Based on this concept, we will now develop（提出）the principle of virtual work so that it can be used to determine the displacement of a point on a structure. To do this, we will consider the structure (or body) to be of arbitrary（任意）shape as shown in Fig. 5-1-(b). Suppose it is necessary to determine the displacement of point A on the body caused by the “real loads” P1, P2, and P3. It is to be understood that these loads cause no movement of the supports; in general, however, they can strain(使..产生应变) the material beyond the elastic limit.
基于这个概念，现在我们将提出虚功原理，以便用以确定结构上某一点的位移。为此，我们考虑图5-1-(b)显示的具有任意形状的结构或物体。假定有必要确定由实际荷载P1、P2 和 P3 引起的A点的位移。可以被理解为这些荷载不引起支座的移动；但是一般而言，它们能使材料的应变超过弹性极限。
Since no external load acts on the body at A and in the direction of , the displacement can be determined by first placing on the body a “virtual” load such that this force P’ acts in the same direction as , Fig.5-1-(a). For convenience（方便起见）, which will be apparent（显而易见）later, we will choose P’ to have a “unit” magnitude（单位值）, that is P’ = 1. The term “virtual” is used to describe the load, since it is imaginary（想象的）and does not actually exist as part of the real loading. The unit load (P’) does, however, create an internal virtual load u in a representative（典型的）element or fiber of the body, as shown in Fig.5-1-(a).
由于没有外力作用在物体的A点以及 的方向上，则位移 可以通过在物体上先设置一个虚力，即如图5-1-(a)中作用一个与方向 相同的力P’来确定。为了方便起见（后面会显而易见），我们选择P’有一个单位值，即P’=1。用术语“虚”来描述荷载，因为它是想象的，非实际存在的真实荷载的部分。但是，单位荷载P’在物体的典型单元或纤维中产生了一个内部虚力u，如图5-1-(a)所示。
Here it is required that P’ and u be related by the equation of equilibrium. As a result of（通过）these loadings, the body and the element will each undergo a virtual displacement due to the load P’, although we will not be concerned with（关心）its magnitude. Once the virtual loadings are applied and then the body is subjected to the real loads P1, P2, and P3, Fig.5-1-(b), point A is displaced an amount , causing the element to deform dL.
这里要求P’和u通过平衡方程联系起来。通过这些（虚）荷载，物体和单元各自由于荷载P’而经历一个虚位移，尽管我们不会关心它的数值。一旦施加虚荷载，然后物体承受图5-1-(b)中的实际荷载P1、P2和P3，则点A产生位移值 ，导致单元发生变形dL。
As a result（因此）, the external virtual force P’ and internal virtual load u “ride along” by（乘上） and dL, respectively（分别地）, and therefore perform external virtual work of 1 on the body and internal virtual work of u dL on the element. Realizing that the external virtual work is equal to the internal virtual work done on all the elements of the body, we can write the virtual work equation as
1 = u dL （5-2）
因此，外部虚力P’和内部虚力u分别与 和dL“乘在一起”，因此在物体上形成外部虚功，在单元上形成内部虚功。了解到外部虚功等于对物体所有单元作的内部虚功，我们可以写出虚功方程
1 = u dL （5-2）
Where P’=1=external virtual unit load acting in the
direction of
u = internal virtual load acting on the element in = external displacement caused by the real loads dL = internal deformation of the element caused the direction of dL
by the real loads
By choosing P’ = 1, it can be seen that the solution for follows directly, since = u dL .
这里，P’等于1，也等于作用在 方向上的外部虚单位力；
u等于以dL方向作用在单元上的内部虚力； 等于真实荷载引起的外部位移； dL等于真实荷载引起的单元的内部变形。
可以看到通过选择P’=1能直接得到解，因为 = u dL 。
In a similar manner, if the rotational displacement（转动位移）or slope（转角）of the tangent at a point on a structure is to be determined, a virtual couple moment（力偶矩）M’ having a “unit” magnitude is applied at the point. As a consequence（因此）, this couple moment cause a virtual load in one of the element of the body. Assuming that the real loads deform