Shear Flow in Open and Closed Sections Questions and Answers

Answer: C

A closed cell resists torque by a shear flow circulating round the enclosed area, which is enormously effective. An open section can only resist by the St Venant mechanism of shear stresses varying through the wall thickness, and the difference in stiffness is commonly two or three orders of magnitude.

Answer: D

The St Venant result for a narrow rectangle carries the cube on the thickness, and an open section is treated as a sum of such strips. The cube is what makes an open section so soft in torsion: halving the wall thickness cuts the stiffness by eight.

Answer: A

Because the skin is assumed to carry no direct stress, there is nothing to change the flow between booms, so it is constant along each panel. Each boom carries direct load and produces a discrete step in the flow, which turns the integration into a running sum.

Answer: B

The step is the shear force over the second moment of area, times the boom's first moment about the neutral axis. A boom on the neutral axis therefore produces no step at all, whatever its area, which is a useful check on a calculation.

Answer: C

The rate of twist follows from integrating the shear strain round the cell, giving an expression containing the flow divided by thickness and the shear modulus. It is what allows a multi-cell problem to be closed, by requiring every cell to twist at the same rate.

Answer: D

The cells are joined along their common webs and cannot twist independently, so equating the rates of twist supplies the extra equations needed. Together with the torque equilibrium equation, the system can then be solved for each cell's shear flow.

Answer: A

Correct answer: The difference of the two cell flows

Each cell carries its own circulating flow, and in the shared web those circulations run in opposite directions - one clockwise, the other anticlockwise as seen from the web. The net flow is therefore \( q_1 - q_2 \), which is why interior webs of a multi-cell box are typically the least stressed and can be made light.

Answer: B

Resolving the offset load into a load through the shear centre together with a torque separates the problem into a pure bending and shear case and a pure torsion case. The two are solved independently and superposed, which is the standard route for combined loading.

Answer: C

Two axes of symmetry fix both coordinates of the shear centre at their intersection, which for such a box is also the centroid. This is one reason a symmetric torsion box is such a convenient structural form: bending and torsion decouple naturally.

Answer: D

The cut turns the indeterminate problem into a determinate one that can be solved by the open-section method, after which the constant redundant flow is added back using a compatibility condition. Where the cut is placed makes no difference to the final answer.