P x L P P
P
M ( x, y ) = Py
②Approximate differential equation of the deflection curve
y
xM P P 2 2 y ′′ + y = y ′′ + k y = 0 , where : k = EI EI 19
M P y ′′= = y EI EI
EIy′′= M ( x)= Py+M
x M0 Let
k2 = P EI
y ′′ + k 2 y = k 2
M0 P P
M0
M y = c cos kx + d sin kx + ′ = d cos kx c sin kx P y
The boundary conditions are:
M P
x=0, y= y′=0;x=L, y= y′=0 27
③Solution of the differential equation： ④Determine the integral constants：
y = A sin x + B cos x
y ( 0 )= y ( L )= 0
A× 0 + B = 0 That is A sin kL + B cos kL = 0
Instable equilibrium
15
二、压杆失稳与临界压力 ： 1.理想压杆：材料绝对理想；轴线绝对直；压力绝对沿轴线作用。 2.压杆的稳定平衡与不稳定平衡：
稳 定 平 衡
不 稳 定 平 衡
16
3).loss of stability of compressed column：
4).Critical pressure of compressed columns
C: Inflection point
Pcr =
π 2 EI
l
2
Pcr ≈
π 2 EI
(0.7l )
2
Pcr ≈
π 2 EI
(0.5l )
2
Pcr ≈
π 2 EI
(2l )
2
Pcr =
π 2 EI
l2
25
=1
≈0.7
=0.5
=2
=1
0.5l
of stability
l
C
2l l
表10–1 各种支承约束条件下等截面细长压杆临界力的欧拉公式 支承情况 两端铰支 一端固定 两端固定 另端铰支 Pcr 失 稳 时 挠 曲 线 形 状 A C— 临界力Pcr 欧拉公式 长 B Pcr B Pcr B 一端固定 另端自由 Pcr 两端固定但可沿 横向相对移动 Pcr
Suppose the pressure has reached the critical value and the column has been in tiny bending state as shown in the figure. Start to determine the critical force with the deflective curve. ①bending moment：
by the approximate differential equation of the deflective curve.
Solution：The deformation of the column P P M0 x L P
is shown in the figure. The approximate differential equation of its deflection curve is:
Critical state
corresponding Stable intermediate state Instable
equilibrium equilibrium
pressure
Critical pressure: Pcr
17
3.压杆失稳：
4.压杆的临界压力 临界状态 稳 定 过 平 衡
对应的
Pcr
The shape of the deflective curve in lost
Pcr B
Pcr B
Pcr
Pcr
0.7 0.7l
0.5l
D
l
B
l
l
A
C A
A
C: Inflection C、D: Inflection point point Euler’s formula of the critical forcePcr Length coefficient
P EI
1 cos kL
=0
临界力 Pcr 是微弯下的最小压力，故，只能取n=1 ；且 杆将绕惯性矩最小的轴弯曲。
∴ Pcr =
π 2 EI min
L2
22
P = cr
π 2EI m in
L2
Euler’s formula of the critical pressure for the compressive column with two hinged ends
Pcr =
π 2 EI min
L2
21
③微分方程的解： ④确定积分常数：
y = A sin x + B cos x
y ( 0 )= y ( L )= 0
A× 0 + B = 0 即: A sin kL + B cos kL = 0
∴ sin kL = 0
∴
nπ ∴ k= = L
0 sin kL
5
§10–1 压杆稳定性的概念 构件的承载能力： ①强度 ②刚度 ③稳定性 工程中有些构 件具有足够的强度、 刚度，却不一定能 安全可靠地工作。
6
P
7
P
8
1、Stable and instable equilibrium ： 、 1). instable equilibrium
9
一、稳定平衡与不稳定平衡 ： 1. 不稳定平衡
①Strength ②Rigidity ③Stability
Some structure members in engineering have enough strength and rigidity but they are unable to work safely and reliably.
Supports
Two hinged ends
One free end and one hinged end
Two fixed ends
One fixed end and one free end
Two fixed ends but one of them is movable laterally.
0 sin kL
nπ k = = L P EI
1 cos kL
=0
sin kL = 0
so
The critical force Pcr is the smallest pressure under tiny bending，therefore we only take n=1 and the column will bend about the axis with the smallest moment of inertia.
Mechanics of Materials
1
2
CHAPTER 10 STABILIZATION OF COMPRESSIVE COLUMNS
§10–1 §10–2
CONCEPTS OF STABILITY OF COMPRESSED COLUMNS EULER’S FORMULA OF THE CRITICAL FORCE OF SLENDER
边界条件为:
M y ′′ + k y = k P
π 2EI m in P = cr (L)2
General form of Euler’s formula of the critical pressure
—Length coefficient（or constraint coefficient）
23
P = cr
π 2 EImin
2 L
两端铰支压杆临界力的欧拉公式
COMPRESSED COLUMNS §10–3 CRITICAL STRESS OF COMPRESSED COLUMNS AS STRESS
EXCEEDS PROPORTIONAL LIMIT §10-4 STABILITY CHECK AND REASONABLE SECTION OF COMPRESSED COLUMNS
压力
临界压力: 临界压力:
不 稳 度 定 平 衡 Pcr
18
§10–2
EULER’S FORMULA OF THE CRITICAL PRESSURE OF SLENDER COMPRESSED COLUMNS
1、Critical pressure for the column with two hinged ends:
2、Application range of the formula： 、
1).Ideal compressive columns； 2).In linear elastic range； 3).The ends of the column are supported by hinges.
3、Euler’s formula of the critical pressure for the column with 、 other end conditions:
二、此公式的应用条件： 1.理想压杆； 2.线弹性范围内； 3.两端为球铰支座。