第1章 轴向拉伸与压缩
Chapter 1 Axial Tension and Compression
1.1、 轴向拉伸与压缩的概念
The Concept of Axial Tension and Compression
1. 拉杆：Tensile Rod 2. 压杆: Compressive Rod
3. 受力特点：外力合力的作用线与杆轴线重合
Characteristic of the External Forces: The acting line of the resultant of external forces is coincided with the axis of the rod.
4. 变形特点：杆沿轴向伸长或缩短
Characteristic of Deformation: Rod will elongate or contract along the axis of the rod.
图1.1 轴向拉伸与压缩受力和变形示意图
Fig. 1.1 the axial tension and compression stress and deformation diagram 轴向拉伸：杆的变形是轴向伸长，横向缩短。
Axial tension ：Deformation of the rod is axial elongation and lateral shortening. 轴向压缩：杆的变形是轴向缩短，横向变粗。
Axial compression：Deformation of the rod is axial shortening and lateral
enlargement.
1.2、轴向拉伸和压缩时的内力、轴力图
The internal force, the axial force diagram of Axial tension and compression （1）内力的概念：物体内部一部分与另一部分的相互作用力，构件受到外力作用的同时，在内部产生相应内力（外力作用引起的内力改变量）。 Internal force：Internal force is the resultant of internal forces, which is acting mutually between two neighbour parts inside the body,caused by the external forces.
内力的计算是分析构件强度、刚度、稳定性等问题的基础。求内力的一般方法是截面法。
Calculation of the internal forces is the foundation to analyze the problems of
strength、rigidity、stability etc. The general method to determine internal forces is the method of section.
（2）截面法的基本步骤：
① 截开：在所求内力的截面处，假想地用截面将杆件一分为二。 ② 代替：任取一部分，其弃去部分对留下部分的作用，用作用在截
开面上相应的内力（力或力偶）代替。
③ 平衡：对留下的部分建立平衡方程，根据其上的已知外力来计算
杆在截开面上的未知内力（此时截开面上的内力对所留部分而言是外力）。
Basic steps of the method of section：
① Cut off：Assume to separate the rod into two distinct parts in the section in which the internal forces are to be determined.
②Substitute：Take arbitrary part and substitute the action of another part on it by the corresponding internal force in the cut-off section.
③Equilibrium：Set up equilibrium equations for the remained part and determine the unknown internal forces according to the external forces acted on it.（Here the internal forces in the cut-off section are the external forces for the remained part）
（3）轴力N的正负号规定：
N 与外法线同向，为正轴力(拉力)，N 与外法线反向，为负轴力(压力) Sign conventions for the axial force:
axial force N (tensile force)is positive when its direction point to the
outward direction of the normal line of the section, (compressive force)negative inward
（4）轴力图—— N (x) 的图象表示。①反映出轴力与横截面位置变化关系，较直观；②确定出最大轴力的数值及其所在横截面的位置，即确定危险截面位置，为强度计算提供依据。
Diagram of the axial force— sketch expression of N (x). ①Reflected the
variety relation between the corresponding axial force and the position of the
section. ②Find out value of the maximum axial force and the position of the section in which the maximum axial force act. That is to determine the position of the critical section and supply the information for the calculation of strength. 1.3、横截面上的应力 Stress on the section
（1）应力的概念 Concept of stress 应力：由外力引起的内力集度
Stress ：Intensity of the internal force due to the external forces.
工程构件，大多数情形下，内力并非均匀分布，集度的定义不仅准确而且重要，因为“破坏”或“失效”往往从内力集度最大处开始。
Under most cases distribution of the internal force inside engineering members is not uniform. Definition of intensity is neither accurate and important because breakage or failure often begins from the point at which intensity of the internal force is maximum.
应力可分为正应力s和切应力t（剪应力）。 Stress can be dived into normal stress and shear stress
?N（垂直于作用截面）
?A?0?A?NNormal stress: ??lim(perpendicular to the section)
?A?0?A?Q切应力??lim（位于作用截面）
?A?0?A?QShear stress: ??lim(Stress lying in the section)
?A?0?A正应力：??lim（2）轴向拉压时的应力计算
Calculation of Stress in the cross section of the rod in tension or compression 平面假设：原为平面的横截面在变形后仍为平面。纵向纤维变形相同。 Hypothesis of plane section：Cross sections remain planes before and after deformations . Deformations of longitudinal fibers are the same
根据平截面假设和圣维南原理，在离加力点一定距离之外，横截面上各点的纵向变形是均匀的，内力分布也是均匀的，并且垂直于横截面。
According to hypothesis of plane section and Saint-Venant principle, In the distance from the loading point on the cross section, longitudinal deformation of each
point is uniform, the internal force distribution is uniform, and the cross section perpendicular to the cross section.
L拉压杆横截面上的应力：设横截面积为A，则有拉伸（或压缩）正应力：
P A??Stress in the cross section of the rod in tension or compression: Supposes the section area is A, then has the tension (or compression) the normal stress: ??1.4、拉压变形与胡克定律
Deformation of the rod in tension or compression and Hooke's law (1) 弹性变形: Elastic Deformation (2) 塑性变形: Plastic Deformation (3) 纵向应变: Longitudinal Strain (4) 横向应变: Lateral Strain
(5) 线弹性变形:Linear Elastic Deformation (6) 泊松比:Poisson’s ratio
(7)弹性模量-E：表示材料抵抗拉压变形的 能力
E- modulus of elasticity：Indicates the capability of materials for resisting tension or compression
(8) 抗拉刚度-EA:表示构件抵抗拉压变形的能力
EA-the axial rigidity: Indicates the capability of constructive members for resisting tension or compression
(9)胡克定律:当应力不超过材料的比例极限时，应力与应变成正比. Hooke’s Law: The stress is proportional to the strain within the elastic region. 1.5、材料拉压时的力学性能
Mechanical Properties of Materials with Tensile Load (1) 标准试件: Specimen
(2) 低碳钢(C≤0.3%): Low Carbon Steel
P A
(3) 弹性阶段:Elastic Region (4) 屈服阶段:Yielding Stage
(5) 强化阶段:Hardening Stage (6) 颈缩阶段: Necking Stage
(7) σp----比例极限: Proportional Limit (8) σe----弹性极限: Elastic Limit (9)σs----屈服极限: Yielding Stress (10)σb----强度极限: Ultimate Stress (11) 延伸率: Percent Elongation
?n?l1?l0?100% l0(12) 断面收缩率: Percent Reduction of Area ??A0?A1?10%0 A0(13) 塑性材料: Ductile Materials (14) 脆性材料: Brittle Materials (15) 铸铁:Cast iron
1.6、轴向拉伸和压缩时的强度计算
Strength calculation of axial tension and compression (1) 许用应力、极限应力、安全系数 permissible stress、limit stress、 Safety factor
第1章轴向拉伸与压缩
