第十三章能量法 Chapter13 Energy Method
r (Energy Method) 第十三章能量法( Energy Methods) D§13-1概述( Introduction) D§13-2杆件变形能的计算( Calculation of strain energy for various types of loading D§13-3互等定理( Reciprocal theorems) §13-4单位荷载法·莫尔定理(Unit-oad method mohr's theorem) D§13-5卡氏定理( Castigliano’ s Theorem) D§13-6计算莫尔积分的图乘法( The meth. od of moment areas for mohr's integration)
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r (Energy Method) §13-1概述( Introduction) 、能量方法( Energy methods) 利用功能原理V=W来求解可变形固体的位移变形和内力 等的方法 二、外力功( Work of the external force) 固体在外力作用下变形,引起力作用点沿力作用方向位移, 外力因此而做功,则成为外力功 、变形能( Strain energy) 在弹性范围内弹性体在外力作用下发生变形而在体内积蓄 的能量称为弹性变形能简称变形能
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能量滤( Energy Method) 四、功能原理(Work- energy principle) 可变形固体在受外力作用而变形时,外力和内力均将作功对 于弹性体不考虑其他能量的损失外力在相应位移上作的功在数 值上就等于积蓄在物体内的应变能 The formula: V=w(Work-Energy Principle We will not consider other forms of energy such as thermal energy, chemical energy, and electromagnetic energy. Therefore, if the stresses in a body do not exceed the elastic limit, all of work done on a body by external forces is stored in the body as elastic strain energy. 用
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r (Energy Method) §13-2杆件变形能的计算 Calculation of strain energy for various types of loading 杆件变形能的计算( Calculation of strain energy for various types of loading) 1.轴向拉压的变形能( Strain energy for axial loads) 当拉力为F1时杆件的伸长为△1 当再增加一个dF1时,相应的变形增量为d(△M1) 此外力功的增量为: dw=fd(al) d(Al)=dFL :EA
5 d d(Δ ) 1 1 W F l EAF l l 1 1 d d(Δ )
r (Energy Method) F dF △1 d△ △ △ F 积分得:W=[dW FF de 2E42
6 F F l F l F O l l1 dl1 dF1 F1 l F EA F l F EA l W W F F Δ 2 2 d d 2 1 0 1
r (Energy Method) 根据功能原理 V=W,可得以下变形能表达式 =W=FMl=F、Al F FMl EA EA F-l F4l 2E42E4 当轴力或截面发生变化时:V2=∑ 用
7 V W F l F Δl 21 Δ 21 ε N EAF l EAFl l N Δ EA F l EA F l V 2 2 2N 2 ε ni i i i i E A F l V 1 2N ε 2
r (Energy Method) 当轴力或截面连续变化时:V I FN(x)dx 02E4(x) 比能( strain energy density) 单位体积的应变能记作U U F△l V Al 2 o= Ea 2E8 0=-8 (单位J/m3) 22E2
8 l EA x F x x V 0 2 N ε 2 ( ) ( )d σε Al F l V U 2 1 Δ 2 1 ε σ Eε 2 2 2 1 2 2 ε Eε E σ σε
r (Energy Method) 2.扭转杆内的变形能( Strain energy for torsional loads) M E=W=M△φ=M MI MI TZI P 2GI 2GI T() dx 或V=∑ 2GI(x) i=1 2G: pi
9 或 p 2 p 2 e p e ε e e 2 2 2 1 Δ 2 1 GI T l GI M l GI M l V W M M l x GI x T x V d 2 ( ) ( ) p 2 ε n i i i i i G I T l V 1 p 2 ε 2
r (Energy Method) 3.弯曲变形的变形能 (Strain energy for flexural loads) e 纯弯曲( pure bending) - MI MI VW=M·0=-M=e 2 EI 2EI 横力弯曲( nonuniform bending) e dx E-J2EI(x) 10
10 θ EI M l EI M l V W M θ M 2 2 1 2 1 2 e ε e e x EI x M x V l d 2 ( ) ( ) 2 e ε