Discussion
A simply-supported beam of rectangular cross-section (width \(w\) and height \(h\)) is subjected to the loads as shown in the figure (i). The enlarged view of the beam cross-section is shown in figure (ii). The coordinate system is indicated in the figures. Assuming Euler-Bernoulli beam approximation, the shear stress \(\tau_{xz}\) and normal stress \(\sigma_{xx}\) at the origin, O are respectively given by
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.
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