xviliSYMBOLSTshear stressshear stresses on planes perpendicularto thex,y,and zaxes and actingTryeTyeTexparallel to the y, z, and x axesshearstressonaplaneperpendiculartothexaxis andactingparalleltoTyythey,axis (rotated axes)Teshear stress on an inclined planeallowablestress (orworkingstress)inshearTallowultimate stress in shear;yield stressin shearTu,TyΦangle,angleoftwistofabar intorsiontangle,angleofrotationwangularvelocity,angularfrequency(w=2f)*A star attached to a section number indicates a specialized or advanced topic.One or more stars attached to a problem number indicate an increasing level ofdifficulty in the solution.GreekAlphabetANAlphaVNua三BβXiBetaT0YGammaOmicron04IIPiDeltaTEPERhoEpsilonpZZNZetaaSigmaHTEtaTauTn0Y6ThetaUUpsilon1d$IotaPhitKXKappaChiKXΛ亚4PsiLambdaM2MuOmegauw
xviii SYMBOLS t shear stress txy, tyz, tzx shear stresses on planes perpendicular to the x, y, and z axes and acting parallel to the y, z, and x axes tx1y1 shear stress on a plane perpendicular to the x1 axis and acting parallel to the y1 axis (rotated axes) tu shear stress on an inclined plane tallow allowable stress (or working stress) in shear tU, tY ultimate stress in shear; yield stress in shear f angle, angle of twist of a bar in torsion c angle, angle of rotation v angular velocity, angular frequency (v 2pf ) Greek Alphabet a Alpha n Nu b Beta j Xi g Gamma o Omicron d Delta p Pi e Epsilon r Rho z Zeta s Sigma h Eta t Tau u Theta y Upsilon i Iota f Phi k Kappa x Chi l Lambda c Psi ! m Mu " v Omega ★A star attached to a section number indicates a specialized or advanced topic. One or more stars attached to a problem number indicate an increasing level of difficulty in the solution.
Mechanics ofMaterials
Mechanics of Materials
Thistelecommunicationstowerisanassemblageofmanymembersthatactprimarilyintensionorcompression
NPR. Used with permission. This telecommunications tower is an assemblage of many members that act primarily in tension or compression
1Tension,Compression,and ShearCHAPTEROVERVIEWIn Chapter 1,weare introduced to mechanics ofmaterials, which exam-ines the stresses, strains, and displacements in bars of various materialsacted on by axial loads applied at the centroids of their cross sections.Wewilllearnaboutnormal stress(o)andnormalstrain(e)inmaterialsused for structural applications,then identifykeyproperties of variousmaterials, such as themodulus of elasticity (E)and yield (o)and ulti-mate()stresses,fromplotsofstress(o)versusstrain(e).Wewillalsoplot shear stress ()versus shear strain () and identify the shearingmodulus of elasticity (G).If these materials perform only in the linearrange,stress and strain are related by Hooke's Law fornormal stress andstrain (α =E.e)and also for shear stress and strain (T =G·).Wewill see that changes in lateral dimensions and volume depend uponPoisson's ratio ().Material properties E, G,and v,in fact, are directlyrelatedtooneanotherandarenotindependentpropertiesofthematerialAssemblage of bars to form structures(such as trusses)leadsto consideration of average shear ()and bearing (op) stresses intheir connections as well asnormal stresses acting onthenet area of thecross section (if in tension)or on the full cross-sectional area (ifin compression).If we restrictmaximum stresses at anypointtoallow-able values by use of factors of safety,we can identify allowable levelsofaxial loadsfor simple systems,such ascables and bars.Factorsofsafety relate actual to required strength of structural members andaccount for a variety of uncertainties, such as variations in materialproperties and probability of accidental overload. Lastly, we will con-sider design: the iterative process by which the appropriate size ofstructuralmembersisdeterminedtomeetavarietyofbothstrengthandstiffiness requirements for a particular structure subjected to a variety ofdifferentloadings
CHAPTER OVERVIEW In Chapter 1, we are introduced to mechanics of materials, which examines the stresses, strains, and displacements in bars of various materials acted on by axial loads applied at the centroids of their cross sections. We will learn about normal stress (#) and normal strain ($) in materials used for structural applications, then identify key properties of various materials, such as the modulus of elasticity (E) and yield (#y) and ultimate (#u) stresses, from plots of stress (#) versus strain ($). We will also plot shear stress (%) versus shear strain (&) and identify the shearing modulus of elasticity (G). If these materials perform only in the linear range, stress and strain are related by Hooke’s Law for normal stress and strain (# E . $) and also for shear stress and strain (% G . &). We will see that changes in lateral dimensions and volume depend upon Poisson’s ratio (v). Material properties E, G, and v, in fact, are directly related to one another and are not independent properties of the material. Assemblage of bars to form structures (such as trusses) leads to consideration of average shear (%) and bearing (#b) stresses in their connections as well as normal stresses acting on the net area of the cross section (if in tension) or on the full cross-sectional area (if in compression). If we restrict maximum stresses at any point to allowable values by use of factors of safety, we can identify allowable levels of axial loads for simple systems, such as cables and bars. Factors of safety relate actual to required strength of structural members and account for a variety of uncertainties, such as variations in material properties and probability of accidental overload. Lastly, we will consider design: the iterative process by which the appropriate size of structural members is determined to meet a variety of both strength and stiffness requirements for a particular structure subjected to a variety of different loadings. 3 1 Tension, Compression, and Shear
4CHAPTER1Tension,Compression,andSheaChapterisorganized asfollows:1.1Introduction toMechanics of Materials51.2NormalStressandStrain71.3MechanicalPropertiesofMaterials15Elasticity,Plasticity,andCreep241.41.5Linear Elasticity,Hooke's Law,and Poisson's Ratio27Shear Stress and Strain321.61.7AllowableStressesandAllowableLoads431.8Design forAxial Loads andDirect Shear49ChapterSummary&Review55Problems57
Chapter 1 is organized as follows: 1.1 Introduction to Mechanics of Materials 5 1.2 Normal Stress and Strain 7 1.3 Mechanical Properties of Materials 15 1.4 Elasticity, Plasticity, and Creep 24 1.5 Linear Elasticity, Hooke’s Law, and Poisson’s Ratio 27 1.6 Shear Stress and Strain 32 1.7 Allowable Stresses and Allowable Loads 43 1.8 Design for Axial Loads and Direct Shear 49 Chapter Summary & Review 55 Problems 57 4 CHAPTER 1 Tension, Compression, and Shear