.1 Generalized Forces and dAlembertsPrinciple 1.121.5.2 The Lagrange Equations 1.141.5.3 The Euler Angles 1.151.5.4 Small Oscillations of a System nearEquilibrium 1.171.5.5 Hamiltons Principle 1.19The aim of this chapter is to present the concepts and results of newtonian dynamicswhich are required in a discussion of rigid-body motion. The detailed analysis of par-ticularrigid-body motions is not included. The chapter contains a few topics which,while not directly needed in the discussion, either serve to round out the presentationor are required elsewhere in this handbook.1.1 INTRODUCTIONThe study of classical dynamics is founded on Newtons three laws of motion and onthe accompanying assumptions of the existence of absolute space and absolute time.In addition, in problems in which gravitational effects are of importance, Newtonslaw of gravitation is adopted. The objective of the study is to enable one to predict,given the initial conditions and the forces which act, the evolution in time of amechanical system or, given the motion, to determine the forces which produce it.The mathematical formulation and development of the subject can be approached in twoways. The vectorial method, that used by Newton, emphasizes the vector quantities forceand acceleration. The analytical method, which is largely due to Lagrange, utilizes thescalar quantities work and energy. The former method is the more physical and generallypossesses the advantage in situations in which dissipative forces are present. The latter ismore mathematical and accordingly is very useful in developing powerful general results.1.2 THE BASIC LAWS OF DYNAMICSThe first law of motion states that a body which is under the action of no forceremains at rest or continues in uniform motion in a straight line. This statement is alsoCopyright 2006, 1996 by The McGraw-Hill Companies, Inc. Click here for terms of use. 21. 1.4 MECHANICAL DESIGN FUNDAMENTALSknown as the law of inertia, inertia being that property of a body which demandsthat a force is necessary to change its motion. Inertial mass is the numerical measureof inertia. The conditions under which an experimental proof of this law could be carriedout are clearly not attainable.In order to investigate the motion of a system it is necessary to choose a frame of refer-ence,assumed to be rigid, relative to which the displacement, velocity, etc., of the system areto be measured. The law of inertia immediately classifies the possible frames of referenceinto two types. For, suppose that in a certain frame S the law is found to be true; then it mustalso be true in any frame which has a constant velocity vector relative to S. However, the lawis found not to be true in any frame which is in accelerated motion relative to S. A frame ofreference in which the law of inertia is valid is called an inertial frame, and any frame inaccelerated motion relative to it is said to be noninertial. Any one of the infinity of inertialframes can claim to be at rest while all others are in motion relative to it. Hence it is notpossible to distinguish, by observation, between a state of rest and one of uniform motion ina straight line. The transformation rules by which the observations relative to two inertialframes are correlated can be deduced from the second law of motion.Newtons second law of motion states that in an inertial frame the force acting ona mass is equal to the time rate of change of its linear momentum. Linear momentum,a vector, is defined to be the product of the inertial mass and the velocity. The law canbe expressed in the formddt(mv)F (1.1)which, in the many cases in which the mass m is constant, reduces tomaF (1.2)where a is the acceleration of the mass.The third law of motion, the law of action and reaction, states that the force withwhich a mass mi acts on a mass mj is equal in magnitude and opposite in direction tothe force which mj exerts on mi. The additional assumption that these forces arecollinear is needed in some applications, e.g., in the development of the equations govern-ingthe motion of a rigid body.The law of gravitation asserts that the force of attraction between two pointmasses is proportional to the product of the masses and inversely proportional to thesquare of the distance between them. The masses involved in this formula are thegravitational masses. The fact that falling bodies possess identical accelerations leads,in conjunction with Eq. (1.2), to the proportionality of the inertial mass of a body toits gravitational mass. The results of very precise experiments by Eotvs and othersshow that inertial mass is, in fact, equal to gravitational mass. In the future the wordmass will be used without either qualifying adjective.If a mass in motion possesses the position vectors r1 and r2 relative to the originsof two inertial frames S1 and S2, respectively, and if further S1 and S2 have a relativevelocity V, then it follows from Eq. (1.2) thatr1r2Vt2const(1.3)t1t2constin which t1 and t2 are the times measured in S1 and S2. The transformation rules Eq. (1.3),in which the constants depend merely upon the choice of origin, are called galileantransformations. It is clear that acceleration is an invariant under such transformations.The rules of transformation between an inertial frame and a noninertial frame areconsiderably more complicated than Eq. (1.3). Their derivation is facilitated by theapplication of the following theorem: a frame S1 possesses relative to a frame S an angularvelocitypassing through the common origin of the two frames. The time rate of change 22. CLASSICAL MECHANICS 1.5of any vector A as measured in S is related to that measured in S1 by the formula(dAdt)S(dAdt)S1A (1.4)The interpretation of Eq. (1.4) is clear. The first term on the right-hand side accountsfor the change in the magnitude of A, while the second corresponds to its change indirection.If S is an inertial frame and S1 is a frame rotating relative to it, as explained in thestatement of the theorem, S1 being therefore noninertial, the substitution of the posi-tionvector r for A in Eq. (1.4) produces the resultvabsvrelr (1.5)In Eq. (1.5) vabs represents the velocity measured relative to S, vrel the velocity relativeto S1, and r is the transport velocity of a point rigidly attached to S1. The law oftransformation of acceleration is found on a second application of Eq. (1.4), in whichA is replaced by vabs. The result of this substitution leads directly to(d2rdt2)S(d2rdt2)S1 (r)r2vrel (1.6)in which is the time derivative, in either frame, of . The physical interpretation ofEq. (1.6) can be shown in the formaabsarelatransacor (1.7)where acor represents the Coriolis acceleration 2vrel. The results, Eqs. (1.5) and(1.7), constitute the rules of transformation between an inertial and a nonintertialframe. Equation (1.7) shows in addition that in a noninertial frame the second law ofmotion takes the formmarelFabs macor matrans (1.8)The modifications required in the above formulas are easily made for the case in whichS1 is translating as well as rotating relative to S. For, if D(t) is the position vector of theorigin of the S1 frame relative to that of S, Eq. (1.5) is replaced byVabs(dDdt)Svrelrand consequently, Eq. (1.7) is replaced byaabs(d2Ddt2)SarelatransacorIn practice the decision as to what constitutes an inertial frame of reference dependsupon the accuracy sought in the contemplated analysis. In many cases a set of axes rigidlyattached to the earths surface is sufficient, even though such a frame is noninertial to theextent of its taking part in the daily rotation of the earth about its axis and also its yearlyrotation about the sun. When more precise results are required, a set of axes fixed at thecenter of the earth may be used. Such a set of axes is subject only to the orbital motion ofthe earth. In still more demanding circumstances, an inertial frame is taken to be onewhose orientation relative to the fixed stars is constant.1.3 THE DYNAMICS OF A SYSTEM OF MASSESThe problem of locating a system in space involves the determination of a certainnumber of variables as functions of time. This basic number, which cannot be reducedwithout the imposition of constraints, is characteristic of the system and is known as 23. 1.6 MECHANICAL DESIGN FUNDAMENTALSits number of degrees of freedom. A point mass free to move in space has threedegrees of freedom. A system of two point masses free to move in space, but subjectto the constraint that the distance between them remains constant, possesses fivedegrees of freedom. It is clear that the presence of constraints reduces the number ofdegrees of freedom of a system.Three possibilities arise in the analysis of the motion-of-mass systems. First, thesystem may consist of a small number of masses and hence its number of degrees offreedom is small. Second, there may be a very large number of masses in the system,but the constraints which are imposed on it reduce the degrees of freedom to a smallnumber; this happens in the case of a rigid body. Finally, it may be that the constraintsacting on a system which contains a large number of masses do not provide an appreciablereduction in the number of degrees of freedom. This third case is treated in statisticalmechanics, the degrees of freedom being reduced by statistical methods.In the following paragraphs the fundamental results relating to the dynamics of mass sys-temsare derived. The system is assumed to consist of n constant masses mi (i1, 2, . . ., n).The position vector of mi, relative to the origin O of an inertial frame, is denoted by ri. Theforce acting on mi is represented in the form(1.9)in which Fine is the external force acting on mi, Fij is the force exerted on mi by mj, andFii is zero.1.3.1 The Motion of the Center of MassThe motion of mi relative to the inertial frame is determined from the equation(1.10)ea nOn summing the n equations of this type one finds(1.11)Fea ni1a nj1Fija ni1midvidtwhere Fe is the resultant of all the external forces which act on the system. ButNewtons third law states thatFijFjiand hence the double sum in Eq. (1.11) vanishes. Further, the position vector rc of thecenter of mass of the system relative to O is defined by the relation(1.12)mrca ni1miriin which m denotes the total mass of the system. It follows from Eq. (1.12) that(1.13)and therefore from Eq. (1.11) thatmvca ni1miviFem d2rc dt2 (1.14)Fij1FijmidvidtFiFei aj1Fij 24. CLASSICAL MECHANICS 1.7which proves the theorem: the center of mass moves as if the entire mass of the systemwere concentrated there and the resultant of the external forces acted there.Two first integrals of Eq. (1.14) provide useful results [Eqs. (1.15) and (1.16):(1.15)The integral on the left-hand side is called the impulse of the external force.Equation (1.15) shows that the change in linear momentum of the center of mass isequal to the impulse of the external force. This leads to the conservation-of-linear-momentumtheorem: the linear momentum of the center of mass is constant if noresultant external force acts on the system or, in view of Eq. (1.13), the total linearmomentum of the system is constant if no resultant external force acts:(1.16)which constitutes the work-energy theorem: the work done by the resultant externalforce acting at the center of mass is equal to the change in the kinetic energy of thecenter of mass.In certain cases the external force Fie may be the gradient of a scalar quantity Vwhich is a function of position only. ThenFeV/rcand Eq (1.16) takes the form(1.17)If such a function V exists, the force field is said to be conservative and Eq. (1.17) providesthe conservation-of-energy theorem.1.3.2 The Kinetic Energy of a SystemThe total kinetic energy of a system is the sum of the kinetic energies of the individualmasses. However, it is possible to cast this sum into a form which frequently makesthe calculation of the kinetic energy less difficult. The total kinetic energy of the massesin their motion relative to O isnbut rirciwhere i is the position vector of mi relative to thesystem center of mass C (see Fig. 1.1).HenceT 12nai1mir .2cn ai1mir .c #.i 12nai1mi .i2T 12ai1miv2ic12mv2c Vd21 021Fe # rc 12mv2cd21t2t1Fe dtmvcst2dmvcst1d0rimiircCFIG. 1.1 25. 1.8 MECHANICAL DESIGN FUNDAMENTALSbutby definition, and so(1.18)2 nnwhich proves the theorem: the total kinetic energy of a system is equal to the kinetic energyof the center of mass plus the kinetic energy of the motion relative to the center of mass.1.3.3 Angular Momentum of a System (Moment of Momentum)Each mass mi of the system has associated with it a linear momentum vector mivi. Themoment of this momentum about the point O is rimivi. The moment of momentumof the motion of the system relative to O, about O, isIt follows thatwhich, by Eq. (1.10), is equivalent to(1.19)neannnIt is now assumed that, in addition to the validity of Newtons third law, the force Fij iscollinear with Fji and acts along the line joining mi to mj, i.e., the internal forces arecentral forces. Consequently, the double sum in Eq. (1.19) vanishes and(1.20)nwhere M(O) represents the moment of the external forces about the point O. The followingextension of this result to certain noninertial points is useful.Let A be an arbitrary point with position vector a relative to the inertial point O(see Fig. 1.2). If i is the position vector of mi relative to A, then in the notationalready developeda .nThus (ddt) H(A)(ddt)H(O)mvcam(dvcdt), which reduces on application of Eqs. (1.14)and (1.20) tosd/dtdHsAdMsAda .The validity of the result mvc(ddt)H(A)M(A) (1.21)nHsAdai1imidridtai1sriadmidridt HsOdamvcddt HsOdai1riFei MsOdddt HsOdai1riFini1riaj1FijddtHsOdai1rimid2ridt2HsOdai1rimiviT 12 mr .c12 ai1mi .i2ai1mii00Amiri iFIG. 1.2 26. is assured if the point A satisfies either of the conditions1.0; i.e., the point A is fixed relative to O.2. is parallel to vc; i.e., the point A is moving parallel to the center of mass of thesystem.A particular, and very useful case of condition 2 is that in which the point A is thecenter of mass. The preceding results [Eqs. (1.20) and (1.21)] are contained in thetheorem: the time rate of change of the moment of momentum about a point is equalto the moment of the external forces about that point if the point is inertial, is movingparallel to the center of mass, or is the center of mass.As a corollary to the foregoing, one can state that the moment of momentum of asystem about a point satisfying the conditions of the theorem is conserved if themoment of the external forces about that point is zero.The moment of momentum about an arbitrary point A of the motion relative to A is(1.22)nnnIf the point A is the center of mass C of the system, Eq. (1.22) reduces toHrel(C)H(C) (1.23)which frequently simplifies the calculation of H(C).Additional general theorems of the type derived above are available in the litera-ture.The present discussion is limited to the more commonly applicable results.1.4 THE MOTION OF A RIGID BODYAs mentioned earlier, a rigid body is a dynamic system that, although it can be consideredto consist of a very large number of point masses, possesses a small number of degrees offreedom. The rigidity constraint reduces the degrees of freedom to six in the most generalcase, which is that in which the body is translating and rotating in space. This can be seenas follows: The position of a rigid body in space is determined once the positions of threenoncollinear points in it are known. These three points have nine coordinates, amongwhich the rigidity constraint prescribes three relationships. Hence only six of the coordi-natesare independent. The same result can be obtained otherwise.Rather than view the body as a system of point masses, it is convenient to consider it tohave a mass density per unit volume. In this way the formulas developed in the analysis ofthe motion of mass systems continue to be applicable if the sums are replaced by integrals.The six degrees of freedom demand six equations of motion for the determinationof six variables. Three of these equations are provided by Eq. (1.14), which describesthe motion of the center of mass, and the remaining three are found from moment-of-momentumconsiderations, e.g., Eq. (1.21). It is assumed, therefore, in what followsthat the motion of the center of mass is known, and the discussion is limited to therotational motion of the rigid body about its center of mass C.Letbe the angular velocity of the body. Then the moment of momentum about Cis, by Eq. (1.3),HsCd rsrd dV(1.24) VHrelsAdai1imididtai1imisr .ia .dHsAda . ai1miia . a .CLASSICAL MECHANICS 1.9Rotational motion about any fixed point of the body is treated in a similar way. 27. 1.10 MECHANICAL DESIGN FUNDAMENTALSxzwhere r is now the position vector of the element of volume dV relative to C (see Fig. 1.3), is the density of the body, and the integral is taken over the volume of the body. By adirect expansion one findsr(r)r2r(r )r2rr (1.25) r2I rr (r2Irr) and hence H(C)I(C) (1.25)where (1.26)IsCdVsr2Irrd dVis the inertia tensor of the body about C.In Eq. (1.26), I denotes the identity tensor. The inertia tensor can be evaluated oncethe value of and the shape of the body are prescribed. We now make a short digres-sionto discuss the structure and properties of I(C).For definiteness let x, y, and z be an orthogonal set of cartesian axes with origin atC (see Fig. 1.3). Then in matrix notationwhereIsCdIxx 2Ixy 2Ixz2Iyx Iyy 2Iyz2Izx 2Izy IzzIxxVIxyVsy2z2d dVxy dV. . . . . . . . . . . . .It is clear that:1. The tensor is second-order symmetric with real elements.2. The elements are the usual moments and products of inertia.yCrdVFIG. 1.3 28. CLASSICAL MECHANICS 1.113. The moment of inertia about a line through C defined by a unit vector e ise I(C) e4. Because of the property expressed in condition 1, it is always possible to determineat C a set of mutually perpendicular axes relative to which I(C) is diagonalized.Returning to the analysis of the rotational motion, one sees that the inertia tensorI(C) is time-dependent unless it is referred to a set of axes which rotate with the body.For simplicity the set of axes S1 which rotates with the body is chosen to be theorthogonal set in which I(C) is diagonalized. A space-fixed frame of reference withorigin at C is represented by S. Accordingly, from Eqs. (1.4) and (1.21),[(d/dt)H(C)]S[(d/dt)H(C)]S1 H(C)M(C) (1.27)which, by Eq. (1.25), reduces toI(C)(d/dt)I(C)M(C) (1.28)where H(C)iIxxxjIyyykIzzz (1.29)In Eq. (1.29) the x, y, and z axes are those for whichIsCdIxx 0 00 Iyy 00 0 Izzand i, j, k are the conventional unit vectors. Equation (1.28) in scalar form suppliesthe three equations needed to determine the rotational motion of the body. These equa-tions,the Euler equations, are(1.30)Ixxsd x dtd 1y zsIzz 2 Iyyd 5 MxIyysd y dtd 1z xsIxx 2 Izzd 5 MyIzzsd z dtd 1x ysIyy 2 Ixxd 5 MzThe analytical integration of the Euler equations in the general case defines a problemof classical difficulty. However, in special cases solutions can be found. The sources of thesimplifications in these cases are the symmetry of the body and the absence of some com-ponentsof the external moment. Since discussion of the various possibilities lies outsidethe scope of this chapter, reference is made to Refs. 1, 2, 6, and 7 and, for a survey ofrecent work, to Ref. 3. Of course, in situations in which energy or moment of momentum,or perhaps both, are conserved, first integrals of the motion can be written without employ-ingthe Euler equations. To do so it is convenient to have an expression for the kinetic ener-gyT of the rotating body. This expression is readily found in the following manner.The kinetic energy iswhich, by Eqs. (1.24), (1.25), and (1.26), is12 # IsCd # T(1.31)12 V# [rsrd] dVT 12 Vsrd2 dV 29. 1.12 MECHANICAL DESIGN FUNDAMENTALSor, in matrix notation,2Ts x y zd Ixx 0 00 Iyy 00 0 Izz Equation (1.31) can be put in a simpler form by writing122s d # IsCd # s d12I 2Tx y zand hence T 5(1.32)In Eq. (1.32) I is the moment of inertia of the body about the axis of the angularvelocity vector .1.5 ANALYTICAL DYNAMICSThe knowledge of the time dependence of the position vectors ri(t) which locate an n-masssystem relative to a frame of reference can be attained indirectly by determining the depen-denceupon time of some parameters qj ( j1, . . ., m) if the functional relationshipsriri(qj, t) i1, . . ., n; j1, . . ., m (1.33)are known. The parameters qj which completely determine the position of the systemin space are called generalized coordinates. Any m quantities can be used as general-izedcoordinates on condition that they uniquely specify the positions of the masses.Frequently the qj are the coordinates of an appropriate curvilinear system.It is convenient to define two types of mechanical systems:1. A holonomic system is one for which the generalized coordinates and the timemay be arbitrarily and independently varied without violating the constraints.2. A nonholonomic system is such that the generalized coordinates and the timemay not be arbitrarily and independently varied because of some (say s) noninte-grableconstraints of the form(1.34)mIn the constraint equations [Eq. (1.34)] the Aji and Aj represent functions of the qkand t. Holonomic and nonholonomic systems are further classified as rheonomicor scleronomic, depending upon whether the time t is explicitly present or absent,respectively, in the constraint equations.1.5.1 Generalized Forces and dAlemberts PrincipleA virtual displacement of the system is denoted by the set of vectors ri. The workdone by the forces in this displacement isnWa (1.35)i1Fi # riai1Aji dqiAj dt0 j1, 2, . . ., s 30. CLASSICAL MECHANICS 1.13If the force Fi, acting on the mass mi, is separable in the sense thatFi FiaFic (1.36)in which the first term is the applied force and the second the force of constraint, then(1.37)nd caThe generalized applied forces and the generalized forces of constraint are defined by(1.38)nnand (1.39)respectively. Hence, Eq. (1.37) assumes the form(1.40)mmnd # 'ri't tIf the virtual displacement is compatible with the instantaneous constraints t0,and if in such a displacement the forces of constraint do work, e.g., if sliding frictionis absent, then(1.41)Wa mj1QajqjThe assumption that a function V(qj, t) exists such thatleads to the result(1.42)Qaj5 2'V/'qjW 5 2VIn Eq. (1.42), V(qj, t) is called the potential or work function.The first step in the introduction of the kinetic energy of the system is taken byusing dAlemberts principle. The equations of motion [Eq. (1.10)] can be written asand consequently(1.43)nFi 2 mir $i 5 0The principle embodied in Eq. (1.43) constitutes the extension of the principle of vir-tualwork to dynamic systems and is named after dAlembert. When attention is con-finedto ri which represent virtual displacements compatible with the instantaneousconstraints and to forces Fi which satisfy Eqs. (1.36) and (1.41), the principle statesthatmna (1.44)j1Qajqjai1mir $i # riai1sFimir $id # ri0Waj1Qajqjaj1Qcjqjai1sFai FciQcj ai1Fci# 'ri'qjQaj ai1Fai# 'ri'qjWai1sFai Fcimj1'ri'qjqj 'ri't t d 31. 1.14 MECHANICAL DESIGN FUNDAMENTALS1.5.2 The Lagrange EquationsThe central equations of analytical mechanics can now be derived. These equations,which were developed by Lagrange, are presented here for the general case of a rheonomicnonholonomic system consisting of n masses mi, m generalized coordinates qi, and s constraintequations(1.45)maj1Akj dqjAk dt0 k1, 2, . . ., sThe equations are found by writing the acceleration terms in dAlemberts principle[Eq. (1.43)] in terms of the kinetic energy T and the generalized coordinates. By definitionwhereThusAccordingly,(1.46)T dqjdtnmianr .j1'ri'qjn12 a1'ridtand by summing over all values of j, one finds(1.47)becausemir .i2i1, 2, . . ., nnmriaj1'ri'qjqjmnfor instantaneous displacements. From Eqs. (1.44) and (1.47) it follows that(1.48)mThe qj which appear in Eq. (1.48) are not independent but must satisfy the instanta-neousconstraint equations(1.49)mThe elimination of s of the qj between Eqs. (1.48) and (1.49) is effected, in theusual way, by the introduction of s Lagrange multipliers 32. k(k1, 2, . . ., s). This stepleads directly to the equations(1.50)ddt'T'q .j'T'qj Qajs ak1 33. kAkj j1, 2, . . ., maj1Akj qj0 k1, 2, . . ., saj1addt'T'q .j'T'qj Qajb qj0aj1addt'T'q .j'T'qjb qjai1mi r $i # riddt'T'q .j'T'qj ai1mrr $i #'ri'qjj1, 2, . . ., m'T'qjai1mir .i #ddt'ri'qjand'T'q .j ai1mir .i #'ri'qj'r .i'q .j 5 'ri'qj 'r .i'qj 5 sddtds'ri'qjd 34. CLASSICAL MECHANICS 1.15These m second-order ordinary differential equations are the Lagrange equations ofthe system. The general solution of the equations is not available. For a holonomicsystem with n degrees of freedom, Eq. (1.50) reduces to(1.51)ddt'T'q .j'T'qj QajIn the presence of a function V such thatQaj'V'q .andEqs. (1.51) can be written in the form(1.52)ddt'l'q .j'l'qjj1, . . ., n 2'V'qjj0 0 j1, 2, . . ., nin which lTVThe scalar function lthe lagrangianwhich is the difference between the kinetic andpotential energies is all that need be known to write the Lagrange equations in this case.The major factor which contributes to the solving of Eq. (1.52) is the presence ofignorable coordinates. In fact, in dynamics problems, generally, the possibility of find-inganalytical representations of the motion depends on there being ignorable coordi-nates.A coordinate, say qk, is said to be ignorable if it does not appear explicitly in thelagrangian, i.e., if(1.53)'l'qk0If Eq. (1.53) is valid, then Eq. (1.52) leads to'l'q .kconstckand hence a first integral of the motion is available. Clearly the more ignorable coordi-natesthat exist in the lagrangian, the better. This being so, considerable effort hasbeen directed toward developing systematic means of generating ignorable coordinatesby transforming from one set of generalized coordinates to another, more suitable, set.This transformation theory of dynamics, while extensively developed, is not generallyof practical value in engineering problems.1.5.3 The Euler AnglesTo use lagrangian methods in analyzing the motion of a rigid body one must choose a setof generalized coordinates which uniquely determines the position of the body relative toa frame of reference fixed in space. It suffices to examine the motion of a body rotatingabout its center of mass.An inertial set of orthogonal axes, , andwith origin at the center of mass and anoninertial set x, y, and z fixed relative to the body with the same origin are adopted.The required generalized coordinates are those which specify the position of the x, y,and z axes relative to the, , andaxes. More than one set of coordinates whichachieves this purpose can be found. The most generally useful one, viz., the Eulerangles, is used here.Nonholonomic problems are frequently more tractable by vectorial than by lagrangian methods. 35. The frame, , andcan be brought into coincidence with the frame x, y, and z bythree finite rigid-body rotations through angles , , and , in that order, defined asfollows (see Fig. 1.4):1. A rotation about theaxis through an angleto produce the frame x1, y1, z12. A rotation about the x1 axis through an angleto produce the frame x2, y2, z23. A rotation about the z2 axis through an angleto produce the frame x3, y3, z3,which coincides with the frame x, y, zEach rotation can be represented by an orthogonal matrix operation so that theprocess of getting from the inertial to the noninertial frame is(1.54a)(1.54b)(1.54c)Consequently,(1.55)wherecoscos cossinsincossin coscossin 2sincos cossincos2sinsin coscoscos D5CBA5 sinsin2sincos sinsin cossin cos xyz 5 CBA 5 D x3y3z3cossin02sincos00 0 1 x2y2z2Cx2y2z2x2y2z21 0 00 cossin 0 2sincos x1y1z1Bx1y1z1x1y1z1cossin02sincos00 0 1 A 1.16 MECHANICAL DESIGN FUNDAMENTALSy1z1x1z2 z3y2y3x2 x3z1 z2x1 x2y2y1FIG. 1.4This notation is not universally adopted. See Ref. 5 for discussion. 36. CLASSICAL MECHANICS 1.17Since A, B, and C are orthogonal matrices, it follows from Eq. (1.55) that(1.56)where the prime denotes the transpose of the matrix. From Eq. (1.55) one sees that, ifthe time dependence of the three angles , ,is known, the orientation of the x, y, zand axes relative to the, , andaxes is determined. This time dependence is soughtby attempting to solve the Lagrange equations.The kinetic energy T of the rotating body is found from Eq. (1.31) to be2TIxx 2xIyy 2xIzz 2z(1.57)in which the components of the angular velocity are provided by the matrix equation(1.58)It is to be noted that if(1.59)none of the angles is ignorable. Hence considerable difficulty is to be expected inattempting to solve the Lagrange equations if this inequality, Eq. (1.59), holds. A simi-larinference could be made on examining Eq. (1.30). The possibility of there beingignorable coordinates in the problem arises if the body has axial, or so-called kinetic,symmetry about (say) the z axis. ThenIxxIyyIand, from Eq. (1.57),(1.60).d2The anglesanddo not occur in Eq. (1.60). Whether or not they are ignorabledepends on the potential energy V(, , ).1.5.4 Small Oscillations of a System near EquilibriumThe Lagrange equations are particularly useful in examining the motion of a systemnear a position of equilibrium. Let the generalized coordinates q1, q2, . . ., qntheexplicit appearance of time being ruled outrepresent the configuration of the system.It is not restrictive to assume the equilibrium position atq1 and q2qn0q .and, since motion near this position is being considered, the qand may be taken toi ibe small.The potential energy can be expanded in a Taylor series about the equilibrium pointin the formni1 a'V'qib0Vsq1 cqndVs0daqi (1.61) 12 aiaja '2V'qi 'qjb0qi qjc2TIs2 .sin2 2 .dIzzs .cos Ixx 2 Iyy 2 Izz x y zCB00 .C .0000 . 5 D21 xyz 5 Dr xyz 37. 1.18 MECHANICAL DESIGN FUNDAMENTALSIn Eq. (1.61) the first term can be neglected because it merely changes the potentialenergy by a constant and the second term vanishes because is zero at the equi-libriumpoint. Thus, retaining only quadratic terms in qi, one finds(1.62)Vij 5 s'2V'qi 'qjd0 5 Vjiin which (1.63)are real constants.The kinetic energy T of the system is representable by an analogous Taylor series(1.64).d .i cqTsq12 aiwhere TijTji (1.65)are real constants. The quadratic forms, Eqs. (1.62) and (1.64), in matrix notation, aprime denoting transposition are(1.66)12 qrvq12 q .rtq .V 5and T 5(1.67)In these expressions v and t represent the matrices with elements Vij and Tij, respec-tively,and q represents the column vector (q1, . . ., qn). The form of Eq. (1.67) is neces-sarilypositive definite because of the nature of kinetic energy. Rather than create theLagrange equations in terms of the coordinates qi, a new set of generalized coordi-nagesi is introduced in terms of which the energies are simultaneously expressible asquadratic forms without cross-product terms. That the transformation to such coordi-natesis possible can be seen by considering the equationsvbj 38. jtbj j1, 2, . . ., n (1.68)in which 39. j, the roots of the equation|v 40. t|0are the eigenvaluesassumed distinctand bj are the corresponding eigenvectors.The matrix of eigenvectors bj is symbolized by B, and the diagonal matrix of eigenvalues 41. j by . One can writebrkvbj 42. jbrktbjbrkvbj 43. kbrktbjandbecause of the symmetry of v and t. Thus, if 44. j 45. k, it follows thatbrktbj0 k 2 jand, since the eigenvectors of Eq. (1.68) are each undetermined to within an arbitrarymultiplying constant, one can always normalize the vectors so that(1.69)britbi 5 1Hence BtBIajTij q .i q .jVsq1cqnd 12 aiajVij qi qj'V'qi 46. where I is the unit matrix. ButCLASSICAL MECHANICS 1.19vBtB (1.70)and so BvBBtB (1.71)Furthermore, denoting the complex conjugate by an overbar, one hasvbj 5 47. j tbjbrjvbj 48. j brj tbjand (1.72)since v and t are real. However,(1.73)brjvbj 49. jbrjtbjbecause v and t are symmetric. From Eqs. (1.72) and (1.73) it follows that(1.74)brjtbjs 50. j 2 51. jdbrjtbj 5 0The symmetry and positive definiteness of t ensure that the form is real andpositive definite. Consequently the eigenvalues 52. j, and eigenvectors bj, are real.Finally, one can solve Eq. (1.68) for the eigenvalues in the form(1.75) 53. j 5 brjvbjbrjtbjThe transformation from the qi to the i coordinates can now be made by writingqB12qrvq 512rBrvB 512rfrom which (1.76)12q .r tq .12 .rBrtB .12 .rI .V 5and T (1.77)It is seen from Eqs. (1.76) and (1.77) that V and T have the desired forms and that thecorresponding Lagrange equations (1.52) are(1.78)where2d2idt2 2ii0 i1, . . ., ni 54. i. If the equilibrium position about which the motion takes place is stable, the2iare positive. The eigenvalues 55. i must then be positive, and Eq. (1.75) shows that V is positivedefinite. In other words, the potential energy is a minimum at a position of stable equilibrium.In this case, the motion of the system can be analyzed in terms of its normal modesthe nharmonic oscillators Eq. (1.78). If the matrix V is not positive definite, Eq. (1.75) indicates thatnegative eigenvalues may exist, and hence Eqs. (1.78) may have hyperbolic solutions. Theequilibrium is then unstable. Regardless of the nature of the equilibrium, the Lagrange equa-tions(1.78) can always be arrived at, because it is possible to diagonalize simultaneously twoquadratic forms, one of which (the kinetic-energy matrix) is positive definite.1.5.5 Hamiltons PrincipleIn conclusion it is remarked that the Lagrange equations of motion can be arrived atby methods other than that presented above. The point of departure adopted here isHamiltons principle, the statement of which for holonomic systems is as follows. 56. 1.20 MECHANICAL DESIGN FUNDAMENTALSProvided the initial (t1) and final (t2) configurations are prescribed, the motion ofthe system from time t1 to time t2 occurs in such a way that the line integralt23t1l dt 5 extremumwhere lTV. That the Lagrange equations [Eq. (1.52)] can be derived from thisprinciple is shown here for the case of a single-mass, one-degree-of-freedom system.The generalization of the proof to include an n-degree-of-freedom system is madewithout difficulty.The lagrangian islsq, q ., td 5 T 2 Vin which q is the generalized coordinate and q(t) describes the motion that actuallyoccurs. Any other motion can be represented by(1.79)q# std 5 qstd 1 fstdin which f(t) is an arbitrary differentiable function such that f (t1) and f (t2)0 and isa parameter defining the family of curves . The conditionis tantamount to(1.80)for all f(t). Butt2which, by Eq. (1.79), is(1.81)t2q# std.#1, td dt 5 0 5 0t2t2t2t2Its second term having been integrated by parts, Eq. (1.81) reduces to., td dtt2because f(t1)f(t2)0. Hence Eq. (1.80) is equivalent to(1.82)for all f(t). Equation (1.82) can hold for all f(t) only ifddt'l'q .2'l'qwhich is the Lagrange equation of the system.5 03t2t1fstd a'l'q2ddt'l'q .b dt 5 0'' t2t1lsq# , q#t1fstd a'l'qddt'l'q# .b dt'' 3t1sq#1, q# .1, td dt 5 3t1cfstd'l'q#1 f .std'l'q# .d dt'' 3t1lsq#1, q# .1, td dt 5 3t1a'l'q#'q#'1'l.'q#'q# . 'b dt'' 3t1sq#1, q3t1lsq1, q .1, td dt 5 extremum 57. The extension to an n-degree-of-freedom system is made by employing n arbitrarydifferentiable functions fk(t), k1, . . ., n such that fk(t1)fk(t2)0. For the general-izationsof Hamiltons principle which are necessary in treating nonholonomic systems,the references should be consulted.The principle can be extended to include continuous systems, potential energiesother than mechanical, and dissipative sources. The analytical development of theseand other topics and examples of their applications are presented in Refs. 4 and 8through 12.REFERENCES1. Routh, E. J.: Advanced Dynamics of a System of Rigid Bodies, 6th ed., Dover Publications,Inc., New York, 1955.2. Whittaker, E. T.: A Treatise on Analytical Dynamics, 4th ed., Dover Publications, Inc.,New York, 1944.3. Leimanis, E., and N. Minorsky: Dynamics and Nonlinear Mechanics, John WileySons,Inc., New York, 1958.4. Corben, H. C., and P. Stehle: Classical Mechanics, 2d ed., John WileySons, Inc., New York,1960.5. Goldstein, H.: Classical Mechanics, 2d ed., Addison-Wesley Publishing Company, Inc.,Reading, Mass, 1980.6. Milne, E. A.: Vectorial Mechanics, MethuenCo., Ltd., London, 1948.7. Scarborough, J. B.: The Gyroscope, Interscience Publishers, Inc., New York, 1958.8. Synge, J. L., and B. A. Griffith: Principles of Mechanics, 3d ed., McGraw-Hill BookCompany, Inc., New York, 1959.9. Lanczos, C.: The Variational Principles of Mechanics, 4th ed., University of Toronto Press,Toronto, 1970.10. Synge, J. L.: Classical Dynamics, in Handbuch der Physik, Bd III/I, Springer-Verlag,Berlin, 1960.11. Crandall, S. H., et al.: Dynamics of Mechanical and Electromechanical Systems, McGraw-HillBook Company, Inc., New York, 1968.12. Woodson, H. H., and J. R. Melcher: Electromechanical Dynamics, John WileySons,Inc., New York, 1968.CLASSICAL MECHANICS 1.21 58. This page intentionally left blank 59. CHAPTER 2MECHANICS OF MATERIALSStephen B. Bennett, Ph.D.Manager of Research and Product DevelopmentDelaval Turbine DivisionImo Industries, Inc.Trenton, N.J.Robert P. Kolb, P.E.Manager of Engineering (Retired)Delaval Turbine DivisionImo Industries, Inc.Trenton, N.J.2.12.1 INTRODUCTION 2.22.2 STRESS 2.32.2.1 Definition 2.32.2.2 Components of Stress 2.32.2.3 Simple Uniaxial States of Stress2.42.2.4 Nonuniform States of Stress 2.52.2.5 Combined States of Stress 2.52.2.6 Stress Equilibrium 2.62.2.7 Stress Transformation: Three-Dimensional Case 2.92.2.8 Stress Transformation: Two-Dimensional Case 2.102.2.9 Mohrs Circle 2.112.3 STRAIN 2.122.3.1 Definition 2.122.3.2 Components of Strain 2.122.3.3 Simple and Nonuniform States ofStrain 2.122.3.4 Strain-Displacement Relationships2.132.3.5 Compatibility Relationships 2.152.3.6 Strain Transformation 2.162.4 STRESS-STRAIN RELATIONSHIPS 2.172.4.1 Introduction 2.172.4.2 General Stress-Strain Relationship2.182.5 STRESS-LEVEL EVALUATION 2.192.5.1 Introduction 2.192.5.2 Effective Stress 2.192.6 FORMULATION OF GENERAL MECHAN-ICS-OF-MATERIAL PROBLEM 2.212.6.1 Introduction 2.212.6.2 Classical Formulations 2.212.6.3 Energy Formulations 2.222.6.4 Example: Energy Techniques 2.242.7 FORMULATION OF GENERAL THERMO-ELASTICPROBLEM 2.252.8 CLASSIFICATION OF PROBLEM TYPES2.262.9 BEAM THEORY 2.262.9.1 Mechanics of Materials Approach2.262.9.2 Energy Considerations 2.292.9.3 Elasticity Approach 2.382.10 CURVED-BEAM THEORY 2.412.10.1 Equilibrium Approach 2.422.10.2 Energy Approach 2.432.11 THEORY OF COLUMNS 2.452.12 SHAFTS, TORSION, AND COMBINEDSTRESS 2.482.12.1 Torsion of Solid Circular Shafts2.482.12.2 Shafts of Rectangular Cross Section2.492.12.3 Single-Cell Tubular-Section Shaft2.492.12.4 Combined Stresses 2.502.13 PLATE THEORY 2.512.13.1 Fundamental Governing Equation2.512.13.2 Boundary Conditions 2.522.14 SHELL THEORY 2.562.14.1 Membrane Theory: Basic Equation2.562.14.2 Example of Spherical Shell Subjectedto Internal Pressure 2.582.14.3 Example of Cylindrical ShellSubjected to Internal Pressure 2.582.14.4 Discontinuity Analysis 2.582.15 CONTACT STRESSES: HERTZIANTHEORY 2.622.16 FINITE-ELEMENT NUMERICAL ANALYSIS2.632.16.1 Introduction 2.632.16.2 The Concept of Stiffness 2.66Copyright 2006, 1996 by The McGraw-Hill Companies, Inc. Click here for terms of use. 60. 2.2 MECHANICAL DESIGN FUNDAMENTALS2.16.3 Basic Procedure of Finite-ElementAnalysis 2.682.16.4 Nature of the Solution 2.752.16.5 Finite-Element Modeling Guidelines2.762.16.6 Generalizations of the Applications2.762.16.7 Finite-Element Codes 2.782.1 INTRODUCTIONThe fundamental problem of structural analysis is the prediction of the ability ofmachine components to provide reliable service under its applied loads and tempera-ture.The basis of the solution is the calculation of certain performance indices, suchas stress (force per unit area), strain (deformation per unit length), or gross deforma-tion,which can then be compared to allowable values of these parameters. The allow-ablevalues of the parameters are determined by the component function (deformationconstraints) or by the material limitations (yield strength, ultimate strength, fatiguestrength, etc.). Further constraints on the allowable values of the performance indicesare often imposed through the application of factors of safety.This chapter, Mechanics of Materials, deals with the calculation of performanceindices under statically applied loads and temperature distributions. The extension ofthe theory to dynamically loaded structures, i.e., to the response of structures to shockand vibration loading, is treated elsewhere in this handbook.The calculations of Mechanics of Materials are based on the concepts of forceequilibrium (which relates the applied load to the internal reactions, or stress, in thebody), material observation (which relates the stress at a point to the internal deforma-tion,or strain, at the point), and kinematics (which relates the strain to the gross defor-mationof the body). In its simplest form, the solution assumes linear relationshipsbetween the components of stress and the components of strain (hookean materialmodels) and that the deformations of the body are sufficiently small that linear rela-tionshipsexist between the components of strain and the components of deformation.This linear elastic model of structural behavior remains the predominant tool usedtoday for the design analysis of machine components, and is the principal subject ofthis chapter.It must be noted that many materials retain considerable load-carrying ability whenstressed beyond the level at which stress and strain remain proportional. The modifica-tionof the material model to allow for nonlinear relationships between stress andstrain is the principal feature of the theory of plasticity. Plastic design allows moreeffective material utilization at the expense of an acceptable permanent deformation ofthe structure and smaller (but still controlled) design margins. Plastic design is oftenused in the design of civil structures, and in the analysis of machine structures underemergency load conditions. Practical introductions to the subject are presented inRefs. 6, 7, and 8.Another important and practical extension of elastic theory includes a materialmodel in which the stress-strain relationship is a function of time and temperature.This creep of components is an important consideration in the design of machinesfor use in a high-temperature environment. Reference 11 discusses the theory of creepdesign. The set of equations which comprise the linear elastic structural model do nothave a comprehensive, exact solution for a general geometric shape. Two approachesare used to yield solutions:The geometry of the structure is simplified to a form for which an exact solution isavailable. Such simplified structures are generally characterized as being a levelsurface in the solution coordinate system. Examples of such simplified structures 61. MECHANICS OF MATERIALS 2.3include rods, beams, rectangular plates, circular plates, cylindrical shells, andspherical shells. Since these shapes are all level surfaces in different coordinatesystems, e.g., a sphere is the surface rconstant in spherical coordinates, it is agreat convenience to express the equations of linear elastic theory in a coordinateinvariant form. General tensor notation is used to accomplish this task.The governing equations are solved through numerical analysis on a case-by-casebasis. This method is used when the component geometry is such that none of theavailable beam, rectangular plate, etc., simplifications are appropriate. Althoughseveral classes of numerical procedures are widely used, the predominant procedurefor the solution of problems in the Mechanics of Materials is the finite-elementmethod.2.2 STRESS2.2.1 Definition2Stress is defined as the force per unit area acting on an elemental plane in thebody. Engineering units of stress are generally pounds per square inch. If the force isnormal to the plane the stress is termed tensile or compressive, depending uponwhether the force tends to extend or shorten the element. If the force acts parallel tothe elemental plane, the stress is termed shear. Shear tends to deform by causingneighboring elements to slide relative to one another.2.2.2 Components of Stress2A complete description of the internal forces (stress distributions) requires that stressbe defined on three perpendicular faces of an interior element of a structure. In Fig. 2.1a small element is shown, and, omitting higher-order effects, the stress resultant onany face can be considered as acting at the center of the area.The direction and type of stress at a point are described by subscripts to the stresssymbolor . The first subscript defines the plane on which the stress acts and thesecond indicates the direction in which itacts. The plane on which the stress acts isindicated by the normal axis to thatplane; e.g., the x plane is normal to the xaxis. Conventional notation omits thesecond subscript for the normal stressand replaces theby afor the shearstresses. The stress components canthus be represented as follows:Normal stress:xxxyyy (2.1)zzzShear stress:xyxy yzyzFIG. 2.1 Stress components. 62. 2.4 MECHANICAL DESIGN FUNDAMENTALSxzxz zxzx (2.2)yxyz zyzyIn tensor notation, the stress components are xxyxz (2.3)ij yxyyz zxzyzStress is positive if it acts in the positive-coordinate direction on those elementfaces farthest from the origin, and in the negative-coordinate direction on thosefaces closest to the origin. Figure 2.1 indicates the direction of all positive stresses,wherein it is seen that tensile stresses are positive and compressive stresses negative.The total load acting on the element of Fig. 2.1 can be completely defined by thestress components shown, subject only to the restriction that the coordinate axes aremutually orthogonal. Thus the three normal stress symbols x, y, z and six shear-stresssymbols xy, xz, yx, yz, zx, zy define the stresses of the element. However, fromequilibrium considerations, xyyx, yzzy, xzzx. This reduces the necessarynumber of symbols required to define the stress state to x, y, z, xy, xz, yz.2.2.3 Simple Uniaxial States of Stress1Consider a simple bar subjected to axial loads only. The forces acting at a transversesection are all directed normal to the section. The uniaxial normal stress at the sectionis obtained fromP/A (2.4)where Ptotal force and Across-sectional area.Uniaxial shear occurs in a circular cylinder, loaded as in Fig. 2.2a, with a radiuswhich is large compared to the wall thickness. This member is subjected to a torquedistributed about the upper edge:TPr (2.5)FIG. 2.2 Uniaxial shear basic element. 63. MECHANICS OF MATERIALS 2.5Now consider a surface element (assumed plane) and examine the stresses acting. Thestresseswhich act on surfaces a-a and b-b in Fig. 2.2b tend to distort the originalrectangular shape of the element into the parallelogram shown (dotted shape). Thistype of action of a force along or tangent to a surface produces shear within the ele-ment,the intensity of which is the shear stress.2.2.4 Nonuniform States of Stress1In considering elements of differential size, it is permissible to assume that the forceacts on any side of the element concentrated at the center of the area of that side, andthat the stress is the average force divided by the side area. Hence it has been impliedthus far that the stress is uniform. In members of finite size, however, a variable stressintensity usually exists across any given surface of the member. An example of a bodywhich develops a distributed stress pattern across a transverse cross section is a simplebeam subjected to a bending load as shown in Fig. 2.3a. If a section is then taken ata-a, F1 must be the internal force acting along a-a to maintain equilibrium. Forces F1and F1 constitute a couple which tends to rotate the element in a clockwise direction,and therefore a resisting couple must be developed at a-a (see Fig. 2.3b). The internaleffect at a-a is a stress distribution with the upper portion of the beam in tension andthe lower portion in compression, as in Fig. 2.3c. The line of zero stress on the trans-versecross section is the neutral axis and passes through the centroid of the area.FIG. 2.3 Distributed stress on a simple beam subjected to a bending load.2.2.5 Combined States of StressTension-Torsion. A body loaded simultaneously in direct tension and torsion, suchas a rotating vertical shaft, is subject to a combined state of stress. Figure 2.4a depictssuch a shaft with end load W, and constant torque T applied to maintain uniform rota-tionalvelocity. With reference to a-a, considering each load separately, a force systemFIG. 2.4 Body loaded in direct tension and torsion. 64. 2.6 MECHANICAL DESIGN FUNDAMENTALSas shown in Fig. 2.2b and c is developed at the internal surface a-a for the weight loadand torque, respectively. These two stress patterns may be superposed to determine thecombined stress situation for a shaft element.Flexure-Torsion. If in the above case the load W were horizontal instead of vertical,the combined stress picture would be altered. From previous considerations of a simplebeam, the stress distribution varies linearly across section a-a of the shaft of Fig. 2.5a.The stress pattern due to flexure then depends upon the location of the element in ques-tion;e.g., if the element is at the outside (element x) then it is undergoing maximumtensile stress (Fig. 2.5b), and the tensile stress is zero if the element is located on thehorizontal center line (element y) (Fig. 2.5c). The shearing stress is still constant at agiven element, as before (Fig. 2.5d). Thus the combined or superposed stress statefor this condition of loading varies across the entire transverse cross section.FIG. 2.5 Body loaded in flexure and torsion.2.2.6 Stress EquilibriumEquilibrium relations must be satisfied by each element in a structure. These are sat-isfiedif the resultant of all forces acting on each element equals zero in each of threemutually orthogonal directions on that element. The above applies to all situations ofstatic equilibrium. In the event that some elements are in motion an inertia termmust be added to the equilibrium equation. The inertia term is the elemental mass mul-tipliedby the absolute acceleration taken along each of the mutually perpendicularaxes. The equations which specify this latter case are called dynamic-equilibriumequations (see Chap. 4).Three-Dimensional Case.5,13 The equilibrium equations can be derived by separatelysumming all x, y, and z forces acting on a differential element accounting for the incre-mentalvariation of stress (see Fig. 2.6). Thus the normal forces acting on areas dz dyare x dz dy and [x(x/x) dx] dz dy. Writing x force-equilibrium equations, andby a similar process y and z force-equilibrium equations, and canceling higher-orderterms, the following three cartesian equilibrium equations result:x/xxy/yxz/z0 (2.6)y/yyz/zyx/x0 (2.7)z/zzx/xzy/y0 (2.8) 65. MECHANICS OF MATERIALS 2.7FIG. 2.6 Incremental element (dx, dy, dz) withincremental variation of stress.or, in cartesian stress-tensor notation,ij, j0 i, jx,y,z (2.9)and, in general tensor form,gikij,k0 (2.10)where gik is the contravariant metric tensor.Cylindrical-coordinate equilibrium considerations lead to the following set ofequations (Fig. 2.7):r/r(1/r)(r/)rz/z(r)/r0 (2.11)r/r(1/r)(/)z/z2r/r0 (2.12)rz/r(1/r)(z/)z/zrz/r0 (2.13)The corresponding spherical polar-coordinate equilibrium equations are (Fig. 2.8) rr1r 1in rr s1rr(2rr cot )0 (2.14) rr 1r1in sr1r [() cot 3r]0 (2.15)FIG. 2.7 Stresses on a cylindrical element. FIG. 2.8 Stresses on a spherical element. 66. 2.8 MECHANICAL DESIGN FUNDAMENTALS rr 1r1in r s1r (3r2 cot )0 (2.16) The general orthogonal curvilinear-coordinate equilibrium equations are h1h2h3 h2h3 hh 31 67. h1h2 68. h h1 21 h1 69. h1h3 1h1h2 h1 1 70. h1h3 h2 10 (2.17) h3h1h2h3 h 3h1 71. h h 1 72. 2 73. h2h3 h h23 74. 1 h2 h2h1 1 75. h2h3 h21 h2h1 h310 (2.18) h1h1h2h3 76. h 77. 1h2 78. h h 2 3 79. h3h1 h h3 80. 11h3 81. h3h2 1 h3h1 h3 82. 1h3h2 h1 83. 10 (2.19) h2where the, , 84. specify the coordinates of a point and the distance between twocoordinate points ds is specified by(ds)2(d /h1)2(d/h2)2(d 85. /h3)2 (2.20)which allows the determination of h1, h2, and h3 in any specific case.Thus, in cylindrical coordinates,(ds)2(dr)2(r d)2(dz)2 (2.21)so thatr h11 h21/r 86. z h31In spherical polar coordinates,(ds)2(dr)2(r d)2(r sind)2 (2.22)so thatr h11 h21/r 87. h31/(r sin )All the above equilibrium equations define the conditions which must be satisfied by eachinterior element of a body. In addition, these stresses must satisfy all surface-stress-boundaryconditions. In addition to the cartesian-, cylindrical-, and spherical-coordinatesystems, others may be found in the current literature or obtained by reduction from thegeneral curvilinear-coordinate equations given above. 88. MECHANICS OF MATERIALS 2.9In many applications it is useful to integrate the stresses over a finite thickness andexpress the resultant in terms of zero or nonzero force or moment resultants as in thebeam, plate, or shell theories.Two-Dimensional CasePlane Stress.2 In the special but useful case where thestresses in one of the coordinate directions are negligibly small (zxzyz0)the general cartesian-coordinate equilibrium equations reduce tox/xxy/y0 (2.23)y/yyx/x0 (2.24)The corresponding cylindrical-coordinate equi-libriumequations becomer/r(1/r)(r/)(r)/r0 (2.25)r/r(1/r)(/)2(r/r)0 (2.26)This situation arises in thin slabs, as indicatedin Fig. 2.9, which are essentially two-dimensionalproblems. Because these equations are used informulations which allow only stresses in theplane of the slab, they are classified as plane-stressequations.FIG. 2.9 Plane stress on a thin slab.2.2.7 Stress Transformation: Three-Dimensional Case4,5It is frequently necessary to determine the stresses at a point in an element which isrotated with respect to the x, y, z coordinate system, i.e., in an orthogonal x, y, z sys-tem.Using equilibrium concepts and measuring the angle between any specific origi-naland rotated coordinate by the direction cosines (cosine of the angle between thetwo axes) the following transformation equations result:x[x cos (xx)xy cos (xy)zx cos (xz)] cos (xx) [xy cos (xx)y cos (xy)yz cos (xz)] cos (xy) [zx cos (xx)yz cos (xy)z cos (xz)] cos (xz) (2.27)y[x cos (yx)xy cos (yy)zx cos (yz)] cos (yx) [xy cos (yx)y cos (yy)yz cos (yz)] cos (yy) [zx cos (yx)yz cos (yy)z cos (yz)] cos (yz) (2.28)z[x cos (zx)xy cos (zy)zx cos (zz)] cos (zx) [xy cos (zx)y cos (zy)yz cos (zz)] cos (zy) [zx cos (zx)yz cos (zy)z cos (zz)] cos (zz) (2.29)xy[x cos (yx)xy cos (yy)zx cos (yz)] cos (xx) [xy cos (yx)y cos (yy)yz cos (yz)] cos (xy) [zx cos (yx)yz cos (yy)z cos (yz)] cos (xz) (2.30) 89. 2.10 MECHANICAL DESIGN FUNDAMENTALSyz[x cos (zx)xy cos (zy)zx cos (zz)] cos (yx) [xy cos (zx)y cos (zy)yz cos (zz)] cos (yy) [zx cos (zx)yz cos (zy)z cos (zz)] cos (yz) (2.31)zx[x cos (xx)xy cos (xy)zx cos (xz)] cos (zx) [xy cos (xx)y cos (xy)yz cos (xz)] cos (zy) [zx cos (xx)yz cos (xy)z cos (xz)] cos (zz) (2.32)In tensor notation these can be abbreviated asklAlnAkmmn (2.33)where Aijcos (ij) m,n x,y,z k,l x,y,zA special but very useful coordinate rotation occurs when the direction cosines areso selected that all the shear stresses vanish. The remaining mutually perpendicularnormal stresses are called principal stresses.The magnitudes of the principal stresses x, y, z are the three roots of the cubicequations associated with the determinantx xyzx0 (2.34)xyy yz zxyzz where x,, xy, are the general nonprincipal stresses which exist on an element.The direction cosines of the principal axes x, y z with respect to the x, y, z axesare obtained from the simultaneous solution of the following three equations consider-ingseparately the cases where nx, y z:xy cos (xn)(yn) cos (yn)yz cos (zn)0 (2.35)zx cos (xn)yz cos (yn)(zn) cos (zn)0 (2.36)cos2 (xn)cos2 (yn)cos2 (zn)1 (2.37)2.2.8 Stress Transformation: Two-Dimensional Case2,4Selecting an arbitrary coordinate direction in which the stress components vanish, itcan be shown, either by equilibrium considerations or by general transformation for-mulas,that the two-dimensional stress-transformation equations becomen[(xy)/2][(xy)/2] cos 2 xy sin 2(2.38)nt[(xy)/2] sin 2 xy cos 2(2.39)where the directions are defined in Figs. 2.10 and 2.11 (xy nt,0).The principal directions are obtained from the condition thatnt0 or tan 2 2xy/( xy) (2.40)where the two lowest roots of (first and second quadrants) are taken. It can be easilyseen that the first and second principal directions differ by 90. It can be shown thatthe principal stresses are also the maximum or minimum normal stresses. Theplane of maximum shear is defined by 90. MECHANICS OF MATERIALS 2.11FIG. 2.10 Two-dimensional plane stress. FIG. 2.11 Plane of maximum shear.tan 2 (xy)/2xy (2.41)These are also represented by planes which are 90 apart and are displaced from theprincipal stress planes by 45 (Fig. 2.11).2.2.9 Mohrs CircleMohrs circle is a convenient representation of the previously indicated transformationequations. Considering the x, y directions as positive in Fig. 2.11, the stress condition onany elemental plane can be represented as a point in the Mohr diagram (clockwise sheartaken positive). The Mohrs circle is constructed by connecting the two stress points anddrawing a circle through them with center on theaxis. The stress state of any basic ele-mentcan be represented by the stress coordinates at the intersection of the circle with anarbitrarily directed line through the circle center. Note that point x for positive xy is belowtheaxis and vice versa. The element is taken as rotated counterclockwise by an anglewith respect to the x-y element when the line is rotated counterclockwise an angle 2withrespect to the x-y line, and vice versa (Fig. 2.12).FIG. 2.12 Stress state of basic element. 91. 2.12 MECHANICAL DESIGN FUNDAMENTALS2.3 STRAIN2.3.1 Definition2Extensional strain is defined as the extensional deformation of an element divided bythe basic elemental length,u/l0.In large-strain considerations, l0 must represent theinstantaneous elemental length and the definitions of strainmust be given in incremental fashion. In small strain consid-erations,to which the following discussion is limited, it isonly necessary to consider the original elemental length l0and its change of length u. Extensional strain is taken posi-tiveor negative depending on whether the element increasesor decreases in extent. The units of strain are dimensionless(inches/inch).Shear strain 92. is defined as the angular distortion of anoriginal right-angle element. The direction of positive shearstrain is taken to correspond to that produced by a positiveshear stress (and vice versa) (see Fig. 2.13). Shear strain 93. FIG. 2.13 Shear-strain-deformedelement.is equal to 94. 1 95. 2. The units of shear strain are dimensionless (radians).2.3.2 Components of Strain2A complete description of strain requires the establishment of three orthogonal exten-sionaland shear strains. In cartesian stress nomenclature, the strain components areExtensional strain: xx x yy y (2.42) zz zShear strain: xy yx12 96. xy yz zy12 97. yz (2.43) zx xz12 98. zxwhere positivex,y, orz corresponds to a positive stretching in the x, y, z directionsand positive 99. xy, 100. yz, 101. zx refers to positive shearing displacements in the xy, yz, and zxplanes. In tensor notation, the strain components are 12 102. 12 103. xxyzx 12 104. 1 105. (2.44)ij xyy2yz12 106. 1 107. zx2yzz2.3.3 Simple and Nonuniform States of Strain2Corresponding to each of the stress states previously illustrated there exists either asimple or nonuniform strain state. 108. MECHANICS OF MATERIALS 2.13In addition to these, a state of uniform dilatation exists when the shear strainvanishes and all the extensional strains are equal in sign and magnitude. Dilatation isdefined as x y z (2.45)and represents the change of volume per increment volume.In uniform dilatation,3 x3 y3 z (2.46)2.3.4 Strain-Displacement Relationships4,5,13Considering only small strain, and the previous definitions, it is possible to express thestrain components at a point in terms of the associated displacements and their deriva-tivesin the coordinate directions (e.g., u, v, w are displacements in the x, y, z coordi-natesystem).Thus, in a cartesian system (x, y, z), xu/x 109. xyv/xu/y yv/y 110. yzw/yv/z (2.47) zw/z 111. z xu/zw/xor, in stress-tensor notation,2 ijui, juj, i i,j x,y,z (2.48)In addition the dilatationu/xv/yw/z (2.49)or, in tensor form,ui,j i x,y,z (2.50)Finally, all incremental displacements can be composed of a pure strain involvingall the above components, plus rigid-body rotational components. That is, in generalU xX12 112. xyY12 113. zxZzYyZ (2.51)V12 114. xyX yY12 115. yzZxZzX (2.52)W12 116. zxX12 117. yzY zZyXxY (2.53)where U, V, W represent the incremental displacement of the point xX, yY, zZin excess of that of the point x, y, z where X, Y, Z are taken as the sides of the incre-mentalelement. The rotational components are given by2xw/yv/z2yu/zw/x (2.54)2zv/xu/y 118. 2.14 MECHANICAL DESIGN FUNDAMENTALSor, in tensor notation,2ijui, juj, i i,jx,y,z (2.55)zyx, xzy, yxzIn cylindrical coordinates, rur/r 119. z(1/r)(uz/u/z (1/r)(u/)ur/r 120. zrur/zuz/r (2.56) zuz/z 121. ru/ru/r(1/r)(ur/)The dilatation is(1/r)(/r)(rur)(1/r)(u/)uz/z (2.57)and the rotation components are2r(1/r)(uz/)u/z2ur/zuz/r (2.58)2z(1/r)(/r)(ru)(1/r)(ur/)In spherical polar coordinates, r urr1r 122. u u cotr1ins u1r uur r1in 123. rrs ururur (2.59)1in rs uur urcot r 124. r urur 1r urThe dilatation is(1/r2 sin )[(/r)(r2ur sin )(/)(ru sin )(/)(ru)] (2.60)The rotation components are2r(1/r2 sin )[(/)(ru sin )(/)(ru)]2(1/r sin )[ur/(/r)(ru sin )] (2.61)2(1/r)[(/r)(ru)ur/]In general orthogonal curvilinear coordinates, h1(u / )h1h2u(/)(1/h1)h3h1u 125. (/ 126. )(1/h1) h2(u/)h2h3u 127. (/ 128. )(1/h2)h1h2u (/ )(1/h2) 129. h3(u 130. / 131. )h3h1u (/ )(1/h3)h2h3u(/)(1/h3) (2.62) 132. (h2/h3)(/)(h3u 133. )(h3/h2)(/ 134. )(h2u) 135. (h3/h1)(/ 136. )(h1u )(h1/h3)(/ )(h3u 137. ) 138. (h1/h2)(/ )(h2u)(h2/h1)(/)(h1u ) 139. MECHANICS OF MATERIALS 2.15h1h2h3[(/ )(u /h2h3)(/)(u/h3h1)(/ 140. )(u 141. /h1h2)] (2.63)2 h2h3[(/)(u 142. /h3)(/ 143. )(u/h2)]2h3h1[(/ 144. )(u /h1)(/ )(u 145. /h3)] (2.64)2 146. h1h2[(/ )(u/h2)(/)(u /h1)]where the quantities h1, h2, h3 have been discussed with reference to the equilibriumequations.In the event that one deflection (i.e., w) is constant or zero and the displacementsare a function of x, y only, a special and useful class of problems arises termed planestrain, which are analogous to the plane-stress problems. A typical case of planestrain occurs in slabs rigidly clamped on their faces so as to restrict all axial deforma-tion.Although all the stresses may be nonzero, and the general equilibrium equationsapply, it can be shown that, after combining all the necessary stress and strain relation-ships,both classes of plane problems yield the same form of equations. From this, onesolution suffices for both the related plane-stress and plane-strain problems, providedthat the elasticity constants are suitably modified. In particular the applicable strain-displacementrelationships reduce in cartesian coordinates to xu/x yv/y (2.65) 147. xyv/xu/yand in cylindrical coordinates to rur/r (1/r)(u/)ur/r (2.66) 148. ru/ru/r(1/r)(ur/)2.3.5 Compatibility Relationships2,4,5In the event that a single-valued continuous-displacement field (u, v, w) is not explicitlyspecified, it becomes necessary to ensure its existence in solution of the stress, strain,and stress-strain relationships. By writing the strain-displacement relationships andmanipulating them to eliminate displacements, it can be shown that the following sixequations are both necessary and sufficient to ensure compatibility:2 y/z22 z/y22 149. yz/y z2(2 x/y z)(/x)( 150. yz/x 151. zx/y 152. xy/z) (2.67)2 z/x22 x/z22 153. zx/x z2(2 y/z x)(/y)( 154. yz/x 155. zx/yyxy/z) (2.68)2 x/y22 y/x22 156. xy/x y2(2 z/x y)(/z)( 157. yz/x 158. zx/y 159. xy/z) (2.69)In tensor notation the most general compatibility equations are ij,kl kl,ij ik,jl jl,ik0 i,j,k,lx,y,z (2.70)which represents 81 equations. Only the above six equations are essential. 160. 2.16 MECHANICAL DESIGN FUNDAMENTALSIn addition to satisfying these conditions everywhere in the body under considera-tion,it is also necessary that all surface strain or displacement boundary conditions besatisfied.2.3.6 Strain Transformation4,5As with stress, it is frequently necessary to refer strains to a rotated orthogonal coordi-natesystem (x, y, z). In this event it can be shown that the stress and strain tensorstransform in an identical manner.x x x x xy 12 161. xy xy 12 162. xyy y y y yz 12 163. yz yz 12 164. yzz z z z zx 12 165. zx zx 12 166. zxIn tensor notation the strain transformation can be written asm, n x, y, zlk x, y zeklAlnAkm mn (2.71)As a result the stress and strain principal directions are coincident, so that all remarksmade for the principal stress and maximum shear components and their directionsFIG. 2.14 Strain transformation. 167. MECHANICS OF MATERIALS 2.17apply equally well to strain tensor components. Note that in the use of Mohrs circle inthe two-dimensional case one must be careful to substitute 12 168. forin the ordinateand forin the abscissa (Fig. 2.14).2.4 STRESS-STRAIN RELATIONSHIPS2.4.1 Introduction2It can be experimentally demonstrated that a one-to-one relationship exists betweenuniaxial stress and strain during a single loading. Further, if the material is alwaysloaded within its elastic or reversible range, a one-to-one relationship exists for allloading and unloading cycles.For stresses below a certain characteristic value termed the proportional limit, thestress-strain relationship is very nearly linear. The stress beyond which the stress-strainrelationship is no longer reversible is called the elastic limit. In most materialsthe proportional and elastic limits are identical. Because the departure from linearity isvery gradual it is often necessary to prescribe arbitrarily an apparent or offset elas-ticlimit. This is obtained as the intersection of the stress-strain curve with a line par-allelto the linear stress-strain curve, but offset by a prescribed amount, e.g., 0.02 per-cent(see Fig. 2.15a). The yield point is the value of stress at which continueddeformation of the bar takes place with little or no further increase in load, and theultimate limit is the maximum stress that the specimen can withstand.Note that some materials may show no clear difference between the apparent elas-tic,inelastic, and proportional limits or may not show clearly defined yield points(Fig. 2.15b).The concept that a useful linear range exists for most materials and that a simplemathematical law can be formulated to describe the relationship between stress andstrain in this range is termed Hookes law. It is an essential starting point in thesmall-strain theory of elasticity and the associated mechanics of materials. In theabove-described tensile specimen, the law is expressed asE(2.72)FIG. 2.15 Stress-strain relationship. 169. 2.18 MECHANICAL DESIGN FUNDAMENTALSas in the analogous torsional specimenG 170. (2.73)where E and G are the slope of the appropriate stress-strain diagrams and are calledthe Youngs modulus and the shear modulus of elasticity, respectively.2.4.2 General Stress-Strain Relationship2,4,5The one-dimensional concepts discussed above can be generalized for both small andlarge strain and elastic and nonelastic materials. The following discussion will be lim-itedto small-strain elastic materials consistent with much engineering design. Basedupon the above, Hookes law is expressed as x(1/E)[x(yz)] 171. xyxy/G y(1/E)[y(zx)] 172. yzyz/G (2.74) z(1/E)[z(xy)] 173. zxzx/Gwhereis Poissons ratio, the ratio between longitudinal strain and lateral contrac-tionin a simple tensile test.In cartesian tension form Eq. (2.74) is expressed as ij[(1)/E]ij(v/E)ijkk i, j,kx,y,z (2.75)where ij{0 i j1 ijThe stress-strain laws appear in inverted form asx2G xy2G yz2G zxyG 174. xy(2.76)yzG 175. yzzxG 176. zxwhere (1)(12v) x y zGE/2(1)In cartesian tensor form Eq. (2.76) is written asij2G ijij i, jx,y,z (2.77)and in general tensor form asij2G ij gij (2.78) 177. MECHANICS OF MATERIALS 2.19where gij is the covariant metric tensor and these coefficients (stress modulus) are oftenreferred to as Lams constants, and gmn mn.2.5 STRESS-LEVEL EVALUATION2.5.1 Introduction1,6The detailed elastic and plastic behavior, yield and failure criterion, etc., are repeatableand simply describable for a simple loading state, as in a tensile or torsional specimen.Under any complex loading