Shear Stresses in Beams Questions and Answers

Answer: C

For a rectangular section the maximum shear stress is 1.5 times the average shear stress, and it acts at the neutral axis:

This is 4.5 MPa. The average shear stress is 3.0 MPa, which is not among the options, and the choices 3.5, 4.0 and 5.0 MPa do not follow from the factor of 1.5.

Answer: C

The transverse shear stress is maximum at the neutral axis.

For a rectangular section of depth h the shear stress is \(\tau = \frac{VQ}{Ib}\), and the first moment Q of the area beyond the level considered varies parabolically with depth:

\[ \tau(y) = \frac{6V}{bh^{3}}(h^{2}/4 - y^{2}) \]

It is zero at the top and bottom faces (\(y = \pm h/2\)) and largest at the neutral axis, \(\tau_{max} = 1.5V/(bh)\). So it is neither linear nor constant through the depth, and it is the bending stress, not the shear stress, that is zero at the neutral axis.

Answer: D

Both stresses are zero: 0 , 0.

The origin O is at mid-span and at the centroid of the section. The loading is symmetric (equal loads P at a quarter span from each support), so each support carries P, and between the two loads the shear force is \(P - P = 0\). The shear stress \(\tau_{xz}\) at O is therefore zero.

The bending moment at mid-span is \(PL/4\), which is not zero, but O lies on the neutral axis, where the bending stress \(\sigma_{xx} = Mz/I\) vanishes because z = 0.

\(\frac{3P}{2wh}\) is the peak shear stress where the shear force is P, near the supports, and \(\frac{3PL}{2wh^{2}}\) is the outer-fibre bending stress at mid-span; neither occurs at O.

Answer: D

The distribution is parabolic.

The shear stress at a distance y from the neutral axis of a rectangular section is \(\tau = \frac{F}{2I}\left(\frac{d^{2}}{4} - y^{2}\right)\), a quadratic in y. It is zero at the top and bottom fibres and largest at the neutral axis, where it is 1.5 times the average shear stress.

It is not uniform or linear, and 'elliptic' does not describe the quadratic curve. Bending stress, not shear stress, is the one that varies linearly across the depth.

Answer: C

The equivalent moment of inertia is \(I_{t} + mI_{s}\).

In a flitched beam the steel plate is converted into an equivalent area of timber. Since steel is m times stiffer, with \(m = E_{s}/E_{t}\), the steel's width is multiplied by m, and so is its moment of inertia about the common neutral axis:

\[ I_{eq} = I_{t} + m I_{s} \]

Dividing by m (\(I_{s}/m\)) would be right only if the section were expressed in terms of steel. \(m/I_{s}\) has the wrong dimensions, and \(I_{t} + 2mI_{t}\) applies the modular ratio to the timber itself.

Answer: B

The Board's key is 1.5, the standard textbook ratio for a rectangular section.

For a rectangular beam of width \(b\) and depth \(d\) carrying a shear force \(V\), the shear stress varies parabolically across the depth. It is zero at the top and bottom faces and largest at the neutral axis.

\[ \begin{aligned} \tau_{avg} &= \frac{V}{bd} \\ \tau_{max} &= \frac{3V}{2bd} \end{aligned} \]

\[ \frac{\tau_{max}}{\tau_{avg}} = 1.5 \]

The stem says the beam is subjected to torsion, but the 1.5 ratio belongs to transverse shear in bending; a twisted rectangular bar has a different stress pattern with no such simple average. The item is therefore best read as the transverse shear case, and 1.5 is the only option that matches it. A solid circular section gives 4/3 and a thin-walled circular tube gives 2, so those values do not describe a rectangle.

Answer: B

The shear centre lies outside the section.

The shear centre is the point through which a transverse load must act for the beam to bend without twisting. For a channel with its axis of symmetry horizontal, it lies on that axis but beyond the web, on the side away from the flanges, because the shear flows in the two flanges form a couple that has to be balanced by moving the load line back past the web.

It does not coincide with the centroid, which is inside the section between the web and the flange tips. It is also not at the junction of the axis and the web centre line; it is a definite distance behind the web.