Show that p must be directed through centroids of cross sections if axial stress ? is not to vary over a cross section.

Assuming P (usually written in upper case) represents a force normal to a given cross section.

If a point load is applied to any point of the section, stress concentration will cause axial stress to vary.

The context of the question considers the uniformity of axial stress at a certain distance away from the point of application (thus stress concentration can be neglected).

If a force P is applied through the centroid, sections will be stressed uniformly.  However, if the force P is applied at a distance "e" from the centroid, the equivalent load on the section equals an axial force and a moment Pe.  The latter causes bending of the member, causing non-uniform stress.

If we assume A=(uniform) cross sectional area, and I=moment of inertia of the section, then stress varies with the distance y from the centroid equal to
stress=sigma=P/A + My/I
where P=axial force, M=moment = Pe.
Therefore when e>0, the stress varies across the section.

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