《材料力学》课程PPT教学课件（双语版）第六章 弯曲变形

Mechanics of materials CHAPTER6 DEPORMIATIOI OFBEAMS DUETOBENDNG

1 Mechanics of Materials

树料力 叫变形

2

CHAPTER 6 DEFORMATION IN BENDING §6-1 Summary 86-2 Approximate differential equation of the deflection curve of the beam and its integration 86-3 Method of conjugate beam to determine the deflection and the rotational angle of the beam >6-4 Determine deflections and angles of rotation of the beam by the principle of superposition d$6-5 Ckeck the rigidity of the beam 国§6-6 Strain energy of the beam in bending D 86-7 Method to solve simple statically indeterminate problems of the beam D86-8 How to increase the load-carrying capacity of the beam

3 §6–4 Determine deflections and angles of rotation of the beam by the principle of superposition §6–5 Ckeck the rigidity of the beam CHAPTER 6 DEFORMATION IN BENDING §6–6 Strain energy of the beam in bending §6–7 Method to solve simple statically indeterminate problems of the beam §6–8 How to increase the load-carrying capacity of the beam §6–1 Summary §6–2 Approximate differential equation of the deflection curve of the beam and its integration §6–3 Method of conjugate beam to determine the deflection and the rotational angle of the beam

第六章弯曲变形 §6-1概述 □§6-2梁的挠曲线近似微分方程及其积分 §63求梁的挠度与转角的共轭梁法 §64按叠加原理求梁的挠度与转角 回§65梁的刚度校核 §66梁内的弯曲应变能 □§6-7简单超静定梁的求解方法 回§68如何提高梁的承载能力

4 §6–1 概述 §6–2 梁的挠曲线近似微分方程及其积分 §6–3 求梁的挠度与转角的共轭梁法 §6–4 按叠加原理求梁的挠度与转角 §6–5 梁的刚度校核 第六章 弯曲变形 §6–6 梁内的弯曲应变能 §6–7 简单超静定梁的求解方法 §6–8 如何提高梁的承载能力

DEFORMATIONOF BEAMS DUE TO BENDING §6-1 SUMMARY 桥式吊梁在自重及 重量作用下发生弯曲变形 Study range: Calculation of the displacement of the straight beam with equal sections in symmetric bending Study object: Checking rigidify of the beam; 2Solving problems about statically indeterminate beams to provide complementary equations for the geometric-deformation conditions of the beam

§6－１ SUMMARY Study range：Calculation of the displacement of the straight beam with equal sections in symmetric bending. Study object：①checking rigidify of the beam；②Solving problems about statically indeterminate beams（to provide complementary equations for the geometric-deformation conditions of the beam )

§6-1概述 桥式吊梁在自重及 重量作用下发生弯曲变形 研究范围:等直梁在对称弯曲时位移的计算 研究目的:①对梁作刚度校核; ②解超静定梁(为变形几何条件提供补充方 程)

§6－１ 概 述 研究范围：等直梁在对称弯曲时位移的计算。 研究目的：①对梁作刚度校核； ②解超静定梁（为变形几何条件提供补充方 程）

DEFORMATIONOF BEAMS DUE TO BENDING 1, Two basic displacement quantities of to measure deformation of the beam ) Deflection: The displacement of the centroid of a section in a direction perpendicular to the axis of the beam it is designated by y. it is positive if its direction is the same as f, otherwise it is negative P 2). Angle of rotation: The angle by which cross section turns with respect X to its original position about the neutral axis. it is designated by 0. It is ● positive if the angle of rotation rotates in the clockwise direction otherwise it 1 Is negative. 2 deflection curve: The smooth curve that the axis of the beam is transformed into after deformation is called the deflection curve. Its equation is v=f(r) Small deflection 3 The relation between the angle of rotation and the defection curve: tg 0= df →b= f dx

1).Deflection：The displacement of the centroid of a section in a direction perpendicular to the axis of the beam. It is designated by v . It is positive if its direction is the same as f，otherwise it is negative. 3、The relation between the angle of rotation and the deflection curve： 1、Two basic displacement quantities of to measure deformation of the beam (1) d d tg f x f  =   =  Small deflection P x v C  C1 f 2). Angle of rotation：The angle by which cross section turns with respect to its original position about the neutral axis .it is designated by . It is positive if the angle of rotation rotates in the clockwise direction, otherwise it is negative. 2、deflection curve：The smooth curve that the axis of the beam is transformed into after deformation is called the deflection curve. Its equation is v =f (x)

度量梁变形的两个基本位移量 1.挠度:横截面形心沿垂直于轴线方向的线位移。用ν表示。 与f同向为正,反之为负。 P x2.转角:横截面绕其中性轴转 动的角度。用表示,顺时 ●● 针转动为正,反之为负。 1 二、挠曲线:变形后,轴线变为光滑曲线,该曲线称为挠曲线。 其方程为: v=fer) 小变形 三、转角与挠曲线的关系: tg0 df →b= f dx

1.挠度：横截面形心沿垂直于轴线方向的线位移。用v表示。 与 f 同向为正，反之为负。 2.转角：横截面绕其中性轴转 动的角度。用 表示，顺时 针转动为正，反之为负。 二、挠曲线：变形后，轴线变为光滑曲线，该曲线称为挠曲线。 其方程为： v =f (x) 三、转角与挠曲线的关系： 一、度量梁变形的两个基本位移量 (1) d d tg f x f  =   =  小变形 P x v C  C1 f

DEFORMATIONOF BEAMS DUE TO BENDING 86-2 APPROXIMATE DIFFERENTIAL EQUATION OF THE DEFLECTION CURVE OF THE BEAM AND ITS INTEGTION I\ Approximate differential equation of the deflection curve x M>0 1M(x) El f"(x)0 Formula (2) is the approximate differential equation of the deflection curve

§6-2 APPROXIMATE DIFFERENTIAL EQUATION OF THE DEFLECTION CURVE OF THE BEAM AND ITS INTEGTION z z EI 1 M (x) =  1、Approximate differential equation of the deflection curve z z EI M x f x ( )  ( ) =  Formula (2) is the approximate differential equation of the deflection curve. EI M x f x ( )   ( ) = − …… （2） ( ) (1 ) 1 ( ) 2 3 2 f x f f x    +   =   Small deformation f x M>0 f (x)  0 f x M<0 f (x)  0 (1)

§6-2梁的挠曲线近似微分方程及其积分 、挠曲线近似微分方程 1M(x) x p EI (1 M>0 f"(x)0 式(2)就是挠曲线近似微分方程

§6-2 梁的挠曲线近似微分方程及其积分 z z EI 1 M (x) =  一、挠曲线近似微分方程 z z EI M x f x ( )  ( ) =  式（2）就是挠曲线近似微分方程。 EI M x f x ( )   ( ) = − …… （2） ( ) (1 ) 1 ( ) 2 3 2 f x f f x    +   =   小变形 f x M>0 f (x)  0 f x M<0 f (x)  0 (1)