[PPT]材料力学的结构稳定性英文版

1、1CHAPTER 10 STABILITY OF COMPRESSED COLUMNS2CHAPTER 10 STABILIZATION OF COMPRESSIVE COLUMNS101 CONCEPTS OF STABILITY OF COMPRESSED COLUMNS102 EULERS FORMULA OF THE CRITICAL FORCE OF SLENDER COMPRESSED COLUMNS103 CRITICAL STRESS OF COMPRESSED COLUMNS AS STRESS EXCEEDS PROPORTIONAL LIMIT 10-4 STABILIT

2、Y CHECK AND REASONABLE SECTION OF COMPRESSED COLUMNS3101 CONCEPTS OF STABILITY OF COMPRESSED 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.STABIL

3、ITY OF COMPRESSED COLUMN4PSTABILITY OF COMPRESSED COLUMN51、Stable and instable equilibrium ：1). instable equilibrium STABILITY OF COMPRESSED COLUMN62). Stable equilibriumSTABILITY OF COMPRESSED COLUMN73). Stable and instable equilibrium STABILITY OF COMPRESSED COLUMN82、Loss of stability and the crit

4、ical pressure of the column：1).Ideal compressive columns：the material is absolutely ideal；the axis is absolutely straight；the compressive force is absolutely along the axis of the column.2). Stable and instable equilibrium of compressive columns：STABILITY OF COMPRESSED COLUMNInstable equilibriumStab

5、le equilibrium93).loss of stability of compressed column：4).Critical pressure of compressed columnsStable equilibriumInstable equilibriumCritical stateCritical pressure: Pcr correspondingpressureSTABILITY OF COMPRESSED COLUMNintermediate state10102 EULERS FORMULA OF THE CRITICAL PRESSURE OF SLENDER

6、COMPRESSED COLUMNS1、Critical pressure for the column with two hinged ends:bending moment：Approximate differential equation of the deflection curvePPxyPMSTABILITY OF COMPRESSED COLUMN Suppose the pressure has reached the critical value and the column has been in tiny bending state as shown in the fig

7、ure. Start to determine the critical force with the deflective curve.PxL11Solution of the differential equation：Determine the integral constants： The critical force Pcr is the smallest pressure under tiny bending，therefore we only take n=1 and the column will bend about the axis with the smallest mo

8、ment of inertia.STABILITY OF COMPRESSED COLUMNThat is122、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 elastic range；3).The ends of the column are supported by hinges.Length coefficient（or

9、 constraint coefficient）Eulers formula of the critical pressure for the compressive column with two hinged ends General form of Eulers formula of the critical pressureSTABILITY OF COMPRESSED COLUMN130.5lTable101 Eulers formula of the slender compressive column under various constraint conditionsSupp

10、ortsTwo hinged endsOne free end and one hinged endTwo fixed endsOne fixed end and one free endTwo fixed ends but one of 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:

11、Inflection point C、D: Inflection point 0.5lPcrPcrl2llC: Inflection point STABILITY OF COMPRESSED COLUMN14Solution：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 deduce the critical f

12、orce of the following slender column by the approximate differential equation of the deflective curve.PLxPM0PM0PM0 xPM0STABILITY OF COMPRESSED COLUMNLet15In order to determine the minimum critical pressure“k”must be the minimum value except zero, that is：Therefore the critical pressure is： = 0.5STAB

13、ILITY OF COMPRESSED COLUMNand16Critical pressure of the column Example 2 Determine the critical pressure of the following slender compressive column.=1.0，Solution：About the axis y，two ends are hinged：=0.7，About the axis z，the left end is fixed and the right end is hinged：yzL1L2yzhbxSTABILITY OF COMP

14、RESSED COLUMN17Example 3 Determine the critical pressure of the following slender compressive columns.Knowing:L=0.5m and E=200GPa.Solution：Fig.(a)Fig.(b)Fig.(a)Fig.(b)5010PLPL(4545 6)Equal-leg angle steelyzSTABILITY OF COMPRESSED COLUMN18103 CRITICAL STRESS OF COMPRESSED COLUMNS AS STRESS EXCEEDS TH

15、E PROPORTIONAL LIMIT 1、 Basic concepts1).Critical stress：Mean stress in the cross section of the compressive column in the critical state.3).Flexibility：2).Critical stress for slender compressed columns：22lpsEcr=That is:STABILITY OF COMPRESSED COLUMNRadius of inertiaFlexibility (or slenderness ratio

16、) of the column194).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 ：STABILITY OF COMPRESSED COLUMNis one of the middle flexibility column .The column withIts critical stress

17、 may be determined by the 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 col

18、umn can not be calculated by Eulers formula.20Total figure of the critical stressAs S ：bss=a -sl PPEspl2 =STABILITY OF COMPRESSED COLUMN The critical stress of this column is its yield limit.is one of the small flexibilityThe column withcolumn. 212)Empirical formula with the parabolic variationThe c

19、ommon form in the structural engineering is：As Ps ：As s ：STABILITY OF COMPRESSED COLUMNFor A3steel, A5steel and 16Mn steel,AsThis formula is used to determine the critical stress.22Solution：For one piece of angle steel：After two pieces of angle steel is combined as shown in the figure,Therefore the

20、critical pressure should be determined by the empirical formulayzSTABILITY OF COMPRESSED COLUMNExample 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 determine t

21、he critical pressure and the safety coefficient of stability nst by Eulers formula or the empirical formula with the parabolic variation.with the parabolic variation.23Safety coefficientSTABILITY OF COMPRESSED COLUMN24104 STABILITY CHECK AND REASONABLE SECTIONS OF COMPRESSED COLUMNS1、Permissible str

22、ess 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 coefficient:2、Stability condition of the compressed columns:STABILITY OF COMPRESSED COLUMNReduction coeffici

23、ent. It is related to material properties and the flexibility of columns,and . 25 Example 5 The column AB in the simple crane shown 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：App

24、ly the method of reduction coefficient. Maximum flexibility In plane xy， =1.0In plane zy， =2.0T1ABWT2xyzOSTABILITY OF COMPRESSED COLUMN26Determine the reduction coefficientDetermine the permissible pressureSTABILITY OF COMPRESSED COLUMNFor wooden column:as273、Reasonable section of the column:It is r

25、easonable.STABILITY OF COMPRESSED COLUMN保国寺大殿的拼柱形式Sakyamni wooden pagodaYingzhou wooden pagodaIt was built in 1056 with the double-sleeve structure.The pagoda plane is octagon.It underwent the level 8 earthquake of 1305.28Solution：For each channel steel of number 10,the centroid is at the point C1.

26、After two channel steels are combined，PLz0yy1z1C1aSTABILITY OF COMPRESSED COLUMNExample 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 when the critical pressure of the upright column is the maximum?That is it is reasonable asso29Determine the critical force： It i