能量原与实分法 §10-2位移变分方程与极小势能原理 变分及其性质 高等数学我们学过微分的概念,微分是变量的 增量。那么什么是变分呢?变分是函数的增量,通 常用δ表示。变分具有以下的性质 8(l+)=8a+。v u 6∫ds=JndS 16
16 一 变分及其性质 高等数学我们学过微分的概念，微分是变量的 增量。那么什么是变分呢？变分是函数的增量，通 常用δ表示。变分具有以下的性质： = = + = + udS u S u x x u u w u w δ δ d δ δ δ ( ) δ δ §10-2 位移变分方程与极小势能原理
II. The variational equation of displacement Suppose that the elastic body is under the status of equilibrium when imposed some external forces. It undergoes the true displacement u,v,w which satisfy the equations of equilibrium and satisfy the boundary conditions of displacement and the stress boundary conditions expressed by the displacement. Now suppose that the components of displacement undergo minute changes(virtual displacement) 8u, Sv, Sw which the boundary conditions of displacement allow. Now the extermal forces do virtual work during the virtual displacement which should be equal to the increment of the functional of the strain energy This equation is the so-called variational equation of displacement, where X,Y, Z are the components of body forces, and the x Y. z are the components of the surface faces 17
17 II. The variational equation of displacement Suppose that the elastic body is under the status of equilibrium when imposed some external forces. It undergoes the true displacement u,v,w, which satisfy the equations of equilibrium and satisfy the boundary conditions of displacement and the stress boundary conditions expressed by the displacement. Now suppose that the components of displacement undergo minute changes(virtual displacement) du,dv,dw which the boundary conditions of displacement allow. Now the extermal forces do virtual work during the virtual displacement which should be equal to the increment of the functional of the strain energy. U (X u Y v Z w)dxdydz (X u Y v Z w)dS d = d + d + d + d + d + d This equation is the so-called variational equation of displacement, where X,Y, Z are the components of body forces, and the are the components of the surface faces. X ,Y, Z
熊量原与分法 二位移变分方程 设弹性体在一定外力作用下,处于平衡状态。发 生的真实位移为u,ν,1,它们满足位移分量表示的平 衡方程,并满足位移边界条件和用位移表示的应力边 界条件。现在假设位移分量发生了位移边界条件所容 许的微小改变(虛位移)δu、δv、δw,这时外力 在虚位移上作虚功,虛功应和变形能泛函的增加相等, 即 dU= 这个方程就是所谓位移变分方程。其中X,Y,Z为体力分 量,X,Y,Z为面力分量。 18
18 二 位移变分方程 设弹性体在一定外力作用下，处于平衡状态，发 生的真实位移为 u,v,w，它们满足位移分量表示的平 衡方程，并满足位移边界条件和用位移表示的应力边 界条件。现在假设位移分量发生了位移边界条件所容 许的微小改变（虚位移）δu 、δv、δw，这时外力 在虚位移上作虚功，虚功应和变形能泛函的增加相等， 即 U (X u Y v Z w)dxdydz (X u Y v Z w)dS d = d + d + d + d + d + d 这个方程就是所谓位移变分方程。其中X,Y,Z为体力分 量， X ,Y, Z 为面力分量
IIL. The principle of the minimum potential energy Because the virtual displacement is very small, the value and the direction of the extemal forces can be regarded as invariable during the virtual displacement, and only its acting points are changed. According to the properties of the variation, the variational equation of displacement can be rewrite as: s(-J(Yu++Ew)dxdyd=-Slu u+Yy+ Zw aS=o Suppose the potential energy of external forces is Ther (+)=0 The meaning of this equation is that under the action of given external forces, among all groups of displacements that satisfy the boundary conditions of displacement, the group of displacement that exist actually should make the total potential energy a minimum. If considering the second order variation, we can further analyze it and prove that this extremum is a minimum for the stable equilibrium status. So this equation is also called the principle of minimum potential energy 19
19 III. The principle of the minimum potential energy Because the virtual displacement is very small, the value and the direction of the external forces can be regarded as invariable during the virtual displacement, and only its acting points are changed. According to the properties of the variation, the variational equation of displacement can be rewrite as: − ( + + ) − ( + + ) = 0 d U Xu Yv Zw dxdydz Xu Yv Zw dS V (Xu Yv Zw)dxdydz (Xu Yv Zw)dS = − + + − + + Suppose the potential energy of external forces is: Then d (U +V) = 0 The meaning of this equation is that under the action of given external forces, among all groups of displacements that satisfy the boundary conditions of displacement, the group of displacement that exist actually should make the total potential energy a minimum. If considering the second order variation, we can further analyze it and prove that this extremum is a minimum for the stable equilibrium status. So this equation is also called the principle of minimum potential energy
熊量原与分法 三极小势能原理 由于虛位移是微小的,因此在虚位移的过程中 外力的大小和方向可以当做保持不便,只是作用点有 了改变。利用变分的性质,位移变分方程可改写为 沙-(Xx+5+2h-(+51+2]=0 设外力势能为 =(+2)(+1y+21s 则(+)=0 该式的意义是:在给定的外力作用下,在满足位 移边界条件的各组位移中,实际存在的一组位移应使 总势能为极值。如果考虑二阶变分,进一步的分析证 明,对于稳定平衡状态,这个极值是极小值。因此, 该式又称为极小势能原理
20 三 极小势能原理 由于虚位移是微小的，因此在虚位移的过程中， 外力的大小和方向可以当做保持不便，只是作用点有 了改变。利用变分的性质，位移变分方程可改写为： − ( + + ) − ( + + ) = 0 d U Xu Yv Zw dxdydz Xu Yv Zw dS V (Xu Yv Zw)dxdydz (Xu Yv Zw)dS = − + + − + + d (U +V) = 0 设外力势能为 则 该式的意义是：在给定的外力作用下，在满足位 移边界条件的各组位移中，实际存在的一组位移应使 总势能为极值。如果考虑二阶变分，进一步的分析证 明，对于稳定平衡状态，这个极值是极小值。因此， 该式又称为极小势能原理