Chapter 2: Tension, Compression & Direct Shear¶

2.1 Axial Tension and Compression¶

Historical perspective

Before Industrial Revolution:

Timber, Brick, & Mortar
\(\Rightarrow\) Compressive Loads

After Industrial Revolution:

Metals, Polymers, Plastics, etc
\(\Rightarrow\) Tensile Loads

Internal force on cross section¶

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Force feature: 外力沿着轴线方向
Deformation feature: 杆件沿轴线方向拉伸或压缩

Axial force diagram（轴力图）

Example 2-1

切两个截面 1-1 和 2-2

Solution 2-1

轴力图：

轴力图

2.2 Internal Force & Stress¶

Stress on Cross Section¶

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\[N = \int_A \sigma \, dA\]

Plane cross-section assumption（平截面假设）

If the cross section is initially plane, it remains plane after deformation and perpendicular to the axial line.

在截面处，应力分布是均匀的，所以有

\[\sigma = \mathrm{const.}\]

由平截面假设，可以得到

\[\begin{equation} \tag{2-1} \sigma = \frac{N}{A} \end{equation}\]

若 \(N(x), A(x)\) 变化不剧烈，则下式也成立：

\[\sigma(x) = \frac{N(x)}{A(x)}\]

Saint-Venant’s Principle

如果作用在弹性体一小块表面上的力被静力等效力系替代，这种替换仅仅会使局部表面产生显著的应力变化，而在比应力变化表面的线性尺寸更远的地方，其影响可忽略不计。

影响的区域：圣维南区域

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Example 2-2

Internal Force on Inclined Section¶

\(A_{\alpha}\) 为斜截面面积，\(P_{\alpha}\) 为斜截面上的内力

\[A_{\alpha} = \frac{A}{\cos \alpha}, \quad P_{\alpha} = P\]

得到

\[p_{\alpha} = \frac{P_{\alpha}}{A_{\alpha}} = \frac{P}{A} \cos \alpha = \sigma \cos \alpha\]

正应力 \(\sigma_{\alpha} = p_{\alpha} \cos \alpha = \sigma \cos^2 \alpha\)，剪应力 \(\tau_{\alpha} = p_{\alpha} \sin \alpha = \sigma \sin \alpha \cos \alpha\)

Discussions:

\(\alpha = 0\)，\(\max \sigma_{\alpha} = \sigma\)
\(\alpha = 45^{\circ}\)，\(\max \tau_{\alpha} = \frac{\sigma}{2}\)
\(\alpha = 90^{\circ}\)，\(\sigma_{\alpha} = \tau_{\alpha} = 0\)（自由面）

2.3 Mechanical Properties of Materials Under Tension/Compression¶

Strength
Hardness
Toughness（韧性）
Elasticity
Plasticity
Brittleness（脆性）
Ductility, Malleability（延展性）

应力和应变：

\[\sigma = \frac{P}{A}, \quad \varepsilon = \frac{\Delta l}{l}\]

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Elastic Range¶

Hooke's Law:

\[\sigma = E \varepsilon\]

\(E\): Elastic Modulus / Young's Modulus（弹性模量/杨氏模量）

属于本构关系

Proportional Limit \(\sigma_{\mathrm{p}}\)
Elastic Limit \(\sigma_{\mathrm{e}}\)

Yielding Range¶

Upper yielding limit: unstable
Lower yielding limit: stable \(\sigma_{\mathrm{s}}\) reflects the strength of materials.（屈服应力）
Yielding is related to the maximum sheering stress.

材料进入屈服即失效。

Hardening Range¶

在 d 处卸载，应力退为零，但应变不为零，从弹性转变为了塑性

Ultimate Strength \(\sigma_{\mathrm{b}}\)

Necking Range¶

颈缩：几何缺陷导致失稳

f 处断裂

Elongation & Reduction of Area¶

Residual relative elongation \(\delta = \frac{l - l_0}{l_0}\)
\(\delta < 5\%\)：Brittle materials
\(\delta > 5\%\)：Ductile materials
Permanent relative reduction of area \(\psi = \frac{A_0 - A}{A_0}\)

Unloading and Cold-hardening¶

d 处卸载，斜率也为 \(E\)
重新加载，和卸载的曲线重合，但是强度提高（\(\sigma_{\mathrm{p}}\) 增大）

工艺：冷硬化

Mechanical Properties of other Materials in Tension¶

对于塑性材料，若没有明显的屈服，工程上约定 \(\sigma_{\mathrm{p0.2}}\)（offset strain = 0.2）为屈服应力
对于脆性材料，线性部分不明显，用割线代替

Mechanical Properties of Materials under Compression¶

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2.4 Criterion of Strength Design¶

Safety Factor and Allowable Stress¶

Failure
- Ductile: Plastic deformation
- Brittle: Fracture
Limit stress \(\sigma_{\mathrm{u}}\)
- Ducitle: \(\sigma_{\mathrm{s}}\)
- Brittle: \(\sigma_{\mathrm{b}}\)
Allowable stress \(\left[\sigma \right] = \frac{\sigma_{\mathrm{u}}}{n}\)（许用应力）
- \(\frac{\sigma_{\mathrm{s}}}{n_{\mathrm{s}}}\): Ductile, \(n_{\mathrm{s}} = 1.2 \sim 2.5\)
- \(\frac{\sigma_{\mathrm{b}}}{n_{\mathrm{b}}}\): Brittle, \(n_{\mathrm{b}} = 2 \sim 3.5\)

Criterion of Strength Design¶

\[\sigma = \frac{F_N}{A} \leq \left[\sigma \right]\]

Strength checking: \(\sigma \leq \left[\sigma \right]\)
Cross-section designing: \(A \geq \frac{F_N}{\left[\sigma \right]}\)
Allowable load determining: \(F_N \leq A \left[\sigma \right]\)

How to determine the safety factor?

用途越关键，安全系数越大（保守）
辅助作用，安全系数较小（经济）

Example 2-3

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Example 2-4

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Example 2-5

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注意图中的 ┘└ 记号，表示有两块角钢！槽钢同理也有两块。横截面积要 \(\times 2\)。

2.5 Deformation of Bar under Tension and Compression¶

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Axial Strain¶

\[ \begin{aligned} \varepsilon &= \frac{\Delta l}{l} \\ \sigma &= E \varepsilon \\ \sigma &= \frac{F_N}{A} \\[2ex] \Rightarrow \Delta l &= \frac{F_N l}{E A} \end{aligned} \]

\(EA\): tensile rigidity（拉伸刚度）

可以看到，这就是胡克定律。

Lateral Strain¶

\[ \Delta b = b_1 - b, \, \varepsilon' = \frac{\Delta b}{b} \]

在线弹性范围内，\(\varepsilon' / \varepsilon = \mathrm{const.}\) 称为泊松比 \(\mu\)。

\[\mu = -\frac{\varepsilon'}{\varepsilon}\]

从热力学上可以证明，\(\mu \in (-1, 0.5)\)。

\(\mu = 0.5\)：变形后总体积不变
\(\mu < 0\)：材料拉长，横截面还变大（负泊松比材料）

For steel, \(E \sim 200 \text{ GPa}\), \(\, \mu \sim 0.3\)

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找两根铰接杆变形后的交点

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\(C'\)：精确的交点
\(C''\)：近似的交点

Example 2-6

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Non-uniform bars¶

横截面积分段变化

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\(\(\Delta l = \sum \Delta l_i = \sum \frac{N_i l_i}{E_i A_i}\)\)

通常情况下轴力 \(N_i\) 相同 - 横截面积连续变化

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\(\(\Delta l = \int \frac{N(x) dx}{E A(x)}\)\)

2.6 Strain Energy of Bar under Tension and Compression¶

应变能：物体在外力作用下发生形变时，吸收的能量

\[dW = F d(\Delta l)\]

\[ \begin{aligned} U = W = \int_0^{\Delta l} F d(\Delta l) &= \int_0^{\Delta l} \frac{EA \Delta l}{l} d(\Delta l) = \frac{1}{2} F \Delta l \\ &= \frac{F^2 l}{2EA} = \frac{EA (\Delta l)^2}{2l} \end{aligned} \]

应变能密度

\[u = \frac{U}{V} = \frac{P \Delta /2}{Al} = \frac{1}{2}\sigma \varepsilon = \frac{1}{2}E \varepsilon^2 = \frac{\sigma^2}{2E}\]

2.7 Statically Indeterminate Structures¶

约束反力多于独立平衡方程数，需要增加几何变形约束方程和本构方程求解。

静不定结构：结构的强度和刚度均得到提高

求解：

Equilibrium equation
Constitutive equation
Compatibility equation
- 变形协调方程，几何约束