Shear flow and shear centre in thin-wall beams

Shear flow

For a thin-walled section, the shear stress is assumed constant across the thickness and parallel to the wall. Shear flow $q$ is the shear force per unit length of the wall:

$$q = \tau\,t\quad (\text{N/mm})$$

Derivation

Consider a beam element of length $dz$ under shear force $V$. Cut the section at a point and consider the part of the area $A^*$ beyond the cut, with first moment $Q = \int_{A^*} y\,dA$ about the neutral axis.

The bending moments on the two faces are $M$ and $M + dM$, so the normal forces on the cut part of the faces differ by

$$dF = \frac{dM}{I}\,Q$$

This is balanced by the shear force on the longitudinal cut, $q\,dz$:

$$q\,dz = \frac{dM}{I}\,Q \quad\Rightarrow\quad \boxed{q = \frac{V\,Q}{I}}\quad\left(\text{since } V = \frac{dM}{dz}\right)$$

and $\tau = \dfrac{VQ}{It}$. The shear flow is zero at a free edge and changes linearly in a flange of constant thickness (flange shear flow $q = V\,t\,s\,y_f/I$), and parabolically in a web.

Shear centre

The shear centre (flexural centre) $S$ is the point on the cross-section through which a transverse load must act so that the beam bends without twisting. Its position is found by equating the moment of the shear flow resultants about any point to the moment of the applied shear force $V$ about the same point.

Importance

If the load does not pass through $S$, the section twists as well as bends. Thin open sections have very low torsional stiffness, so the twist can be large and the stresses higher than for pure bending. Designers therefore place the load line through $S$ or add torsion allowance.

Location