影響線及其應用PPT（雙語98頁）

1、第 十 一 章 Chapter 11影響線及其應用Influence lines and their application本章主要內容Main contents1、影響線概念 The concept of influence lines2、靜力法繪制影響線 Static method 3、機動法繪制影響線 mechanismic method 4、影響線應用 Application of influence lines（1）利用影響線求固定荷載作用下的內力Determine internal forces under fixed loads（2）確定最不利荷載位置 Determine the

2、 unfavorable locations of loads（3）簡支梁的絕對最大彎矩Determine the maximum bending moment（4）簡支梁的包絡圖 Draw envelopes 6-1 影響線的概念 The concept of influence lines1、移動荷載 Moving loads移動荷載:方向、大小不變，僅作用位置變化的荷載 direction and magnitude dont change, only positions change (moving load)如橋梁上行駛的車輛，工業(yè)廠房中吊車梁上開動的行車等, for instanc

3、e the moving cars on bridges, the cranes in factories。 結構承受移動荷載作用時，其反力、內力以及位移等均隨荷載作用位置的變化而改變。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 為解決千變萬化的移動荷載作用下結

4、構的設計問題，基于線彈性結構的疊加原理，如果掌握了單位移動荷載下結構的反力、內力與位移變化規(guī)律，則任意移動荷載下的反力、內力與位移可用疊加方法獲得。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 定義Definition: 單位移動荷載作用下，結構反力、內力或位移等隨荷載位置變化的函數(shù)關系，分別稱為反力、內力、位移的影響系數(shù)方程，對應的函數(shù)圖形分別稱為反力、內力、位移的影響線（Influence Line 縮寫為 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. 為了方便，以下把反力、內力和位移均稱為量值或物理量。For the convenie

7、nce the reactions , internal forces and displacements of structures called parameters 注意：定義中“單位移動荷載”下的量值稱為影響系數(shù)，可以理解為：在單個移動荷載作用下某指定物理量與荷載P 的比值，習慣上以在結構上移動的P=1（即量綱一或無量綱）表示“單位移動荷載”。因此，物理量影響系數(shù)的量綱是，物理量的量綱和移動荷載量綱之比。例如單位移動荷載P是集中力，則彎矩、反力偶影響線的量綱為L(長度)，剪力、反力影響線的量綱為(或無量綱)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.作影響線有兩種基本方法 Two basic methods：靜力法（static method）, 機動法(或者稱虛功法)（mechanismic or virtual displacement method）。6-2 用靜力法

9、作單跨靜定梁的影響線Constructing influence lines of statically determinate single span beams by static method靜力法思路the procedure：利用靜力求解方法（對靜定結構用平衡條件）求結構在P=1移動荷載下所求某物理量與荷載P=1 的位置間的函數(shù)關系式，即影響系數(shù)方程，然后由方程作出影響線。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、確定坐標系，以坐標 x 表示荷載P=1的位置define the coordinate of the unit load by x,；2、對于給定的x，P=1 看成是固定的荷載，確定所求量的值即可 得影響系數(shù)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、出影響線并標明符號和控制點的縱坐標值construct influence lines by these functions. 。注：正確的影響線應該具有正確的外形、必要的控制點縱坐標值和正負號（簡稱三要素）。內力正負號規(guī)定與前幾章相同。 習慣上將縱標為正的影響 線畫于基線上方，但因為要標正負號，所以這并不是規(guī)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). 具體步驟為

12、procedure： 1、簡支梁的影響線I.L.for simple beam（1） 支座反力影響線 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）內力影響線I.L.for internal forcesABlCabP=1RARBI.L.MCI.L.QCCharact

13、eristics 特點：Bending moment and shearing force ILs ABlCabP=1RA1I .L.RA1I .L.RBI.L.MCI.L.QCRB2.懸臂梁的影響線 I.L.for cantilever beamsP=1CABlabbI.L.MC1I.L.QC3、伸臂梁的影響線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、間接荷載作用下的影響線 I.L.for for girders with floor systems 主梁Girder橫梁 Floor beam縱梁stringerP設：縱梁簡支在橫梁上，橫梁簡支在主梁上。If stringers lie on floor beam, and floor beams lie on girder, 主梁只在橫梁處（結點處）受集中力作用。荷載loads縱梁stringer橫梁floor beams主梁Girder對主梁而言，這種荷載稱為間接荷載或結點荷載 panel loads。Panel pointsThe transformation of the loads

15、間接荷載作用下主梁某量值的影響線的繪制The I.L . For one desired quantity for girder due to panel loadsP=1以D截面彎矩MD為例 take the I.L.for MD on the section D：1、當P=1位于橫梁（結點）處時，與直接荷載相同，既間接荷載作用下的影響線在結點處的值與直接荷載作用下的影響線在結點處的值相同。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直接荷載下影響線間接影響線值與直接影響線值相同2、當P=1在橫梁（結點）之間時when P=1 is located within stringer containing the location of the response function：DP=1DP=1xdAB直接荷載下影響線yAyBI.L.MD間接荷載作用下AB段影響線結點（橫梁）間影響線為直線，且在兩端與直接影響線值相同。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間接荷載作用下影響線繪制過程procedure for constructing I.L. due to panel-point loads：（1）作出直接荷載作用下的影響線Construct I.L. when P=1 is applied directly to the girder；（2）取各結點出的豎標，并在各豎標間連直線 Connect all the ordin

18、ates at panel points by straight lines。6-4 用機動法作單跨靜定梁的影響線 Mechanismic method for constructing I.L.for statically determinate single-span beams(Muller-Breslau method)思路：利用虛位移原理將靜力學問題轉化為幾何作圖問題Using virtual displacement method the problem of construction of I.L. is transformed into geometric problem以簡支

19、梁支座A的反力RA影響線為例 Take the construction of I.L. for reaction RA at support Aas an example.ABP=1ABP=1RAAP虛位移圖由虛功原理：P 為P=1作用點沿P=1方向的虛位移。由于P=1是移動的，因此各點的P連線構成圖形，稱為（豎向）虛位移圖。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，則 Ra= P ，此時虛位移圖 P 便代表 Ra 的影響線。1+I.L .RA沿RA方向單位位移引起沿P=1方向的虛位移圖機動法：1.將與量值X相應的聯(lián)系拆除，并代之以相應的力X，得到釋放約束結構 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.使體系沿X正方向產生單位位移所得荷載作用點的豎向位移圖即代表X的影響線。注意：位移是單位力作用點處的位移Attention: The displ

22、acement is the displacement of the point where the unit load is applied to.機動法作影響線 I.L.by mechanismic method6-5 多跨靜定梁的影響線Influence lines of statically determinate multi-span beams求Mk，QB左，及RF當F=1在量值本身所在段時等同于單跨梁；當F=1在對于量值本身所在段來講為基本部分時量值為零；當F=1在對于量值本身所在段來講為附屬部分時量值影響線為直線。根據(jù)在鉸處量值已知及支座處為零定出影響線。利用機動法求Mk, Q

23、B左,RF影響線.機動法作影響線I.L.by mechanismic method試作圖示多跨梁的影響線。 機動法作影響線 I.L.by mechanismic method試作圖示結點傳荷主梁的影響線。6-6 桁架的影響線 I.L. for trusses只討論單跨梁式桁架的影響線 Here only I.L.for single span girder trusses are discussed. 梁式單跨桁架的支座反力計算與相應單跨梁的反力計算相同，故兩者反力影響線相同。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. 主要討論桁架桿件內力影響線 Here we will mainly discuss the I.L for the members of trusses.桁架承受的是結點荷載間接荷載 The loads are applied on the joints of truss, they can be treat

25、ed as panel point loads.桁架桿內力影響線與間接荷載作用下的影響線特征相同 在各結間為直線段。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. 將P=1作用于各結點上，求出內力值，得結點處影響線值，再連以直線，即得所求內力影響線。Determine the ordinates at joi

26、nts, and then connect the ordinates at the points, we obtain the I.L. 注意attentions：1、區(qū)分P=1在上弦或下弦移動 It is necessary to identify the I.L. for unit load applied on top chords and on bottom chords. 2、對斜桿，可繪出內力的水平或豎向分力影響線for inclined members we may construct the I.L.for vertical or horizontal components

27、of the internal forces. 3、對于求內力截開的區(qū)間應該給予特別的關注, 截開節(jié)間兩端的豎標需要連直線。 Particular attention should be paid to the cut panel，the ordinates at the ends of panel should be connected by straight line.桁架內力計算 結點法、截面法(力矩法、投影法)I.L. by joint method and section methodP=1abcdABRARB11ACDACFEDEF求F12、 F45、F15、F25的影響線（1）F

28、12的影響線用I-I面截開結構，以5為矩心列平衡方程即得F12的表達式（2）F45的影響線用I-I面截開結構，以1為矩心列平衡方程即得F45的表達式（3）F15的影響線用I-I面截開結構，以o為矩心列平衡方程即得F15的表達式（5）F4A的影響線以結點A為隔離體，列平衡方程即得F4A的表達式（4）F25的影響線用II-II面截開結構，列豎向平衡方程即得F25的表達式 單位荷載作用于上下弦所引起的指定量值影響線是不同的when the unit value load acts on upper and bottom chords the I L of the desired quantity a

29、re different.1.利用結點K求出2個 斜桿內力的關系。2.然后利用截面I-I截開結構，利用豎向平衡方程求出b桿的內力。3.c桿內力類似求出。4.最后利用結點3的平衡來求a桿的內力。6-7 利用影響線求量值Determine quantities by I.L.本節(jié)解決的問題是The problem to be solved：當若干個集中荷載或分布荷載作用于已知位置時，利用影響線來求某一量值。When the positions of several concentrated or distributed loads are given, determine the desired

30、quantities by I.L.1、集中荷載作用 For concentrated forces當荷載處于影響線為直線段時loads within straight segment of the I.L.2、均布荷載作用 for uniformly distributed loads-+6-8 最不利荷載位置The most unfavorable location of the loads 本節(jié)要點：為了進行設計，必須確定某量值的最大（最小）值。為此，必須先確定使該量值發(fā)生最大（最小）值時的荷載位置。位置一確定，就可用上節(jié)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.定義：稱

32、使某量值為最大（最?。┲档暮奢d位置為該量值的最不利荷載位置。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. 本節(jié)就是討論如何利用影響線來確定最不利荷載位置 In this section we will use I.L. to determine the most u

33、nfavorable positions。1、單個集中荷載single concentrated force將單位載放在影響線最大豎標處產生最大量值。Place the load at the location of the maximum positive ordinate of I.L. to develop maximum positive value 將單位載放在影響線最小豎標處產生最小量值Place the load at the location of the negative maximum ordinate of I.L. to develop maximum negative

34、 value 。2、可任意斷續(xù)分布的均布荷載 Arbitrary uniformly distributed loads荷載布滿影響線所有正面積部分產生最大量值The load distributing over all the sections with positive ordinate will develop maximum positive value；荷載布滿影響線所有負面積部分產生最小量值The load distributing over all the sections with negative ordinate will develop maximum negative

35、value ；3、一系列間距不變的移動荷載組（行列荷載）A series of moving concentrated loads with fixed distances between them或or（1）、當S為極大時，when S is maximum thenS0，則x 0（荷載左移when loads move slightly to the right ） （2）、當S為極小時when S is minimum then,S0,則x 0（荷載左移when loads move slightly to the right ） 即，當荷載左、右移動微小距離時，Ritgi 必須變號，才

36、能是極值I.e. when loads move slightly to the left and right, Ritgi changes its sign, then S reaches its extreme values.若荷載從左向右移動， Ritgi 由正變負，取極大。 Ritgi 由負變正，取極小。When loads move from left to right, Ritgi changes its sign from negative to positive. has its minimum value條件conditions：tgi 為常數(shù)，故只有Ri 變化， Ritgi

37、 才有可能變號。 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.只有當某一集中荷載恰好位于影響線的某一頂點時，才有可能。This is possible when one of the series of loads locates at the vertex point of the I.L.最不利荷載位置的大致估計The approxima

38、te identification of the most unfavorable position：（1）、荷載中數(shù)值較大且較密集的部分置于影響線的最大豎標附近；the part of loads with greater magnitudes and density should be placed on the vicinity of position with maximum ordinates（2).同符號的影響線范圍內的荷載應盡可能多The diapason of I.L. with the same sign should be placed as more loads as p

39、ossible。通過試算，來確定最不利荷載位置By trials to determine the most unfavorable positions當某荷載位于影響線某頂點時，使Ritgi 變號，稱該荷載為臨界荷載，此時的荷載位置稱為臨界位置。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.當

40、影響線為三角形時，臨界荷載Pcr滿足When the I.L. has the Triangular shape, critical load Pcr satisfies the following condition：（極大值）均布荷載作用時for uniformly distributed loads：Determine the most unfavorable location of the loads and absolute maximum value of ZExample ICExample II因為有無窮大斜率，需要試算6-9 簡支梁的絕對最大彎矩The absolute ma

41、ximum moment定義：各截面的最大彎矩中的最大值稱為絕對最大彎矩。The maximum moment among the maximum moments at every sections is termed as the absolute maximum moment.對一般的移動荷載，理論上應將各截面的最大彎矩求出，來獲得絕對最大彎矩。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.實際中，截面無窮多，不可能這樣作，因此可取有限個截面，來近似計算。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確定絕對最大彎矩要解決以下兩個問題 2 problems need to be solved：（1）絕對最大彎矩發(fā)生在哪一截面；Identify the section at

43、which the absolute maximum moment is reached（2）此截面發(fā)生最大彎矩時的荷載位置The positions on which the absolute maximum moment is reached .對由集中荷載組成的移動荷載，問題可以簡化,因為彎矩的最大值點必定是某個集中載的作用點：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 作用點處的彎矩Mx為 The moment at the section of application of FK is 式中MK 表示FK以左梁上荷載對于FK點處的力矩總和 Where MK is the sum of moments induced by all the loads in the left of FK about the application point of FK支座反力為The reaction at support A is 極值條件為the condition for extreme value i

45、s 即當FK 與合力FR的位置對稱于梁的中點時FK之下截面的彎矩達到最大值 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當FR 位于FK 的左邊時 When FR is located at the left of FK we have 在荷載多時，上述方法非常煩雜。When there are many loads, the previou

46、s method is very complicated. 可以利用這樣的事實：通常簡支粱的絕對最大彎矩發(fā)生在粱的中點附近。But we can use the following fact: generally the maximum bending moment arises at the middle of the beam. 使粱中點截面發(fā)生最大彎矩的臨界荷載通常也是發(fā)生最大彎矩的荷載。利用這一事實可得下列簡化方法。 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.計算最大彎矩的基本方法(basic method for determination of maximum bending moment1、首先確定使粱的中點發(fā)生最大彎矩的臨界荷載3、計算 作用處的彎矩，即為絕對最大彎矩。2、計算合力 ，把 與 對稱于梁的中點布置，這時要注意是否有力移出粱跨。如果有力移出粱跨，需要重新計算合力，重復上述步驟.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并計算此時粱中點的最大彎矩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 簡支梁的內力包絡圖 The envelopes of internal forces of simple beam定義： 聯(lián)結各截面最大、最小內力的圖