Apparently, the displacement existing exactly should satisfy the equations of equilibrium and the boundary conditions of stress expressed by displacement besides boundary conditions of displacement. Now we can see also that the displacement existing exactly still satisfy the variational equations of displacement besides boundary conditions of displacement Furthermore, through calculating, we can deduce differential equations of equilibrium and boundary conditions of stress expressed by displacement from variational equations of displacement. So it is obvious that variationa equations of displacement can substitute for differential equations of equilibrium and boundary conditions of stress 21
21 Apparently, the displacement existing exactly should satisfy the equations of equilibrium and the boundary conditions of stress expressed by displacement besides boundary conditions of displacement. Now we can see also that the displacement existing exactly still satisfy the variational equations of displacement besides boundary conditions of displacement. Furthermore, through calculating, we can deduce differential equations of equilibrium and boundary conditions of stress expressed by displacement from variational equations of displacement. So it is obvious that variational equations of displacement can substitute for differential equations of equilibrium and boundary conditions ofstress
熊量原与分法 显然,实际存在的位移,除了满足位移边界杀件 以外,还应当满足位移表示的平衡方程和应力边界条 件;现在又看到,实际存在的位移,除了满足位移边 界条件外,还满足位移变分方程。而且,通过运算, 还可以从位移变分方程导出用位移表示的平衡微分方 程和应力边界条件。于是可见:位移变分方程可以代 替平衡微分方程和应力边界条件。 22
22 显然，实际存在的位移，除了满足位移边界条件 以外，还应当满足位移表示的平衡方程和应力边界条 件；现在又看到，实际存在的位移，除了满足位移边 界条件外，还满足位移变分方程。而且，通过运算， 还可以从位移变分方程导出用位移表示的平衡微分方 程和应力边界条件。于是可见：位移变分方程可以代 替平衡微分方程和应力边界条件
8 10-3 The Variational Method of Displacement I. The Ritz method We set expressions of components of displacement that satisfy the boundary conditions of displacement, which contain several undetermined coefficients. Then we decide these coefficients according to the principle of minimum potential energy. Take the expression of components of displacement as following: =1+∑4m m,V=vo +>Bmw= +∑ where uo, vo, wo are undetermined coefficients whose boundary values equal to the known displacement on the boundaries. um, vm, Wm are the set functions whose boundary conditions equal to zero. Am Bm. Cm are undetermined coefficients, by whose variation the variation of displacement are realized
23 §10-3 The Variational Method of Displacement = + = + = + m m m m m m m u u0 Am um ,v v0 B v ,w w0 C w where u0 ,v0 ,w0 are undetermined coefficients whose boundary values equal to the known displacement on the boundaries. um 、vm、wm are the set functions whose boundary conditions equal to zero. Am、Bm、Cm are undetermined coefficients, by whose variation the variation of displacement are realized. We set expressions of components of displacement that satisfy the boundary conditions of displacement , which contain several undetermined coefficients. Then we decide these coefficients according to the principle of minimum potential energy. Take the expression of components of displacement as following: I. The Ritz method
能量原与实分法 §10-3位移变分法 一瑞次法 先设定满足位移边界条件的位移分量的表达式, 其中包含若干个待定的系数,再根据极小势能原理, 决定这些系数。取位移分量的表达式如下 n=6+∑44m,1=+∑Bnm"=V+2Cmvm 其中砌V2W为设定的函数,它们的边界值等于边界 上的已知位移;l、Wn为边界值等于零的设定 函数,A、B、C为待定的系数,位移的变分由它们 的变分来实现。 24
24 §10-3 位移变分法 = + = + = + m m m m m m m u u0 Am um ,v v0 B v ,w w0 C w 其中u0 ,v0 ,w0 为设定的函数，它们的边界值等于边界 上的已知位移；um 、vm、wm 为边界值等于零的设定 函数，Am、Bm、Cm为待定的系数，位移的变分由它们 的变分来实现。 先设定满足位移边界条件的位移分量的表达式， 其中包含若干个待定的系数，再根据极小势能原理， 决定这些系数。取位移分量的表达式如下： 一 瑞次法
The variations of components of displacement are 6=∑un8An。8v=∑vn8Bn。8W=∑wn8Cm The variation of strain energy is 6=∑(M8A+m28Bm+ The variation of potential energy of external forces is 6=∑∫(Xx5Am+Ym8Bm+2n3Cm) dxd yd ∑』(Xun3An+nsBm+2n3Cm)dS 25
25 = = = 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 The variation of strain energy is The variation of potential energy of external forces is − + + = − + + 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 The variations of components of displacement are