10.1. de Saint Venant beam

10.1.1. Assumptions

10.1.2. Internal actions

10.1.2.1. Axial

10.1.2.2. Shear

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The axial equilibrium of an infinitesimal section of the beam between \(z\) and \(z+dz\) and \(y = y^*\), under linear axial stress, \(\sigma_z = \sigma_{z/y} \, y = \frac{M_x}{J_x} \, y\), reads for beams with constant section (todo is this assumption really required?)

\[\begin{split}\begin{aligned} 0 & = - \int_{x \in b(y*)} \int_{dz} \tau_{zy} \, dz \, dx + \int_{A^*(z+dz)} \sigma_z \, dz \, dy - \int_{A^*(z)} \sigma_z \, dz \, dy = \\ & \simeq - dz \int_{x \in b(y*)} \tau_{zy} + M_x(z+dz) \int_{A^*(z+dz)} \dfrac{y}{J_x} - M_x(z) \int_{A^*(z)} \dfrac{y}{J_x} = \\ & \simeq dz \left[ - \int_{x \in b(y*)} \tau_{zy} + M'_x(z) \dfrac{S^*(z)}{J_x} \right] \ , \end{aligned}\end{split}\]

with \(S^*(z) = \int_{A^*(z)} y\). Expliting the rotational equilibrium, \(0 = M'_x(z) - T(z)\), and the definition of the average shear stress \(\overline{\tau}_{zy} = \frac{1}{b(y^*)} \int_{x \in b(y^*)} \tau_{zy}\), it follows that

\[\overline{\tau}_{zy}(z,y) = \dfrac{S^*(y)}{b^*(y) J_x} T_y \ .\]

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Shear stiffness. With \(\gamma_{zy} = 2 \varepsilon_{zy} = \partial_y s_z + \partial_z s_y = \frac{\tau_{zy}}{G}\), and an equilibrated shaer load \(\widetilde{T}(z) = \widetilde{T}(z+dz) = 1\), so that \(\widetilde{M}(z+dz) = \widetilde{M}(z) + \widetilde{T}(z) d z\) with \(\widetilde{M}(z) = 0\), and \(\widetilde{\tau} = \frac{S^*}{b^* J} \widetilde{T}\), and \(\widetilde{\sigma} = \frac{\widetilde{M}}{J} y\), it follows

\[\begin{split}\begin{aligned} 0 & = \int_V \widetilde{\sigma}_{ij} \varepsilon_{ij} - \int_{S_D} n_i \widetilde{\sigma}_{ij} s_j = \dots \\ & = \int_V 2 \widetilde{\tau}(z) \frac{\tau(z)}{2 G} + \widetilde{T}_y(z) s_y(z) - \widetilde{T}_y(z+dz) s_y(z+dz) - \widetilde{M}_x(z+dz) \theta_x (z + d z) = \\ & = \int_{\ell} \int_A \dfrac{S^*}{b^* J} \widetilde{T} \dfrac{1}{G} \dfrac{S^*}{b^* J} T \, dA \, d\ell - \widetilde{T}_y(z) s'_y(z) \, dz - \widetilde{T}(z) dz \, \left( \theta(z) + \theta'(z) d z \right) \simeq \\ & = dz \left[ \dfrac{1}{GA} \ \underbrace{A \int_{A} \dfrac{S^{* 2}}{b^{* 2} J^2}}_{ \chi } \, T(z) - \left( s'_y(z) + \theta_x(z) \right) \right] \end{aligned}\end{split}\]

and thus

\[s'_y(z) + \theta_x(z) = \dfrac{\chi_y}{GA} T_y(z) \ ,\]

having introduced the definition of the shear factor \(\chi\) into the shear stiffness \(\frac{GA}{\chi}\).

Example 10.1 (Shear factor of a rectangular section)

As the static moment \(S^*(y)\) of a rectangular section with base \(a\) and height \(b\) reads

\[S^*(y) = a \left( \frac{b}{2} - y \right) \cdot \frac{1}{2} \left( \frac{b}{2} + y \right) = \frac{1}{2} \left( \frac{b^2}{4} - y^2 \right) a \ .\]

the shear factor \(\chi\) of a rectangular section is

\[\begin{split}\begin{aligned} \chi & = ab \, \int_{x = -a/2}^{a/2} \int_{y = -b/2}^{b/2} \dfrac{ \frac{1}{4} \left( \frac{b^4}{16} - \frac{b^2 y^2}{2} + y^4 \right)}{ b^2 \left(\frac{1}{12} a b^3\right)^2 a^2 } = \\ & = \frac{36 \, ab}{b^8} a \left( \dfrac{1}{16} - \dfrac{1}{6}\dfrac{1}{4} + \frac{1}{5} \dfrac{1}{16} \right) b^5 = \\ & = \frac{ab}{b^8} a \left( \dfrac{9}{4} - \dfrac{3}{2} + \frac{9}{20} \right) b^5 = \\ & = \dfrac{6}{5} \dfrac{a^2}{b^2} \ . \end{aligned}\end{split}\]

10.1.2.3. Bending

10.1.2.4. Torsion