材料力学-弯曲变形课件.ppt

* Chapter 6 Deformations of Beams 6.1 弯曲变形的概念 x y o x y y(x) o x x y y (x) θ (x) o x 挠度以 w 轴正向为正 Concept of deformations for a beam There are two forms of deformation in a beam : (1) 挠度 ( deflection ) x y L ---called the equation of deflection curve or elastic curve 梁有两种变形形式 挠度函数或挠曲线方程 Acoording to the assumpation of small deformations we can get : 转角以 x 轴正向逆时针旋转为正 (2) 转角 ( slope ) x y L Relation between and : ---called the equation of slope 转角函数或转角方程 注意: 材料力学所说的梁的变形实质上指的是梁的位移! x y y (x) θ (x) o x 6.2 挠曲线的近似微分方程 Approximately differential equation of the deflection curve 中性层曲率与弯矩的关系: 曲率计算公式 : Relation between bending moment and curvature of neutral surface The formula of curvature in mathematics is shown follows : EI ：抗弯刚度 ( bending stiffness ) The positive and negative symbol of curvature is just coincident with that of bending moment , so we have the result shown follows : It is called the approximately differential equation of the deflection curve . 挠曲线近似微分方程 曲率的正负号与弯矩的正负号一致 6.3 积分法求梁的变形 Using integration method to determine the deformations of a beam Basic equation is : M(x)/EI L x Single function (1) M(x)/EI 在梁中为单一函数 ( a single function in whole beam ) L x 0 积分常数C和D由约束条件确定： A Fixed pin There are two boundary conditions for a beam . (2) M(x)/EI 在梁中是分段函数 ( is not same function in one part of the beam from another ) M(x)/EI x A A Movable pin Fixed end Spring Another bar A l A k R Another beam A i a. 梁的分段图 ： x y b. 梁的连续条件(Continuous conditions of the beam) At the divided points of the beam the deflection and slope must be continuous , so we can get : There are 2n-2 equations . ( i = 2,3,….n ) Continuous conditions : 2n-2 Boundary conditions : 2 (i=1,2,…n) 在梁的分界点处 , 梁的挠度和转角都应连续 分析和讨论: 线弹性小变形条件下,梁的内力,应力及变形与下述哪些因素有关 (1) 梁的受力情况 (2) 梁材料的力学性能 (3) 梁的截面形状 (4)