静定结构位移计算双语

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,*,第 六 章,Chapter VI,静定结构位移计算,The computation of displacements of,statically determinate structures,6-1,概述,Introduction,结构位移、变形的实例,The examples of displacements and deformations,1.,结构的位移,Displacements of structures,线位移：结构某点移动的距离（,）,Linear displacements or translation: the distance of movement of a point alone a line;,角位移（转角）：结构构件某截面转的角度（,）,Angular displacement or rotation:sectional rotation of a member of a structure.,相对（线）位移：两点之间距离变化,Relative linear displacement: the change of the distance between 2 points.,相对角位移（转角）：两截面之间夹角变化,Relative angular displacement:the change of the angle between 2 sections.,2.,广义位移的概念,Generalized displacements,为便于进行理论分析和公式推导，将所讨论的各种结构位移用统一的符号来表示,，称为广义位移。,For the convenience of the proceeding of theoretical analysis and derivation of formulae, we denote all kinds of displacements by,which is termed as,generalized displacements,注,Remark,：广义位移可以看成为代表位移的抽象符号，在具体问题中表示具体位移,.,Generalized displacements,may be looked at as abstract sign, which represents concrete displacement in concrete problems,.,讨论,Discussion,：位移与变形,displacements and deformation,位移,变形的宏观集成表现,Displacement- macroscopic integration of deformation,.,变形,相对位移,Deformation- relative displacement per unit length or angle.,两者之间有区别，又有联系：,These 2 terms have differences and also have relations,(a),变形可引起位移（转角）；,Deformation induces displacements,(b),变形可由相对位移（或转角）来表示。,Deformation may be represented by relative displacements.,3,、引起位移的原因：,Causes inducing displacements,荷载；温度变化；支座移动（沉降）；材料收缩；制作误差等,Loads, temperature change, support settlement, contraction of material, manufacture errors etc.,4,、位移计算的目的,The,purposes of calculation of displacements,（,1,）、校核结构的刚度,checking rigidity of structures,；,（,2,）、为超静定结构计算打基础,preparation for the calculation of statically indeterminate structures,。,6-2,变形体的虚功原理,Principle of virtual work of deformable bodies,1,、功、虚功,work and virtual work,功：,力与沿力方向的位移的乘积，,W=F,Work: the product of force and the distance in the direction of force.,当位移与力方向相同时，力作的功为正。,If the directions of force and distance coincide, then the work is positive.,力偶作功,work done by a couple of forces,：,W=M,M,为力偶矩,M,is,couple of forces,，,为角位移,angular displacement,。,力在线位移上作功，力偶在角位移上作功；,实功,real work,：力在与力有关的位移上作的功,work done by force in related displacement,。,线弹性体上外力的实功,for elastic bodies,：,虚功,virtual work,：力在与之无关的位移上作的功,work done by force in unrelated displacement,。,作功的过程中，力保持不变,force remains constant during the process of doing work,。,广义力,generalized forces,：用符号,F,表示集中力、力偶、力对、力偶对等，,广义力在相应的广义位移上作功,generalized,forces correspond to definite generalized displacements.,功的表达式写成,work can be expressed as,：,W=F,广义力，,广义位移。,、虚功原理,Principle of virtual work,：,(a),刚体的虚功原理,：刚体体系处于平衡的充要条件是对任何虚位移，所有外力作的虚功总和为零。,Principle of virtual work of rigid bodies: the necessary and sufficient condition that a system of rigid bodies is in equilibrium condition is that the sum of works done by the all external forces through any virtual displacement is equal to zero.,（外力包括荷载和约束力（如支座反力）,External forces include external loads and reaction forces.,(b),变形体的虚功原理：变形体处于平衡的充要条件是对任何虚位移，外力所作虚功总和等于各微段上内力在其变形上所作虚功总和。,Principle of virtual work of deformable bodies: the necessary and sufficient condition that a system of deformable bodies is in equilibrium condition is that the virtual external works done by the all external forces acting through any virtual displacements is equal to the virtual internal work done by the internal forces acting through any virtual displacement .,虚位移为满足约束条件，变形连续，微小的协调位移,virtual displacements are small and consistent with the support and continuity conditions,。,注,remark,：虚功原理中包括两个状态：力状态和位移状态，且两个状态是彼此无关的,Principle of virtual work includes 2 states:force state and displacement state , these 2 states have no relationship with each other,。,即：,外力虚功等于内力虚功,I.e.the external force work is equal to internal force work,虚功原理,the principle of virtual work,：,两个独立的状态,2 independent states,：,（平衡）,（协调）,1,、按外力虚功与内力虚功角度来计算总虚功：,取微段,ds,，其上所有力状态中的力在位移状态的位移上作的虚功,dW=dW,e,+dW,i,则总虚功,W,为：,the total virtual work is,或,如图,：,（截面两侧位移相同，而作用力方向相反）,At the,2 sides of the section the displacements are identical, and the forces are opposite,.,故,所以,(,dW,e,外力虚功,，,dW,i,内力虚功,),2,、按刚体虚功和变形虚功来计算总虚功：,du,dv =,ds,d,轴向变形,剪切变形,弯曲变形,（两截面之间相对位移）,而,故,即,于是有,（外力虚功等于变形虚功）,（刚体位移引起）,（变形引起）,虚功原理的数学表达,mathematical expression,：,W,e,外力虚功总和,the sum of external force works,W,V,微段变形虚功的总和,the sum of internal force works,注,remark,：在证明中没有涉及材料的本构关系，因此对弹性、非弹性，线性、非线性的变形体，虚功原理都适用。,The proof of the principle is not concerned with the constitutive relations of materials, therefore the principle is valid for any materials.,或,虚功原理的应用,the application of principle of virtual work,：,、对于给定力状态，另虚设位移状态，求解力状态中的未知力,虚位移原理,For given force state, assume virtual displacement state, we can determine unknown forces ,principle of virtual displacement.,、对于给定的位移状态，另虚设力状态，求解位移状态中的未知位移,虚力原理。,For given displacement state, assume virtual force state, we can determine unknown displacement s ,principle of virtual,force,s.,（本章将应用这一原理计算结构位移,In this chapter we will use principle of virtual,force,s to determine unknown displacements,）,6-, 位移计算的一般公式,-,单位荷载法（单位力法）,The general equation and unit load method for determination of displacements,实际问题,计算,K,沿,K,施加单位力,得,：,实际荷载引起的位移状态,displacement state induced by actual loads,单位力荷载引起的力状态,force state induced by unite load,即,：,得：,对一般情况，上式可写成,In general case the above equation can be written as,：,称之为：单位力（荷载）法,unit load method,弯曲变形,bending,剪切变形,shear,轴向变形,elongation,支座移动,support settlements,对所求位移,的贡献,the contributions,注：,1,、公式中，,实际上是,F=1,所作的虚功，因此，,计算结果为正时，,与,F=1,方向相同；计算结果为负时，与,F=1,方向相反。,Remark: is actually the virtual work done by,F=1,if the result is positive, then the direction of coincides with that of,F=1,;,otherwise the direction of is opposite to that of,F=1.,2,、是广义位移,is generalized displacement,。,3,、,F=1,是在上作功的广义力,is the force doing work in,公式为虚设单位力,F=1,在,上作的功，因此,F=1,要与相应（,F=1,为广义力）,The correspondence,between generalized unit force and generalized displacement,。,下面举例说明,F=1,的类型：,Ax,A,设置单位力可以求线位移,设置单位力偶可以求线位移,设置一对单位力可以求相对线位移,设置一对力偶矩可以求相对线转角,6-4,荷载作用下的位移计算,Calculation of displacements caused by loads,由材料力学,from material mechanics,：,；,；,实际状态,real system,位移问题转化为内力计算问题。,Therefore the problem of the determination transforms into the problem of the calculation of the internal forces.,虚设（单位力）状态,Virtual unit force state,梁、刚架,For beams and rigid frame,：（忽略轴向与剪切变形,axial deformation and shear are neglected,）,桁架,for trusses,：,积分运算转化为代数运算,。,组合结构,for composite structures,：,实体拱，一般只取弯矩项即可；扁平拱，压力较大，应该考虑轴力的影响；计算曲杆时可以忽略曲率的影响；,例题,求图示刚架,C,点的竖向位移,AH,及,A,点,角位移,Determine the horizontal displacement and rotation of point C,，,各杆抗弯刚度,EI,相同。,EI is identical for every members.,AB,段,：,BC,段：,AB,段,：,BC,段：,，,【,例,】,求图示桁架,D,结点的竖向位移。,6-5,图乘法,Graph-multiplication method,条件,condtions,：,1.,杆件为直杆,the members are straight,；,2,、,EI,沿杆件不变,EI=constant,；,3,、,与,M,图中至少有一个为直线图形,At least one of moment diagrams is straight line,。,则,*,c,形心,M,P,图,A,y,C,对于图示弯矩图,也即上述积分等于弯矩图的面积乘以形心对应的另一图形上的竖标。此法为图乘法。此法为莫斯科建筑学院的学生,Veryschagin,发现。,The integral Equals to the product of the area and the ordinate of the centroid of another area,Graph-multiplication method,，,which was discovered by,Veryschagin.,抛物线的面积计算,the areas of parabolas,：,形心位置,the position of centroid,离高边,阴影部分面积,形心位置离高边,基本图形的面积与形心位置,The areas and centroids of some commonly used graphs,：,顶点,顶点,顶点,顶点：,M,图切线斜率为零,2.,分解,The separation of graphs,（,1,） 梯形图形,trapezoids,例,求图示刚架,B,点的水平位移,Bx,。（各杆长均为：,l,）,C,A,B,EI,2EI,P,P,+,+,+,（,2,） 抛物线图形,Parabolas,将叠加法绘制,M,图的过程，反方向进行，即可将抛物线型的,M,图分解为简单的抛物线和梯形（或三角形）。,By backward process of superposition we can separate the M graphs into simple parabolas and trapezoids.,+,3,、分段,常见错误,Common mistakes,：,错误,错误,3,、标准抛物线的方程为,equation of standard parabola,4.,正负号,the signs,1.,2.,5.,正负三角形,positive and negative triangles,注意事项,Remarks,：,1.EI,整段相同,EI remains constant over the segment,；,2.,若 、 位于不同的侧，则乘积为负；,If,、,lie on different sides of the base line, then the product is negative;,3.,若,2,图形都为直线，则竖标可以在任何以图形上取；,If all 2 moment diagrams are straight-lined, then the ordinate of the centroid can be taken at any graph,.,4.,若为折线,应该分段处理,If one graph is polygon, and another is a curved trapezoid, then calculation should be performed over several segments,；,5.,若,2,图为梯形，则可以分解成三角形,If all 2 moment diagrams are,trapezoids, then they can be separated into 2 triangles,；,6.,由均布载引起的弯矩图可以分解成抛物线、三角形或者梯形,The M diagram induced by distributed loads may be separated into parabola, triangle or trapezoid,。,例,求简支梁,B,端截面的角位移,B,。,Determine the rotation of end B,例,求悬臂梁,B,端的竖向位移,By,Determine the vertical displacement of end B,。,q,l,l,EI=,常数,B,M,P,1,例,求梁,A,点处的转角 及,C,点竖向位移,Cy,。,Determine the vertical displacement of C and rotation of point A,，,例,求,C,、,D,两点的相对位移,CD,Determine the relative displacement of C and D.,例,求静定梁,C,点处的竖向位移,Cy,。,Determine vertical displacement of end C.,例,求刚架,B,点处的水平位移,Dx,和,C,截面转角,C,。,Determine horizontal displacement of point B and the rotation of section C.,20kN/m,40kN,6,m,EI,2EI,B,A,C,2,m,2,m,80,80,80,90,M,P,(kNm),求,Dx,：,1,4,4,80,80,80,90,M,P,(kNm),1,1,求,C,：,例,求下面三铰刚架铰,C,两边截面之间的相对角位移（转角）,C,。,(,各杆,EI,为常数,),Find the relative rotation of left and right of joint C .,l/,2,l/,2,l,A,C,B,q,1,1,1,1,例,求图示组合结构,C,点处的竖向位移,Cy,。,Determine the vertical displacement of C of the following composite structure.,6-6,温度变化时的位移计算,Displacements caused by temperature change,对于无支座位移得情况，应用虚功原理得单位荷载法求位移得公式,For case without support displacements, using virtual work principle we obtain,其中,为实际状态中,dx,微段上由于温度改变所引起的变形,实际状态,real state,虚拟状态,virtual state,当杆截面对称于形心时,when the section is symmetrical with respect to centroid we have,当杆截面不对称于形心时,when the section is not symmetrical with respect to centroid we have,对于 的确定：比较实际状态与虚拟状态，若二者相同，则取正号，否则取负号。,对于桁架温度改变引起的位移为,制造误差,引起的位移,例,求图示刚架,C,点的竖向位移,cy,。,Determine vertical displacement of C,6-7,支座移动时的位移计算,Calculation of displacement caused by support movement,无外荷载时上式变为：,实际位移状态,虚拟力状态,例：求,Determine the relative rotation of left and right of joint C,实际位移状态,虚拟力状态,例,求图示刚架,A,端的转角,Ay,。,已知：,l,=12m, h=8m,By,=0.06m, ,Bx,=0.04m,。,实际位移状态,虚拟力状态,6-8,线弹性结构的互等定理,Reciprocal laws for linear elastic system,1,、功的互等定理,Law of reciprocal work,当先施加 再施加 时， 在 上所作的功为,二、位移互等定理,Law of reciprocal,displacements,当先施加 再施加 时， 在 上所作的功为,三、反力互等定理,Law of reciprocal,reactions,四、反力位移互等定理,Law of reciprocal of reaction and displacement,小 结,Summary,1.,基本概念,Basic concepts,功、虚功,work, virtual work,；广义力与广义位移,generalized forces an displacements and their several cases,。几种广义力及广义位移,。,2.,变形体的,虚功原理,Principle of virtual work,基本物理意义,basic physical meaning,虚位移原理,Principle of virtual displacement,虚力原理,Principle of virtual,force,3.,单位荷载法求位移,对于不同的结构，不同的受力采用上式中的一项或者多项。,For different structures and different loadings, we can take one or more terms.,4.,图乘法,graph-multiplication method,3,个条件,3 conditions,常犯的错误,common mistakes,5.,温度及支座位移引起的位移,Displacements caused by temperature change and support movements,6.,互等定理,Reciprocal laws,1),、功的互等定理,law of reciprocal work,2),、位移互等定理,Law of reciprocal,displacements,3),、反力互等动力,Law of reciprocal,reactions,4),、位移反力互等定理,Law of reciprocal of reaction and displacement,Popular vocabulary,位移、变形,displacements and deformations,线位移,Linear displacements or translation,角位移（转角,),Angular displacement or rotation,相对（线）位移：,Relative linear displacement,相对角位移,Relative angular displacement,广义位移,Generalized displacements,广义力,Generalized forces,荷载；温度变化；支座移动（沉降）；材料收缩；制作误差等,Loads, temperature change, support settlement, contraction of material, manufacture errors etc,变形体的虚功原理,Principle of virtual work of deformable bodies,虚功,virtual work,Popular vocabulary,单位荷载法（单位力法）,unit load method,虚设（单位力）状态,Virtual unit force state,图乘法,Graph-multiplication method,线弹性结构的互等定理,Reciprocal laws for linear elastic,systems,功的互等定理,Law of reciprocal work,位移互等定理,Law of reciprocal,displacements,反力互等定理,Law of reciprocal,reactions,反力位移互等定理,Law of reciprocal of reaction and displacement,