Combined Axial and Bending Stress

Superposition is the quiet workhorse of elastic analysis: because both stresses are linear in load, the axial and bending parts simply add. On one face they reinforce, on the other they cancel. A 100 × 300 mm post with $A = 0.03$ m², $I = 2.25 \times 10^{-4}$ m⁴ and $c = 0.15$ m under 300 kN and 15 kN·m carries $P/A = 10$ MPa and $Mc/I = 10$ MPa, so one face sees 20 MPa and the other sees exactly zero.

Zero is not a coincidence. That load sits precisely on the kern boundary: the eccentricity $e = M/P = 0.05$ m is exactly $h/6$, the classic middle-third rule. Push the load any further out and the far face goes into tension, which masonry, plain concrete and an unbolted column base cannot supply — the joint opens instead, the contact area shrinks, and the real stresses shoot past what this linear formula predicts. Every retaining wall and spread footing is checked against that same middle-third line, and it is the reason a chimney survives its own weight but not a modest sideways push.

Two things to watch. Enter $c$ or $M$ as negative to look at the relieved face; the relation stays linear either way, so the solver will happily rearrange for any variable. And superposition only holds while the geometry does not change under load. A slender member deflects, the axial force then acts through that deflection and generates extra moment — the P-delta effect — and beyond a slenderness of roughly 40 this formula understates the stress badly. Codes handle it with a moment magnifier for exactly that reason.