Chapter 5: Stresses in Bending¶
5.1 Pure Bending¶
横轴面上只有弯矩 \(M\)，没有剪力 \(Q\)。
Experiment¶
- Lateral lines remain straight and rotate
- longitudinal straight lines change into curves and are still normal to the lateral lines
Concepts¶
- Neutral layer：中性面
- Neutral axis：中性轴
Hypothesis¶
- Hypothesis of plane section: The cross sections remain planes and only rotate through some angles around their neutral axes after deformation. 横截面变形后仍为平面。
- No normal stress between longitudinal fibers. 纵向层纤维之间无正应力。
5.2 Normal Stress On The Cross Sections Of The Beam In Pure Bending¶
- Geometric eq.
\(OO_1\) 为中性面。
\[ \begin{aligned} \varepsilon_x &= \frac{A_1 B_1 - AB}{AB} = \frac{A_1 B_1 - OO_1}{OO_1} \\ &= \frac{(\rho + y)d\theta - \rho d\theta}{\rho d\theta} = \frac{y}{\rho} \end{aligned} \]
中性面以下受拉，为正；上方受压，为负。
- Physical relation
纵向纤维之间无正应力，每一层相当于拉压杆
\[ \begin{equation} \tag{2} \sigma_x = E\varepsilon_x = \frac{Ey}{\rho} \end{equation} \]
- Static relations
\[\sum N_x = \int_A \sigma dA = \int_A \frac{Ey}{\rho} dA = \frac{E}{\rho} \cdot \underset{\text{静矩, Static moment}}{\boxed{\int_A y\,dA}} = \frac{ES_z}{\rho} = 0\]
\(\Longrightarrow S_z = 0\), so \(z\)-axis (neutral axis) is through the center of the section. 中性轴过形心。
\[\sum M_y = \int_A (\sigma dA)z = \int_A \frac{Eyz}{\rho} dA = \frac{E}{\rho} \cdot \underset{\text{惯性矩}}{\boxed{\int_A yz\,dA}} = \underset{\text{(Symmetric plane)}}{\frac{E I_{yz}}{\rho} \equiv 0}\]
\[\sum M_z = \int_A (\sigma dA)y = \int_A \frac{Ey^2}{\rho} dA = \frac{E}{\rho} \int_A y^2\,dA = \frac{E I_{z}}{\rho} = M\]
得到
\[\begin{equation} \tag{3} \frac{1}{\rho} = \frac{M}{E I_z} \end{equation}\]
- \(E I_z\) 为梁的抗弯刚度（flexural rigidity）
\[\begin{equation} \tag{4} \sigma_x = \frac{My}{I_z} \end{equation}\]
5.3 Normal Stress On The Cross Sections Of The Beam In Nonuniform Bending¶
5.4 Shearing Stress On The Cross Sections Of The Beam¶
Beam with rectangular sections¶
两个假设：
- 剪力和切应力平行
-
在横截面上，离中轴线距离相同的位置，切应力相同
-
梁上截取一小段 \(\mathrm{d}x\)
- \(y\) 方向截取小方块（Fig. b 中蓝色部分）
\[\sum X = N_2 - N_1 - \tau_1 b\, dx = 0\]
- \(N_1\) 为 \(\sigma\) 正应力的合力
\[ \left \{ \begin{aligned} N_1 &= \int_{A^*} \sigma\, dA = \frac{M}{I_z} \int_{A^*} y\, dA = \frac{M S_z^*}{I_z} \\ N_2 &= \frac{(M + dM)S_z^*}{I_z} \\ \tau_1 &= \frac{dM}{dx} \frac{S_z^*}{b I_z} = \frac{Q S_z^*}{b I_z} \end{aligned} \right. \]
From the theory of the conjugate shearing stress（切应力互等定理），
\[\tau = \tau_1 = \frac{Q S_z^*}{b I_z}\]
\[S_z^* = y_c^* A^* = \frac{\frac{h}{2}+y}{2} b (\frac{h}{2} - y) = \frac{b}{2} (\frac{h^2}{4} - y^2)\]
So
\[\tau = \frac{Q}{2 I_z} (\frac{h^2}{4} - y^2)\]
\[\tau_{\max} = \tau|_{y = 0} = \frac{3}{2} \frac{Q}{A} = \frac{3}{2} \bar{\tau}\]
剪应力的大小分布是抛物线，最大剪应力为平均剪应力的 \(1.5\) 倍。
Beam with other shapes of sections¶
I-section（工字梁）¶
\[\tau_{\max} \approx \frac{Q}{A_f}\]
-
\(A_f\) 为腹板的面积
-
翼缘（flange）：几乎只承受正应力
- 腹板（web）：几乎只承受铅垂剪应力
Circular section¶
\[\tau_{\max} = \frac{4}{3} \frac{Q}{A} = \frac{4}{3} \bar{\tau}\]
Thin-walled cirque¶
\[\tau_{\max} = 2 \frac{Q}{A} = 2 \bar{\tau}\]
5.5 Strength Conditions Of \(\sigma\) And \(\tau\)¶
\[\begin{equation} \tag{5} \sigma_{\max} = \frac{M_{\max}}{W_z} \leq [\sigma], \quad \tau_{\max} = \frac{Q_{\max} S_{z\max}^*}{b I_z} \leq [\tau] \end{equation}\]
同时有正应力和切应力时，上面的强度条件可能会失效。第七章将介绍新的强度条件。
Appendix I: Geometrical Properties Of Plane Areas¶
I.1 Intro¶
- Tension \(\(\sigma = \frac{N}{A}\)\)
- Torsion:
I.2 Centroid & Static Moment¶
\[\begin{aligned} \bar{y} &= \frac{\int y\, dA}{A} \\ \bar{z} &= \frac{\int z\, dA}{A} \\ S_z &= \int y\, dA \\ S_y &= \int z\, dA \end{aligned}\]
Notes
\[S_z = \bar{y} A, \, S_y = \bar{z} A\]
\[S_y = 0, \, S_z = 0 \iff \bar{z} = 0, \, \bar{y} = 0\]
- Centroid and static moment of the composite area
\[S_z = \int_A y\, dA = \sum_i \int_{A_i} y\, dA_i = \sum_i A_i \bar{y}_i\]
I.3 Moment of Inertia of an Area¶
-
惯性积（Product of inertia）
-
极惯性矩（Polar moment of inertia）