材料力学第五版刘鸿文主编第六章 弯曲变形ppt

Mechanics of Materials
Chapter6 Deflection of Beams
(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
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
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)
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
Hale Waihona Puke 曲线向上凸时, w 0M 0w
M
M 0 w 0
M
§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
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
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
(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
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 )
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
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 )
(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 )
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
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
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;
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 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
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)
§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)
(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
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
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)