Beam Shear Stress Calculator (tau = VQ/Ib)
The horizontal shear stress in a beam at any cross-sectional plane is given by the shear formula tau = VQ / (Ib), attributed to the mechanics of materials derivation by Grashof and standardised in structural engineering practice. Here V is the transverse shear force, Q is the first moment of area of the part of the section on one side of the plane (about the neutral axis), I is the full cross-section's moment of inertia about the neutral axis, and b is the width of the section at the plane being examined. The formula is valid for prismatic beams of homogeneous elastic material loaded in the plane of symmetry. Shear stress varies parabolically across a rectangular section and peaks at the neutral axis.
Shear stress formula
tau = V × Q / (I × b)
Units: if V is in kips, Q in in^3, I in in^4, and b in inches, then tau is in kips/in^2 (ksi). For SI units: N, mm^3, mm^4, mm gives N/mm^2 (MPa).
Shear stress distributions in common sections
- Rectangular section: tau varies parabolically; maximum at neutral axis = 1.5V/A where A = full area.
- Circular solid section: maximum at neutral axis = 4V/(3A).
- Wide-flange (I) section: shear is nearly uniform in the web; the flanges carry very little shear.
- In wood beams, horizontal shear at the neutral axis often governs the design because wood is weak in shear parallel to grain.
Frequently asked questions
What is the shear formula tau = VQ/(Ib)?
This is the flexural shear stress formula for prismatic elastic beams. tau is the shear stress at a horizontal plane a distance y from the neutral axis. V is the shear force at the section, Q is the first moment of area of the portion above y, I is the full section moment of inertia, and b is the width of the section at the plane of interest.
What is the first moment Q?
Q = A' * y-bar', where A' is the area of the cross-section above (or below) the plane of interest and y-bar' is the distance from the centroid of A' to the neutral axis of the full section. For a rectangular section of width b and depth d at mid-depth, Q = (b * d/2) * (d/4) = b*d^2/8.
Where is shear stress maximum in a rectangular beam?
Shear stress is parabolic and reaches its maximum at the neutral axis (mid-depth) for a rectangular section. The maximum value is tau_max = 1.5 V/A, where A is the full cross-sectional area.
Is the formula valid for I-beams and T-beams?
Yes, but Q and b vary at each horizontal plane. The maximum shear stress in an I-beam occurs at the neutral axis in the web, where b is the web thickness. Always use the correct b and Q for the specific plane being checked.
What is the allowable shear stress for steel beams?
AISC ASD allows Fv = 0.40 Fy for beam webs (no tension field action). For A36 steel, Fv = 14.4 ksi. AISC 360 LRFD checks phi*Vn against Vu, with phi = 1.00 for webs satisfying h/tw limits.
Official sources
- American Institute of Steel Construction: AISC Steel Construction Manual, Chapter G (Shear).
- American Wood Council: NDS for Wood Construction, Horizontal Shear.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.