Shear flow in a thin-walled section is defined as
Answer: A
Shear flow is a force per unit length along the wall. It is the natural variable for thin-walled work because it stays constant along a wall of varying thickness, whereas the stress does not, so equilibrium arguments become much simpler.
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In an open thin-walled section under a transverse shear force, the shear flow
Answer: B
A free edge carries no complementary shear, so the flow starts at zero there and accumulates as the first moment of area is swept up. The problem is statically determinate, which is what distinguishes it from the closed-section case.
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The shear centre of a section is the point through which a transverse load must act if
Answer: C
Applying the load anywhere else leaves a residual torque equal to the load times its offset from the shear centre, and the section twists as well as bends. Locating the shear centre is therefore essential before any pure-bending analysis can be trusted.
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For a section with one axis of symmetry, the shear centre
Answer: D
Symmetry forces the shear flows either side of the axis to mirror one another, so their resultant must lie on it. One axis of symmetry therefore fixes one coordinate of the shear centre and only the other has to be computed.
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The shear centre of a thin-walled angle section lies at
Answer: A
Each leg's shear flow produces a force acting along that leg's centreline, so both resultants pass through their intersection and can generate no moment about it. The same argument applies to any section built from legs meeting at a single point, such as a tee.
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For a channel section loaded parallel to the web, the shear centre lies
Answer: B
The flange shear flows form a couple that the web flow alone cannot balance about any point inside the section, so the resultant falls outside it. A load applied at the web will therefore twist a channel, which is why channels are rarely used alone as beams.
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A closed thin-walled section under shear is statically indeterminate because
Answer: C
Cutting the section produces an open one that can be solved directly, but the cut has to be restored by a constant redundant shear flow. Finding that redundancy requires a compatibility condition on top of equilibrium, which is what makes the problem indeterminate.
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The redundant shear flow in a closed section is found by imposing
Answer: D
Equilibrium alone cannot fix the constant, so a deformation condition is added: for a load through the shear centre the twist is zero, and for pure torsion the rate of twist is the same for every cell. That extra equation closes the problem.
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The Bredt-Batho theory gives the shear flow in a single-cell closed section under pure torsion as
Answer: A
The shear flow is constant round the cell and equals the torque over twice the area enclosed by the wall centreline. Note that it is the enclosed area that matters, not the area of material, which is why closing a section is so much more effective than thickening it.
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For a closed section under pure torsion, the shear flow around the wall is
Answer: B
Correct answer: Constant, even where the thickness varies
Equilibrium of a small element of wall requires the shear flow \( q = \tau t \) to be the same everywhere round the cell, since any variation would leave an unbalanced axial force. The stress is therefore not constant: it is \( q/t \), and it is largest exactly where the wall is thinnest.
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