真空-堆载联合预压蠕变机理及微观结构特征试验研究课件.ppt

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1、,CHAPTER 4 INTERNAL FORCESIN BENDING,第四章 弯曲内力,材料力学,41 Concepts of planar bending and calculation sketch of the beam42 The shearing force and bending moment of the beam43 The shearing-force and bending-moment equations the shearing-force and bending-moment diagrams44 Relations among the shearing forc

2、e、the bending moment and the density of the distributed load and their applications 45 Plot the bending-moment diagram by the theorem of superpositiom46 The internal-force diagrams of the planar rigid theorem frames and curved rods Exercise lessons about the internal force of bending,CHAPTER 4 INTER

3、NAL FORCES IN BENDING,41 平面弯曲的概念及梁的计算简图42 梁的剪力和弯矩43 剪力方程和弯矩方程 剪力图和弯矩图44 剪力、弯矩与分布荷载集度间的关系及应用45 按叠加原理作弯矩图46 平面刚架和曲杆的内力图 弯曲内力习题课,第四章 弯曲内力,41 CONCEPTS OF PLANAR BENDING AND CALCULATION SKETCH OF THE BEAM,1、CONCEPTS OF BENDING,1).BENDING:The action of the external force or external the couple vector perp

4、endicular to the axis of the rod makes the axis of the rod change into curve from original straight lines,this deformation is called bending.,2).BEAM：The member of which the deformation is mainly bending is generally called beam.,INTERNAL FORCES IN BENDING,弯曲内力,41 平面弯曲的概念及梁的计算简图,一、弯曲的概念,1.弯曲:杆受垂直于轴线

5、的外力或外力偶矩矢的作用时,轴 线变成了曲线，这种变形称为弯曲。,2.梁：以弯曲变形为主的 构件通常称为梁。,3).Practical examples in engineering about bending,INTERNAL FORCES IN BENDING,3.工程实例,弯曲内力,INTERNAL FORCES IN BENDING,弯曲内力,4).Planar bending：After deformation the curved axis of the beam is still in the same plane with the external forces.,Symmet

6、ric bending（as shown in the following figure）a special example of the planar bending.,INTERNAL FORCES IN BENDING,弯曲内力,4.平面弯曲：杆发生弯曲变形后，轴线仍然和外力在同一 平面内。,对称弯曲（如下图）平面弯曲的特例。,Unsymmetrical bending if a beam does not possess any plane of symmetry,or the external forces do not act in a plane of symmetry of t

7、he beam with symmetric planes,this kind of bending is called unsymmetrical bending.In later chapters we will mainly discuss the bending stresses and deformations of the beam under symmetric bending.,INTERNAL FORCES IN BENDING,弯曲内力,非对称弯曲 若梁不具有纵对称面，或者，梁虽具有纵 对称面但外力并不作用在对称面内，这种 弯曲则统称为非对称弯曲。下面几章中，将以对称弯曲为

8、主，讨论梁的应力和变形计算。,2、Calculation sketch of the beam,In general supports and external forces of the beam are very complex.We should do some necessary simplification for them for our convenient calculation and obtain the calculation sketch.,1).Simplification of the beams In general case we take the place

9、of the beam by its axis.,2).Simplification of the loads The loads(including the reaction)acting on the beam may be reduced into three types：concentrated force、concentrated force couple and distributed force.,3).Simplification of the supports,INTERNAL FORCES IN BENDING,弯曲内力,二、梁的计算简图,梁的支承条件与载荷情况一般都比较复

10、杂，为了便于分析计算，应进行必要的简化，抽象出计算简图。,1.构件本身的简化 通常取梁的轴线来代替梁。,2.载荷简化 作用于梁上的载荷（包括支座反力）可简化为三种类型：集中力、集中力偶和分布载荷。,3.支座简化,Fixed hinged support 2 constraints，1 degree of freedom.Such as the fixed hinged support under bridges，thrust ball bearing etc.,Movable hinged support 1 constraint，2 degree of freedom.Such as the

11、 movable hinged support under the bridge，ball bearing etc.,INTERNAL FORCES IN BENDING,弯曲内力,固定铰支座 2个约束，1个自由度。如：桥梁下的固定支座，止推滚珠轴承等。,可动铰支座 1个约束，2个自由度。如：桥梁下的辊轴支座，滚珠轴承等。,Rigidly fixed end 3 constraints，0 degree of freedom.Such as the support of diving board at the swimming pool，support of the lower end of

12、a wooden pole.,4)Three basis types of beams,Simple beam(or simply supported beam),Cantilever beam,INTERNAL FORCES IN BENDING,A,弯曲内力,固定端 3个约束，0个自由度。如：游泳池的跳水板支座，木桩下端的支座等。,4.梁的三种基本形式,简支梁,悬臂梁,Overhanging beam,5).Statically determinate and statically indeterminate beams,Statically determinate beams：React

13、ions of the beam can be determined only by static equilibrium equations，such as the above three kinds of basic beams.Statically indeterminate beams：Reactions of the beam cannot be determined or only part of reactions can be determined by static equilibrium equations.,INTERNAL FORCES IN BENDING,弯曲内力,

14、外伸梁,5.静定梁与超静定梁,静定梁：由静力学方程可求出支反力，如上述三种基本 形式的静定梁。超静定梁：由静力学方程不可求出支反力或不能求出全 部支反力。,Example 1 A stock tank is shown in the figure.Its length is L=5m，its inside diameter is D=1m，thickness of its wall is t=10mm.Density of steel is 7.8g/cm.Density of the liquid is 1g/cm.Height of the liquid is 0.8m.Length of

15、 overhanging end is 1m.Try to determine the calculation sketch of the stock tank.,Solution：,INTERNAL FORCES IN BENDING,弯曲内力,例1贮液罐如图示，罐长L=5m，内径 D=1m，壁厚t=10mm，钢的密度为：7.8g/cm，液体的密度为：1g/cm,液面高 0.8m，外伸端长 1m，试求贮液罐的计算简图。,解：,INTERNAL FORCES IN BENDING,弯曲内力,42 THE SHEARING FORCE AND BENDING MOMENT OF THE BEAM

16、,1、Internal force in bending：,Example Knowing conditions are P，a，l,as shown in the figure.Determine the internal forces on the section at the distance x to the end A.,l,A,A,B,B,Solution：Determine external forces,INTERNAL FORCES IN BENDING,42 梁的剪力和弯矩,一、弯曲内力：,弯曲内力,举例已知：如图，P，a，l。求：距A端x处截面上内力。,l,A,A,B,B

17、,解：求外力,Determine internal forces method of section,A,Q,M,M,Q,Internal forces of the beam in bending,1).Bending moment：M Moment of the internal force couple with the acting plane in the cross-section perpendicular to the section when the beam is bending.,C,C,INTERNAL FORCES IN BENDING,弯曲内力,求内力截面法,A,Q

18、,M,M,Q,弯曲构件内力,1.弯矩：M 构件受弯时，横截面上其作用面垂直于截面的内力偶矩。,C,C,2).Shearing force：Q Internal force which the acting line in the cross-section parallel to the section,when the beam is bending.,3).Sign conventions for the internal forces:,Shearing force Q:It is positive when it results in a clockwise rotation with

19、 respect to the object under consideration,otherwise it is negative.,Bending moment M：It is positive when it tends to bend the portion concave upwards,otherwise it is negative.,Q(+),Q(),Q(),Q(+),M(+),M(+),M(),M(),INTERNAL FORCES IN BENDING,弯曲内力,2.剪力：Q 构件受弯时，横截面上其作用线平行于截面的内力。,3.内力的正负规定:,剪力Q:绕研究对象顺时针转

20、为正剪力；反之为负。,弯矩M：使梁变成凹形的为正弯矩；使梁变成凸形的为负弯矩。,Q(+),Q(),Q(),Q(+),M(+),M(+),M(),M(),Example 2：Determine the internal forces acting on sections 11 and 22 section as shown in fig.(a).,Solution：Determine internal forces by the method of section.Free body diagram of the left portion of section 11 is shown in fi

21、g.（b）.,Fig.（a）,2、Examples,Q1,A,M1,Fig.（b）,INTERNAL FORCES IN BENDING,例2：求图（a）所示梁1-1、2-2截面处的内力。,解：截面法求内力。1-1截面处截取的分离体 如图（b）示。,图（a）,二、例题,Q1,A,M1,图（b）,弯曲内力,Free body diagram of the left portion of section 22 is shown in fig.（b）.,图（a）,q,Q2,B,M2,图（c）,INTERNAL FORCES IN BENDING,2-2截面处截取的分离体如图（c）,图（a）,q,Q2

22、,B,M2,弯曲内力,图（c）,1.Internal-force equations:Expressions that show the internal forces as functions of the position x of the section.,2.The shearing-force and bending-moment diagrams:,Shearing-force diagram,sketch of the shearing-force equation,Bending Moment diagram,sketch of the bending-moment equat

23、ion,43 THE SHEARING-FORCE AND BENDING-MOMENT EQUATIONS THE SHEARING-FORCE AND BENDING-MOMENT DIAGRAMS,INTERNAL FORCES IN BENDING,弯曲内力,1.内力方程：内力与截面位置坐标（x）间的函数关系式。,2.剪力图和弯矩图：,43 剪力方程和弯矩方程 剪力图和弯矩图,Example 3 Determine the internal-force equations and plot the diagrams of the beam shown in the following

24、figure.,Solution：Determine the reactions of the supports,Write out the internal-force equations,P,Plot the internal-force diagrams,Q(x),M(x),x,x,P,PL,INTERNAL FORCES IN BENDING,弯曲内力,例3 求下列各图示梁的内力方程并画出内力图。,解：求支反力,写出内力方程,P,YO,L,根据方程画内力图,Q(x),M(x),x,x,P,PL,Solution：Write out the internal-force equation

25、s,Plot the internal-force diagram,L,q,Q(x),x,qL,INTERNAL FORCES IN BENDING,弯曲内力,解：写出内力方程,根据方程画内力图,L,q,Q(x),x,qL,Solution：Determine the reactions of the supports,Write out the internal-force equations,q0,RA,Plot the internal-force diagrams,RB,x,INTERNAL FORCES IN BENDING,弯曲内力,解：求支反力,内力方程,q0,RA,根据方程画内

26、力图,RB,x,1、Relations among the shearing force、the bending moment and the the distributed load,By analysis of the equilibrium of the infinitesimal length dx，we can get,44 RELATIONS AMANG THE SHEARING FORCE,THE BENDING MOMENT AND THE INDENSITY OF THE DISTRIBUTED LOAD AND THEIR APPLICATIONS,q(x),q(x),M(

27、x)+d M(x),Q(x)+d Q(x),Q(x),M(x),dx,A,y,INTERNAL FORCES IN BENDING,Slope of the tangential line at a point in the shearing-force diagram is equal to the intensity of the distributed load at the same point.,弯曲内力,一、剪力、弯矩与分布荷载间的关系,对dx 段进行平衡分析，有：,44 剪力、弯矩与分布荷载集度间的关系及应用,q(x),q(x),M(x)+d M(x),Q(x)+d Q(x),Q

28、(x),M(x),dx,A,y,剪力图上某点处的切线斜率等于该点处荷载集度的大小。,q(x),M(x)+d M(x),Q(x)+d Q(x),Q(x),M(x),dx,A,y,Slope of the tangential line at a point in the bending-moment diagram is equal to the magnitude of the shearing force at the same point.,Relation between the bending moment and the indensity of the distributed lo

29、ad：,INTERNAL FORCES IN BENDING,弯曲内力,q(x),M(x)+d M(x),Q(x)+d Q(x),Q(x),M(x),dx,A,y,弯矩图上某点处的切线斜率等于该点处剪力的大小。,弯矩与荷载集度的关系是：,2、Relations between the shearing force、the bending moment and the external load,External force,No external-force segment,Uniform-load segment,Concentrated force,Concentrated couple,

30、Characteristics of Q-diagram,Characteristics of M-diagram,Horizontal straight line,Inclined straight line,Sudden change from the left to right,No change,Inclined straight line,curves,Flex from the left to the right,Sudden change from the left to the right,Opposite tom,INTERNAL FORCES IN BENDING,二、剪力

31、、弯矩与外力间的关系,外力,无外力段,均布载荷段,集中力,集中力偶,Q图特征,M图特征,水平直线,斜直线,自左向右突变,无变化,斜直线,曲线,自左向右折角,自左向右突变,与m反,弯曲内力,Simple method to plot the diagram:The method to plot the diagrams by using the relation between the internal forces and the external forces and values of the internal forces at some special points.,Example

32、4 Plot the internal force diagrams of the beams shown in the following figures by the simple method to plot the diagram.,Solution:,Special points:,Plot the diagram by using the relation between the internal forces and the external forces and the internal force values at some special points of the be

33、am.,End point、partition point（the point at which external forces changed）and stationary point etc.,INTERNAL FORCES IN BENDING,弯曲内力,简易作图法:利用内力和外力的关系及特殊点的内力值来作 图的方法。,例4 用简易作图法画图示梁的内力图。,解:利用内力和外力的关系及 特殊点的内力值来作图。,特殊点:端点、分区点（外力变化点）和驻点等。,Left end：,Shape of the curve is determined according to,；,；,And the

34、law of the point acted by concentrated force.,Partition point A：,Stationary point of M：,Right end：,Q,x,x,M,INTERNAL FORCES IN BENDING,弯曲内力,左端点：,分区点A：,M 的驻点：,右端点：,Q,x,x,M,Example 5 Plot the internal-force diagrams of the beams shown in the following figures by the simple method to plot the diagram.,S

35、olution：Determine reactions,Left end A：,Right of point B：,Left of point C：,Stationary point of M：,Right of point C：,Right end D：,q,qa2,qa,RA,RD,Q,x,qa/2,qa/2,qa/2,A,B,C,D,qa2/2,x,M,qa2/2,qa2/2,3qa2/8,Left of point B：,INTERNAL FORCES IN BENDING,弯曲内力,例5 用简易作图法画下列各图示梁的内力图。,解：求支反力,左端点A：,B点左：,B点右：,C点左：,M

36、 的驻点：,C点右：,右端点D：,q,qa2,qa,RA,RD,Q,x,qa/2,qa/2,qa/2,A,B,C,D,qa2/2,x,M,qa2/2,qa2/2,3qa2/8,45 PLOT THE DIAGRAM OF BENDING MOMENT BY THE THEOREM OF SUPERPOSITIOM,1、Theorem of superposition：Internal forces in the structure due to simultaneous action of many forces are equal to algebraic sum of the intern

37、al forces due to separate action of each force.,Applying condition：Relation between the parameters(internal forces、stresses、displacements）and the external forces must be linear,that is they satisfy Hookes law.,INTERNAL FORCES IN BENDING,弯曲内力,45 按叠加原理作弯矩图,一、叠加原理：多个载荷同时作用于结构而引起的内力等于每个载荷单独作用于结构而引起的内力的代

38、数和。,适用条件：所求参数（内力、应力、位移）必然与荷载满 足线性关系。即在弹性限度内满足虎克定律。,2、Structural members in mechanics of material is of small deformation and linear elasticity,and must obey this principle method of superposition,Steps：Plot respectively the diagram of the bending moment of the beam under the separate action of each

39、external load；Sum up the corresponding longitudinal coordinates(Attention:do not simply piece together figures.）,INTERNAL FORCES IN BENDING,弯曲内力,二、材料力学构件小变形、线性范围内必遵守此原理 叠加方法,步骤：分别作出各项荷载单独作用下梁的弯矩图；将其相应的纵坐标叠加即可（注意：不是图形的简单拼凑）。,Example 6 Plot the diagram of bending moment by the principle of superpositi

40、on.(AB=2a，force P is acting at the middle point of the beam AB.）,P,q,P,=,+,A,A,A,B,B,B,=,+,INTERNAL FORCES IN BENDING,弯曲内力,例6按叠加原理作弯矩图(AB=2a，力P作用在梁AB的中点处）。,q,P,P,=,+,A,A,A,B,B,B,=,+,3、Applications of symmetry and antisymmetry：For the symmetric structure under the action of symmetric loads the diagra

41、m of its shearing stress Q is antisymmetric and the diagram of the bending moment M is symmetric.For the symmetric structure under the action of antisymmetric loads the diagram of its shearing stress Q is symmetric and the diagram of the bending moment M is antisymmetric.,INTERNAL FORCES IN BENDING,

42、弯曲内力,三、对称性与反对称性的应用：对称结构在对称载荷作用下，Q图反对称，M图对称；对称结构在反对称载荷作用下，Q图对称，M图反对称。,Example 7 Plot internal-force diagrams of the beams shown in the following figure.,P,PL,PL,0.5P,0.5P,0.5P,0.5P,P,0,0.5P,0.5P,0.5P,P,INTERNAL FORCES IN BENDING,弯曲内力,例7 作下列图示梁的内力图。,P,PL,PL,0.5P,0.5P,0.5P,0.5P,P,0,0.5P,0.5P,0.5P,P,P,P

43、L,PL,0.5P,0.5P,0.5P,0.5P,P,0,M,x,M1,x,M2,x,0.5PL,PL,0.5PL,0.5PL,INTERNAL FORCES IN BENDING,弯曲内力,P,PL,PL,0.5P,0.5P,0.5P,0.5P,P,0,M,x,M1,x,M2,x,0.5PL,PL,0.5PL,0.5PL,Example 8 Correct the mistakes in the following internal-force diagrams.,a,2a,a,q,qa2,A,B,Q,x,x,M,qa/4,qa/4,3qa/4,7qa/4,qa2/4,49qa2/32,3q

44、a2/2,5qa2/4,INTERNAL FORCES IN BENDING,RA,RB,弯曲内力,例8 改内力图之错。,a,2a,a,q,qa2,A,B,Q,x,x,M,qa/4,qa/4,3qa/4,7qa/4,qa2/4,49qa2/32,3qa2/2,5qa2/4,Example 9 Knowing Q-diagram,determine external loads and M-diagram(Therefore no concentrated force couples acted on the beam).,M(kNm),INTERNAL FORCES IN BENDING,弯曲

45、内力,例9 已知Q图，求外载及M图（梁上无集中力偶）。,Q(kN),x,1m,1m,2m,2,3,1,5kN,1kN,q=2kN/m,M(kNm),x,1,1,1.25,46 THE INTERNAL-FORCE DIAGRAMS OF THE PLANAR RIGID FRAMES AND CURVED RODS,1、Planar rigid frame,1).Planar rigid frame：Structure made from rods of different direction that are mutually connected in rigidity at their e

46、nds in the same plane.Characteristics：There are internal forces Q,M and N in each rod.,2).Conventions to plot diagram of internal forces：Bending-moment diagram：Plot it at the side where fibers are elongated and not mark the sign of positive or negative.Shearing-force and axial-force diagrams：May be

47、plotted at any side of the frame（In common the diagram with positive value is plotted outside the frame），but must mark the signs of positive and negative.,INTERNAL FORCES IN BENDING,弯曲内力,46 平面刚架和曲杆的内力图,一、平面刚架,1.平面刚架：同一平面内，不同取向的杆件，通过杆端相 互刚性连接而组成的结构。特点：刚架各杆的内力有：Q、M、N。,2.内力图规定：弯矩图：画在各杆的受拉一侧，不注明正、负号。剪力图

48、及轴力图：可画在刚架轴线的任一侧（通常正值画在刚架的外侧），但须注明正、负号。,Example 10 Try to plot the internal-force diagrams of the rigid frame shown in the figure.,P1,P2,a,l,A,B,C,N-diagram,Q-diagram,P1,M-diagram,INTERNAL FORCES IN BENDING,弯曲内力,例10 试作图示刚架的内力图。,P1,P2,a,l,A,B,C,N 图,Q 图,M 图,P1a+P2 l,Example 11 As shown in the figure,P

49、 and R are known,try to plot internal force diagrams of Q,M and N.,P,Solution：set up polar coordinates，O is the pole and OB is polar axis，q denotes the position of the section m-m.,A,B,2、Planar rod：Rod that the axis is of a planar curve.Method to plot internal-force diagram of a curved rod is the sa

50、me as that of the planar rigid frame.,INTERNAL FORCES IN BENDING,弯曲内力,二、平面曲杆：轴线为一平面曲线的杆件。内力情况及绘制方法与平面刚架相同。,例11 已知：如图所示，P及R。试绘制Q、M、N 图。,P,解：建立极坐标，O为极点，OB 极轴，q表示截面mm的位置。,A,B,P,A,B,M图,Q图,N图,2PR,P,P,INTERNAL FORCES IN BENDING,弯曲内力,P,A,B,M-diagram,Q-diagram,N-diagram,2PR,P,P,1、Method to determine directl