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会计学12第1页/共64页91 SUMMARY92 SKEW BENDING93 COMBINATION OF BENDING AND TORSION9-4 9-4 BENDING AND TENSION OR COMPRESSION ECCENTRIC TENSION OR COMPRESSION KERNEL OF THE SECTION3第2页/共64页91 概述概述92 斜弯曲斜弯曲93 弯曲与扭转的组合弯曲与扭转的组合9-4 9-4 拉拉(压压)弯组合弯组合 偏心拉（压）偏心拉（压）截面核心截面核心4第3页/共64页MPRzxyPP1 1、Composite deformation：Structural members will produce several types of simple deformations when subjected to complex external loads.The stress corresponding to each simple deformation can not be neglected when the magnitude of each stress has the same order.This kind of deformation is called the composite deformation.5第4页/共64页一、组合变形一、组合变形：在复杂外载作用下，构件的变形会包含几种简单变形，当几种变形所对应的应力属同一量级时，不能忽略之，这类构件的变形称为组合变形。MPRzxyPP6第5页/共64页Phg g7第6页/共64页Phg g8第7页/共64页DamqPhg g9第8页/共64页水坝水坝qPhg g10第9页/共64页 2 2、Methods to study composite deformation sPrinciple of superpositionAnalysis of external forces：External forces are reduced along the centroid of section and resolved along principal axes of inertia.Analysis of internal forces：Determine the internal-force equation and its diagram corresponding to each external force component and the critical section.Analysis of stresses，do the superposition of the stresses and establish the strength condition of the critical point.11第10页/共64页二、组合变形的研究方法二、组合变形的研究方法 叠加原理叠加原理外力分析：外力向形心(或弯心)简化并沿形心主惯性轴分解内力分析：求每个外力分量对应的内力方程和内力图，确 定危险面。画危险面应力分布图，叠加，建立危险点的强 度条件。12第11页/共64页92 SKEW BENDING 1 1、Skew bending：After bending deformation,the deflection curve and the external forces（transversal forces）of the rod are not in the same plane 2 2、Methods to study the skew bending：1).Resolve：Resolve the external load along two centroid principal axes of inertia of the cross section and get two perpendicular planar bending.PzPyyzPj jxyzPPyPz13第12页/共64页92 斜弯曲斜弯曲一、斜弯曲一、斜弯曲：杆件产生弯曲变形，但弯曲后，挠曲线与外力（横 向力）不共面。二、斜弯曲的研究方法二、斜弯曲的研究方法：1.分解：将外载沿横截面的两个形心主轴分解，于是得到两个正交的平面弯曲。PyPzPzPyyzPj j14xyzPPyPz13第13页/共64页2).Sum：Analyze bending in two perpendicular planes and sum the results of the calculation.xyzPyPzPPzPyyzPj j15第14页/共64页2.叠加：对两个平面弯曲进行研究；然后将计算结果叠加起来。xyzPyPzPPzPyyzPj j第15页/共64页jsinPPyjcosPPzSolution：1.Resolve the external force along the centroid principal axis of inertia of the cross section2.Study the bending in two planes:jjsinsin)()(MxLPxLPMyzjcosMMyxyzPyPzPPzPyyzPj jLmmx17第16页/共64页jsinPPyjcosPPz解：1.将外载沿横截面的形心主轴分解2.研究两个平面弯曲jjsin sin)()(MxLPxLPMyzjcosMMy18PzPyyzPj j第17页/共64页jcos yyyIMIzMz jsin zzzIMIyMy)sincos(jjzyIyIzM Stress due to My:Stress due to M z：Resultant stress：LPzPyyzPj jxyzPyPzPLmmx19第18页/共64页jcos yyyIMIzMz jsin zzzIMIyMy )sincos(jjzyIyIzM My引起的应力：M z引起的应力：合应力：20LPzPyyzPj jxyzPyPzPLmmx第19页/共64页0)sincos(00jjzyIyIzMjctgtg00yzIIzyIt is obvious that only as Iy=Iz the neutral axis is perpendicular to the external force.2maxDL1maxDy22zyfffzyfftgAs As j j=，it is the planar bending.it is the planar bending.PzPyyzPj jD1D2 Neutral axisffzfy The maximum normal stress of tension or compression occurs in the points that lie in two sides of the neutral axis and have the farthest distance to the neutral axis.21第20页/共64页0)sincos(00jjzyIyIzMjctgtg00yzIIzy可见：只有当Iy=Iz时，中性轴与外力才垂直。在中性轴两侧，距中性轴最远的点为拉压最大正应力点。22zyfffzyfftg当当j j=时，即为平面弯曲。时，即为平面弯曲。D1D2 中性轴中性轴222maxDL1maxDyPzPyyzPj jD1D2 ffzfy 第21页/共64页Example 1 1 Force P is through the center of section and makes an angle the j with axis z in the beam as shown in the figure.Determine the maximum stress and deflection of the beam.2maxmax1maxDyyzzDLWMWM232322)3()3(yzzyzyEILPEILPfffjtgtgzyzyIIffAs Iy=Iz the beam produce planar bending.Solution：Analysis of the critical point is shown in the figureffzfy yzLxPyPzPhbPzPyyzPj jD2D1 Neutral axis23第22页/共64页 例例11结构如图，P过形心且与z轴成j角，求此梁的最大应力与挠度。232322)3()3(yzzyzyEILPEILPfffjtgtgzyzyIIff当Iy=Iz时，即发生平面弯曲。解：危险点分析如图242maxmax1maxDyyzzDLWMWM中性轴中性轴ffzfy yzLxPyPzPhbPzPyyzPj jD2D1 第23页/共64页 Example 2 A wood purline is shown in the figure.Its span is L=3m and a uniformly distributed load q=800N/m is acting on it.The permissible stress and deflection of it is respectively =12MPa and L/200，E=9GPa，Try to determine the dimension of the section and check the rigidity of the beam.N/m358447.0800sinqqySolution：Analysis of external forceresolve q yyzzWMWMmaxN/m715894.0800cosqqzNm40383358822maxLqMyzNm80483715822maxLqMzy 2634 hbyzqqLAB25第24页/共64页 例例2 矩形截面木檩条如图，跨长L=3m,受集度为q=800N/m的均布力作用，=12MPa，许可挠度为：L/200，E=9GPa，试选择截面尺寸并校核刚度。N/m358447.0800sinqqy解：外力分析分解q yyzzWMWMmaxN/m715894.0800cosqqzNm40383358822maxLqMyzNm80483715822maxLqMzy 2634 hbyzqqLAB26第25页/共64页 93 COMBINATION OF COMBINATION OF BENDING AND TORSION80P2zyxP1150200100ABCD27第26页/共64页 93 弯曲与扭转的组合弯曲与扭转的组合80P2zyxP1150200100ABCD28第27页/共64页Solution：Reduce the external force to the centroid of the section and resolve itEstablish the strength condition of the rod shown in the figure.Bending and torsion80P2zyxP1150200100ABCD150200100ABCDP1MxzxyP2yP2zMx29第28页/共64页解：外力向形心 简化并分解建立图示杆件的强度条件弯扭组合变形80P2zyxP1150200100ABCD30150200100ABCDP1MxzxyP2yP2zMx第29页/共64页Sum the bending moments and plot the diagram of the resultant bending moment.)()()(22xMxMxMzy)(;)(;)(xMxMxMnzyM Z (N m)X(Nm)MzxMy (N m)XMy(Nm)x（Nm）xMnMnMn(Nm)xM (N m)XMmaxM(Nm)MmaxxInternal equations and diagrams corresponding to each external force component31第30页/共64页每个外力分量对应的内力方程和内力图叠加弯矩，并画图)()()(22xMxMxMzy确定危险面)(;)(;)(xMxMxMnzyM Z (N m)X(Nm)MzxMy (N m)XMy(Nm)x（Nm）xMnMnMn(Nm)xM (N m)XMmaxM(Nm)Mmaxx32第31页/共64页WMxBmax1PnBWM12231)2(2223134r2222max4PnWMWM1xB1B1xB1B33xB1B2MyMzMnMx1xB2xBM1B第32页/共64页画危险面应力分布图，找危险点WMxBmax1PnBWM12231)2(2建立强度条件2222max4PnWMWM1xB1B1xB1B34xB1B2MyMzMnMx1xB2xBM1B223134r第33页/共64页213232221421r1xB1B223WMMn2275.0WMMMnzy22275.0WMMMnzyr222475.0WMMMnzyr2223223134r2222max4PnWMWMWMMMnzy22235第34页/共64页36223WMMn2275.0WMMMnzy22275.0WMMMnzyr222475.0WMMMnzyr2223223134r2222max4PnWMWMWMMMnzy222351xB1B213232221421r第35页/共64页Analysis of external forces：Reduce the external forces to the centroid of section and resolve them.Analysis of stressesEstablish strength conditions.Steps of solving the problem of composite deformation of bending and torsion:Analysis of internal forces：Determine the internal equation and its diagram corresponding toeach external force component and critical section.37WMMMnzyr2223WMMMnzyr222475.0第36页/共64页外力分析：外力向形心简化并分解。内力分析：每个外力分量对应的内力方程和内力图，确定危 险面。建立强度条件。WMMMnzyr2223WMMMnzyr222475.0弯扭组合问题的求解步骤：弯扭组合问题的求解步骤：38第37页/共64页 Example 3 A hollow circular shaft is shown in the figure.Its inside diameter is d=24mm and its outside diameter is The diameters of pulley B and D are respectively D=30mm D1=400mm and D2=600mm，P1=600N，=100MPa.Try to check the strength of the shaft with the third strength.Analysis of external forces：Bending and torsion80P2zyxP1150200100ABCD150200100ABCDP1MxzxyP2yP2zMxSolution:39第38页/共64页 例例3 图示空心圆轴，内径d=24mm，外径D=30mm，B 轮直径D 1 400mm，D轮直径 D 2600mm，P1=600N，=100MPa，试用第三强度理论校核此轴的强度。外力分析：弯扭组合变形80P2zyxP1150200100ABCD150200100ABCDP1MxzxyP2yP2zMx解：40第39页/共64页Analysis of internal forces：Internal forces in the critical section are：WMMnr22max3Nm3.71maxMNm120nM)8.01(03.014.31203.71324322 MPa5.97It is safeM Z (N m)X(Nm)MzxMy (N m)XMy(Nm)x（Nm）xMnMnMn(Nm)xM (N m)XMmaxM(Nm)71.3x71.25407.051205.540.641第40页/共64页内力分析：危险面内力为：Nm3.71maxMNm120nM)8.01(03.014.31203.71324322 MPa5.97安全M Z (N m)X(Nm)MzxMy (N m)XMy(Nm)x（Nm）xMnMnMn(Nm)xM (N m)XMmaxM(Nm)71.3x71.25407.051205.540.642WMMnr22max3第41页/共64页94 BENDING AND TENSION OR COMPRESSION ECCENTRIC TENSION OR COMPRESSION KERNEL OF THE SECTION PRPxyzPMyxyzPMyMz1 1、Composite deformation of Composite deformation of bending and tension or compression：Deformation of the rod due to simultaneous action of transversal and axial forces.43第42页/共64页94 拉拉(压压)弯组合弯组合 偏心拉（压）偏心拉（压）截面核心截面核心一、拉一、拉(压压)弯组合变形：弯组合变形：杆件同时受横向力和轴向力的作用而产 生的变形。PR44PxyzPMyxyzPMyMz第43页/共64页APxPzzxMIyMzyyxMIzMyyyzzxIzMIyMAP2、Analysis of stress：45PMyMzPMZMyxyzz y第44页/共64页APxPzzxMIyMzyyxMIzMyyyzzxIzMIyMAP二、应力分析二、应力分析：46PMyMzPMZMyxyzz y第45页/共64页000yyzzxIzMIyMAP4、Critical point（Farthest point from the neutral axis）3、Equation of the neutral axisFor the problem of eccentric tension or compression0)1(20202020yPzPyPzPizziyyAPAizPzAiyPyAPyyzzLWMWMAPmaxyyzzyWMWMAPmax012020 yPzPizziyy47P（zP，yP）yzyz),(PPyzPNeutral axis第46页/共64页000yyzzxIzMIyMAP四、危险点四、危险点（距中性轴最远的点）三、中性轴方程三、中性轴方程对于偏心拉压问题0)1(20202020yPzPyPzPizziyyAPAizPzAiyPyAP012020 yPzPizziyy中性轴中性轴48yyzzLWMWMAPmaxyyzzyWMWMAPmaxP（zP，yP）yzyz),(PPyzP第47页/共64页yz5、Kernel of section in the problem of the eccentric tension、compression：ayaz012zyPiay012yzPiazAfter knowing ay and az,The action range of the compressive force.As the compressive force is acted in this range there are no tensile stresses in the section.May determine an action point of the force P.),(PPyz012020yPzPizziyyNeutral axis),(PPyzPKernel of section49第48页/共64页yz五、（偏心拉、压问题的）截面核心：五、（偏心拉、压问题的）截面核心：ayaz012zyPiay012yzPiaz已知 ay，az 后，压力作用区域。当压力作用在此区域内时，横截面上无拉应力。可求P力的一个作用点),(PPyz012020yPzPizziyy中性轴中性轴),(PPyzP截面核心50第49页/共64页MPa75.82.02.0350000 max2AP11max1zWMAPMPa7.113.02.06503503.02.03500002Solution：The stresses in the cross sections of the two poles are both compressive ones.Example 4 4 Two poles subjected to the force P=350kN are shown in the figure.The section of one pole is unequal and the section of the other pole is equal.Try to determine the normal stress with maximum absolute value in the poles.MPPd51Fig.Fig.P300200200P200200第50页/共64页MPa75.82.02.0350000 max2AP11max1zWMAPMPa7.113.02.06503503.02.03500002解：两柱横截面上的最大正应力均为压应力 例例4 4 图示不等截面与等截面柱，受力P=350kN，试分别求出两柱内的绝对值最大正应力。图（1）图（2）MPPd52.P300200200P200200第51页/共64页mm5102010100201020Cz235100101210010CyI4523mm1027.7252010122010Solution：Analysis of the internal force is shown in the figure.Centroid of the slot in the coordinates shown in the figureNm5001053PMPPSlot Example 5 A steel plate shown in the figure is subjected to forces P=100kN.Try to determine the maximum normal stress；If the slot is moved to the middle of the plate and the maximum normal stress is kept constant,how much should the width of the slot be?53PPMN2010020yzyC10第52页/共64页mm5102010100201020Cz235100101210010CyI4523mm1027.7252010122010例例5 图示钢板受力P=100kN，试求最大正应力；若将缺口移至板宽的中央，且使最大正应力保持不变，则挖空宽度为多少？解：内力分析如图坐标如图，挖孔处的形心Nm5001053PMPP54PPMN2010020yzyC10第53页/共64页PPMNycIzMANmaxmaxMPa8.1628.37125 Analysis of stress is shown in the figure.73631027.710555001080010100As the hole is moved to the middle of the plate)100(10mm9.631108.16210100263maxxNAmm8.36 sox2010020yzyC55第54页/共64页PPMNMPa8.1628.37125应力分析如图73631027.710555001080010100孔移至板中间时)100(10mm9.631108.16210100263maxxNAmm8.36 x2010020yzyC56ycIzMANmaxmax第55页/共64页MPa7.351.07000163nWTMPa37.6101.050432APSolution：For the composite deformation of tension and torsion,the stressed state at the critical point is shown in the figure.Example 6 A circular rod which the diameter d=0.1m is subjected to forces T=7kNm and P=50kN as shown in the figure,=100MPa.Try to check the strength of the rod with the third strength theory.Therefore,the rod is safe.2234r MPa7.717.35437.622AAPPTT57第56页/共64页MPa7.351.07000163nWTMPa37.6101.050432AP解：拉扭组合,危险点应力状态如图 例例6 直径为d=0.1m的圆杆受力如图,T=7kNm,P=50kN,=100MPa,试按第三强度理论校核此杆的强度。故，安全。MPa7.717.35437.622AAPPTT582234r第57页/共64页59 Chapter 9 Exercises 1.A circular shaft of steel is deformed under tension and torsion.Try to write out the strength conditions.If it shows the tension-torsion-bending composite deformation,try to write out the strength conditions.2.The cross-section area of the square-section rod is reduced half at the section mn.Try to determine the maximum tensile stress at the section mn induced by the axial force P.Solution:3.A rectangular-section beam is shown in the figure.Knowing b=50mm and h=75mm.Try to determine the maximum normal stress of the beam.If the beam is changed to have circular sections with the diameter d=65mm，what is the maximum normal stress?WMANmax222864/42/aPaaaPaP第58页/共64页60 第九章第九章 练习题练习题 一、钢圆轴为拉伸与扭转的组合变形，试写一、钢圆轴为拉伸与扭转的组合变形，试写出其强度条件。若为拉伸、扭转和弯曲的组合变出其强度条件。若为拉伸、扭转和弯曲的组合变形，试写出其强度条件。形，试写出其强度条件。二、方形截面杆的横截面面积在二、方形截面杆的横截面面积在 mn 处减少处减少一半，试求由轴向载荷一半，试求由轴向载荷 P 引起的引起的 mn 截面上的最截面上的最大拉应力。大拉应力。解：解：三、矩形截面梁如图。已知三、矩形截面梁如图。已知 b=50mm,h=75 mm，求梁内的最大正应力。如改为，求梁内的最大正应力。如改为 d=65mm 的圆截面，最大正应力为多少？的圆截面，最大正应力为多少？WMANmax222864/42/aPaaaPaP第59页/共64页61 Soluion:If the beam is changed to have circular sections,yyzzWMWMmaxmaxmaxMPa9605.0075.061021075.005.06105.12323MPaoWMMyz8.9265.25.1323222max2maxmax第60页/共64页62 解：解：如改为圆截面如改为圆截面 yyzzWMWMmaxmaxmaxMPaoWMMyz8.9265.25.1323222max2maxmaxMPa9605.0075.061021075.005.06105.12323第61页/共64页63第62页/共64页64第63页/共64页
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