材料力学第五版刘鸿文主编第六章 弯曲变形ppt

B
x
C C'
转角
w挠度
挠曲线

B
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
5、挠度和转角符号的规定
（Sign convention for deflection and slope)
§6–2 挠曲线的微分方程
（ Differential equation of the deflection curve) 一、推导公式（Derivation of the formula)
1、纯弯曲时曲率与弯矩的关系（Relationship between the curvature of beam and the bending moment)
一、微分方程的积分 （Integrating the differential equation )
M ( x) w EI
若为等截面直梁, 其抗弯刚度EI为一常量上式可改写成
EIw M ( x )
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
第六章
弯曲变形 (Deflection of Beams)
§6-1 工程中的弯曲变形问题 (Engineering problems of beam deflection) §6-2 挠曲线的微分方程（Differential equation of the deflection curve)
§6-3 用积分法求弯曲变形 （Beam deflection by integration )
3、挠曲线 (Deflection curve)
梁变形后的轴线称为挠曲线 .
挠曲线方程(Equation of deflection curve) w f ( x )
式中,x 为梁变形前轴线上任一点的横坐标,w 为该点的挠度. w
A
C'
挠曲线
C
B
x
w挠度（

B
转角
(Chapter 6: Deflection of Beams)
因此, w 与 M 的正负号相同
O
M 0 w 0
x
x
O
w 0 M 0
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
Mechanics of Materials
Chapter6 Deflection of Beams
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
M EI
横力弯曲时, M 和 都是x的函数.略去剪力对梁的位移的影响, 则
1
1 M ( x) ( x) EI
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELEC百度文库RICAL ENGINEERING Mechanics of Materials 韩光平
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
在规定的坐标系中,x 轴水平向右 为正, w轴竖直向上为正. 曲线向下凸时:
w
M
M
曲线向上凸时, w 0
M 0w
M
M 0 w 0
M
(3)
tg w w( x )
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
§6–3 用积分法求弯曲变形 （Beam deflection by integration )
横截面形心 C (即轴线上的点)在垂直于 x 轴方向 的线位移,称为该截面的挠度.用w表示.
w A C B
x
w挠度
C'
B'
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
A B
wA 0
A 0
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
在弯曲变形的对称点上，转角应等于零。
F
0
C
A
C
l/2 l/2
挠度 向上为正,向下为负.
转角 自x 转至切线方向,逆时针转为正,顺时针转为负.
w
A
C C' B
x
w挠度
挠曲线

转角
B
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
解： (1) 弯矩方程为
w
F
A B
x
x
M ( x ) F (l x )
(2) 挠曲线的近似微分方程为
(1)
l
EIw '' M ( x ) Fl Fx (2)
对挠曲线近似微分方程进行积分
2
Fx EIw ' Flx C1 (3) 2 2 3 Flx Fx EIw C 1x C 2 2 6
w (1 w )
2
2
3
2
M ( x) EI
w ' 与 1 相比十分微小而可以忽略不计,故上式可近似为 M ( x) w" EI
此式称为 梁的挠曲线近似微分方程（Differential equation
of the deflection curve)
2 w 近似原因 : (1) 略去了剪力的影响 ; (2) 略去了 项;
§6-4 用叠加法求弯曲变形 ( Beam deflections by superposition )
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
但在另外一些情况下,有时却要求构件具有较大的 弹性变形,以满足特定的工作需要. 例如,车辆上的板弹簧,要求有足够大的变形,以缓解 车辆受到的冲击和振动作用. F 2
§6–1 工程中的弯曲变形问题
（Basic concepts and example problems)
一. 工程实例（Example problem)
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
x 0, w 0
将边界条件代入(3) (4)两式中,可得 梁的转角方程和挠曲线方程分别为
F 2
F
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
二、基本概念（basic concepts) 1、挠度（ Deflection ）
EIw M ( x )dxdx C1 x C 2
二、积分常数的确定 （Evaluating the constants of integration)
1、边界条件（Boundary conditions) 2、连续条件 (Continue conditions)
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
4、挠度与转角的关系
（ Relationship between deflection and slope)： w
A
tg w ' w '( x )
(4)
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
Fx 2 EIw Flx C1 (3) 2 2 3 Flx Fx EIw C 1x C 2 2 6 边界条件 x 0, w 0
例题1 图示一抗弯刚度为 EI 的悬臂梁, 在自由端受一 集中力 F 作用.试求梁的挠曲线方程和转角方程, 并确定 其最大挠度 wmax和最大转角 max
w
A
F
B x
l
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
§6-5 简单超静定梁 (Simply and statically indeterminate beams) §6-6 提高弯曲刚度的措施 (The measures to strengthen rigidity)
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
2、由数学得到平面曲线的曲率 (The curvature from the mathematics )
1 | w | 3 2 2 ( x) (1 w )

| w | (1 w )
2 3 2
M ( x) EI
(Chapter 6: Deflection of Beams)
2、转角 (slope) 横截面对其原来位置的角位移,称为该截面的转角. 用 表示。
w
A C' C B x
w挠度（

B
转角
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
B
挠曲线是一条连续光滑的曲线，即在挠曲线的任意 点上，有唯一确定的挠度和转角。
A
B
A
B
(Chapter 6: Deflection of Beams)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
1、积分一次得转角方程 (The first integration gives the equation for the slope )
EIw M ( x )d x C1
2、再积分一次, 得挠度方程
(Integrating again gives the equation for the deflection)
SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING Mechanics of Materials 韩光平
在简支梁中, 左右两铰支座处的
挠度 w A 和
w B 都等于0.
A
B
wA 0
在悬臂梁中,固定端处的挠度 和转角 A 都应等于零.
wB 0
wA