《弹性力学》（双语版） 第十章 能量原理与变分法

熊量原与分法 位移分量的变分是 8a=∑ ∑ vδB.61 ∑wn8C 应变能的变分为 U=∑ (,8An+ a CB 8 Bm+8 Cm) CA d 外力势能的变分为 6=∑∫( (Xu 8 A+Yvm8 B+ Zwm 8 Cm)dxdydz ∑∫/xmn5An+Y125Bn+2nCm)dS

26 =  =  =  m m m m m m m δ u um δ Am ,δ v v δ B ,δ w w δ C δ = ( δ + δ + δ ) m m m m m m C C U B B U A A U U       应变能的变分为 外力势能的变分为   − + + = − + + m m m m m m m m m m m m m m Xu A Yv B Zw C S V Xu A Yv B Zw C x y z ( δ δ δ )d δ ( δ δ δ )d d d 位移分量的变分是

Substitutinginto 8(U+v)=0 yields B0 Xu dxdyd=+l Xum ds yy dxd ydz+lYy dS 2 -=Zwm dxdydz+ZwmdS C Above system of linear algebraic equations includes 3m equations After solving these equations, we introduce the results into the expressions of the components of displacement to obtain the approximate solutions of the components of displacement. This method is called the Ritz method 27

27 yields Above system of linear algebraic equations includes 3m equations. After solving these equations, we introduce the results into the expressions of the components of displacement to obtain the approximate solutions of the components of displacement. This method is called the Ritz method. Substituting into       = + = + = + Zw x y z Zw S C U Yv x y z Yv S B U Xu x y z Xu S A U m m m m m m m m m d d d d d d d d d d d d       d (U +V) = 0

代入δ(+1)=0 中。得至 = Xum dxdydz+ Xu. ds CA aB ∫ Y1m dxd yd=+』yv v ds J5 Zwm dxd yd=+[Zwm d S 上面是个数为3m的线性代数方程组,求解后,代回 位移分量的表达式,得到位移分量的近似解。这种 方法称为瑞次法。 28

28 中，得到 上面是个数为3m的线性代数方程组，求解后，代回 位移分量的表达式，得到位移分量的近似解。这种 方法称为瑞次法。 代入       = + = + = + Zw x y z Zw S C U Yv x y z Yv S B U Xu x y z Xu S A U m m m m m m m m m d d d d d d d d d d d d       d (U +V) = 0

Ⅱ. The galerkin method If we regard variation as the function of the components of strain en dU=ll su, dxdvdz= Because =0x0y 6=-n,…,6v.=Cn+-6 a8 ax SO +…+r-8+|+… dxdydz aya Applying the Ostrogradsky-Gauss formula to the first term, we have 29

29 II. The Galerkin method If we regard variation as the function of the components of strain, then dxdydz U U U U dxdydz yz yz x x           +   + +   = =  d    d   d d 1 1 1 Because , , , ; , , , . 1  1   v  z w y u x U U yz x yz yz x x  d d d  d d      +   =   = =   =   so        +          +   + +   = v dxdydz z w y u x dU  x d   yz d d  Applying the Ostrogradsky—Gauss formula to the first term, we have

熊量原与分法 二伽辽金法 将变分看做形变分量的函数,则 aU aU au=SU, dxdydz 6;+…+6+…dz y 由于 …;δ,=i…,n=-n+-Cm;… da ax 所以 况U=b+…+r|bn+0b+…tdhe Oy 应用奥高公式,对上式中的第一项,我们有

30 二 伽辽金法 将变分看做形变分量的函数，则 dxdydz U U U U dxdydz yz yz x x           +   + +   = =  d    d   d d 1 1 1 由于 , , , ; , , , . 1  1   v  z w y u x U U yz x yz yz x x  d d d  d d      +   =   = =   =   所以        +          +   + +   = v dxdydz z w y u x dU  x d   yz d d  应用奥高公式，对上式中的第一项，我们有