10.9 Shear flow in multicells with stringer-web beams

We have established that multiwalled sections have junctions where three or more shear flows meet.

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The shear flows for torque sum such that: \[q_3=q_1-q_2\]

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However, if there are concentrated areas, the equation is not valid. Consider:

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Then: \[\begin{align} \displaystyle \sum F_z = 0 \\ ({\sigma_{zz}}+ \Delta {\sigma_{zz}}) A - {\sigma_{zz}}A + q_1 \Delta z - q_2 \Delta z - q_3 \Delta z = 0 \end{align}\] Thus: \[q_1 = q_2 + q_3 - A \frac{d {\sigma_{zz}}}{dz}\]

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Consider that a bending force may be present.

For example, if \(S_y\) existed: \[\begin{align} {\sigma_{zz}}= \frac{M_x y}{I_{xx}} \\ \frac{d M_x}{d z} = S_y \\ \end{align}\] \(y_i\) is the position of the stringer.

Substitute this into the above equation and we see the shear continuity equation requires modification. \[q_1 = q_2 + q_3 - \frac{S_y A y}{I_{xx}}\]