Shear Flow in Open and Closed Sections Questions and Answers

Answer: A

The centroids coincide, while the shear centres do not.

A sharp cut removes a negligible amount of wall, so the material of the section, and hence its centroid, is unchanged. The closed rectangular tube is doubly symmetric, so its shear centre lies at the centroid. Cutting the bottom wall at its centre leaves an open section that is still symmetric about the vertical axis, so its shear centre lies on that axis, but the shear flow now has to start from zero at the cut. The resulting shear flow distribution is different, and the shear centre of the open section moves along the axis of symmetry, away from the centroid.

So the options saying the shear centres coincide (with or without the centroids) are wrong, and so is the option saying the centroids move.

Answer: B

The force causes both bending and twisting in case (ii).

A shear force produces bending only if its line of action passes through the shear centre of the section; otherwise it also twists the beam. For sections made of thin walls meeting at one point, the shear centre is that meeting point.

• (i) Angle: the shear centre is at the corner where the legs meet, and P is at the corner, so there is no twist.
• (iii) T: the shear centre is at the web-flange junction, where P is, so there is no twist.
• (iv) Cross: the shear centre is at the centre, where P is, so there is no twist.
• (ii) Channel: the shear centre lies outside the web, on the side away from the flanges. The force acts along the web, offset from the shear centre, so it twists the channel as well as bending it.

Answer: A

Both [a] and [r] are true and [r] is the correct reason for [a].

A wing carries large torsional loads from the aerodynamic pitching moment, from the offset of the lift from the shear centre, and from control-surface loads. A closed section, such as the box formed by the front and rear spars with the upper and lower skins, carries torque through a constant shear flow around the closed cell (the Bredt-Batho relation), which makes it very stiff and strong in torsion. Open sections of the same material are far weaker in twist.

So the box beam is used in wings precisely because it resists torsion, and [r] explains [a]. That rules out the options saying either statement is false, and the option that accepts both but denies the link.

Answer: B

The beam undergoes only bending.

The shear centre is, by definition, the point in the cross-section through which a transverse load must act for the section to bend without twisting. For an unsymmetric section it generally does not coincide with the centroid, which is why a load through the centroid of a channel, for example, makes it twist. Here P is applied at the shear centre, so no torque about the shear centre arises.

The beam still bends because P is a transverse force on a cantilever, so 'neither torsion nor bending' is wrong. Torsion alone, or torsion with bending, would require the load to be offset from the shear centre.