Beam Shear Stress

Shear stress in a beam

On a solid rectangular section, the peak shear stress sits at the neutral axis and works out to τ_max = 3V / (2·A), with V the shear force and A = b·h the gross area. That's 1.5 times the average shear stress V/A. With an I or H profile the web does almost all the work, so in practice you estimate τ_max ≈ 1.5·V / A_w, where A_w is the web area. To see it in numbers: V = 20 kN, b = 200 mm, h = 400 mm → τ_max = 3·20 000 / (2·200·400) = 0.375 MPa.

In concrete beams, NBR 6118 art. 17.4 caps the shear stress through τ_Rd2 = 0.27·α_v2·f_cd and calls for stirrups once τ_Sd goes past τ_Rd1. For steel, NBR 8800 holds the shear to V_R = 0.60·f_y·A_w / γ. If you need the general case, the Jourawski formula τ = V·Q / (I·b) handles any cross-section.

Applications

You'll use it to size stirrups in reinforced-concrete beams and to verify the webs of steel I and H profiles. It also drives the design of bolted and welded connections, where shear usually governs, and the checks on short beams and corbels, where bending stays small but shear runs high. On top of that, it keeps the ultimate-limit-state design in line with NBR 6118 and NBR 8800.

FAQ

Why is τ_max 1.5× the average for a rectangle? The shear stress isn't uniform across the depth; it follows a parabola and reaches its peak at the neutral axis, exactly where the first moment of area Q is at its largest.

Why use only the web area for I sections? The flanges are busy carrying the bending moment but contribute almost nothing to shear, since their first moment Q at the neutral axis is small. The web ends up taking roughly 95% of V.

When are stirrups required? For concrete beams, the trigger is the design shear V_Sd climbing above V_Rd1, the resistance the concrete provides on its own. That threshold is set in NBR 6118 §17.4.2.

Is shear ever critical compared with bending? It can be. In short, deep beams (L/h < 4), in corbels and in connections, shear may govern the design well before the bending stresses ever get near the yield limit.