位移法8-1、8-2___概述、等截面直杆的转角位移方程

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（超静定次数）; these extra forces are called redundants（多余未知力）.ReviewForce MethodAnalysis by the Force Method involves:Choosing the necessary number, locations, and directions of the redundants.Releasing the redundants to produce a stable and determinate structure referred

4、 to as the primary structure（基本结构）（基本结构）.Calculating deformations on the primary structure at and in the directions of the redundants （在多余未知力作用处沿着多余未知力方向的位移）（在多余未知力作用处沿着多余未知力方向的位移）due to external loads and each redundant.Using superposition to formulate a set of linear algebraic equations. Each equa

5、tion is based on the deformation compatibility（位移条件）（位移条件） of the original structure at a particular redundant location in the primary structure.Solving the deformation compatibility equations for the redundants.Using the computed redundants, to determine all reactions and internal forces of the mem

6、ber.8.1 Introduction of Displacement MethodDisadvantages of Force Method用位移法计算却只有1个基本未知量 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 displacement method,the unk

7、owns to be solved are joint displacements(rotations and translations).缺点：缺点：对于高次超静定结构的求解，力法的计算工作量大而不便应用。用力法计算有8个基本未知量 优点：优点：对于高次超静定结构的求解，宜采用位移法求解。力法是以多余未知力作为基本未知量，位移法以结点位移作为力法是以多余未知力作为基本未知量，位移法以结点位移作为基本未知量，这是两者的基本区别之一。基本未知量，这是两者的基本区别之一。Consider a plane frame loaded by a force and the deflected shape

8、 is shown as the dashed line. Ignoring the axial deformation, we can discompose the structure as shown figure b、c。将结点将结点1的角位移的角位移Z1 作作为基本未知量，求出为基本未知量，求出Z1，进而求出各杆内力。，进而求出各杆内力。需解决的问题需解决的问题：（1）用力法算出单跨超静定梁在各种外因作用）用力法算出单跨超静定梁在各种外因作用下的内力下的内力 （2）确定哪些位移作为基本未知量）确定哪些位移作为基本未知量 （3）如何求出这些位移）如何求出这些位移8.1 Introduct

9、ion 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 结点结点B B的平衡方程为：的平衡方程为：8.1 Introduction of Displacement Method For this purpose, it is necessary at first为此需要求出为此需要求

10、出： 1）To express the member end foces in terms of external loads and member end displacements;单跨超静定梁在荷载及杆端单跨超静定梁在荷载及杆端位移作用下的内力表达式；位移作用下的内力表达式； 2）To determine the primary unknowns. 确定取何种位移为基确定取何种位移为基本的未知量。本的未知量。1Discrete the structure into several single-span statically indeterminate beams, express th

11、e member end forces in terms of external loads and member end displacements;把荷载在可动结点拆为相应的单跨超静定梁，列出各杆把荷载在可动结点拆为相应的单跨超静定梁，列出各杆在荷载及杆端位移作用下杆端的内力表达式；在荷载及杆端位移作用下杆端的内力表达式；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： 2Restore the original structure, de

12、velop equilibrium equations, solve the equilibrium equations and determine the displacements. 恢复原结构，建立平衡方程，求解位移。恢复原结构，建立平衡方程，求解位移。Keynotes of Displacement Method位移法解题要点：位移法解题要点： （1） the primary unkowns to be solved are joint displacements(rotations and translations) ；（3） equilibrium equation is base

13、d on the Force equilibrium of the joints；（2）the primary structure- single-span statically indeterminate beams ；8.1 Introduction of Displacement Method位移法应用的前提条件位移法应用的前提条件1. 杆端力与杆端位移的关系需已知杆端力与杆端位移的关系需已知位移法的基础是以单根杆件的分析，杆端力与杆端位移以及荷载的关系（称为转角位移方程）需要事先导出，杆件分析杆件分析是结构分析的基础。是结构分析的基础。详见下节。2. 先满足变形协调条件先满足变形协调条

14、件在力法中，先满足平衡条件然后再满足变形协调条件。而位移法不同在于，位移法在选择基本未知量的同时就需要满足变形协调条件，然后由平衡条件将基本未知量求出。3. 结构变形的假定结构变形的假定（1）结构的变形是微小的；（2）受弯杆忽略轴向变形。8.1 Introduction of Displacement MethodISeveral definitions（1）3 types of single-span indeterminate prismatic beams三种类型的等截面直杆超静定梁三种类型的等截面直杆超静定梁beam with 2 (b) with one fixed (c) with

15、 one fixed support fixed supports and one hinged and one double-link support两端固定两端固定 一端固定一端简支一端固定一端简支 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 等截面直杆的转角位移方程等截面直杆的转角位移方程（2 2） Sign convention符号规定：符号规定： The member end moments M, end rotations , a

16、nd chord rotation are posotive when clockwise; the shears are considered to be positive when they tend to make the member rotate clockwise. 对于杆端来讲对于杆端来讲杆端弯矩杆端弯矩 M , , 及两端连线转角及两端连线转角 以以顺时针方向为正；而对于支座及结点来讲逆时针方向为正。杆顺时针方向为正；而对于支座及结点来讲逆时针方向为正。杆端端剪力剪力以使整个杆顺时针方向转动为正。以使整个杆顺时针方向转动为正。BA,BA,lABlABII. The determ

17、ination of the fixed end forces under external loads 在外载作用下固端力的确定在外载作用下固端力的确定 abEIPaEIPaEIlEIlEIlXXXXPPPP236;23;2;002221322221121122221211212111023632022223122221abPaXlXlPaXllXallPbFlPabMbllPaFlbPaMbllPaXlbPaXSBAABSBABA2;2;2;32223222322221 The fixed end forces under other external loads are determin

18、ed Similarly. The calculation results are tabulated in the textbook.在其它外载作用下固端力的求法类似。书中有列表。在其它外载作用下固端力的求法类似。书中有列表。Internal force diagramIII. Member end forces of various single-span indeterminate prismatic beams due to their displacements. 在杆端位移作用下在杆端位移作用下的杆端力的确定的杆端力的确定 ;13;2;00213222211211222212112

19、12111AAAAllcRcREIlEIlEIlXXXX(1). Member end forces duo to end rotation由杆端角位移由杆端角位移引起的杆端力引起的杆端力 AABQBAAABABAABQBAAABBAAABAAABAAAliFiMliFiMlilELXilELXlXEIlXEIlXEIlXEIl6;46;266;22032022212312221 Diagrams of bending moment and shearing force(2). Member end forces duo to relative translation of ends 在杆端相

20、对位移作用下的杆端力在杆端相对位移作用下的杆端力The chord rotation is （弦转角为）（弦转角为） 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最终的弹性曲线由下列两步完成。首先把体系整体旋转最终的弹性曲线由下列两步完成。首先把体系整体旋转 ，然，然后把杆端后把杆端A A及及B B

21、端反时针旋转端反时针旋转 。这样得到。这样得到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.在杆端位移作用下在杆端位移作用下一端固定一端定向支一端固定一端定向支

22、座的杆的杆端力的确定座的杆的杆端力的确定 BAAABABABABMiiiM2224V. Member end forces of single-span indeterminate prismatic beams with one fixed and one hinged due to their displacements. 在杆端位移作用下在杆端位移作用下一端固定一端铰支杆的杆端力的确定一端固定一端铰支杆的杆端力的确定 AABABABABiiiM32224232221212322266lililiFlililiMABABABAAAASABABABAAABVII. Slope-deflect

23、ion equation转角位移方程转角位移方程 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.转角位移方程转角位移方程把杆端弯矩与杆端转角及位移联系了起来把杆端弯矩与杆端转角及位移联系了起来。 图图a所示两端固定的等截面梁，所示两端固定的等截面梁，两端支座发生了位移。取基本结构如两端支座发生了位移。取基本结构如

24、图图b。 X3对梁的弯矩无影响，可不考虑，对梁的弯矩无影响，可不考虑，只需求解只需求解X1、X2。符号规定：杆端弯矩以对杆端顺时针方向为正；符号规定：杆端弯矩以对杆端顺时针方向为正； 均以顺时针方向为正；均以顺时针方向为正； AB 以使整个杆件顺时针方向转动为正。以使整个杆件顺时针方向转动为正。BA、力法典型方程为力法典型方程为BAXXXX222212112121118.2 The Slope-deflection equation of single-span indeterminate prismatic beams 等截面直杆的转角位移方程等截面直杆的转角位移方程作作X1、X2分别等于分

25、别等于1时的弯矩图如图时的弯矩图如图c、d。EIlEIlEIl63,321122211由图由图e可得可得lABAB21AB弦转角，顺时针方向为正。弦转角，顺时针方向为正。解典型方程得解典型方程得ABABABBAlEIlEIlEIXlEIlEIlEIX22216246248.2 The Slope-deflection equation of single-span indeterminate prismatic beams 等截面直杆的转角位移方程等截面直杆的转角位移方程令令杆件的线刚度杆件的线刚度lEIi MAB=X1，MBA=X2，可得，可得固端弯矩固端弯矩 ：单跨梁在荷载作用及温度变化时

26、产生的：单跨梁在荷载作用及温度变化时产生的 杆端弯矩。杆端弯矩。FFBAABMM、ABABBAABBAABliiiMliiiM624624 当单跨梁除支座位移外，还有荷载作用及温度变化时，当单跨梁除支座位移外，还有荷载作用及温度变化时，其杆端弯矩为其杆端弯矩为FBAABABBAFABABBAABMliiiMMliiiM624624转角位移方程转角位移方程8.2 The Slope-deflection equation of single-span indeterminate prismatic beams 等截面直杆的转角位移方程等截面直杆的转角位移方程对于一端固定另一端铰支的等截面梁，设对

27、于一端固定另一端铰支的等截面梁，设B端为铰支，则有端为铰支，则有0624FBAABABBAMliiiM)213(21FBAABABMilB不是独立的不是独立的FFFF2133BAABABABABAABMMMMliiM杆端弯矩杆端弯矩杆端剪力杆端剪力8.2 The Slope-deflection equation of single-span indeterminate prismatic beams 等截面直杆的转角位移方程等截面直杆的转角位移方程对于一端固定另一端铰支的等截面梁，设对于一端固定另一端铰支的等截面梁，设B端为铰支，则有端为铰支，则有0624FBAABABBAMliiiMF33

28、ABABAABMliiM8.2 等截面直杆的转角位移方程等截面直杆的转角位移方程转角位移方程转角位移方程BFQBAMABAABFQABFQABMABAABMBAFQBAFBAABAFABAABMLEIMMLEIM8.2 等截面直杆的转角位移方程等截面直杆的转角位移方程对于一端固定另一端滑动的等截面梁的转角位移方程对于一端固定另一端滑动的等截面梁的转角位移方程转角位移方程转角位移方程FQABMABAABMBAFQBAFBAABAFABAABMLEIMMLEIM对于一端固定另一端滑动的等截面梁的转角位移方程对于一端固定另一端滑动的等截面梁的转角位移方程转角位移方程转角位移方程4422ABAABAAAEIMiLEIMiL由力法求得由力法求得由力法求得由力法求得4422BABBABBBEIMiLEIMiL1.1.两端固定单元，在两端固定单元，在A A端发生一个顺时针的转角端发生一个顺时针的转角 。AAABMABMBA2.2.两端固