英汉双语材料力学10资料讲解.ppt

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1、英汉双语材料力学10CHAPTER 10 STABILIZATION OF CHAPTER 10 STABILIZATION OF CHAPTER 10 STABILIZATION OF COMPRESSIVE COLUMNS COMPRESSIVE COLUMNS COMPRESSIVE COLUMNS101 CONCEPTS OF STABILITY OF COMPRESSED COLUMNS102 EULERS FORMULA OF THE CRITICAL FORCE OF SLENDER COMPRESSED COLUMNS103 CRITICAL STRESS OF COMPRES

2、SED COLUMNS AS STRESS EXCEEDS PROPORTIONAL LIMIT 10-4 STABILITY CHECK AND REASONABLE SECTION OF COMPRESSED COLUMNS2第十章第十章第十章第十章 压杆稳定压杆稳定压杆稳定压杆稳定101 压杆稳定性的概念压杆稳定性的概念102 细长压杆临界力的欧拉公式细长压杆临界力的欧拉公式103 超过比例极限时压杆的临界应力超过比例极限时压杆的临界应力10-4 10-4 压杆的稳定校核及其合理截面压杆的稳定校核及其合理截面3101 CONCEPTS OF STABILITY OF COMPRESSED

3、 COLUMNSThe load-carrying capacity of structure members：StrengthRigidityStability Some structure members in engineering have enough strength and rigidity but they are unable to work safely and reliably.4101 压杆稳定性的概念压杆稳定性的概念构件的承载能力：强度刚度稳定性 工程中有些构件具有足够的强度、刚度，却不一定能安全可靠地工作。5P6P71、Stable and instable equ

4、ilibrium：1).instable equilibrium 8一、稳定平衡与不稳定平衡一、稳定平衡与不稳定平衡 ：1.不稳定平衡92).Stable equilibrium102.稳定平衡113).Stable and instable equilibrium 123.稳定平衡和不稳定平衡132、Loss of stability and the critical pressure of the column：1).Ideal compressive columns：the material is absolutely ideal；the axis is absolutely strai

5、ght；the compressive force is absolutely along the axis of the column.2).Stable and instable equilibrium of compressive columns：Instable equilibriumStable equilibrium14二、压杆失稳与临界压力二、压杆失稳与临界压力 ：1.理想压杆：材料绝对理想；轴线绝对直；压力绝对沿轴线作用。2.压杆的稳定平衡与不稳定平衡：稳稳稳稳定定定定平平平平衡衡衡衡不不不不稳稳稳稳定定定定平平平平衡衡衡衡153).loss of stability of c

6、ompressed column：4).Critical pressure of compressed columnsStable Stable equilibriumequilibriumInstable Instable equilibriumequilibriumCritical stateCritical pressure:Pcr correspondingcorrespondingpressurepressureintermediate stateintermediate state163.压杆失稳：4.压杆的临界压力稳稳稳稳定定定定平平平平衡衡衡衡不不不不稳稳稳稳定定定定平平平平衡

7、衡衡衡临界状态临界状态临界压力临界压力:Pcr17102 EULERS FORMULA OF THE CRITICAL PRESSURE OF SLENDER COMPRESSED COLUMNS1 1、Critical pressure for the column with two hinged ends:bending momentbending moment：Approximate differentialApproximate differential equation of the deflection curve equation of the deflection curveP

8、PxyPM Suppose the pressure has reached the critical value and the column has been in tiny bending state as shown in the figure.Start to determine the critical force with the deflective curve.PxL18102 细长压杆临界力的欧拉公式细长压杆临界力的欧拉公式一、两端铰支压杆的临界力一、两端铰支压杆的临界力:假定压力已达到临界值，杆已经处于微弯状态，如图，从挠曲线入手，求临界力。弯矩：弯矩：弯矩：弯矩：挠曲线

9、近似微分方程：挠曲线近似微分方程：挠曲线近似微分方程：挠曲线近似微分方程：PxLPxyPM19Solution of the differential equationSolution of the differential equation：Determine the integral constantsDetermine the integral constants：The critical force Pcr is the smallest pressure under tiny bending，therefore we only take n=1 and the column will

10、 bend about the axis with the smallest moment of inertia.That is20微分方程的解：微分方程的解：确定积分常数：确定积分常数：临界力 Pcr 是微弯下的最小压力，故，只能取n=1；且杆将绕惯性矩最小的轴弯曲。212、Application range of the formula：3、Eulers formula of the critical pressure for the column with other end conditions:1).Ideal compressive columns；2).In linear ela

11、stic range；3).The ends of the column are supported by hinges.Length coefficient（or constraint coefficient）Eulers formula of the critical pressure for the compressive column with two hinged ends General form of Eulers formula of the critical pressure22二、此公式的应用条件：三、其它支承情况下，压杆临界力的欧拉公式1.理想压杆；2.线弹性范围内；3.

12、两端为球铰支座。长度系数（或约束系数）。两端铰支压杆临界力的欧拉公式两端铰支压杆临界力的欧拉公式 压杆临界力欧拉公式的一般形式压杆临界力欧拉公式的一般形式230.5lTable101 Eulers formula of the slender compressive column under various constraint conditionsSupportsTwo hinged endsOne free end and one hinged endTwo fixed endsOne fixed end and one free endTwo fixed ends but one of

13、them is movable laterally.The shape of the deflective curve in lost of stabilityPcrABlEulers formula of the critical forcePcr Length coefficient=10.7=0.5=2=1PcrABlPcrABl0.7lCCDC:Inflection point C、D:Inflection point 0.5lPcrPcrl2llC:Inflection point 240.5l表101 各种支承约束条件下等截面细长压杆临界力的欧拉公式支承情况两端铰支一端固定另端铰支

14、两端固定一端固定另端自由两端固定但可沿横向相对移动失稳时挠曲线形状PcrABl临界力Pcr欧拉公式长度系数=10.7=0.5=2=1PcrABlPcrABl0.7lCCDC 挠曲线拐点C、D 挠曲线拐点0.5lPcrPcrl2llC 挠曲线拐点25Solution：The deformation of the column is shown in the figure.The approximate differential equation of its deflection curve is:The boundary conditions are:Example 1 Try to dedu

15、ce the critical force of the following slender column by the approximate differential equation of the deflective curve.PLxPM0PM0PM0 xPM0Let26解：变形如图，其挠曲线近似微分方程为：边界条件为:例例1 1 试由挠曲线近似微分方程，导出下述细长压杆的临界力 公式。PLxPM0PM0PM0 xPM027In order to determine the minimum critical pressure“k”must be the minimum value e

16、xcept zero,that is：Therefore the critical pressure is：=0.5and28为求最小临界力，“k”应取除零以外的最小值，即取：所以，临界力为：=0.529Critical pressure of the columnCritical pressure of the column Example Example 2 Determine the critical pressure of the following slender compressive column.=1.0，Solution：About the axisAbout the axi

17、s y，two ends are hinged：=0.7，About the axisAbout the axis z，the left end is fixed and the right end is hinged：yzL1L2yzhbx30压杆的临界力 例例2 求下列细长压杆的临界力。=1.0，解：绕 y 轴，两端铰支：=0.7，绕 z 轴，左端固定，右端铰支：yzL1L2yzhbx31Example Example 3 Determine the critical pressure of the following slender compressive columns.Knowing

18、:L=0.5m and E=200GPa.Solution：Fig.(a)Fig.(b)Fig.(a)Fig.(b)5010PLPL(4545 6)Equal-leg angle steelyz32 例例3 求下列细长压杆的临界力。已知：L=0.5m ，E=200GPa.图(a)图(b)解：图(a)图(b)5010PLPL(4545 6)等边角钢yz33103 CRITICAL STRESS OF COMPRESSED COLUMNS AS STRESS EXCEEDS THE PROPORTIONAL LIMIT 1、Basic concepts1).Critical stress：Mean

19、 stress in the cross section of the compressive column in the critical state.3).Flexibility：2).Critical stress for slender compressed columns：22l lp ps sEcr=That is:That is:Radius of inertiaFlexibility(or slenderness ratio)of the column34103 超过比例极限时压杆的临界应力超过比例极限时压杆的临界应力一、一、基本概念基本概念1.临界应力：压杆处于临界状态时横截

20、面上的平均应力。3.柔度：2.细长压杆的临界应力：354).Division of large flexibility column：2、Calculation of the critical stress for the columns with middle or small flexibility1).Empirical formula with the linear variationAsPS：is one of the middle flexibility column.The column withIts critical stress may be determined by t

21、he empirical formula.The columns with is called the large flexibility column(or slender clumn).The critical pressure of this kind of column is calculated by Eulers formula.The columns with is called middle or small flexibility column.Its critical pressure of this kind of column can not be calculated

22、 by Eulers formula.364.大柔度杆的分界：二、中小柔度杆的临界应力计算二、中小柔度杆的临界应力计算1.直线型经验公式PS 时：37Total figure of the critical stressAs S：bss=a-s-sl l PPEs sp pl l2 =The critical stress of this column is its yield limit.is one of the small flexibilityThe column withcolumn.38临界应力总图S 时：sbas-=s sl l PPEs sp pl l2 =392)Empiri

23、cal formula with the parabolic variationThe common form in the structural engineering is：As Ps：As s：For A3steel,A5steel and 16Mn steel,AsThis formula is used to determine the critical stress.402.抛物线型经验公式我国建筑业常用：Ps 时：s 时：41Solution：For one piece of angle steel：After two pieces of angle steel is combi

24、ned as shown in the figure,Therefore the critical pressure should be determined by the empirical formulayzExample 4 A compressive column is made from two equal-leg angle steel of 56568.Its two ends are both hinged.Its length is L=1.5m.The pressure is P=150kN.The angle steel is A3 steel.Try to determ

25、ine the critical pressure and the safety coefficient of stability nst by Eulers formula or the empirical formula with the parabolic variation.with the parabolic variation.42 例例4 一压杆长L=1.5m，由两根 56568 等边角钢组成，两端铰支，压力P=150kN，角钢为A3钢，试用欧拉公式或抛物线公式求临界压力和稳定安全系数nst。解：一个角钢：两根角钢图示组合之后所以，应由抛物线公式求临界压力。yz43Safety

26、coefficient44安全系数45104 STABILITY CHECK AND REASONABLE SECTIONS OF COMPRESSED COLUMNS1、Permissible stress for the compressed columns in stabilization:1).Determine the permissible stress by the method of the safety coefficient:2).Determine the permissible stress by the method of the reduction coeffici

27、ent:2、Stability condition of the compressed columns:Reduction coefficient.It is related to material properties and the flexibility of columns,and .46104 压杆的稳定校核及其合理截面压杆的稳定校核及其合理截面一、压杆的稳定许用应力一、压杆的稳定许用应力:1.安全系数法确定许用应力:2.折减系数法确定许用应力:二、压杆的稳定条件二、压杆的稳定条件:47 Example 5 The column AB in the simple crane show

28、n in the figure is made from circular deal.Its length is L=6m and its diameter is d=0.3m.=11MPa.Try to determine the permissible pressure of the column.Solution：Apply the method of reduction coefficient.Maximum flexibility In plane xy，=1.0In plane zy，=2.0T1ABWT2xyzO48例例5 图示起重机，AB 杆为圆松木，长 L=6m，=11MPa

29、，直径：d=0.3m，试求此杆的许用压力。解：折减系数法最大柔度x y面内，=1.0z y面内，=2.0T1ABWT2xyzO49Determine the reduction coefficientDetermine the permissible pressureFor wooden column:as50求折减系数求许用压力513 3、Reasonable section of the column:It is reasonable.保国寺大殿的拼柱形式保国寺大殿的拼柱形式Sakyamni wooden pagodaYingzhou wooden pagodaIt was built i

30、n 1056 with the double-sleeve structure.The pagoda plane is octagon.It underwent the level 8 earthquake of 1305.52三、压杆的合理截面三、压杆的合理截面:合 理保国寺大殿的拼柱形式保国寺大殿的拼柱形式1056年建，年建，“双筒体双筒体”结构，塔身平面结构，塔身平面为八角形。经历了为八角形。经历了1305年的八级地震。年的八级地震。53Solution：For each channel steel of number 10,the centroid is at the point C1

31、.After two channel steels are combined，PLz0yy1z1C1aExample 6 The upright column shown in the figure is made from two channel steels.Its length is L=6m.Knowingthe material is A3steel,E=200GPa and .Its lower end is fixed and its upper end is supported by a spherical hinge.How much is the value of a wh

32、en the critical pressure of the upright column is the maximum?That is it is reasonable asso54 例例6 图示立柱，L=6m，由两根10号槽钢组成，材料为A3钢E=200GPa，下端固定，上端为球铰支座，试问 a=？时，立柱的临界压力最大，值为多少？解：对于单个10号槽钢，形心在C1点。两根槽钢图示组合之后，y1PLz0yz1C1a55Determine the critical force：It is a large flexibility rod.The critical force is deter

33、minedby Eulers formula.56求临界力：大柔度杆，由欧拉公式求临界力。57 Chapter 10 Exercises1.How to distinguish the stable equilibrium and the unstable equilibrium of compressed columns?2.A compressed rod produces the bending deformation due to the loss of stability.A beam produces the bending deformation due to the actio

34、n of transverse forces.What are the differences of the two in nature?3.Three circular rods with the same diameter d=16cm are shown in the figure.Knowing their materials are A3 steel,E=200GPa and .Try to determine:Which compressed rod loses stability easily?The maximum critical value of the compressi

35、ve forces for the three rods.58 第十章第十章 练习题练习题 一、如何区别压杆的稳定平衡和不稳定平衡？一、如何区别压杆的稳定平衡和不稳定平衡？二二、压压杆杆因因失失稳稳而而产产生生弯弯曲曲变变形形，与与梁梁在在横横向向力作用下产生弯曲变形，在性质上有何区别？力作用下产生弯曲变形，在性质上有何区别？三三、三三根根直直径径均均为为 d=16cm 的的圆圆杆杆如如图图所所示示，材料均为材料均为A3钢，钢，E=200GPa，。试求试求:哪一根压杆最容易失稳？哪一根压杆最容易失稳？三杆中最大的临界压力值。三杆中最大的临界压力值。59 Solution:Rod(a)：Rod(b)：Rod(a)loses stability more easily.Rod(c)：The critical force of Rod(c)is maximum.60 解：解：杆杆(a)(a)：杆杆(b)(b)：杆杆(a)(a)最易失稳最易失稳 杆杆(c)(c)：杆杆(c)(c)的临界力最大的临界力最大 616263此课件下载可自行编辑修改，仅供参考！此课件下载可自行编辑修改，仅供参考！感谢您的支持，我们努力做得更好！谢谢感谢您的支持，我们努力做得更好！谢谢