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1、2第1页/共64页CHAPTER 9 COMPOSITE DEFORMATIONSCHAPTER 9 COMPOSITE DEFORMATIONS91 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页第九章第九章 组合变形组合变形91 概述92 斜弯曲93 弯曲与扭转的组合9-4 9-4 拉(压)弯组合 偏心拉（压）
2、截面核心4第3页/共64页91 SUMMARYMPRzxyPP1 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 ki
3、nd of deformation is called the composite deformation.5第4页/共64页一、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。91 概 述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qPhg g10第9页/共64页 2 2、Methods to study composite deformation sPrinciple of superpositionAnalysis o
4、f 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：Plot
5、 the distribution diagram of the stress in the critical Plot the distribution diagram of the stress in the critical sectionsection，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 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
6、和内力图，确 定危险面。应力分析：应力分析：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 t
7、wo centroid principal axes of inertia of the cross section and get two perpendicular planar bending.PzPyyzP xyzPPyPz13第12页/共64页92 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：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。PyPzPzPyyzP 14xyzPPyPz13第13页/共64页2).Sum：Analyze bending in two perpendicular planes
8、and sum the results of the calculation.xyzPyPzPPzPyyzP 15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。xyzPyPzPPzPyyzP 第15页/共64页Solution：1.Resolve the external force along the centroid principal axis of inertia of the cross section2.Study the bending in two planes:Internal forcesInternal forcesxyzPyPzPPzPyy
9、zP Lmmx17第16页/共64页解：1.将外载沿横截面的形心主轴分解2.研究两个平面弯曲内内力力18PzPyyzP 第17页/共64页StressesStressesStress due to My:Stress due to M z：Resultant stress：LPzPyyzP xyzPyPzPLmmx19第18页/共64页应应力力My引起的应力：M z引起的应力：合应力：20LPzPyyzP xyzPyPzPLmmx第19页/共64页Maximum normal stressMaximum normal stressCalculation of the deformationCa
10、lculation of the deformationEquation of the neutral axisEquation of the neutral axisIt is obvious that only as Iy=Iz the neutral axis is perpendicular to the external force.As As =，it is the planar bending.it is the planar bending.PzPyyzP D1D2a aNeutral axisffzfy The maximum normal stress of tension
11、 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页最大正应力最大正应力变形计算变形计算中性轴方程中性轴方程可见：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。当 =时，即为平面弯曲。D1D2a a中性轴22PzPyyzP D1D2a affzfy 第21页/共64页Example 1 1 Force P is throug
12、h the center of section and makes an angle the with axis z in the beam as shown in the figure.Determine the maximum stress and deflection of the beam.Maximum normal stressMaximum normal stressCalculation of deformationCalculation of deformationAs Iy=Iz the beam produce planar bending.Solution：Analys
13、is of the critical point is shown in the figureffzfy yzLxPyPzPhbPzPyyzP D2D1a aNeutral axis23第22页/共64页 例11结构如图，P过形心且与z轴成角，求此梁的最大应力与挠度。当Iy=Iz时，即发生平面弯曲。解：危险点分析如图24最大正应力最大正应力变形计算变形计算中性轴ffzfy yzLxPyPzPhbPzPyyzP D2D1a a第23页/共64页 Example 2 A wood purline is shown in the figure.Its span is L=3m and a unifo
14、rmly 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.Solution：Analysis of external forceresolve qa a=2634hbyzqqLAB25第24页/共64页 例2 矩形截面木檩条如图，跨长
15、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。解：外力分析分解qa a=2634hbyzqqLAB26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80P2zyxP1150200100ABCD28第27页/共64页Solution：Reduce the external force to the centroid of the section and
16、 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 re
17、sultant bending moment.Determine the Determine the critical sectioncritical section(Nm)MzxMy(Nm)xMn(Nm)xM(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，并画图确定危险面(Nm)MzxMy(Nm)xMn(Nm)xM(Nm)Mmaxx32第31页/共64页Plot the diagram of stres
18、s distributionPlot the diagram of stress distribution in the critical section and determine in the critical section and determine the critical point.the critical point.Establish the strength conditionEstablish the strength condition33xB1B2MyMzMnMxM第32页/共64页画危险面应力分布图，找危险点建立强度条件34xB1B2MyMzMnMxM第33页/共6
19、4页35第34页/共64页3635第35页/共64页Analysis of external forces：Reduce the external forces to the centroid of section and resolve them.Analysis of stresses:Establish strength conditions.Steps of solving the problem of composite deformation of bending and torsion:Analysis of internal forces：Determine the inter
20、nal equation and its diagram corresponding toeach external force component and critical section.37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。应力分析：应力分析：建立强度条件。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 dia
21、meter 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3 图示空心圆轴，内径d=24
22、mm，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：Analysis of stressAnalysis of stress：It is safe(Nm)MzxMy(Nm)xMn(Nm)xM(Nm)71.
23、3x71.25407.051205.540.641第40页/共64页内力分析：危险面内力为：应力分析：应力分析：安 全(Nm)MzxMy(Nm)xMn(Nm)xM(Nm)71.3x71.25407.051205.540.642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 te
24、nsion or compression：Deformation of the rod due to simultaneous action of transversal and axial forces.43第42页/共64页94 拉(压)弯组合 偏心拉（压）截面核心一、拉(压)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2、Analysis of stress：45PMyMzPMZMyxyzz y第44页/共64页二、应力分析：46PMyMzPMZMyxyzz y第45页/共64页4、Critical point（Far
25、thest point from the neutral axis）3、Equation of the neutral axisFor the problem of eccentric tension or compression47P（zP，yP）yzyzNeutral axis第46页/共64页四、危险点（距中性轴最远的点）三、中性轴方程对 于 偏 心 拉 压 问 题中性轴48P（zP，yP）yzyz第47页/共64页yz5、Kernel of section in the problem of the eccentric tension、compression：ayazAfter kno
26、wing 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.Neutral axisKernel of section49第48页/共64页yz五、（偏心拉、压问题的）截面核心：ayaz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
27、力的一个作用点中性轴截面核心50第49页/共64页Solution：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 stres
28、s with maximum absolute value in the poles.MPPd51Fig.Fig.P300200200P200200第50页/共64页解：两柱横截面上的最大正应力均为压应力 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Solution：Analysis of the internal force is shown in the figure.Centroid of the slot in the coordinates shown in
29、 the figurePPSlot 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
30、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PP54PPMN2010020yzyC10第53页/共64页PPMN Analysis of stress is shown in the figure.As the hole is moved to the middle of the plate2010020yzyC55第54页/共64页PPMN应力分析如图孔移至板中间时2010020yzyC56第55页/共64页Solution：For the composite deformation
31、 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
32、safe.AAPPTT57第56页/共64页解：拉扭组合,危险点应力状态如图 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。AAPPTT58第57页/共64页58 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 d
33、eformation,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=5
34、0mm 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?第58页/共64页59 第九章 练习题 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形，试写出其强度条件。二、方形截面杆的横截面面积在 mn 处减少一半，试求由轴向载荷 P 引起的 mn 截面上的最大拉应力。解：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 d=65mm 的圆截面，最大正应力为多少？第59页/共64页60 Soluion:If the beam is changed to have circular sections,第60页/共64页61 解：如改为圆截面 第61页/共64页62第62页/共64页63第63页/共64页64感谢您的观看！第64页/共64页