User:Eas4200c.f08.blue.b/Lecture 36

A plan for attacking the problem of solving for the shear flow in a multi-cell section with stringers was introduced.

This miniPLAN is shown below. S) Singe-cell sections                   S.1)  without stringers S.2) with stringers               M)  Multi-cell sections M.1) without stringers                   M.2)  with stringers

First, a single-cell section with no stringers(as in Figure 1) is analyzed to find the shear flow q through the entire section (S.1 from the miniPLAN). The section can be viewed as a thin walled cross section. As is shown below, a thin walled cross section cannot resist transverse shear (Vz).



Figure 1: Shear flow (q) in a single-cell section

Next, a single-cell section with stringers is analyzed as is shown in. The solution to this problem (P) can be analyzed by decomposing the problem into two parts: A section with stringers must be considered without stringers (as in problem P1 in Figure 3) and an open cross section with stringers (as in problem P2 in Figure 4).



Figure 2: Single-cell section with stringers



Figure 3: Single-cell section without stringers



Figure 4: Open single-cell section with stringers

P=P1+P2

It is known that the sum of shear forces in P is not equal to zero and that the shear flows between stringers in P are not equal but the shear flow is constant in the within each parcel. Therefore, the presence of stringers results in a non-constant shear flow.

$$R^{z}=\sum{V_{z}}\neq 0$$

$$q_{12}\neq q_{23}\neq q_{31}$$

$$q_{ij}(s)=const$$

Problem P1 is solved just as in S.1 of the miniPLAN and the resulting shear flows between stringers can be solved with the superposition of P1 and P2.

$$q_{ij}=q+\tilde{q}_{ij} $$

The method for solving the problem consists of the following steps:

1) Solve P2 for $$\tilde{q}_{12},\tilde{q}_{23},\tilde{q}_{31} $$

2) Take the moment about any point in the plane of the section

2.1) Superposition $$q_{ij}=q+\tilde{q}_{ij} $$

2.2) Selection point in plane and find moment about that point