北京航空航天大学 材料力学课件 Chapt6-2004

Differential equation of deflection curve
挠曲轴微分方程
Curvature of the neutral surface中性层曲率:
1


M （pure bending纯弯） EI
1 M ( x) （推广到非纯弯non-uniform bending） ( x) EI
( x ) －转角方程
Simplification: （ Neglect the effect of shear forces 忽略剪力影响）
dw ' tan ' dx
(small deflection小变形)
（rad）
dw dx
小变形的条件下，水平位移比挠度小得多， 可以略而不计。
约束处位移应满足的条件 －boundary conditions 边界条件
P
A
E
B
C
D
Pin or roller support(铰支座) :
wC=0
Fixed support(固定端): wA=0,
θA =0
Interface continuum conditions(连续条件):
左 C 0 右 C 0 左 右 C C

左 B
右 B

左 E
右 E

左 E
右 E
Integration method to calculate deflections of beam
积分法求梁位移
To find the slope of section A ，EI = const(常数) 计算图示梁截面 A 的转角
Math 数学:
1 w ( x) 1 w2 3/2
w M ( x) －Differential equation of deflection curve 2 3/2 EI 1 w 挠曲轴微分方程
w－deflection挠度 max < p
Approximate Differential equation of deflection curve
§2 Differential equation of beam deflection 梁变形微分方程
Differential equation of deflection curve 挠曲轴微分方程 Approximate Differential equation of deflection curve 挠曲轴近似微分方程
Examples 例题
To find the deflection and slope of section B ，EI = const(常数) 试计算截面 B 的挠度与转角
A 解:弯矩方程
x
M x M 0
0 B A beam in pure bending
M
挠曲轴近似微分方程 w" x
Integration and boundary conditions of the differential equation 微分方程 的积分与边界条件 Integration method to calculate deflections of beam积分法求梁位移 Examples例题
Deflection and slope 挠度与转角
挠曲轴 转角 －挠度
Deflection(挠度)－ vertical displacement of centroid of cross section
(横截面形心在垂直于梁轴方向的位移)
w w( x ) －挠曲轴方程
Slope(转角)－angle of rotation of cross section(横截面绕中性轴的转角 )
挠曲轴近似微分方程
w M ( x) 2 3/2 EI 1 w
Small deflection ： Simplification简化:
w2 << 1
d2w M ( x ) － Approximate differential equation of 2 dx EI deflection curve 挠曲轴近似微分方程
Differential equation 挠曲轴微分方程
w M ( x) EI
Boundary conditions 位移边界条件与连续条件 Protraction of deflection curve Construct M diagram画 M 图 Determine the shape of deflection curve 确定挠曲轴 的形状(由 M 图的正、负、零点或零值区，确定挠 曲轴的凹、凸、拐点或直线区) Determine the position of deflection curve by boundary conditions由位移边界条件确定挠曲轴的空间位置
M0 w' x xC EI
M0 EI
M0 2 wx x Cx D 2 EI
边界条件
w0 0, D 0
w' 0 0, C 0
M 0l 2 wB 2 EI M 0l B EI
M0 2 M0 wx x x x 2 EI EI
d2w M ( x) 2 dx EI
(材力) (数学)
d2w M ( x) 2 dx EI
The conditions of above equation to be valid 应用条件： Within linearly elastic range max p
Under small deflection小变形 Neglect effect of Fs (slender bar)
Chapter 6 Deflections of Beams
梁的变形
§1 Introduction 引言 §2 Differential equation of beam deflection 梁变形微分方程 §3 Integration method to calculate deflections 计算梁位移的积分法 §4 Macaulay's method to calculate deflections 计算梁位移的奇异函数法 §5 Method of superposition to calculate deflections 计算梁位移的叠加法 §6 Statically Indeterminate Beams静不定梁 §7 Stiffness analysis and design of beams 梁的刚度条件与合理设计
w is positive when in the upward direction坐标轴 w 向上 If w is positive when in the downward direction 当坐标轴 w 向下时：
)x ( M w2d - 2 IE xd
§3 Integration method to calculate deflections 计算梁位移的积分法

Baidu Nhomakorabea

w2 x
Examples 例题
To find the deflection and slope of section C ，EI = const(常数) 试计算截面 C 的挠度与转角
qa 2 3qa FBy 2 FAy
解：1. 建立挠曲轴微分方程并积分
d 2 w1 qa x1 2 dx1 2 EI dw1 qa 2 x1 C1 dx1 4 EI qa 3 w1 x1 C1 x1 D1 12 EI d 2 w2 q 2 x2 2 dx2 2 EI dw2 q 3 x2 C 2 dx2 6 EI q 4 w2 x2 C2 x2 D2 24 EI
§1 Introduction 引言
Deflection curve
挠曲轴
Deflection and slope
挠度与转角
deflection curve 挠曲轴
挠曲轴
The axis of a beam that is bent into an arc, is called deflection curve(变弯后的梁轴称为挠曲轴) Neglecting the effect of shear forces, the entire transverse section of the beam , remains plane and normal to the deflection curve忽略剪力对弯曲变形的影响, 因而横 截面仍保持平面, 并与挠曲轴正交 Under the case of symmetric bending, the deflection curve is in the plane of symmetry 对称弯曲时，挠曲轴为 位于纵向对称面的平面曲线
Integration and boundary conditions of the differential equation 挠曲轴微分方程的积分与边界条件
d2w M ( x) 2 dx EI
dw dx

M ( x) dx C EI
w

M ( x) dx Cx D EI
Last chapter:
1 Internal force of beam 2 Stress and strength analysis of beam 3 1
M EI z
Purpose of this chapter:
1 Stiffness of beam 梁刚度 2 Statically indeterminate beam 静不定梁 3 Problems of Stability稳定问题
w2 l 0 (边界条件)
l l l l w'1 w'2 w1 w2 (连续条件) 2 2 2 2
(光滑条件)
M 0 x l 24 EIl
四个方程定4个常数 w1 x
M0x 4x2 l 2 24lEI
由条件 (1), (2) 与式 (b) ，得
Slope转角
Me l D 0, C 6 EI
Me (3 x 2 l 2 ) 6 EIl Ml A (0) - e （） 6 EI
Protraction of deflection curve 挠曲轴的绘制
右c左c右c左c00右e左e右e左e右b左b?integrationmethodtocalculatedeflectionsofbeam积分法求梁位移tofindtheslopeofsectionaeiconst常数计算图示梁截面a的转角?establishandintegratethedifferentialequationofdeflection?establishandintegratethedifferentialequationofdeflection建立挠曲轴微分方程并积分lmffbyayexlmxmexeilmxw2e2ddac2dd2exeilmxwbd63ecxxeilmw?theconstantsofintegrationcanbefoundbyboundaryconditions利用边界条件确定积分常数100wx处在a2dd2ecxeilmxwb63edcxxeilmw由条件12与式b得?slope转角20l处xw在eilm6cd0e?3622elxeilm?eilm6a0e??protractionofdeflectioncurve挠曲轴的绘制?differentialequation挠曲轴微分方程eixmw?boundaryconditions位移边界条件与连续条件protractionofdeflectioncurve?constructmdiagram画m图?determinetheshapeofdeflectioncurve确定挠曲轴?determinethepositionofdeflectioncurvebyboundaryconditions由位移边界条件确定挠曲轴的空间位置的形状由m图的正负零点或零值区确定挠曲轴的凹凸拐点或直线区?examples例题tofindthedeflectionandslopeofsectionbeiconst常数试计算截面b的挠度与转角abeaminpurebending解
Establish and integrate the differential equation of deflection 建立挠曲轴微分方程并积分
FAy FBy Me l
Me M ( x) x l
d2w Me x 2 dx EIl dw M e 2 x C (a) dx 2 EIl Me 3 w x Cx D (b) 6 EIl
C, D: constants of integration
梁段交接处位移应满足的条件 － interface continuum conditions 连续条件 The constants of integration can be found by boundary (and interface) conditions (利用位移边界与连续条件确定积分常数)
求挠曲轴微分方程,列挠曲线的位移边界和连续条件
A B
M0
C
l/2
l/2
AB段：
x
M0 x EI l M 0 x3 w1 C1 x D1 6 EI l w1"
位移边界条件和连续条件
w1 0 0
BC段: M x w2 " 0 1 EI l
M w2 0 EI x3 x 2 C2 x D2 6l 2
dw M e 2 x C (a) dx 2 EIl Me 3 w x Cx D (b) 6 EIl
The constants of integration can be found by boundary
conditions 利用边界条件确定积分常数
在 x 0 处，w 0 (1) 在 x l 处，w 0 (2)