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Shear flow for stringer-web sections

Which led to: \[\begin{align} q =& - \frac{I_{xx} S_x - I_{xy} S_y}{I_{xx}I_{yy}-I_{xy}^2} \int_0^s x t(s) ds \\ & - \frac{I_{yy} S_y - I_{xy} S_x}{I_{xx}I_{yy}-I_{xy}^2} \int_0^s y t(s) ds \\ & + q (s=0) \\ \end{align}\]

• The integrations are related to the first moment of the area measured from the centroid. Recall that an integration is a sum of a series of very small pieces.

• When stringers are present, we make the assumption that the stringers carry all the bending load.

  • Thus, \({\sigma_{zz}}\) is zero in web and non-zero in the stringers. Thus, in the web: \[{\frac{\partial q}{\partial s}} = 0\] and we conclude that the shear flow is approximately constant in these web segments. \[q = \mathrm{constant}\]

The change in shear flow that occurs over a stringer section can be determined by examining the integral. \[\begin{align} \int_0^s x t(s) ds =& \; \displaystyle \sum_{k=1}^i x_k A_k = Q_{xi}\\ \int_0^s y t(s) ds =& \; \displaystyle \sum_{k=1}^i y_k A_k = Q_{yi}\\ \end{align}\] where \(A_k\) is the area of stringer section \(k\), \(x_k\) and \(y_k\) are the distances from the centroid of the area.

Thus, we have “step changes” in the shear flow at each stringer.

Example

Consider:

Assume:

• \(A_4=A_1\), \(A_3=A_2\)

The shear flow, \(q_i\) produced by a vertical shear force \(S_y\) is: \[q_i=\frac{-S_y Q_i}{I_z}\] where \[Q_i = \displaystyle \sum_{k=1}^i y_k A_k\]

\[I_{xx} = 2 h^2 (A_1 + A_2)\] and \(y_k\) is the vertical position of the stringer \(A_k\).

for \(q_1\): \[Q_1 = A_1 h\]

Thus: \[\begin{align} q_1 = - \frac{S_y A_1 h}{2 h^2 (A_1+A_2)} \\ q_1 = - \frac{S_y A_1}{2 h (A_1+A_2)} \end{align}\]

Similarly: \[\begin{align} q_2 =& \, - \frac{S_y h (A_1 +A_2)}{2 h^2 (A_1+A_2)} \\ =& \, - \frac{S_y}{2 h} \\ q_3 =& \, - \frac{S_y h (A_1 + A_2 - A_2) }{2 h^2 (A_1+A_2)} \\ =& \, - \frac{S_y A_1}{2 h (A_1+A_2)} \\ \end{align}\] Note, the direction of shear flow is opposite that shown in the figure.

We could sanity check the compute shear flows sum to the applied shear force resultants.

Shear flow in multicells with stringer-web beams

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

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

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

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: \[q_1 = q_2 + q_3 - A \frac{d {\sigma_{zz}}}{dz}\]

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. \[q_1 = q_2 + q_3 - \frac{S_y A y}{I_{xx}}\]