影响线及其应用PPT（双语98页）

1、第 十 一 章 Chapter 11影响线及其应用Influence lines and their application本章主要内容Main contents1、影响线概念 The concept of influence lines2、静力法绘制影响线 Static method 3、机动法绘制影响线 mechanismic method 4、影响线应用 Application of influence lines（1）利用影响线求固定荷载作用下的内力Determine internal forces under fixed loads（2）确定最不利荷载位置 Determine the

2、 unfavorable locations of loads（3）简支梁的绝对最大弯矩Determine the maximum bending moment（4）简支梁的包络图 Draw envelopes 6-1 影响线的概念 The concept of influence lines1、移动荷载 Moving loads移动荷载:方向、大小不变，仅作用位置变化的荷载 direction and magnitude dont change, only positions change (moving load)如桥梁上行驶的车辆，工业厂房中吊车梁上开动的行车等, for instanc

3、e the moving cars on bridges, the cranes in factories。 结构承受移动荷载作用时，其反力、内力以及位移等均随荷载作用位置的变化而改变。Under the moving loads, the reactions , internal forces and displacements of structures will change with the movement of the loads10kNP=10kNP=10kNP=10kN7.5kNP=10kN5kNP=10kN2.5kNP=10kN0kN10kN 为解决千变万化的移动荷载作用下结

4、构的设计问题，基于线弹性结构的叠加原理，如果掌握了单位移动荷载下结构的反力、内力与位移变化规律，则任意移动荷载下的反力、内力与位移可用叠加方法获得。For the solution of problems of design of structures, based on the superposition principle of linear elastic structures, if we know the variation laws of the reactions , internal forces and displacements , then the reactions ,

5、 internal forces and displacements under any moving loads can be determined by superposition principle 定义Definition: 单位移动荷载作用下，结构反力、内力或位移等随荷载位置变化的函数关系，分别称为反力、内力、位移的影响系数方程，对应的函数图形分别称为反力、内力、位移的影响线（Influence Line 缩写为 I .L.）。 under the action of unit load, the functional relations of the reactions , int

6、ernal forces and displacements of structures with the unit load position are called the influential coefficient equation, and the corresponding function graphs are called the Influence Line of the reactions , internal forces and displacements of structures. 为了方便，以下把反力、内力和位移均称为量值或物理量。For the convenie

7、nce the reactions , internal forces and displacements of structures called parameters 注意：定义中“单位移动荷载”下的量值称为影响系数，可以理解为：在单个移动荷载作用下某指定物理量与荷载P 的比值，习惯上以在结构上移动的P=1（即量纲一或无量纲）表示“单位移动荷载”。因此，物理量影响系数的量纲是，物理量的量纲和移动荷载量纲之比。例如单位移动荷载P是集中力，则弯矩、反力偶影响线的量纲为L(长度)，剪力、反力影响线的量纲为(或无量纲)等等。Remark: influential coefficients : th

8、e ration of the pointed quantity to the unit loads. The dimension of the influential coefficient is equal to the dimension of response function divided by the dimension of the unit load.作影响线有两种基本方法 Two basic methods：静力法（static method）, 机动法(或者称虚功法)（mechanismic or virtual displacement method）。6-2 用静力法

9、作单跨静定梁的影响线Constructing influence lines of statically determinate single span beams by static method静力法思路the procedure：利用静力求解方法（对静定结构用平衡条件）求结构在P=1移动荷载下所求某物理量与荷载P=1 的位置间的函数关系式，即影响系数方程，然后由方程作出影响线。Using static equilibrium equations to determine the functions of the desired parameters with the position o

10、f the unit loads, then construct influence lines by these functions. 1、确定坐标系，以坐标 x 表示荷载P=1的位置define the coordinate of the unit load by x,；2、对于给定的x，P=1 看成是固定的荷载，确定所求量的值即可 得影响系数方程For given x and unit load , determine the functions of the desired parameters with the position of the unit loads x, 3、按方程作

11、出影响线并标明符号和控制点的纵坐标值construct influence lines by these functions. 。注：正确的影响线应该具有正确的外形、必要的控制点纵坐标值和正负号（简称三要素）。内力正负号规定与前几章相同。 习惯上将纵标为正的影响 线画于基线上方，但因为要标正负号，所以这并不是规定。Particular attention should be paid to the form, ordinates of controlling points and the signs of the I.L. graphs ( 3 elements of I .L). 具体步骤为

12、procedure： 1、简支梁的影响线I.L.for simple beam（1） 支座反力影响线 The I.L. for reactionsP=1ABlRARB1I .L.RA1I .L.RBUsing equilibrium equations to determine the reaction force functions with the unit load position x, then construct IL by these functions.（2）内力影响线I.L.for internal forcesABlCabP=1RARBI.L.MCI.L.QCCharact

13、eristics 特点：Bending moment and shearing force ILs ABlCabP=1RA1I .L.RA1I .L.RBI.L.MCI.L.QCRB2.悬臂梁的影响线 I.L.for cantilever beamsP=1CABlabbI.L.MC1I.L.QC3、伸臂梁的影响线I.L.for overhanging beamsll1l2aba1b1ABCDEK1K2P=1RARB11I.L.MCI.L.QCI.L.RAI.L.RBll1l2aba1b1ABCDEK1K2P=1RARBI.L.MK1a1b111I.L.MK2I.L.QK1I.L.QK26-3

14、间接荷载作用下的影响线 I.L.for for girders with floor systems 主梁Girder横梁 Floor beam纵梁stringerP设：纵梁简支在横梁上，横梁简支在主梁上。If stringers lie on floor beam, and floor beams lie on girder, 主梁只在横梁处（结点处）受集中力作用。荷载loads纵梁stringer横梁floor beams主梁Girder对主梁而言，这种荷载称为间接荷载或结点荷载 panel loads。Panel pointsThe transformation of the loads

15、间接荷载作用下主梁某量值的影响线的绘制The I.L . For one desired quantity for girder due to panel loadsP=1以D截面弯矩MD为例 take the I.L.for MD on the section D：1、当P=1位于横梁（结点）处时，与直接荷载相同，既间接荷载作用下的影响线在结点处的值与直接荷载作用下的影响线在结点处的值相同。When P=1 locates at panel points, the ordinates of I.L. is identical to that of I.L. when P=1 is appli

16、ed directly to the girder.DP=1P=1直接荷载下影响线间接影响线值与直接影响线值相同2、当P=1在横梁（结点）之间时when P=1 is located within stringer containing the location of the response function：DP=1DP=1xdAB直接荷载下影响线yAyBI.L.MD间接荷载作用下AB段影响线结点（横梁）间影响线为直线，且在两端与直接影响线值相同。The ordinates between points are connected by straight lines, and the or

17、dinates at 2 ends are identical to that of the I.L. under the action of loads applied directly on girder. I.L.MD间接荷载作用下影响线绘制过程procedure for constructing I.L. due to panel-point loads：（1）作出直接荷载作用下的影响线Construct I.L. when P=1 is applied directly to the girder；（2）取各结点出的竖标，并在各竖标间连直线 Connect all the ordin

18、ates at panel points by straight lines。6-4 用机动法作单跨静定梁的影响线 Mechanismic method for constructing I.L.for statically determinate single-span beams(Muller-Breslau method)思路：利用虚位移原理将静力学问题转化为几何作图问题Using virtual displacement method the problem of construction of I.L. is transformed into geometric problem以简支

19、梁支座A的反力RA影响线为例 Take the construction of I.L. for reaction RA at support Aas an example.ABP=1ABP=1RAAP虚位移图由虚功原理：P 为P=1作用点沿P=1方向的虚位移。由于P=1是移动的，因此各点的P连线构成图形，称为（竖向）虚位移图。P represents the displacement of the point of application of P=1. The track of displacement ordinates constitute virtual displacement g

20、raph 令A=1，则 Ra= P ，此时虚位移图 P 便代表 Ra 的影响线。1+I.L .RA沿RA方向单位位移引起沿P=1方向的虚位移图机动法：1.将与量值X相应的联系拆除，并代之以相应的力X，得到释放约束结构 1. From a given structure release the restraint corresponding to the response function whose I.L. is desired, and impose a corresponding forces X to obtain the released structure. 2. Apply a

21、unit displacement in the direction X, construct deflected shape diagram of the released structure that is consistent with the support and continuity conditions of the released to obtain the general shape of the influence line.2.使体系沿X正方向产生单位位移所得荷载作用点的竖向位移图即代表X的影响线。注意：位移是单位力作用点处的位移Attention: The displ

22、acement is the displacement of the point where the unit load is applied to.机动法作影响线 I.L.by mechanismic method6-5 多跨静定梁的影响线Influence lines of statically determinate multi-span beams求Mk，QB左，及RF当F=1在量值本身所在段时等同于单跨梁；当F=1在对于量值本身所在段来讲为基本部分时量值为零；当F=1在对于量值本身所在段来讲为附属部分时量值影响线为直线。根据在铰处量值已知及支座处为零定出影响线。利用机动法求Mk, Q

23、B左,RF影响线.机动法作影响线I.L.by mechanismic method试作图示多跨梁的影响线。 机动法作影响线 I.L.by mechanismic method试作图示结点传荷主梁的影响线。6-6 桁架的影响线 I.L. for trusses只讨论单跨梁式桁架的影响线 Here only I.L.for single span girder trusses are discussed. 梁式单跨桁架的支座反力计算与相应单跨梁的反力计算相同，故两者反力影响线相同。The method for determination of reactions of single span gi

24、rder trusses is identical to the method of the reactions for single span beams, therefore the I.L for these 2 types of structures are identical. 主要讨论桁架杆件内力影响线 Here we will mainly discuss the I.L for the members of trusses.桁架承受的是结点荷载间接荷载 The loads are applied on the joints of truss, they can be treat

25、ed as panel point loads.桁架杆内力影响线与间接荷载作用下的影响线特征相同 在各结间为直线段。The characteristics of the I.L.for members of truss is identical to that of I.L. for panel point loads- the ordinates at all the points are connected by straight lines. 将P=1作用于各结点上，求出内力值，得结点处影响线值，再连以直线，即得所求内力影响线。Determine the ordinates at joi

26、nts, and then connect the ordinates at the points, we obtain the I.L. 注意attentions：1、区分P=1在上弦或下弦移动 It is necessary to identify the I.L. for unit load applied on top chords and on bottom chords. 2、对斜杆，可绘出内力的水平或竖向分力影响线for inclined members we may construct the I.L.for vertical or horizontal components

27、of the internal forces. 3、对于求内力截开的区间应该给予特别的关注, 截开节间两端的竖标需要连直线。 Particular attention should be paid to the cut panel，the ordinates at the ends of panel should be connected by straight line.桁架内力计算 结点法、截面法(力矩法、投影法)I.L. by joint method and section methodP=1abcdABRARB11ACDACFEDEF求F12、 F45、F15、F25的影响线（1）F

28、12的影响线用I-I面截开结构，以5为矩心列平衡方程即得F12的表达式（2）F45的影响线用I-I面截开结构，以1为矩心列平衡方程即得F45的表达式（3）F15的影响线用I-I面截开结构，以o为矩心列平衡方程即得F15的表达式（5）F4A的影响线以结点A为隔离体，列平衡方程即得F4A的表达式（4）F25的影响线用II-II面截开结构，列竖向平衡方程即得F25的表达式 单位荷载作用于上下弦所引起的指定量值影响线是不同的when the unit value load acts on upper and bottom chords the I L of the desired quantity a

29、re different.1.利用结点K求出2个 斜杆内力的关系。2.然后利用截面I-I截开结构，利用竖向平衡方程求出b杆的内力。3.c杆内力类似求出。4.最后利用结点3的平衡来求a杆的内力。6-7 利用影响线求量值Determine quantities by I.L.本节解决的问题是The problem to be solved：当若干个集中荷载或分布荷载作用于已知位置时，利用影响线来求某一量值。When the positions of several concentrated or distributed loads are given, determine the desired

30、quantities by I.L.1、集中荷载作用 For concentrated forces当荷载处于影响线为直线段时loads within straight segment of the I.L.2、均布荷载作用 for uniformly distributed loads-+6-8 最不利荷载位置The most unfavorable location of the loads 本节要点：为了进行设计，必须确定某量值的最大（最小）值。为此，必须先确定使该量值发生最大（最小）值时的荷载位置。位置一确定，就可用上节的方法求出最大（最小）值。 For the purpose of

31、design, it is necessary to determine the maximum of the desired quantity. For this end we must determine the load locations inducing the maximum of the quantity. Once the most unfavorable locations are determined, we can use the methods in the previous paragraph to determine maximum and minimum.定义：称

32、使某量值为最大（最小）值的荷载位置为该量值的最不利荷载位置。Definition : the critical positions on which the values of the desired quantity of a structure will reach to maximum values (positive or negative) are called the most unfavorable positions. 本节就是讨论如何利用影响线来确定最不利荷载位置 In this section we will use I.L. to determine the most u

33、nfavorable positions。1、单个集中荷载single concentrated force将单位载放在影响线最大竖标处产生最大量值。Place the load at the location of the maximum positive ordinate of I.L. to develop maximum positive value 将单位载放在影响线最小竖标处产生最小量值Place the load at the location of the negative maximum ordinate of I.L. to develop maximum negative

34、 value 。2、可任意断续分布的均布荷载 Arbitrary uniformly distributed loads荷载布满影响线所有正面积部分产生最大量值The load distributing over all the sections with positive ordinate will develop maximum positive value；荷载布满影响线所有负面积部分产生最小量值The load distributing over all the sections with negative ordinate will develop maximum negative

35、value ；3、一系列间距不变的移动荷载组（行列荷载）A series of moving concentrated loads with fixed distances between them或or（1）、当S为极大时，when S is maximum thenS0，则x 0（荷载左移when loads move slightly to the right ） （2）、当S为极小时when S is minimum then,S0,则x 0（荷载左移when loads move slightly to the right ） 即，当荷载左、右移动微小距离时，Ritgi 必须变号，才

36、能是极值I.e. when loads move slightly to the left and right, Ritgi changes its sign, then S reaches its extreme values.若荷载从左向右移动， Ritgi 由正变负，取极大。 Ritgi 由负变正，取极小。When loads move from left to right, Ritgi changes its sign from negative to positive. has its minimum value条件conditions：tgi 为常数，故只有Ri 变化， Ritgi

37、 才有可能变号。 tgi is constant, Only when the value of one of the resultants Ri changes its magnitude that has the possibility to make the sun Ritgi change its sign.只有当某一集中荷载恰好位于影响线的某一顶点时，才有可能。This is possible when one of the series of loads locates at the vertex point of the I.L.最不利荷载位置的大致估计The approxima

38、te identification of the most unfavorable position：（1）、荷载中数值较大且较密集的部分置于影响线的最大竖标附近；the part of loads with greater magnitudes and density should be placed on the vicinity of position with maximum ordinates（2).同符号的影响线范围内的荷载应尽可能多The diapason of I.L. with the same sign should be placed as more loads as p

39、ossible。通过试算，来确定最不利荷载位置By trials to determine the most unfavorable positions当某荷载位于影响线某顶点时，使Ritgi 变号，称该荷载为临界荷载，此时的荷载位置称为临界位置。The load which makes the sum Ritgi change its sign when it moves to the left or to the right of the position of vertex of the I.L. is termed as critical position of the loads.当

40、影响线为三角形时，临界荷载Pcr满足When the I.L. has the Triangular shape, critical load Pcr satisfies the following condition：（极大值）均布荷载作用时for uniformly distributed loads：Determine the most unfavorable location of the loads and absolute maximum value of ZExample ICExample II因为有无穷大斜率，需要试算6-9 简支梁的绝对最大弯矩The absolute ma

41、ximum moment定义：各截面的最大弯矩中的最大值称为绝对最大弯矩。The maximum moment among the maximum moments at every sections is termed as the absolute maximum moment.对一般的移动荷载，理论上应将各截面的最大弯矩求出，来获得绝对最大弯矩。Theoretically for general moving loads we should at first determine the maximum moments at every sections in order to determ

42、ine the absolute moment.实际中，截面无穷多，不可能这样作，因此可取有限个截面，来近似计算。But in reality we can not do so because of the infinity of the sections in structures. Therefore we may take several sections to do approximate calculation确定绝对最大弯矩要解决以下两个问题 2 problems need to be solved：（1）绝对最大弯矩发生在哪一截面；Identify the section at

43、which the absolute maximum moment is reached（2）此截面发生最大弯矩时的荷载位置The positions on which the absolute maximum moment is reached .对由集中荷载组成的移动荷载，问题可以简化,因为弯矩的最大值点必定是某个集中载的作用点：For a series of moving concentrated loads the problem may be simplified, because the section with maximum bending moment is one of t

44、he application points of loadsFK 作用点处的弯矩Mx为 The moment at the section of application of FK is 式中MK 表示FK以左梁上荷载对于FK点处的力矩总和 Where MK is the sum of moments induced by all the loads in the left of FK about the application point of FK支座反力为The reaction at support A is 极值条件为the condition for extreme value i

45、s 即当FK 与合力FR的位置对称于梁的中点时FK之下截面的弯矩达到最大值 I.e. when FK and FR are symmetrical with respect to the middle point of the beam the moment at the application section of FK reaches its maximum value当FR 位于FK 的左边时 When FR is located at the left of FK we have 在荷载多时，上述方法非常烦杂。When there are many loads, the previou

46、s method is very complicated. 可以利用这样的事实：通常简支粱的绝对最大弯矩发生在粱的中点附近。But we can use the following fact: generally the maximum bending moment arises at the middle of the beam. 使粱中点截面发生最大弯矩的临界荷载通常也是发生最大弯矩的荷载。利用这一事实可得下列简化方法。 The critical loads that induce maximum bending moment at the middle of beam are loads

47、 that result in maximum bending moment. Using this fact we can obtain the following simplified method.计算最大弯矩的基本方法(basic method for determination of maximum bending moment1、首先确定使粱的中点发生最大弯矩的临界荷载3、计算 作用处的弯矩，即为绝对最大弯矩。2、计算合力 ，把 与 对称于梁的中点布置，这时要注意是否有力移出粱跨。如果有力移出粱跨，需要重新计算合力，重复上述步骤.At first determine the loa

48、d that induces the maximum moment at the middle of the beam, and calculate the Calculate FR, and place FR and Fk symmetrically with respect to the middle of the beam, in this case need pay attention to whether there are loads outside the span, if there are, then calculate FR again, and repeat the ab

49、ove mentioned procedureCalculate the moment at the position of we obtain the maximum bending moment并计算此时粱中点的最大弯矩Determine the absolute maximum bending moment of the simple beam under the given loads and compare it with the maximum bending moment at the middle of the beamDetermine the maximum bending moment at the middle of the beamDetermine the absolute maximum bending moment6-10 简支梁的内力包络图 The envelopes of internal forces of simple beam定义： 联结各截面最大、最小内力的图