材料力学第二章

τα
=
pα sinα
=
σ0 2
sin2α
σ max = σα =0 = σ 0
τ max
σ = α =45o
=
σ0 2
3
Examples
Known：F = 50 kN，A = 400 mm2 Ask for： Stresses on m-m plane
Solution: FN = −F
σ0
=
FN A
Sign conventions:Positive in tension拉为正, Negative in compression压为负.
例题：
求任一截面上的轴力，并画出轴力图。考虑自重，密度为ρ， 横截面积为A,长度为L。
ρgAL
FN
+
x
ρgA
x
1
¾ 回顾历史：
直
杆
伽
简
利 略 像
单 拉 伸 实
验
伽利略指出：
1.拉伸标准试样
标距
Test specimen
2. Tensile tests
GB/T6397-1986《金属拉伸试验试样》
n Test machine
o拉伸试验与F-Δl 曲线 Tensile test and load-extension curve
低碳钢拉伸应力－应变曲线 Tensile stress-strain curve for mild steel
E= tanα -elastic modulus 弹性模量
Chapter 2 Tensile & Compressive Stresses and Strength Properties of Materials 轴向拉压应力与材料的力学性能
z Stress in axially loaded bar z Strength Properties of Materials z Strength of axially loaded bar z Strength of Joints and connections §1 Introduction引言 §2 Axial Force and Axial Force Diagram轴力与轴力图 §3 Tensile and Compressive Stresses拉压杆的应力 §4~5 Mechanical Properties of Materials材料的力学性能 §6 Stress Concentration 应力集中 §7 Failure Criterion for Axially Loaded Bar轴向拉压强度
圣维南原理Saint-Venant's Principle 例题Examples
拉压杆横截面上的应力Stresses over the cross section 1.试验观察 Experimental observation
变形后横线仍为直线,仍垂直于杆件轴线,只是间距增大. Transversal line after deformation : straight; perpendicular to the axis.
应力均匀区Uniform-stress area
圣维南原理 Saint-Venant's Principle
力作用于杆端的分布方式，只影响杆端局 部范围的应力分布，影响区约为距杆端 1 倍的横向尺寸。在影响区外，应力的分布 与外力的作用方式无关（At a distance equal to, or greater than, the width of the member, the stress distribution may be assumed independent of the actual mode of application of the loads）。
body diagrams for each section；
设正法求轴力The axial forces can be assumed in positive
direction at first.
轴力Axial force ：通过截面形心并沿杆件轴线 (at the centroid of the cross- section, along the direction of the axis);
1. 如果C的重量越来越大，杆件最后总会象绳索一样断开； 2. 同样粗细的麻绳、木杆、石条、金属棒的承载能力各不相同； 3. 相同材料制成的杆件，承载能力与横截面积成正比，与其长度无关。
思考： 杆AB与杆A’B’材料相同， 杆A’B’的截面积大 于杆AB的截面积。
1、若所挂重物的重量相同，哪根杆危险？
2、若C’的重量大于C的重量，哪根杆危险？
A 细 杆
B
A’ 粗 杆
B’
C
Biblioteka Baidu
C’
FN A
<
?
FN′ A′
§3 Tensile and compressive stresses
轴向拉压应力
拉压杆横截面上的应力Stresses over the Cross-Section
拉压杆斜截面上的应力Stresses on an Oblique Plane
轴力图Axial Force Diagram
To find the axial forces （F1=F，F2=2F）
FR = F2 − F1 =F
AB ：
FN1 = F
BC ： FN2 + F = 0 FN2 = −F
要点：逐段分析轴力 The axial force is determined from free-
② no shear stress
τ = 0 σ = const
3.横截面正应力Stresses over the cross section
σ = FN A
FN :轴力Axial force;
横截面正应力公式
A:横截面的面积Cross-sectional area
两端受均匀分布载荷时锥形杆x方向正应力分布情况 α
横截面上 的正应力 均匀分布The
stress distribution on an cross-
section is uniform
横截面间 的纤维变 形相同Fibres between two cross-sections have the same deformation
斜截面间 的纤维变 形相同Fibres between two oblique sections have the same deformation
斜截面上 的应力均
匀分布
The stress distribution on
an oblique section is uniform
2. pα
∑ Fx = 0,
pα
A cos α
−
F
=
0
pα
=
Fcosα A
= σ 0cosα
3. σα 、τα and maximum stress
σα = pαcosα = σ 0cos2α
§4~5 Mechanical Properties of Materials
材料的力学性能 拉伸试验与应力－应变图Tensile Tests and Stress-Strain Diagram 低碳钢拉伸应力－应变曲线Tensile Stress-Strain Curve for Mild Steel 卸载与再加载路径Unloading and Reloading Path 名义屈服极限Conditional Yield Limit 脆性材料拉伸应力－应变曲线Stress-Strain Curves for Brittle Materials 复合与高分子材料的力学性能Strength Properties of Composite Materials
and Polymers 材料压缩时的应力－应变曲线Compressive stress-strain curve 温度对力学性能的影响Temperature Effect to Strength Properties
拉伸试验与应力－应变图Tensile tests and stress-strain diagram
=
−F A
=
− 50 × 103 N = 400 × 10−6 m2
−1.25 ×108
Pa
=
−125
MPa
α = 50o
σ 50o
=σ 0
cos 2α
= σ 0cos2 50o
= -51.6 MPa
τ 50o
=
σ 0 2
sin
2α
=
σ 0 2
sin 100o
=
-61.6 MPa
¾ 思考：
F
F
1、变形后两直线的夹角是否改变 2、如果改变，试定性解释为什么改变 3、如果改变，试定量分析角度的改变量
低碳钢拉伸时的应力－应变图
σ Slip line 硬化
Strain
σb
hardening
屈服Yield σσsp
线弹性
Linear elastic
α
o
Neck ε
σp-proportional limit 比例极限
σs-yield stress
屈服极限
σb-ultimate strength强度极限
在材料力学的习题中，一般假定外力是均匀地加在截面上。
Stresses on an Oblique Plane
1. Stresses on an Oblique Plane
F
斜截面应力 F
低碳钢拉伸时为什么会沿45°出现滑移线？
α: Positive when rotate anticlockwise from the x axis to the normal. 以x 轴为始边，逆时针转向者为正 斜截面上有何应力？What kinds of stresses on an oblique plane? 如何分布？ Distribution of the stress?
1.等直杆或小锥度杆Straight bar(or stepped bar) with uniform section, or with small taper ; 2.外力过轴线 The applied force P acts through the centroid of the cross section; 3.当外力均匀地加在截面上，此式对整个杆件都 适用，否则仅适用于离开外力作用处稍远的截面 The normal stress distribution in an axially loaded member is uniform, except in the near vicinity of the applied load (known as Saint-Venant's Principle) .
杆端镶入底座,横 向变形受阻 The transversal displacement is restricted.
Stresses over the cross section
σ = FN A
公式的适用范围 Necessary conditions for the equation to be valid
条件
§8 Strength of Joints and Connections连接部分的强度
Tension and compression
§1引言Introduction
bar §2轴力与轴力图Axial Force and Axial Force Diagram
杆件受力特点： 外力或其合力的作用线沿杆件轴线 External Force： along the direction of the axis 杆件变形特点： 轴向伸长或缩短 Deformation：extension or contraction, along the direction of the axis
F
F
α x
α=2.8o
α=11.3o
锥度2α ≤15o时σ，max 与 σ av 的相对误差<5%
2
Saint-Venant's Principle 圣维南原理
问题：杆端作用均布力，横截面应力均布. 杆端作用集中力，横截面应力均布吗? 加载点临域的应力分布
Stress distribution in the vicinity of the applied load
平面假设: 横截面仍保持为平面，且仍垂直于杆件轴线；
正应变沿横截面均匀分布
横截面上没有切应变
γ = 0 ε = const
2. Assumption based on deformation observation
(plane assumption)
① uniform distribution of stresses over cross section