Bending Stresses in Beams Questions and Answers

Answer: B

The maximum stress occurs only at the bottom fibres.

In pure bending, normal stress is \(\sigma = My/I\), proportional to the distance y from the neutral axis, which passes through the centroid. A T-section is symmetric only about the vertical axis. Its wide flange is at the top, which pulls the centroid up toward the flange, so the bottom of the web is farther from the neutral axis than the top of the flange. The largest |y|, and so the largest stress, is at the bottom fibres.

Equal stresses at top and bottom would need a section symmetric about the horizontal axis, such as an I or a rectangle. At the centroidal fibres y = 0, so the bending stress there is zero, the smallest, not the largest.

Answer: C

Both statements are true. In pure bending the fibres on one side of the beam are stretched and those on the other side are shortened, so there is a surface between them, the neutral surface, where the length does not change: the bending strain is zero. Since stress is proportional to strain in the elastic range (\(\sigma = E\varepsilon\)), the bending stress there is also zero, as the flexure formula \(\sigma = My/I\) shows when y = 0. The options that call either statement false are wrong.

Answer: D

The neutral axis of a beam in bending is the axis where the bending stress is zero. On one side of it the fibres are in tension and on the other in compression, and the stress varies linearly with the distance from it and is maximum at the outermost fibres. The shear stress is greatest at the neutral axis for common sections, but the statement that it is the axis where shear stress is maximum is not the keyed one, since the question's defining property is the zero bending stress.

Answer: D

In a machine element subjected to bending the maximum bending stress occurs at the outermost fibre from the neutral axis. The bending stress is given by \[ \sigma = \frac{My}{I} \] where y is the distance from the neutral axis, so the stress is largest where y is largest. At the neutral axis and the centroid the stress is zero.