弯曲内力-2.pptx

本章内容小结,本章基本要求,1.,梁的,内力,（第,4,章）,2.,梁的,应力,5.,梁的刚度,3.,梁的强度,（第,5,章）,4.,梁的变形,背景材料,前言,6.,梁的超静定问题,（第,6,章）,F,v,横梁横截面上的应力如何计算？行车移动时，这种应力如何变化？,背 景 材 料,汽车在轮轴上的支承为什么设计为叠板弹簧的形式？这种结构的力学性能特点是什么？,跳杆横截面上的应力与轴线曲率半径有什么关系？玻璃钢材料制成的跳杆是否满足强度要求？,薄壁杆件的弯曲应力具有什么特点？如何避免薄壁杆件的强度失效？,本章基本要求,1.,熟练掌握作梁内力图的方法,.,2.,了解梁的正应力和切应力的推导过程,熟练应用正应力和切应力公式进行应力和强度计算,.,3.,掌握积分法求梁的变形,;,熟练掌握叠加法计算梁的变形,;,正确应用刚度条件计算梁的刚度,.,4.,熟练掌握计算简单梁的超静定问题,.,Preface,一,.,梁的计算简图,(1),梁的计算模型,(The calculated model of beams),Axis of beam,对称面,Symmetric plane,注意,:,本章只考虑横截面是左右对称的梁,而且所有载荷都作用在该对称面内,.,对称轴,Symmetric axis,Symmetric plane,If external forces are in the symmetric plane and are transverse,there are only two internal forces in the beam.,Transverse loadings,横向载荷,(2),梁的截面形状,(Forms of transverse cross section of beam,),(3),梁的形状,(Ordinary forms of beam),Straight beam having same cross section,Step layer beam,Cone beam,等截面直梁,阶梯状梁,小锥度梁,二,.,梁上的载荷,Loadings acted on the beam,(1),梁上的横向载荷,(Transverse loadings acted on the beam),-called concentrated loading(force),(focus loaging or force),F,-called distributed loadings(forces,),q,(,x,),-called uniform distributed loadings(forces),q,-,-called concentrated moment,(couple),m,-called distributed moments,集中载荷,(,力,),分布载荷,均布载荷,集中力偶,分布力偶,Positive and negtive transverse loadings acted on the beam(involve reactions),(+),(-),F,q,(,x,),m,(2),梁上横向载荷的正负号规定,(,包括支反力,),x,y,y,x,三,.,梁的支承形式,Supported forms of beams,(1),铰支承,(,Supported by pins,),(2),弹性支承,(Special supported forms),Supported by a rod,Supported by a spring,Supported by a another beam,拉杆支承,弹簧支承,梁支承,Supported by elastic ground,弹性地基支承,四,.,梁弯曲的基本概念,（,1,）梁的平面弯曲,(Plane bending),在梁的平面弯曲中，梁的轴线保持在一个平面内。,(2),纯弯曲和横力弯曲,纯弯曲,(pure bending),纯弯曲,(pure bending),横力弯曲,(transverse load bending),(3),梁的中性面,(neutral surface),受拉区,受压区,中性面,中性轴,(neutral axis),梁的中性面，是梁的轴向纤维伸长区和缩短区的界面。,悬臂梁,(cantilever beam),简支梁,(simple supported beam),外伸梁,(,Overhanging beam),五,.,梁的分类,(1),静定梁,（,Statically determinate beams,）,All the reactions can be obtained according to the equilibrium equations of the beam.,中间铰梁,（,beam having middle pins),(2),超静定梁,（,Statically indeterminate beams,）,The reactions of the beam can not be obtained according to the equilibrium equations.,Beam fixed at one end and simply supported at the other end,Continuous beam,Fixed beam,(3),特殊梁,（,Special beams,）,Layer beam,刚架,层合梁,曲梁,Curved beam,Rigid frame,苏轼 蝶恋花,花褪残红青杏小,.,燕子飞时，,绿水人家绕,.,枝上柳绵吹又少，,天涯何处无芳草！,墙里秋千墙外道,.,墙外行人，,墙里佳人笑笑渐不闻声渐悄，,多情却被无情恼,.,1.,梁 的 内 力,Internal forces,in a beam,1.1,内力函数,(Function of internal forces),x,y,z,when the beam is acted by transverse loadings,there are only two internal forces on the cross section,one is,shearing force,(,剪力,),and another is,bending moment,(,弯矩,).,1.1,内力函数,(Function of internal forces),x,y,z,x,y,z,x,y,z,F,s,(,x,),M,(,x,),剪力函数,Shearing force,弯矩函数,Bending moment,Ordinarily the function of internal forces in a beam can be obtained by section method.,内力函数一般可以用截 面 法求得,.,q,l,ql,/,2,ql,/,2,q,l,ql,/,2,ql,/,2,q,l,x,x,y,ql,/,2,ql,/,2,q,l,x,x,y,ql,/,2,q,l,x,x,y,ql,/,2,F,s,M,Example :,求简支梁承受均布荷载的内力函数,。,（,Determine the function of internal forces in a simple supported beam shown follows that acted by uniform distributed loadings.),q,l,ql,/,2,ql,/,2,q,l,ql,/,2,ql,/,2,Example :,求简支梁承受均布荷载的内力函数。,（,Determine the function of internal forces in a simple supported beam shown follows that acted by uniform distributed loadings.),注意,:,(1),梁的坐标系可以任意选取,.,(2),梁的坐标系可以采用局部坐标系,.,(3),梁的坐标系一经选定就不再改变,所有计算均在选定坐标系中进行,.,q,l,ql,/,2,ql,/,2,P,F,Example :,求简支梁承受均布荷载的内力函数。（,Determine the function of internal forces in a simple supported beam shown follows that acted by uniform distributed loadings.),注意,:,(1),梁的坐标系可以任意选取,.,(2),梁的坐标系可以采用局部坐标系,.,(3),梁的坐标系一经选定就不再改变,所有计算均在选定坐标系中进行,.,P,F,内力函数的特点,:,梁的,内力与梁材料的力学性能，梁截面的几何形状以及梁的约束类型等无关。只与梁的受力情况有关。,结论,:,在一般情况下梁中内力函数的变化规律是很复杂的,.,内力函数在梁中可能是连续函数也可能是分段函数甚至可能是间断函数,.,1.2,内力的正负号规定,(Positive and negative internal forces),(1),剪力的正负号规定,(Positive and negative shearing force),F,s,(,x,),(+),正剪力,Positive shearing force,负剪力,negative shearing force,F,s,(,x,),(-),x,1.2,内力的正负号规定,(Positive and negative internal forces),M,(,x,),(+),M,(,x,),(-),(2),弯矩的正负号规定,(Positive and negative bending moment),正弯矩,Positive bending moment,负弯矩,negative bending moment,x,1.3,内力图,(Diagrams of internal forces),F,s,x,l,F,s,(,x,),x,M,l,M,(,x,),-called diagram of bending moment,-called diagram of shearing force,剪力图,弯矩图,弯矩图取向下为正是一种特殊的处理方法。,问题,:,怎样作出梁的,剪力图和弯矩图,?,Section method to determine function of internal forces and draw the digram of shearing force and bending moment in a beam,(1),截面法求内力函数并画内力图,由于内力函数可以由截面法求得,所以梁的内力图可以采用截面法求内力函数然后描图的方法来绘制,.,预先声明,:,此种方法绘制梁的内力图是一种十分笨拙的方法,.,x,M,x,F,s,q,l,ql,/,2,ql,/,2,ql,/,2,ql,/,2,ql,2,/,8,Example:,小问题,:,将弯矩和剪力分别对,x,求导。由此能得到什么启示？,按前所述,需要用截面法分段写出剪力函数和弯矩函数,然后再描图,.,qa,q,a,a,a,qa,2,/,2,问题,:,如何画出图示复杂载荷作用下梁的剪力图和弯矩图？,这一过程非常的复杂和烦琐,.,那么是否有更简便的办法作梁的内力图问题,：,能否省略用截面法求内力函数的过程而直接根据梁上的外载荷画出梁的剪力图和弯矩图？,答案是肯定的,!,(2),连续曲线法作梁的内力图,目的：,省略用截面法求内力函数的过程而直接根据梁上外载荷画出梁的剪力图和弯矩图,.,Using Continuous curve to draw the diagram of internal forces,问题,:,怎样根据梁上外载荷来推断梁中内力的变化规律,?,也即梁上外载荷与梁中的内力有什么关系,?,q,F,s,Fs+,d,F,s,M,M+,d,M,d,x,(2.1),梁的平衡微分方程,differential equilibrium equations of beams,a.,分布载荷情况,（,Uniform distributed loadings),b.,集中力和集中力偶矩情况,（,concentrated force and moment),集中力使剪力突变但不影响弯矩,.,集中力矩使弯矩突变但不影响剪力,.,According to the differential equations,we can know the internal forces how to vary in the beam along the axis x,.,Result,:,结论,:,根据梁的平衡微分方程可以知道剪力和弯矩随轴线坐标变化的规律,d剪力弯矩图的规律,均布荷载,A,B,F,s,A,B,F,s,A,B,M,A,B,M,剪力图线穿过横轴：,A,B,q,a,qa,S,S,S,S,凸性与均布载荷方向一致,集中力,集中力偶矩,A,Fs,A,M,A,M,Fs,不受影响,A,F,F,A,m,m,均布力,q,q,=0,F,s,=0,M,=,C,F,s,0,F,s,0,q,0,m,0,F,0,x,M,x,F,s,q,l,ql,/,2,ql,/,2,ql,/,2,ql,/,2,ql,2,/,8,注意,:,梁的完整的内力图除了内力的图形外还包括一些特殊截面上的内力值的大小,.,问题,:,如何只依据作用在梁上的载荷确定梁的某些特殊截面上内力值的大小,?,a.,两个截面,AB,之间只有分布荷载作用,(2.2),梁的积分方程,(Integral equations of beam),结论,1:,直梁,B,截面上的剪力，等于,A,截面上的剪力与,AB,区间内载荷图面积的代数和。,A,B,q,(,x,),结论,2:,直梁,B,截面上的弯矩，等于,A,截面上的弯矩与,AB,区间内剪力图的面积的代数和。,a.,两个截面,AB,之间只有分布荷载作用,(2.2),梁的积分方程,(Integral equations of beam),A,B,q,(,x,),According to the integral equations,we can calculate the values of shearing force and bending moment at the special points of the beam.,Attention:,the area of,q,(,x,)and,F,s,(,x,)can be positive or negative.,Result,:,结论,:,根据积分方程通过计算载荷图和剪力图的面积确定剪力和弯矩在某些特殊点的值,.,载荷图和剪力图的面积可以是正面积也可以是负面积,1*,确定梁的支反力,并将其和外载荷一样看待,.,2*,剪力图和弯矩图在梁的左端从零开始,.,(2.3),连续曲线法作梁内力图的步骤,Determine the reactions of the beam and consider them,like the external,forces,on the beam .,Diagram of Q(x)and M(x),begin from zero at,the left end,of the beam.,3*,根据微分方程判断剪力图和弯矩图的变化规律,用连续曲线画剪力图和弯矩图,.,4*,根据积分方程通过计算载荷图和剪力图的面积确定剪力和弯矩在某些特殊截面的值,.,According to diffrential equations of the beam,we can know the Q(x)and M(x)how to vary along the axis x.Then drawing the diagram of Q(x)and M(x)by,continous curve,.,According to integral equations of the beam,we can,calculate the values of Q(x)and M(x)at,special,points,of the beam,.,5*,剪力图和弯矩图在梁的右端应回复到零值,.,Diagram of Q(x)and M(x)will,return to,zero,at the right end of the beam,.,特别注意,:,（,1,）作剪力图时只看梁上的集中力和分布力，而不必管集中力偶。,（,2,）作弯矩图时只看剪力图和梁上的集中力偶，而不必管集中力和分布力。,x,Fs,x,Fs,M,x,a,a,Pa,P,P,P,a,a,Example:,P,-P,Pa,P,-Pa,-Pa,P,P,M,x,P,Pa,x,Fs,P,Pl/,2,-Pl/,2,（,2.4,）,根据连续曲线法画剪力弯矩图,x,M,l/,2,l/,2,Pl,P,例,画出梁的剪力弯矩图。,P,a/,2,a/,2,q,x,Fs,3,qa/,8,-qa/,8,3,a/,8,9,qa/,128,2,3,qa/,8,qa/,8,x,M,例,画出梁的剪力弯矩图。,qa/,16,2,x,M,Fs,x,a/,2,a/,2,q,qa/,2,qa/,2,3,qa/,8,2,-3,qa,2,/,8,例,画出梁的剪力弯矩图。,-qa,2,/,8,x,Fs,a,a,q,qa,2,qa/,4,-3,qa/,4,-qa/,4,2,qa/,32,2,3,qa/,4,2,a/,4,qa/,4,3,qa/,4,M,x,画出梁的剪力弯矩图。,Fs,x,q,2,a,a,qa,2,3,a/,2,3,qa/,2,-qa/,2,qa,2,9,qa,2,/,8,qa/,2,3,qa/,2,x,M,例,画出梁的剪力弯矩图。,3,分钟练习题,剪力和弯矩的一个物理性质,(A physical character of shearing force and bending moment,),F,s,(,x,),M,(,x,),M,(,x,)is symmetric,F,s,(,x,)is anti-symmetric,结论,:,*,载荷对称的梁，其剪力图反对称，弯矩图对称。,*,载荷反对称的梁，其剪力图对称，弯矩图反对称。,P,P,a,a,a,a,q,q,对称梁,（,Symetrical beam,）,反对称梁,（,Anti-symetrical beam,）,a,a,P,P,载荷,对称梁,（,Loadings symetrical beam,）,载荷,反对称梁,（,Loadings anti-symetrical beam,）,qa,q,q,a,a,a,qa,-qa,-qa/,2,2,q,2,qa,a,a,x,F,s,M,x,例,画出梁的剪力弯矩图。,q,2,qa,a,a,剪力图反对称,弯矩图对称,载荷,对称梁,Laodings symetrical beam,x,Fs,qa,2,/,8,a,a,q,q,qa/,2,-qa/,2,qa/,2,-qa,2,/,8,qa/,2,qa/,2,M,x,例,画出梁的剪力弯矩图。,剪力图对称,弯矩图反对称,载荷,反对称梁,Loadings anti-symetrical beam,-qa,qa,-qa,2,/,2,qa,q,q,a,a,a,qa,Fs,x,x,M,3,分钟练习题,画出梁的剪力弯矩图。,x,M,F,s,x,q,a,a,qa,/,2,qa,/,2,qa/,2,qa,/,2,-qa/,2,qa,2,/,2,-qa/,2,2,qa/,8,2,先考虑右半部的平衡,再考虑左半部的平衡,qa,/,2,例,画出带中间铰的梁的剪力弯矩图。,结论,:,(1),中间铰不影响剪力图,.,(2),中间铰处的弯矩始终为零,.,(3),简单刚架,(frame),的内力图,*,平面刚架的内力图应包含轴力图、剪力图和弯矩图。,*,画轴力图和剪力图时，一般以刚架外侧为正，内侧为负；弯矩图则以刚架外侧为负，内侧为正。应标出图形正负号。,*,画刚架内力图时，观察者进入刚架之内，以刚架轮廓为坐标横轴，从左到右逐段画出。,刚结点,P,a,a,P,P,F,N,F,s,M,P,P,Pa,P,P,P,Pa,P,P,例,P,P,Pa,P,P,F,N,F,s,M,P,P,P,Pa,P,a,a,P,P,P,Pa,P,例,分析和讨论,刚节点处力的平衡有什么特点？,刚节点处弯矩图有什么特点？,F,N,Fs,Fs,F,N,同侧,相等,注意：,这些规律只是在刚节点处无集中荷载时适用。,本节内容结束,谢谢大家,