为解决不同的问题，发现弯距图面积法是一种很便利的方法。但当遇到更先进的结构时，此法会变得非常复杂，需要进一步地展开。对你来说在目前阶段了解此法的基本原理已经足够了。引入长半径曲梁的挠度来举例说明构成弯距图面积法基础的原理的功效，使你能知道直梁与曲梁之间的区别。
This chapter afforded an opportunity to become familiar with singularity functions （奇异函数）, and you have seen that certain problems can be greatly simplified by their use. It must be appreciated（意识到）that merely an introduction to the topic has been given; there is much more to learned by those who have a special interest. To illustrate a serious limitation（缺陷） at our present stage, we can express distributed loads （分布荷载） that are variable and are intermittent, but we cannot write a load function for concentrated loads. If we had taken the next step and dealt with the concentrated load, we would have encountered the source of the expression（表达式）“singularity function”, but having regard for（考虑）the scope of this book we have stopped short of（达不到）that step.
本节使你熟悉了奇异函数，并发现通过利用它们能大大地简化某些问题。但必须意识到仅仅是介绍了题目，对那些有特殊兴趣的人还有很多要学。我们可以表示变化的、间断的分布荷载，但不能写出集中荷载的荷载函数，说明了在我们目前阶段（该函数）还存在着严重的缺陷。如果我们进入下一步去研究集中荷载，便会遇到奇异函数表达式的来源，但是考虑到本书的范围，我们不再进入那一步。 Failure Theories 失效理论
In the design of a member subjected to a uniaxial（单轴的） load, the stress was compared with the stress to cause failure in test specimens（试件）that had also been subjected to uniaxial load. This is the simplest of all design problems; the method is quite adequate（合适的）, since the nature（性能）of the loads and the stresses in the test and in the part being designed are identical. However, we soon encounter cases where the member being designed is not so simple and the stresses are not uniaxial; consider, for example, the stresses in the web of a beam or in a pressure vessel（压力容器）. In these cases we know that the stress is two-dimensional（两向的）or biaxial and it may, in other cases, be three-dimensional, or triaxial.
For a structure having biaxial or triaxial stresses, how should we check the safety of the design? The most obvious way would be to conduct tests（进行） in which specimens are stressed（受力）to failure in the same multiaxial（多轴的）manner as in the structure; the allowable multiaxial stress then be determined by the application of an adequate safety factor. However, this would require a group of tests for every new set of multiaxial stresses that occurred in design. Such tests are difficult to perform, and the cost of performing them in the required numbers would be prohibitive. Consequently, we need a theory by which the results of the standard uniaxial test can be used to predict（预测）the failure of a part made of the same material when the stresses are multiaxial. In other words（换句话说）, we need a failure theory. 在设计承受轴向力的构件时，将其应力与导致同样承受轴向力的试验样本（试件）失效的应力相比。这是所有设计问题中最简单的；该法是非常合适的，因为试验和设计中的荷载和应力性质是完全相同的。但是，不久我们便会遇到正在设计的构件不是那么简单，其应力也不是单轴向的；例如，考虑梁的腹板应力或压力容器中的应力。在这些情况下，我们知道其应力是两向的或两轴的，而在其他情况下可能是三向或三轴的。对一个有着两轴或三轴应力的结构，我们应该如何检查它的设计安全性？最显然的办法是进行试验，即试件以与结构相同的多轴受力方式失效；然后运用适当的安全系数确定许用的多轴应力。但是，对设计中出现的每组新的多轴应力都将需要一组试验。这样的试验很难进行，而且以需要的数量进行试验的费用也是禁止的。因此，我们需要一个理论，根据它可以通过利用标准单轴试验的结果来预测同样材料制作的构件在承受多轴应力时的失效。换句话说，我们需要一个失效理论。
To illustrate the need for a failure theory, let us consider a cylindrical pressure vessel. To avoid unnecessary complications, we will consider that all welds（焊缝）are 100% efficient and that the walls（容器壁）are thin. Under internal pressure the main stresses（主应力） are circumferential and longitudinal, and it was implied（认为）in an earlier case that only the circumferential stress, because it is larger than the longitudinal stress, needs to be considered in judging the adequacy of the design. In this approach we tacitly（默认）assumed that the maximum stress could be treated as（看作为）a uniaxial stress and that it alone determined the safety of the design. The longitudinal stress was not considered although it may, without our knowledge（在我们的知识之外）, have had an influence on strength. It happens that our approach in this case is acceptable, but, in a biaxial state of stress, the second stress is not always inconsequential（不重要）and an understanding of failure theory is necessary in order to avoid making some serious errors.
为了举例说明需要一个失效理论，让我们考虑一个圆柱形的压力容器。为避免不必要的复杂，我们认为焊缝完全有效，容器壁是薄的。在内部压力下，主应力是环向和纵向的，由于环向应力比纵向应力大，因此，在一个较早的例子中认为只有环向应力需要在判断设计的适用性时加以考虑。在这个方法中，我们默认地假定最大的应力（即环向应力）可看作为单轴应力，并由它单独地确定设计的安全性。尽管在我们的知识以外，纵向应力可能会对强度有影响，但不被考虑。正巧，在这种情况下我们的方法能被接受，但是，在两轴应力状态下，第二种应力不总是不重要的，为了避免造成一些严重的错误，对失效理论的理解是必要的。