双语版材料力学第十章

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第十章
压杆稳定
§10–1 压杆稳定性的概念
§10–2 细长压杆临界力的欧拉公式 §10–3 超过比例极限时压杆的临界应力
§10-4
压杆的稳定校核及其合理截面
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§10–1
CONCEPTS OF STABILITY OF COMPRESSED COLUMNS
The load-carrying capacity of structure members：
③Solution of the differential equation： ④Determine the integral constants： That is
y Asin xBcosx
y(0) y( L)0
A 0 B 0 A sin kL B cos kL 0
压力
临界压力:
不 稳 度 定 平 衡 Pcr
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§10–2
EULER’S FORMULA OF THE CRITICAL PRESSURE OF SLENDER COMPRESSED COLUMNS
1、Critical pressure for the column with two hinged ends:
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2. 稳定平衡
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3). Stable and instable equilibrium
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3. 稳定平衡和不稳定平衡
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2、Loss of stability and the critical pressure of the column： 1).Ideal compressive columns：the material is absolutely ideal；the axis is absolutely straight；the compressive force is absolutely along the axis of the column. 2). Stable and instable equilibrium of compressive columns： Stable equilibrium
0
P EI
1
sin kL cos kL
0
sin kL 0
n so k L
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.
①Strength ②Rigidity
③Stability
Some structure members
in engineering have enough strength and rigidity but
they are unable to work
safely and reliably.
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§10–1 压杆稳定性的概念
Mechanics of Materials
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第十章 压杆稳定
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CHAPTER 10 STABILIZATION OF COMPRESSIVE COLUMNS
§10–1 CONCEPTS OF STABILITY OF COMPRESSED COLUMNS §10–2 EULER’S FORMULA OF THE CRITICAL FORCE OF SLENDER COMPRESSED COLUMNS §10–3 CRITICAL STRESS OF COMPRESSED COLUMNS AS STRESS EXCEEDS PROPORTIONAL LIMIT §10-4 STABILITY CHECK AND REASONABLE SECTION OF COMPRESSED COLUMNS
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§10–2
细长压杆临界力的欧拉公式
一、两端铰支压杆的临界力:
假定压力已达到临界值，杆已经处于微弯状态，如图，
从挠曲线入手，求临界力。 P x L P P xM y
①弯矩：
M ( x, y)Py
②挠曲线近似微分方程： M P y y EI EI P y y y k 2 y0 EI P 2 其中k : 20 EI
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。
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Baidu Nhomakorabea P
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P
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1、Stable and instable equilibrium ：
1). instable equilibrium
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一、稳定平衡与不稳定平衡 ： 1. 不稳定平衡
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2). Stable equilibrium
M ( x, y) Py
P
②Approximate differential equation of the deflection curve
P x L P P xM y
M P y y EI EI
P y y y k 2 y 0, EI
P where : k EI
Instable equilibrium
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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
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3).loss of stability of compressed column：
4).Critical pressure of compressed columns
Critical state
corresponding Stable intermediate state Instable
equilibrium
equilibrium
pressure
Critical pressure: Pcr
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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
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：