Shear Flow⚠ unverified

Understanding the Parameters

• Transverse shear force $V$ — the shear the section carries (e.g. the lift shear in a wing, largest at the root). Shear flow scales directly with it.

• First moment of area $Q$ — $Q = \int y\,dA$ taken over the portion of the section beyond the cut where you want the shear flow. It measures how much "disconnected" area the shear at that cut must hold on. $Q$ is zero at a free edge (nothing beyond it) and maximum at the neutral axis — so shear flow (and shear stress) peak at the neutral axis and vanish at the top and bottom fibres, the opposite of bending stress.

• Second moment of area $I$ — the full section's area moment, normalising the distribution just as it does in bending (Wing Bending Stress).

• Wall thickness $t$ — dividing by $t$ converts the formula to shear stress $\tau = q/t$; keeping the $t$ in gives shear flow $q$, the force per unit length, which is what a fastener line actually feels. Because $q$ is force-per-length, a thinner wall carries the same $q$ at higher stress.

• The output $q$ — units N/m. For a riveted joint, the rivet spacing $s$ times $q$ gives the load per rivet ($P = q\,s$); for a bonded joint, $q$ divided by the bond shear allowable gives the required bond width.

Derivation (Approaching a Proof)

Shear flow follows from axial equilibrium of a sliced-off element of a beam under varying bending moment. Consider a short length $dx$ of beam and imagine cutting the section horizontally at height $y_1$, isolating the material above the cut. Because the bending moment changes along the beam, the bending stresses on the two faces of this element differ, leaving a net axial force that must be balanced by a shear force on the cut face.

The bending stress is $\sigma = My/I$ (Wing Bending Stress). The net axial force on the isolated area $A'$ (above the cut) from the moment change $dM$ over $dx$ is

$$dF = \int_{A'} \frac{dM\,y}{I}\,dA = \frac{dM}{I}\int_{A'} y\,dA = \frac{dM}{I}\,Q,$$

where $Q = \int_{A'} y\,dA$ is the first moment of the area beyond the cut. This must be reacted by shear on the cut of length $dx$ and thickness $t$, i.e. by shear flow $q$ acting over $dx$: $dF = q\,dx$. Equating and using the shear–moment relation $dM/dx = V$:

$$q = \frac{dF}{dx} = \frac{Q}{I}\frac{dM}{dx} = \frac{V Q}{I}.$$

The shear stress is this per unit thickness, $\tau = q/t = VQ/(It)$; the calculator reports the flow form $q = VQ/(It)$ scaled to the given wall (equivalently $\tau\,t$). $\blacksquare$

Dimensional check. $$\left[\frac{V Q}{I t}\right] = \frac{(\text{N})(\text{m}^3)}{(\text{m}^4)(\text{m})} = \frac{\text{N}}{\text{m}^2} = \text{Pa (a shear stress)},$$ and the pure flow $VQ/I$ is $\text{N/m}$ as expected for force per unit length. $\checkmark$

History and Development

• Jourawski / Zhuravsky. The shear-stress formula $\tau = VQ/(It)$ is due to the Russian engineer Dmitrii Zhuravsky (Jourawski, 1850s), developed while designing railway bridges where longitudinal shear between timber laminations governed. It complements the flexure formula: bending stress peaks at the fibres, shear at the neutral axis.

• Thin-walled aerospace structure. Aircraft are built almost entirely of thin sheets carrying shear, so shear flow — not shear stress — is the working variable. The analysis of closed sections (torsion boxes) uses the same idea plus a constant torsional shear flow $q = T/(2A_m)$ (Bredt's formula), and multi-cell wing boxes are solved by distributing shear flow among cells.

• Sizing joints. Because $q$ is directly the load per unit length a seam must transfer, shear-flow analysis drives rivet pitch, weld sizing, and bond-line width throughout airframe design — one of the most practically used results in the field.

Related Concepts: Wing Bending Stress, Section Modulus, Von Mises Stress 2D, Composite Wing Stiffness, Fuselage Hoop Stress, Buckling Stress

Notes: Registry calculator shear-flow (unverified). Jourawski shear formula — correct as shipped. $Q$ is the first moment of the area beyond the cut; shear flow peaks at the neutral axis, zero at free edges. $q = \tau\,t$ (force per unit length); rivet load $= q\times$ spacing. Closed torsion boxes add a constant $q = T/(2A_m)$ (Bredt). All defaults $1.0$.