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Is there something similar to the Section Modulus in the Flexure Formula

$$\rho_{max} = \dfrac{M_{max}}{S}$$

where $S = \dfrac{I}{c}$ for shear stress formula?

So in

$$\tau_{max} = \dfrac{VQ}{IT}$$

then there should be a Shear Section Modulus published which would be $Q(\text{of centroid}) / IT$

So you can get $\tau_{max} = V * \text{Shear Section Modulus}$.

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What you're looking for is commonly referred to as the shear area of the section (see here for a list of shear areas).

Consider a solid rectangle with depth 20mm and width 10mm. From this calculator we can see that the section properties are:

$I = 6667 \text{ mm}^4$

$Q = 500 \text{ mm}^3$

$t = 10 \text{ mm}$

So the shear area is:

$A_{s} =\dfrac{It}{Q}=\dfrac{(6667)(10)}{500} = 133.3 \text{ mm}^2$

Therefore the shear stress is:

$\tau_{max}=\dfrac{VQ}{It}=\dfrac{V}{A_s} = \dfrac{V}{133.3 \text{ mm}^2}$

Compare this to the resource I shared earlier and you can see that for a thick walled rectangular section the shear area (denoted by W in the resource) is:

$A_s=\dfrac{2}{3}hb = \dfrac{2}{3}(20)(10) = 133.3 \text{ mm}^2$

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