Mechanics of materials CHAPTERII ENERGYMETHODS
Mechanics of Materials
第十一一方
CHAPTERII ENERGY METHOD D811-1 GENERAL EXPRESSIONS OF THE STRAIN ENERGY S811-2 MOHRS THEOREM(METHOD OF UNIT FORCE □]§11-3 CATIGLIANOS THEOREM
CHAPTER 11 ENERGY METHOD §11–1 GENERAL EXPRESSIONS OF THE STRAIN ENERGY §11–2 MOHR’S THEOREM(METHOD OF UNIT FORCE) §11–3 CATIGLIANO’S THEOREM
第十一章能量方法 □§11-1变形能的普遍表达式 □§11-2莫尔定理(单位力法 §11-3卡氏定理
第十一章 能量方法 §11–1 变形能的普遍表达式 §11–2 莫尔定理(单位力法) §11–3 卡氏定理
ENERGYMETHOD 8 11-1 GENERAL EXPRESSIONS OF THE STRAIN ENERGY 、 Principle of energy Strain energy stored in the elastic body is equal to the work done by external forces, that is: U-W Method to analyze and calculate displacements deformations and internal forces of deformable bodies by this kind of relation is called energy method 2, Calculation of the strain energy of rods 1). Calculation of the strain energy of rods in tension or compression N2(x) dx or U= Density of the 丿2EA =/2E, 4: strain energy. O8
§11–1 GENERAL EXPRESSIONS OF THE STRAIN ENERGY 1、Principle of energy： 2、Calculation of the strain energy of rods： 1). Calculation of the strain energy of rods in tension or compression： = L x EA N x U d 2 ( ) 2 = = n i i i i i E A N L U 1 2 2 2 1 u = Strain energy stored in the elastic body is equal to the work done by external forces，that is: U=W Method to analyze and calculate displacements 、deformations and internal forces of deformable bodies by this kind of relation is called energy method. or Density of the strain energy:
能量方法 §11-1变形能的普遍表达式 能量原理: 弹性体内部所贮存的变形能,在数值上等于外力所作 的功,即 U=w 利用这种功能关系分析计算可变形固体的位移、变形 和内力的方法称为能量方法。 二、杆件变形能的计算: 1.轴向拉压杆的变形能计算: N2(x) xdx或U=∑ 丿2EA NL比能:=2E 2E
§11–1 变形能的普遍表达式 一、能量原理： 二、杆件变形能的计算： 1.轴向拉压杆的变形能计算： = L x EA N x U d 2 ( ) 2 = = n i i i i i E A N L U 1 2 2 或 2 1 比能: u = 弹性体内部所贮存的变形能，在数值上等于外力所作 的功，即 U=W 利用这种功能关系分析计算可变形固体的位移、变形 和内力的方法称为能量方法
ENERGYMETHOD 2. Calculation of the strain energy of rods in torsion U= M(x) m4 or U L 2GIp 2G:/ P Density of the strain energyr =ty 2 3. Calculation of strain energy of rods in bending U= M-(x)dx or U=>il b2EⅠ Density of the strain energy: u=O 2
2. Calculation of the strain energy of rods in torsion： = L P n x GI M x U d 2 ( ) 2 = = n i i Pi ni i G I M L U 1 2 2 2 1 u = 3. Calculation of strain energy of rods in bending： = L x EI M x U d 2 ( ) 2 = = n i i i i i E I M L U 1 2 2 2 1 u = or Density of the strain energy: or Density of the strain energy:
能量方法 2.扭转杆的变形能计算: U= M(x) dx或U=SMn L 2GIp F-I 2G: Ip 比能:u 2 tr 3.弯曲杆的变形能计算: M(x) M U= dx或U=∑ 丿2E 2E/ 比能:=1og 2
2.扭转杆的变形能计算： = L P n x GI M x U d 2 ( ) 2 = = n i i Pi ni i G I M L U 1 2 2 或 2 1 比能: u = 3.弯曲杆的变形能计算： = L x EI M x U d 2 ( ) 2 = = n i i i i i E I M L U 1 2 2 或 2 1 比能: u =
ENERGYMETHOD 3 General expressions of the strain energy Strain energy is independent of the order of loading Deformations due to mutually independent load may be summed up each other 0= N2( dx t M2(x) d x t M2(x) L 2EA .2GⅠ P L 2EI o(x) dx L 2E4 as -> Deflection factor of shear For slender columns, the strain energy due to shearing forces may be neglected M2( U= N2(/~Mx) dx+ 0(×)d L 2EAJL 2GIPJL 2ET
3、General expressions of the strain energy： Strain energy is independent of the order of loading. Deformations due to mutually independent load may be summed up each other. For slender columns，the strain energy due to shearing forces may be neglected. x EI M x x GI M x x EA N x U L L P n L d 2 ( ) d 2 ( ) d 2 ( ) 2 2 2 = + + + L x EA Q x d 2 ( ) 2 S S → x EI M x x GI M x x EA N x U L L P n L d 2 ( ) d 2 ( ) d 2 ( ) 2 2 2 = + + Deflection factor of shear
能量方法 三、变形能的普遍表达式: 变形能与加载次序无关;相互独立的力(矢)引起的变形能 可以相互叠加。 0= N( d x t (x) d x t M2(x) JL 2EA .2GⅠ L 2EI +Jax)dxa3→剪切烧度因子 细长杆,剪力引起的变形能可忽略不计。 U= N2(/~Mx) dx+ M2( )△ L 2EAJL 2GIPJL 2ET
三、变形能的普遍表达式： 变形能与加载次序无关；相互独立的力（矢）引起的变形能 可以相互叠加。 细长杆，剪力引起的变形能可忽略不计。 + L x EA Q x d 2 ( ) 2 S S → 剪切挠度因子x EI M x x GI M x x EA N x U L L P n L d 2 ( ) d 2 ( ) d 2 ( ) 2 2 2 = + + x EI M x x GI M x x EA N x U L L P n L d 2 ( ) d 2 ( ) d 2 ( ) 2 2 2 = + +
