# What is the difference between tau = VQ/It and tau = V/A?

What is the difference between the formulas $\tau = VQ/It$ and $\tau = V/A$ for finding shear stress due to transverse loading? I understand that the use of first moment of area, is the only reason for the more complex formula to account for shape?

For a square cross section the maximum shear stress used for failure analysis is simply 4V/3A, which relates to tau = V/A except it has a multiplication factor of 1.33. What is the point of this safety factor, since the rectangle is such a simple shape, and which formula should I use? The factory of safety ends up being very different due to the extra 1/3.

• V/A is the average shear stress. The stress is not uniform, it is zero at top and bottom and maximum in the middle. For a rectangle the max is actually 3/2 V/A This has nothing to do with factors of safety. – agentp Jan 25 '18 at 21:28
• How about define terms? – paparazzo Jan 25 '18 at 21:38

As mentioned, the formula $\tau_{avg}=\frac{V}{A}$ is called average shear stress. In some calculations it might suffice to calculate $\tau_{avg}$ for a widely used cross-section (let's say a standardized beam,e.g. HEA or IPE in Europe), and apply a safety coefficient to come up for shear flow.

The other formula $\tau=\frac{V\cdot Q}{I\cdot t}$ (beam shear formula) takes into account, that variations of shear stress throughout the cross-section occur. Here, $\tau_{avg}$ might be used to simplify the formula of the true shear stress, that is, as a multiple of $\tau_{avg}$.

Take for example a rectangular cross-section: The shear stress in function of $z$ is: $$\tau(z)=\frac{6Vz(h-z)}{wh^3}=\frac{6z(h-z)}{h^2}\cdot \tau_{avg}$$ which for $z=\frac{h}{2}$ (the center point) yields: $$\tau(z=\frac{h}{2})=\frac{3}{2}\frac{V}{wh}=\frac{3}{2}\cdot\tau_{avg}$$

Thus, it might sometimes be a bit easier to give the formula of the shear stress distribution as a multiple of the average shear stress $\tau_{avg}$

The first one is the shear stress in a beam with rectangular cross section and because of parabolic distribution of shear on such cross sections it's factored by 1.5 to get the average stress on the surface.

The second one is the shear flow at a certain height of a beam cross section, usually considered for composite beams to analyse the shear connection and shear stress at that level. Then one can calculate the shear studs or nails or any mechanical attachment to connect for example the web of a composite beam to its flange!

The first formula takes the stress of the fiber parallel to its axis or perpendicular to the plane of consideration. The second takes the stress perpendicular to the axis or in some cases, parallel to the plane of consideration.

See image below. The red line in top drawing represents the fibers being stressed out in shear, VQ/Ib.

The red line in bottom drawing represents the fibers being stressed out in shear, V/A. There is a difference between Tau=VQ/It and Tau=V/A. The first is the shear stress the corresponded V is perpendicular on the biggest dimension of the beam (the axis of the beam) while the second tau is the tensile or compressive stress, in this case V is along the axis of the beam the stress : if V is positive, it's a tensile stress otherwise it's a compressive stress.