山农双语材力9

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,Chapter,9,Stability of compressed columns,1,Chapter 9 Stability of compressed columns,9,1,Concepts of stability of compressed columns,9,2,Eulers formula of the critical force of slender compressed columns,9,3,Scope of application of,Eulers formula,and empirical formula,9,4,Stability check and,measures of increasing stability,2,9,1,CONCEPTS OF STABILITY OF COMPRESSED COLUMNS,The load-carrying capacity of structure members,：,Strength,Rigidity,Stability,Some structure members in engineering have enough strength and rigidity but they are unable to work safely and reliably.,3,P,4,5,6,易拉罐压缩,失稳,7,1,、,Stable and instable equilibrium,：,1).instable equilibrium,8,2).Stable equilibrium,9,3).Stable and instable equilibrium,10,2,、,Loss of stability and the critical 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,：,Instable equilibrium,Stable equilibrium,11,3).loss of stability of compressed column,：,4).,Critical pressure of compressed columns,Stable equilibrium,Instable equilibrium,Critical state,Critical pressure:,F,cr,corresponding,pressure,intermediate state,12,9,2,Eulers formula of the critical force of slender compressed columns,1,、,Critical pressure for the column with two hinged ends:,bending moment,：,Approximate differential,equation of the deflection curve,F,F,x,y,P,M,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.,F,x,L,13,Solution of the differential equation,：,Determine the integral constants,：,The critical force,F,cr,is the smallest pressure under tiny bending，therefore,we only take,n,=1 and the column will bend about the axis with the smallest moment of inertia,.,That is,14,2,、,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 constraint coefficient,）,Eulers formula of the critical pressure for,the compressive column with two hinged ends,General form of Eulers formula of,the critical pressure,15,若杆端在各个方向的约束情况不同（柱形绞），应分别,计算杆在不同方向失稳时的临界力。,I,为其相应的对,中性轴的惯性矩。,16,0.5,l,Table,9,1,Eulers formula of the slender compressive column under various constraint conditions,Supports,Two hinged ends,One free end and one hinged end,Two fixed ends,One fixed end and one free end,The shape of the deflective curve in lost of stability,P,cr,A,B,l,Eulers formula of the critical force,P,cr,Length coefficient,=,1,0.7,=,0.5,=,2,P,cr,A,B,l,P,cr,A,B,l,0.7,l,C,C,D,C,:Inflection point,C,、,D,:Inflection point,P,cr,l,2,l,17,18,Example 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,),50,10,F,L,F,L,(45,45,6),Equal-leg angle steel,y,z,19,9,3,Scope of application of,Eulers formula,and empirical formula,1,、,Basic concepts,1).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,：,2,2,l,p,s,E,cr,=,That is:,Radius of inertia,Flexibility(or slenderness ratio)of the column,20,4).Division of large flexibility column,：,2,、,Calculation of the critical stress for the columns with middle or small flexibility,1).Empirical formula with the linear variation,As,P,S,：,is one of the middle flexibility column,.,The column with,Its critical stress 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 column can not be calculated by Eulers formula.,21,Total figure of the critical stress,As,S,：,b,s,2,=,a,-s,l,P,1,E,s,p,l,2,=,The critical stress of this column is its yield limit.,is one of the small flexibility,The column with,column.,22,2)Empirical formula with the parabolic variation,The common form in the,structural engineering is,：,As,P,s,：,As,s,：,For,A,3,steel,A,5,steel and 16Mn steel,As,This formula is used to determine the critical stress.,23,9,4,Stability check and,measures of increasing stability,1,、Stability check,24,例题,2,：,图示各杆材料和截面均相同，试问哪一 根杆能承受的压力最大，哪一根的最小？,a,P,(1),P,1.3,a,(2),P,(3),1.6a,因为,又,可知,（,1,）杆承受的压力最小，最先失稳；,（,3,）杆承受的压力最大，最稳定。,25,26,目录,解：,CD,梁,AB,杆,27,AB,杆,AB,为大柔度杆,AB,杆满足稳定性要求,28,END,29,