A shelf that sags.

For a simply supported beam with a point load P, Euler–Bernoulli beam theory gives the true maximum deflection — not just the deflection under the load itself, which is smaller whenever the load sits off-centre — as δ_max = Ps(L²−s²)^3⁄2/(9√3 EIL), where s is the distance from the load to whichever support is nearer. That point sits a little closer to mid-span than the load does, marked on the diagram; the two deflections, and their locations, only coincide exactly when the load is at mid-span. The drawn curve itself uses the exact piecewise deflection equation along the whole span, not a fixed bow shape. For a rectangular section, the second moment of area is I = bh³/12. The material is representative No. 2 spruce–pine–fir: E = 8.27 GPa and reference bending value F_b = 6.03 MPa (1.2×10⁶ and 875 psi), from the American Wood Council's NDS design value tables. The red region begins where σ = Mc/I exceeds F_b, evaluated at the load itself where bending stress always peaks; this is a conservative design threshold, not a prediction of the exact crack load. Real lumber strength varies with grade, knots, moisture, load duration, size, and restraint, and design requires the applicable adjustment factors. The L/360 line is a serviceability comparison for visible sag. Sag is shown at a fixed 15× magnification, capped after failure so extreme elastic predictions do not run off the diagram. Read more on Wikipedia: beam deflection, bending stress, and second moment of area.