Beam¶

5.1 Mechanics of a Beam¶

Sign convention¶

力：沿轴正向为正
力矩：统一逆时针为正

Elementary beam theory¶

挠度记为 \(v\),

\[ \begin{aligned} & EI \frac{\mathrm{d}^2 v}{\mathrm{d} x^2} = M(x) \\ & EI \frac{\mathrm{d}^3 v}{\mathrm{d} x^3} = F(x) \\ & EI \frac{\mathrm{d}^4 v}{\mathrm{d} x^4} = q(x) \\ & \sigma = -\frac{M(x) y}{I}. \end{aligned} \]

5.2 Beam Elements in Local Coordinates¶

Select a Displacement Function¶

\[ \begin{aligned} & v(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 \\[2ex] \text{BCs}: \quad & v(x = 0) = v_i, \quad v(x = L) = v_j, \\[1ex] & \frac{\mathrm{d} v}{\mathrm{d} x} (x = 0) = \theta_i, \quad \frac{\mathrm{d} v}{\mathrm{d} x} (x = L) = \theta_j \end{aligned} \implies \left \{ \begin{aligned} a_0 &= v_i \\ a_1 &= \theta_i \\ a_2 &= - \frac{3}{L^2}(v_i - v_j) - \frac{1}{L} (2 \theta_i + \theta_j) \\ a_3 &= \frac{2}{L^3}(v_i - v_j) + \frac{1}{L^2} (\theta_i + \theta_j) \end{aligned} \right. \]