6CHAPTER1STRESSEquations of Equilibrium.Equilibriumof a bodyrequiresboth abalanceof forces,to prevent the bodyfromtranslating orhaving accelerated motion along a straight or curved path, and abalance of moments,to prevent thebodyfrom rotating.Theseconditionsare expressed mathematically as the equations ofequilibrium:ZF=0In orderto design themembers of this(1-1)buildingframeitisfirstnecessarytofindEMo=0the internal loadings at various pointsalongtheirlength.Here,Frepresentsthesumofalltheforcesactingonthebody,andMo is the sum of the moments of all the forces about any point Oeither on or off the body.Ifanx,y,z coordinate system isestablishedwiththeoriginatpointOtheforceand moment vectors can be resolved into components alongeach coordinateaxis, and theabovetwo equations can be writteninscalarform as six equations,namely,ZF=0ZF=0EF=0(1-2)EM=0EM,=0M=0700 NFr = 400 N200 N/mOfteninengineeringpracticetheloadingonabodycanberepresentedasasystemof coplanarforcesinthex-yplane.Inthis caseequilibriumofthe body can be specified with only three scalar equilibrium equationsthat is,5m(a)ZF =0EF=0(1-3)ZMo=0700N400NSuccessfulapplicationoftheequationsofequilibriummustincludeallthe known and unknown forces that act on the body,and the best wayBtoaccount for these loadings is todrawthebody'sfree-bodydiagram2m1.5mIBbeforeapplyingtheequationsofequilibrium.Forexample,thefree-bodydiagram of thebeam inFig.1-lais shown inFig.1-1b.Hereeachforce(b)is identified by its magnitudeand direction,and the body's dimensionsFig.1-1areincludedinordertosumthemomentsoftheforces
6 Chapter 1 Stress 1 Equations of Equilibrium. Equilibrium of a body requires both a balance of forces, to prevent the body from translating or having accelerated motion along a straight or curved path, and a balance of moments, to prevent the body from rotating. These conditions are expressed mathematically as the equations of equilibrium: ΣF = 0 ΣMO = 0 (1–1) Here, Σ F represents the sum of all the forces acting on the body, and Σ MO is the sum of the moments of all the forces about any point O either on or off the body. If an x, y, z coordinate system is established with the origin at point O, the force and moment vectors can be resolved into components along each coordinate axis, and the above two equations can be written in scalar form as six equations, namely, ΣFx = 0 ΣFy = 0 ΣFz = 0 ΣMx = 0 ΣMy = 0 ΣMz = 0 (1–2) Often in engineering practice the loading on a body can be represented as a system of coplanar forces in the x–y plane. In this case equilibrium of the body can be specified with only three scalar equilibrium equations, that is, ΣFx = 0 ΣFy = 0 ΣMO = 0 (1–3) Successful application of the equations of equilibrium must include all the known and unknown forces that act on the body, and the best way to account for these loadings is to draw the body’s free-body diagram before applying the equations of equilibrium. For example, the free-body diagram of the beam in Fig. 1–1a is shown in Fig. 1–1b. Here each force is identified by its magnitude and direction, and the body’s dimensions are included in order to sum the moments of the forces. In order to design the members of this building frame, it is first necessary to find the internal loadings at various points along their length. 1 m 1 m 1 m 1.5 m 200 N/m (a) A B 700 N FR 400 N Fig. 1–1 1 m 2 m 1.5 m 700 N 400 N (b) Ay By Bx
71.2EQUILIBRIUMOFADEFORMABLEBODYMROF4F3T?F:sectionFFiFF2F2F2(c)(a)(b)Fig.1-2Internal Resultant Loadings.In mechanics of materials,statics is primarily used to determine the resultant loadings that actwithin a body.This is done using the method of sections.For example,consider the body shown in Fig.1-2a, which is held in equilibrium bythe four external forces.*In order to obtain the internal loadingsacting on a specificregion withinthebody,it is necessaryto pass animaginary section or"cut"through the region where the internalloadings are to be determined.The two parts of the body are thenseparated,and afree-bodydiagramof oneof theparts is drawn.Whenthis is done,there will be a distribution of internal force acting on the"exposed"area of the section,Fig.1-2b.These forces actuallyrepresent the effects of the material of the top section of the bodyactingonthebottomsection.Although the exact distribution of this internal loading may beunknown,its resultantsFrandMro,Fig.1-2c,aredeterminedbyapplyingthe equations of equilibrium to the segment shown in Fig.1-2c.Herethese loadings act at point O; however, this point is often chosen at thecentroid ofthesectionedarea.*The body's weight is not shown, since it is assumed to be quite small, and thereforenegligible compared with the otherloads
1.2 Equilibrium of a Deformable Body 7 1 Internal Resultant Loadings. In mechanics of materials, statics is primarily used to determine the resultant loadings that act within a body. This is done using the method of sections. For example, consider the body shown in Fig. 1–2a, which is held in equilibrium by the four external forces.* In order to obtain the internal loadings acting on a specific region within the body, it is necessary to pass an imaginary section or “cut” through the region where the internal loadings are to be determined. The two parts of the body are then separated, and a free-body diagram of one of the parts is drawn. When this is done, there will be a distribution of internal force acting on the “exposed” area of the section, Fig. 1–2b. These forces actually represent the effects of the material of the top section of the body acting on the bottom section. Although the exact distribution of this internal loading may be unknown, its resultants FR and MRO, Fig. 1–2c, are determined by applying the equations of equilibrium to the segment shown in Fig. 1–2c. Here these loadings act at point O; however, this point is often chosen at the centroid of the sectioned area. *The body’s weight is not shown, since it is assumed to be quite small, and therefore negligible compared with the other loads. section F4 F2 (a) F1 F3 F1 F2 (b) FR F1 F2 O MRO (c) Fig. 1–2
8CHAPTER 1STRESSTorsionalMomentMRoMRNormalERForceFRBendingM'MomentShearForce6F2(c)(d)Fig. 1-2 (cont.)Three Dimensions.For later application of the formulas formechanics ofmaterials, wewill consider the components of Fr and Mracting both normal and tangent to the sectioned area, Fig.1-2d.Fourdifferent types ofresultantloadings canthenbedefined asfollows:Normal force, N.This force acts perpendicular to the area.It isdeveloped whenever the external loadstendtopushorpull on thetwosegmentsofthebody.Shearforce,V.The shear force lies in the planeof thearea,and it isdeveloped when the external loads tend to cause the two segments ofthe bodyto slide over one another.Torsional moment ortorque,T.This effectis developed when theexternal loadstend totwist one segment of thebodywith respecttotheotherabout an axis perpendicularto thearea.Bendingmoment, M.The bending moment is caused by theexternal loads that tend tobend thebody about an axislyingwithin theplaneof thearea.The weight of this sign and the windloadings acting on itwill cause normal andshearforcesandbendingandtorsionalNoticethatgraphical representationofamomentortorqueisshowninmoments in the supporting column.threedimensions as avector(arrow)withan associatedcurl aroundit.Bythe right-hand rule,thethumbgivesthearrowhead senseof thisvectorand the fingers or curl indicate thetendencyfor rotation (twisting orbending)
8 Chapter 1 Stress 1 Three Dimensions. For later application of the formulas for mechanics of materials, we will consider the components of FR and MRO acting both normal and tangent to the sectioned area, Fig. 1–2d. Four different types of resultant loadings can then be defined as follows: Normal force, N. This force acts perpendicular to the area. It is developed whenever the external loads tend to push or pull on the two segments of the body. Shear force, V. The shear force lies in the plane of the area, and it is the body to slide over one another. Torsional moment or torque, T. This effect is developed when the external loads tend to twist one segment of the body with respect to the other about an axis perpendicular to the area. Bending moment, M. The bending moment is caused by the external loads that tend to bend the body about an axis lying within the plane of the area. Notice that graphical representation of a moment or torque is shown in three dimensions as a vector (arrow) with an associated curl around it. By the right-hand rule, the thumb gives the arrowhead sense of this vector and the fingers or curl indicate the tendency for rotation (twisting or bending). The weight of this sign and the wind loadings acting on it will cause normal and shear forces and bending and torsional moments in the supporting column. O (c) MRO F1 F2 FR (d) O F1 F2 N T M V Torsional Moment Bending Moment Shear Force MRO FR Normal Force Fig. 1–2 (cont.)
91.2EQUILIBRIUMOFADEFORMABLEBODYsectionF-FShearForceM,BendingMomentNNormalForce(a)(b)Fig.1-3Coplanar Loadings.If thebody is subjectedto a coplanarsystem offorces,Fig.1-3a,then only normal-force,shear-force,and bendingmomentcomponentswill existatthe section,Fig.1-3b.Ifweusethex,y,z coordinate axes, as shown on theleft segment,then N can be obtainedbyapplyingF=O,andVcanbeobtainedfromF,=0.Finally,thebendingmomentMo can be determined by summingmoments aboutpointO(thezaxis),Mo=o,inordertoeliminatethemomentscausedbythe unknowns N and V.IMPORTANTPOINTSMechanics of materials is a study of the relationship betweenthe external loads applied to a body and the stress and straincaused by the internal loads within the body.External forces can be applied to a body as distributed orconcentrated surfaceloadings, or as body forces that actthroughout thevolume ofthebody.Linear distributed loadings produce a resultant force having amagnitude equal to the area under the load diagram, andhaving a location that passes through the centroid of this area.Asupportproduces aforce in a particular direction onitsattached memberif it preventstranslation of themember inthatdirection,and itproducesa couplemomentonthememberifit prevents rotation.Theequations of equilibrium F=0 and EM=0mustbesatisfied in order to prevent a body from translating withacceleratedmotionandfromrotating.The method of sections isused to determine the internalresultant loadings acting on the surface of a sectioned body.Ingeneral,these resultants consist of a normal force,shearforce,torsionalmoment,andbendingmoment
1.2 Equilibrium of a Deformable Body 9 1 Coplanar Loadings. If the body is subjected to a coplanar system of forces, Fig. 1–3a, then only normal-force, shear-force, and bendingmoment components will exist at the section, Fig. 1–3b. If we use the x, y, z coordinate axes, as shown on the left segment, then N can be obtained by applying ΣFx = 0, and V can be obtained from ΣFy = 0. Finally, the bending moment MO can be determined by summing moments about point O (the z axis), ΣMO = 0, in order to eliminate the moments caused by the unknowns N and V. section F4 F3 F2 F1 (a) O V MO N x y Bending Moment Shear Force Normal Force (b) F2 F1 Fig. 1–3 • Mechanics of materials is a study of the relationship between the external loads applied to a body and the stress and strain caused by the internal loads within the body. • External forces can be applied to a body as distributed or concentrated surface loadings, or as body forces that act throughout the volume of the body. • Linear distributed loadings produce a resultant force having a magnitude equal to the area under the load diagram, and having a location that passes through the centroid of this area. • A support produces a force in a particular direction on its attached member if it prevents translation of the member in that direction, and it produces a couple moment on the member if it prevents rotation. • The equations of equilibrium ΣF = 0 and ΣM = 0 must be satisfied in order to prevent a body from translating with accelerated motion and from rotating. • The method of sections is used to determine the internal resultant loadings acting on the surface of a sectioned body. In general, these resultants consist of a normal force, shear force, torsional moment, and bending moment. IMPORTANT POINTS
10CHAPTER 1STRESSPROCEDURE FOR ANALYSISThe resultant internal loadings at a point located on the section of abodycanbeobtained usingthemethodofsections.This requires thefollowing steps.SupportReactions..When the body is sectioned,decide which segmentof the bodyis to be considered.If the segment has a support or connectionto another body,then before the bodyis sectioned, it will benecessary to determine the reactions acting on the chosensegment.To do this, draw the free-body diagram of the entirebodyand then applythe necessaryequations ofequilibriumtoobtainthesereactions.Free-BodyDiagramKeep all external distributed loadings, couple moments,torques, and forces in their exact locations, before passing thesection through the body at the point where the resultantinternal loadingsaretobedetermined.Draw a free-body diagram of one of the"cut" segments andindicate the unknown resultants N, V, M, and T at the section.These resultants are normally placed at the point representingthe geometric center or centroid of the sectioned area.If thememberis subjectedtoa coplanarsystemofforces,onlyN, V,and M act at the centroid. Establish the x, y, z coordinate axes with origin at the centroidand showtheresultant internal loadingsacting alongtheaxes.EquationsofEquilibriumMomentsshouldbesummedatthesection.abouteachofthecoordinate axes where the resultants act.Doing this eliminates theunknown forces Nand V andallows a direct solution for Mand T. If the solution of the equilibrium equations yields a negativevalue for a resultant, the directional sense of the resultant isopposite to that shown on the free-body diagram.Thefollowing examplesillustratethisprocedurenumericallyand alsoprovideareviewof someoftheimportantprinciplesofstatics
10 Chapter 1 Stress 1 PROCEDURE FOR ANALYSIS The resultant internal loadings at a point located on the section of a body can be obtained using the method of sections. This requires the following steps. Support Reactions. • When the body is sectioned, decide which segment of the body is to be considered. If the segment has a support or connection to another body, then before the body is sectioned, it will be necessary to determine the reactions acting on the chosen segment. To do this, draw the free-body diagram of the entire body and then apply the necessary equations of equilibrium to obtain these reactions. Free-Body Diagram. • Keep all external distributed loadings, couple moments, torques, and forces in their exact locations, before passing the section through the body at the point where the resultant internal loadings are to be determined. • Draw a free-body diagram of one of the “cut” segments and indicate the unknown resultants N, V, M, and T at the section. These resultants are normally placed at the point representing the geometric center or centroid of the sectioned area. • If the member is subjected to a coplanar system of forces, only N, V, and M act at the centroid. • Establish the x, y, z coordinate axes with origin at the centroid and show the resultant internal loadings acting along the axes. Equations of Equilibrium. • Moments should be summed at the section, about each of the coordinate axes where the resultants act. Doing this eliminates the unknown forces N and V and allows a direct solution for M and T. • If the solution of the equilibrium equations yields a negative value for a resultant, the directional sense of the resultant is opposite to that shown on the free-body diagram. The following examples illustrate this procedure numerically and also provide a review of some of the important principles of statics