MECHANICSOFMATERIALS
MECHANICS OF MATERIALS
CHAPTER(alexskopje/Fotolia)Theboltsusedfortheconnectionsofthissteelframeworkaresubjectedto stressInthis chapter wewilldiscuss howengineers designthese connections and theirfasteners
CHAPTER 1 The bolts used for the connections of this steel framework are subjected to stress. In this chapter we will discuss how engineers design these connections and their fasteners. (© alexskopje/Fotolia)
STRESSCHAPTEROBJECTIVESIn this chapter wewill review some of the important principles ofstatics and show how they are used to determine the internalresultant loadings in a body.Afterwards the concepts of normal andshear stress will be introduced, and specific applications of theanalysis anddesign ofmembers subjectedtoanaxial loadordirectshearwill bediscussed.1.1INTRODUCTIONMechanicsof materialsis abranchofmechanics thatstudiestheinternaleffects of stress and strain in a solid body.Stress is associated with thestrength of the material from whichthe bodyis made,whilestrain isameasure of the deformation of the body.A thorough understanding ofthe fundamentals of this subject is of vital importanceforthedesign ofany machine or structure,because many of the formulas and rulesof design cited in engineering codes are based upon the principles ofthis subject.3
3 STRESS 1.1 INTRODUCTION Mechanics of materials is a branch of mechanics that studies the internal effects of stress and strain in a solid body. Stress is associated with the strength of the material from which the body is made, while strain is a measure of the deformation of the body. A thorough understanding of the fundamentals of this subject is of vital importance for the design of any machine or structure, because many of the formulas and rules of design cited in engineering codes are based upon the principles of this subject. CHAPTER OBJECTIVES n In this chapter we will review some of the important principles of statics and show how they are used to determine the internal resultant loadings in a body. Afterwards the concepts of normal and shear stress will be introduced, and specific applications of the analysis and design of members subjected to an axial load or direct shear will be discussed
4CHAPTER1STRESSHistorical Development.The origin of mechanics of materialsdates back to thebeginningof the seventeenth century,when GalileoGalileiperformed experiments tostudythe effects of loads on rodsandbeamsmade ofvariousmaterials.However,it was not until thebeginningof thenineteenth century when experimental methods for testingmaterials were vastlyimproved.At that time many experimental andtheoretical studies in this subject wereundertaken,primarily in France,by such notables as Saint-Venant, Poisson, Lame,and Navier.Throughtheyears,aftermanyfundamental problemshadbeen solved,it became necessary to use advanced mathematical and computertechniquesto solvemorecomplexproblems.Asa result,mechanics ofmaterials has expanded into other areas of mechanics,such as thetheoryof elasticity and thetheoryof plasticity.1.2EQUILIBRIUMOFADEFORMABLEBODYSince statics plays an important role in both the development andapplicationofmechanics ofmaterials, it isveryimportantto haveagoodgrasp of its fundamentals.For this reason we will nowreview some of themainprinciples of statics that will be used throughoutthe text.Loads.Abodycanbe subjected to bothsurfaceloadsand bodyforces.Surface loads that act on a small area of contact are reported byconcentrated forces, while distributed loadings act over a larger surfacearea of the body.When the loading is coplanar, as in Fig.1-la, then aresultant forceFr of a distributed loading is equal to the area under thedistributed loadingdiagram,and this resultantactsthroughthegeometriccenter or centroid ofthis area.700NFR=400N200N/mW-1m15m(a)Fig.1-1
4 Chapter 1 Stress 1 Historical Development. The origin of mechanics of materials dates back to the beginning of the seventeenth century, when Galileo Galilei performed experiments to study the effects of loads on rods and beams made of various materials. However, it was not until the beginning of the nineteenth century when experimental methods for testing materials were vastly improved. At that time many experimental and theoretical studies in this subject were undertaken, primarily in France, by such notables as Saint-Venant, Poisson, Lamé, and Navier. Through the years, after many fundamental problems had been solved, it became necessary to use advanced mathematical and computer techniques to solve more complex problems. As a result, mechanics of materials has expanded into other areas of mechanics, such as the theory of elasticity and the theory of plasticity. 1.2 EQUILIBRIUM OF A DEFORMABLE BODY Since statics plays an important role in both the development and application of mechanics of materials, it is very important to have a good grasp of its fundamentals. For this reason we will now review some of the main principles of statics that will be used throughout the text. Loads. A body can be subjected to both surface loads and body forces. Surface loads that act on a small area of contact are reported by concentrated forces, while distributed loadings act over a larger surface area of the body. When the loading is coplanar, as in Fig. 1–1a, then a resultant force FR of a distributed loading is equal to the area under the distributed loading diagram, and this resultant acts through the geometric center or centroid of this area. Fig. 1–1 1 m 1 m 1 m 1.5 m 200 N/m (a) A B 700 N FR 400 N
51.2EQUILIBRIUMOFADEFORMABLEBODYAbodyforceisdevelopedwhenonebodyexertsaforceonanotherbodywithoutdirectphysical contactbetweenthebodies.Examplesincludetheeffectscausedbytheearth'sgravitationoritselectromagnetic field.Although these forces affect all the particlescomposing the body,they are normallyrepresented bya singleconcentrated force acting onthe body.In the case of gravitation,thisforce is called the weight W of the body and acts through the body'scenterofgravitySupportReactions.Forbodiessubjectedtocoplanarforcesystems,the supportsmostcommonlyencounteredareshown inTable1-1.Asageneral rule,if the support prevents translation in a given direction,then a force must be developed on the member in that direction.Likewise, if rotation isprevented, a couple moment must be exerted onthemember.Forexample,therollersupportonlypreventstranslationperpendicularornormaltothesurface.Hence,therollerexertsanormalforceFonthememberatitspointofcontact.SincethemembercanManymachineelementsarepinconnectedfreely rotate about the roller,a couple moment cannot be developed oninordertoenablefreerotationattheirthe member.connections.These supports exert a forceonamember,butnomoment.TABLE1-1ReactionTypeofconnectionReactionType ofconnectionFCableTwo unknowns: F,FOneunknown:FExternalpinAF,RollerOne unknown:FInternal pinTwo unknowns: F,FFMFSmoothsupportOneunknown:FThreeunknowns:Fr,F,MFixed support1FSJournal bearingOneunknown:FThrust bearingTwounknowns:Fx,F
1.2 Equilibrium of a Deformable Body 5 1 A body force is developed when one body exerts a force on another body without direct physical contact between the bodies. Examples include the effects caused by the earth’s gravitation or its electromagnetic field. Although these forces affect all the particles composing the body, they are normally represented by a single concentrated force acting on the body. In the case of gravitation, this force is called the weight W of the body and acts through the body’s center of gravity. Support Reactions. For bodies subjected to coplanar force systems, the supports most commonly encountered are shown in Table 1–1. As a general rule, if the support prevents translation in a given direction, then a force must be developed on the member in that direction. Likewise, if rotation is prevented, a couple moment must be exerted on the member. For example, the roller support only prevents translation perpendicular or normal to the surface. Hence, the roller exerts a normal force F on the member at its point of contact. Since the member can freely rotate about the roller, a couple moment cannot be developed on the member. Many machine elements are pin connected in order to enable free rotation at their connections. These supports exert a force on a member, but no moment. F F Type of connection Reaction Cable Roller One unknown: F One unknown: F F F Smooth support One unknown: F External pin Internal pin Fx Fy Fx Fy Two unknowns: Fx, Fy Fx Fx Fy Fy M Fixed support Three unknowns: Fx, Fy, M Two unknowns: Fx, Fy Type of connection Reaction u u u Journal bearing One unknown: F Thrust bearing Two unknowns: Fx, Fy TABLE 1–1