位移法8-1、8-2___概述、等截面直桿的轉角位移方程

1、第第八八章章 位移法位移法Chapter 8 Displacement methodSection 1 IntroductionSection 1 IntroductionStructural MechanicsReviewA statically determinate structureHas the minimum number of reactions and internal forces required to produce a stable system of members that can be solved by the equations of static equil

2、ibrium alone.A statically indeterminate structureBy adding extra reactions to a determinate structure, an indeterminate structure is created.The reactions and internal forces cannot be determined from the equations of static equilibrium alone. The number of extra reactions and internal forces is the

3、 degree of statical indeterminacy（超靜定次數）; these extra forces are called redundants（多余未知力）.ReviewForce MethodAnalysis by the Force Method involves:1. Choosing the necessary number, locations, and directions of the redundants.2. Releasing the redundants to produce a stable and determinate structure re

4、ferred to as the primary structure（基本結構）（基本結構）.3. Calculating deformations on the primary structure at and in the directions of the redundants （在多余未知力作用處沿著多余未知力方向的位移）（在多余未知力作用處沿著多余未知力方向的位移）due to external loads and each redundant.4. Using superposition to formulate a set of linear algebraic equation

5、s. Each equation is based on the deformation compatibility（位移條件）（位移條件） of the original structure at a particular redundant location in the primary structure.5. Solving the deformation compatibility equations for the redundants.6. Using the computed redundants, to determine all reactions and internal

6、 forces of the member.8.1 Introduction of Displacement MethodDisadvantages of Force Method用位移法計算卻只有1個基本未知量 In the case of linear elastic condition, the relationship between the force and displacement is linear, and we can take displacement as the primary unknownsdisplacement method. In the displacem

7、ent method,the unkowns to be solved are joint displacements(rotations and translations).缺點：缺點：對于高次超靜定結構的求解，力法的計算工作量大而不便應用。用力法計算有8個基本未知量 優(yōu)點：優(yōu)點：對于高次超靜定結構的求解，宜采用位移法求解。注意：注意：力法是以多余未知力作為基本未知量，位移法以結點位移作為力法是以多余未知力作為基本未知量，位移法以結點位移作為基本未知量，這是兩者的基本區(qū)別之一?；疚粗浚@是兩者的基本區(qū)別之一。Consider a plane frame loaded by a force

8、 and the deflected shape is shown as the dashed line. Ignoring the axial deformation, we can discompose the structure as shown figure b、c。將結點將結點1的角位移的角位移Z1 作作為基本未知量，求出為基本未知量，求出Z1，進而求出各桿內力。，進而求出各桿內力。需解決的問題需解決的問題：（1）用力法算出單跨超靜定梁在各種外因作用）用力法算出單跨超靜定梁在各種外因作用下的內力下的內力 （2）確定哪些位移作為基本未知量）確定哪些位移作為基本未知量 （3）如何求出這些

9、位移）如何求出這些位移思考：思考：8.1 Introduction of Displacement MethodThe end moments of member AB、BC areBasic Concepts of Displacement Method=+BMBAMBC8.1 Introduction of Displacement MethodThe equilibrium condition of joint B is 結點結點B B的平衡方程為：的平衡方程為：8.1 Introduction of Displacement Method For this purpose, it is

10、 necessary at first為此需要求出為此需要求出： 1）To express the member end foces in terms of external loads and member end displacements;單跨超靜定梁在荷載及桿端單跨超靜定梁在荷載及桿端位移作用下的內力表達式；位移作用下的內力表達式； 2）To determine the primary unknowns. 確定取何種位移為基確定取何種位移為基本的未知量。本的未知量。1Discrete the structure into several single-span statically i

11、ndeterminate beams, express the member end forces in terms of external loads and member end displacements;把荷載在可動結點拆為相應的單跨超靜定梁，列出各桿把荷載在可動結點拆為相應的單跨超靜定梁，列出各桿在荷載及桿端位移作用下桿端的內力表達式；在荷載及桿端位移作用下桿端的內力表達式；8.1 Introduction of Displacement MethodThe fundamental idea of displacement method位移法的基本思想：位移法的基本思想： 2Rest

12、ore the original structure, develop equilibrium equations, solve the equilibrium equations and determine the displacements. 恢復原結構，建立平衡方程，求解位移。恢復原結構，建立平衡方程，求解位移。Keynotes of Displacement Method位移法解題要點：位移法解題要點： （1） the primary unkowns to be solved are joint displacements(rotations and translations) ；（3

13、） equilibrium equation is based on the Force equilibrium of the joints；（2）the primary structure- single-span statically indeterminate beams ；8.1 Introduction of Displacement Method位移法應用的前提條件位移法應用的前提條件1. 桿端力與桿端位移的關系需已知桿端力與桿端位移的關系需已知位移法的基礎是以單根桿件的分析，桿端力與桿端位移以及荷載的關系（稱為轉角位移方程）需要事先導出，桿件分析桿件分析是結構分析的基礎。是結構分

14、析的基礎。詳見下節(jié)。2. 先滿足變形協(xié)調條件先滿足變形協(xié)調條件在力法中，先滿足平衡條件然后再滿足變形協(xié)調條件。而位移法不同在于，位移法在選擇基本未知量的同時就需要滿足變形協(xié)調條件，然后由平衡條件將基本未知量求出。3. 結構變形的假定結構變形的假定（1）結構的變形是微小的；（2）受彎桿忽略軸向變形。8.1 Introduction of Displacement MethodISeveral definitions（1）3 types of single-span indeterminate prismatic beams三種類型的等截面直桿超靜定梁三種類型的等截面直桿超靜定梁(a) beam

15、with 2 (b) with one fixed (c) with one fixed support fixed supports and one hinged and one double-link support兩端固定兩端固定 一端固定一端簡支一端固定一端簡支 一端固定一端定向支座一端固定一端定向支座8.2 The Slope-deflection equation of single-span indeterminate prismatic beams 等截面直桿的轉角位移方程等截面直桿的轉角位移方程（2 2） Sign convention符號規(guī)定：符號規(guī)定： The membe

16、r end moments M, end rotations , and chord rotation are posotive when clockwise; the shears are considered to be positive when they tend to make the member rotate clockwise. 對于桿端來講對于桿端來講桿端彎矩桿端彎矩 M , , 及兩端連線轉角及兩端連線轉角 以以順時針方向為正；而對于支座及結點來講逆時針方向為正。桿順時針方向為正；而對于支座及結點來講逆時針方向為正。桿端端剪力剪力以使整個桿順時針方向轉動為正。以使整個桿順時

17、針方向轉動為正。BA,BA,lABlABII. The determination of the fixed end forces under external loads 在外載作用下固端力的確定在外載作用下固端力的確定 abEIPaEIPaEIlEIlEIlXXXXPPPP236;23;2;002221322221121122221211212111023632022223122221abPaXlXlPaXllXallPbFlPabMbllPaFlbPaMbllPaXlbPaXSBAABSBABA2;2;2;32223222322221 The fixed end forces under

18、 other external loads are determined Similarly. The calculation results are tabulated in the textbook.在其它外載作用下固端力的求法類似。書中有列表。在其它外載作用下固端力的求法類似。書中有列表。Internal force diagramIII. Member end forces of various single-span indeterminate prismatic beams due to their displacements. 在桿端位移作用下在桿端位移作用下的桿端力的確定的桿端

19、力的確定 ;13;2;0021322221121122221211212111AAAAllcRcREIlEIlEIlXXXX(1). Member end forces duo to end rotation由桿端角位移由桿端角位移引起的桿端力引起的桿端力 AABQBAAABABAABQBAAABBAAABAAABAAAliFiMliFiMlilELXilELXlXEIlXEIlXEIlXEIl6;46;266;22032022212312221 Diagrams of bending moment and shearing force(2). Member end forces duo to

20、 relative translation of ends 在桿端相對位移作用下的桿端力在桿端相對位移作用下的桿端力The chord rotation is （弦轉角為）（弦轉角為） lABThe final elastic curve is obtained by 2 steps. At first we rotate the beam as a whole clockwise, and then rotate the 2 ends the angle anticlockwise. In this way，we have最終的彈性曲線由下列兩步完成。首先把體系整體旋轉最終的彈性曲線由下列兩

21、步完成。首先把體系整體旋轉 ，然，然后把桿端后把桿端A A及及B B端反時針旋轉端反時針旋轉 。這樣得到。這樣得到ABBAABABAiMiM2;4,ABBAABABBiMiM4;2,SBAABABBAABSABBAABABABFlilMMFMliiM1266when whenThereforeWe have IV. Member end forces of single-span indeterminate prismatic beams with one fixed support and one double-link support due to their displacements.

22、在桿端位移作用下在桿端位移作用下一端固定一端定向支一端固定一端定向支座的桿的桿端力的確定座的桿的桿端力的確定 BAAABABABABMiiiM2224V. Member end forces of single-span indeterminate prismatic beams with one fixed and one hinged due to their displacements. 在桿端位移作用下在桿端位移作用下一端固定一端鉸支桿的桿端力的確定一端固定一端鉸支桿的桿端力的確定 AABABABABiiiM32224232221212322266lililiFlililiMABABA

23、BAAAASABABABAAABVII. Slope-deflection equation轉角位移方程轉角位移方程 The slope-deflection equation relates the moments at the ends of a member to the rotation and displacements of its ends and the external loads applied to the members.轉角位移方程轉角位移方程把桿端彎矩與桿端轉角及位移聯系了起來把桿端彎矩與桿端轉角及位移聯系了起來。 圖圖a所示兩端固定的等截面梁，所示兩端固定的等截面

24、梁，兩端支座發(fā)生了位移。取基本結構如兩端支座發(fā)生了位移。取基本結構如圖圖b。 X3對梁的彎矩無影響，可不考慮，對梁的彎矩無影響，可不考慮，只需求解只需求解X1、X2。符號規(guī)定：桿端彎矩以對桿端順時針方向為正；符號規(guī)定：桿端彎矩以對桿端順時針方向為正； 均以順時針方向為正；均以順時針方向為正； AB 以使整個桿件順時針方向轉動為正。以使整個桿件順時針方向轉動為正。BA、力法典型方程為力法典型方程為BAXXXX222212112121118.2 The Slope-deflection equation of single-span indeterminate prismatic beams 等截

25、面直桿的轉角位移方程等截面直桿的轉角位移方程作作X1、X2分別等于分別等于1時的彎矩圖如圖時的彎矩圖如圖c、d。EIlEIlEIl63,321122211由圖由圖e可得可得lABAB21AB弦轉角，順時針方向為正。弦轉角，順時針方向為正。解典型方程得解典型方程得ABABABBAlEIlEIlEIXlEIlEIlEIX22216246248.2 The Slope-deflection equation of single-span indeterminate prismatic beams 等截面直桿的轉角位移方程等截面直桿的轉角位移方程令令桿件的線剛度桿件的線剛度lEIi MAB=X1，MB

26、A=X2，可得，可得固端彎矩固端彎矩 ：單跨梁在荷載作用及溫度變化時產生的：單跨梁在荷載作用及溫度變化時產生的 桿端彎矩。桿端彎矩。FFBAABMM、ABABBAABBAABliiiMliiiM624624 當單跨梁除支座位移外，還有荷載作用及溫度變化時，當單跨梁除支座位移外，還有荷載作用及溫度變化時，其桿端彎矩為其桿端彎矩為FBAABABBAFABABBAABMliiiMMliiiM624624轉角位移方程轉角位移方程8.2 The Slope-deflection equation of single-span indeterminate prismatic beams 等截面直桿的轉角位

27、移方程等截面直桿的轉角位移方程對于一端固定另一端鉸支的等截面梁，設對于一端固定另一端鉸支的等截面梁，設B端為鉸支，則有端為鉸支，則有0624FBAABABBAMliiiM)213(21FBAABABMilB不是獨立的不是獨立的FFFF2133BAABABABABAABMMMMliiM桿端彎矩桿端彎矩桿端剪力桿端剪力8.2 The Slope-deflection equation of single-span indeterminate prismatic beams 等截面直桿的轉角位移方程等截面直桿的轉角位移方程對于一端固定另一端鉸支的等截面梁，設對于一端固定另一端鉸支的等截面梁，設B端為

28、鉸支，則有端為鉸支，則有0624FBAABABBAMliiiMF33ABABAABMliiM8.2 等截面直桿的轉角位移方程等截面直桿的轉角位移方程轉角位移方程轉角位移方程BFQBAMABAABFQABFQABMABAABMBAFQBAFBAABAFABAABMLEIMMLEIM8.2 等截面直桿的轉角位移方程等截面直桿的轉角位移方程對于一端固定另一端滑動的等截面梁的轉角位移方程對于一端固定另一端滑動的等截面梁的轉角位移方程轉角位移方程轉角位移方程FQABMABAABMBAFQBAFBAABAFABAABMLEIMMLEIM對于一端固定另一端滑動的等截面梁的轉角位移方程對于一端固定另一端滑動的等截面梁的轉角位移方程轉角位移方程轉角位移方程4422ABAABAAAEIMiLEIMiL由力法求得由力法求得由力法求得由力法求得4422BABBABBBEIMiLEIMiL1.1.兩端固定單元，在兩端固定單元，在A A端發(fā)生一個順時針的轉角端發(fā)生一個順時針的轉角 。AAABMABMBA2.2.