Lesson Plan Review New Words and Phrases Reading and Translating—Unit Three Classroom Exercises

Review nonrigid body internal forces and deformations stress-strain diagram gray cast iron modulus of elasticity the slope of the straight-line portion normal stress and strain Plastic deformation independent of the time duration of the applied load is known as slip. yield point—yield strength ~ ultimate strength permanent set ductile materials ~ brittle materials

Unit Three Contents of Theory of Elasticity 单 元 3 弹性力学理论内容

The theory of elasticity, often called elasticity for short, is the branch of solid mechanics, which deals with the stresses and deformations in elastic solids produced by external forces or changes in temperature.

弹性力学理论，常简称为弹性力学， 是固体力学的一个分支，它研究弹性固体 中由于外力作用或温度变化而产生的应力 和变形。

For students of various engineering disciplines, the purpose of studying elasticity is to analyze the stresses and displacements of structural or machine elements within the elastic range and thereby to check the sufficiency of their strength, stiffness and stability.

对不同工程学科的学生来说，学习弹性力学 的目的意在分析弹性范围内各种结构或机械构件 的应力和位移，校核它们是否具有足够的强度、 刚度和稳定性。

Although this purpose is the same as that of studying mechanics of materials and structural mechanics, these three branches of solid mechanics do differ from one another both in the objects studied and in the methods of analysis used.

尽管研究弹性力学与研究材料力学和结构 力学的目的相同，但是固体力学的这三个分支 无论在研究对象上还是在分析方法上均互不相 同。

Mechanics of materials deals essentially with the stresses and displacements of a structural or machine element in the shape of a bar, straight or curved, which is subjected to tension, compression, shear, bending, or torsion.

材料力学本质上研究结构或机械构件的应力 和位移，这些构件常以直杆或曲杆的形状出现， 并承受拉伸、压缩、剪切、弯曲和扭转等荷载的 作用。

Structural mechanics, on the basis of mechanics of materials, deals with the stresses and displacements of a structure in the form of a bar system, such as a truss or a rigid frame.

建立在材料力学基础上的结构力学研究结 构物的应力和位移，这种结构物常以杆件体系 形式出现，如桁架或刚架。

As to the structural elements, which are not in the form of a bar, such as blocks, plates, shells, dams and foundations, they are analyzed only in the theory of elasticity. Moreover, in order to analyze a bar element thoroughly and precisely, it is necessary to apply the theory of elasticity.

至于非杆件体系结构，如块、板、壳、堤 坝、地基等实体结构则在弹性力学里进行研究， 而为了全面、精确地分析杆状构件，也必须采 用弹性力学理论。

Although bar-shaped elements are studied both in mechanics of materials and in theory of elasticity, the methods of analysis used in the two subjects are not entirely the same.

虽然材料力学和弹性力学均研究杆状 构件，但二者采用的分析方法不尽相同。

When such an element subjected to external loads is studied in mechanics of materials, some assumptions are usually made on the strain condition or the stress distribution. These assumptions simplify the mathematical derivation to a certain extent, but often inevitably reduce the degree of accuracy of the results obtained.

材料力学研究承受外荷载的杆状构件时， 往往对构件应变状态或应力分布做出假设，这 就在一定程度上简化了数学推导，但常常不可 避免地降低了所获结果的精确程度。

In elasticity, however, the study of a bar- shaped element usually does not need those assumptions. Thus the results obtained are more accurate and may be used to check the approximate results obtained in mechanics of materials.

然而在弹性力学中，对杆状构件研究 往往不需要那些假设，因此，所得结果比 较精确，并可用于校核材料力学里所得到 的近似的解答。

For example, when the problem of bending of a straight beam under transverse loads is analyzed in mechanics of materials, it is assumed that a plane section of the beam remains plane after bending. This assumption leads to the linear distribution of bending stresses.

例如，在材料力学中研究一个在横向荷载 作用下直梁的弯曲问题时，就假定梁的平截面 在弯曲后仍保持平面。这就导致横截面上的弯 应力按直线分布。

In the theory of elasticity, however, one can solve the problem without this assumption and prove that if the depth of the beam is not much smaller than the span length, the stress distribution will be far from linear variation, and the maximum tensile stress is seriously undervalued in mechanics of materials.

但在弹性力学理论里，不需引用这样假定 同样可以解决类似问题，并能证明如果梁的高 度并不远小于梁的跨度时，那么横截面上的弯 应力远非直线分布，而且材料力学大大地低估 了最大拉应力。

Another example is the calculation of stress in a prismatic tension member with a hole. It is assumed in mechanics of materials that the tensile stresses are uniformly distributed across the net section of the member,

又例如，在材料力学里计算有孔的等截面 拉伸构件的应力时，通常假定拉应力在净截面 上均匀分布，

whereas the precise analysis in the theory of elasticity shows that the stresses are by no means uniform, but are concentrated near the hole; the maximum stress at the edge of the hole is far greater than the average stress across the net section.

然而弹性力学理论的精确分析表明净截面 上的拉应力绝非均匀分布，而是集中在孔的附 近，孔边缘的最大应力远远超过净截面上的平 均应力。

Before the twentieth century, bar systems were formally analyzed only in structural mechanics and not in elasticity. In spite of this convention, in this century many engineers used a joint application of the two branches of solid mechanics, with the mutual infiltration of the two as a result.

在 20 世纪以前，杆件体的正式分析只出现 在结构力学而不是弹性力学里。尽管此习惯仍 然存在，但本世纪的许多工程师还是采用相互 渗透的方法，综合应用固体力学中这两个学科 的知识。

The utilization of various methods of analysis in structural mechanics greatly strengthened the theory of elasticity and thus enabled engineers to obtain the solutions of many complicated problems in elasticity. Although these solutions are approximate theoretically, they prove to be scientifically accurate for engineering designs.

结构力学中多种分析方法的应用大大增强 了弹性力学的理论，因而使工程师们能获得弹 性力学中许多复杂问题的解答。尽管这些解答 在理论上有一定的近似性，但对工程设计来讲 是有足够的精确度。

Classroom Exercises For example, using the finite element method developed in the last thirty years, we can solve a problem in elasticity by the discretization of the body concerned and then the application of the displacement method, the force method, or the mixed method in structural mechanics. This is a brilliant example of the joint application of the two branches of solid mechanics. Moreover, in the design of a structure, we can utilize the different branches of solid mechanics for different members of the structure, and even for different parts of a single member, to get the most satisfactory results with the least amount of work.