（使..变形）the element at amount dL, the rotation can be found from the virtual-work equation 1 = u dL . This method for applying the principle of virtual work is often referred to as the method of virtual forces（虚力法）, since a virtual force is applied resulting in the calculation of a real displacement. The equation of virtual work in this case represents a compatibility requirement for the structure.
以相似的方法，如果要确定结构上某一点切线的转动位移或转角，可在该点上施加一个虚的单位力偶矩M’。因此，力偶矩在物体的某一单元中形成一个虚力。假定实际的力使单元的变形值为dL，则转角 可从虚功方程1 = u dL中得到。运用虚功原理的方法通常称为虚力法，因为施加虚力能计算出实际的位移。在这种情况下虚功方程代表着对结构的协调要求。
Although not important here, realize（意识）that we can also apply the principle of virtual work as a method of virtual displacements（虚位移）. In this case virtual displacements are imposed on（强加于）the structure, while the structure is subjected to real loadings. This method can be used to determine a force on or in a structure, so that the equation of virtual work is then expressed as an equilibrium requirement.
了解到我们也能运用虚功原理形成虚位移法，尽管在这儿不太重要。这种情况下，虚位移强加于结构，而结构承受实际的荷载。该法能用以确定结构上或结构中的力，因此虚功方程表示为平衡要求。
In 1879 Alberto Castigliano, an Italian railroad engineer, published a book in which he outline（概述）a method for determining the deflection or slope at a point in a structure, be it a truss, beam, or frame. This method, which is referred to as Castigliano’s second theorem（卡氏第二定理）, or the method of least work（最小功法）, applies only to structures that have constant temperature（恒温）, unyielding（不易弯曲）supports, and linear elastic material response.
1879年Alberto Castigliano，一个意大利的铁路工程师，出版了一本书，书中他概述了确定结构（无论是桁架、梁或框架）中某一点的挠度或转角的方法。这种称为卡氏第二定理或最小功法的方法仅应用于具有恒温、支座不易弯曲和材料线弹性响应的结构中。
If the displacement of a point is to be determined, the theorem states（表明）that it is equal to the first partial derivative（一阶偏导数）of the strain energy in the structure with respect to （关于）a force acting at a point and in the direction of displacement. In a similar manner, the slope at a point in a structure is equal to the first partial derivative of the strain energy in the structure with respect to a couple moment acting at the point and in the direction of rotation. 如果要确定某一点的位移，该定理表明位移等于结构中的应变能对于作用在该点并沿该位移方向的力的一阶偏导数。以相似的方式，结构上某一点的转角等于结构中的应变能对于作用在该点并沿该转角方向的力偶的一阶偏导数。
The derivation（推导）of the theorem requires that only conservative forces（保守力）be considered for the analysis. These forces do work that is independent of（与..无关）the path and therefore create no energy loss（能量损失）. Since forces causing a linear elastic response are conservative, the theorem is restricted to linear elastic behavior of the material. This is unlike the method of virtual force, which applies to both elastic and inelastic behavior.
定理的推导要求分析时只考虑保守力。这些力作功与路径无关，因此不造成能量的损失。由于引起线弹性响应的力是保守的，因此该定理（在运用时）被限制在材料的线弹性状态。它不象虚力法可以运用于弹性和非弹性状态。
When analyzing any indeterminate structure, it is necessary to satisfy equilibrium, compatibility, and force-displacement requirements for the structure. Equilibrium is satisfied when the reactive forces hold the structure at rest（保持结构静止）, and compatibility is satisfied when the various segments（部分）of the structure fit together（配合在一起）without intentional（故意的）breaks or overlaps（断裂或重叠）. The force displacement requirements depend upon the way the material responds, which in this chapter we have assumed linear-elastic response. In general there are two different ways to satisfy these requirements when analyzing a statically indeterminate structure: the force or flexibility method（柔度法）, and the stiffness or displacement method.
当分析任何不确定的结构时，有必要满足结构的平衡、协调和力-位移要求。当反力保持结构的静止时满足平衡（要求），当结构中不同的部分配合在一起而没有故意地断开或重叠时满足协调（要求）。力-位移要求依赖于材料响应的途径，在本篇我们已经假定为线弹性响应。当分析一个超静定结构时，一般有两种不同的方法来满足这些要求：力法或柔度法以及刚度法或位移法。
The force method was originally（最初地）developed by James Clerk Maxwell in 1864 and later refined（提炼）by Otto Mohr and Heinrich Muller-Breslau. This method was one of the first available（最早采用）for the analysis of statically indeterminate structures. As suggested（提示）by the name, the force method consists of writing equations that satisfy the compatibility and force-displacement requirements for the structure and involve（涉及）redundant forces（冗余力）as the unknowns. The coefficients（系数）of these unknowns are called flexibility coefficients. Since compatibility forms（形成）the basis for this method, it has sometimes been referred to as the compatibility method or the method of consistent displacements（位移协调法）. The redundant forces are determined by satisfying the equilibrium requirements for the structure.