# How to correct equation consistency when it includes numbers and quantities with inconsistent units

Consider equation like this (it's from EN 13445-3:2021, (18.11-12)):

$$N = \left(\frac{2.69R_m + 89.72}{\Delta \sigma_R}\right)^{10}$$

where:

• $$N$$ is number of cycles (dimensionless)
• $$R_m$$ is ultimate tensile strength (MPa)
• $$\Delta \sigma_R$$ is stress range from fatigue design curve (MPa)

The problem is, since $$R_m$$ and $$\Delta \sigma_R$$ represent quantities (i.e. products of numbers and units), the equation is not really consistent and some additional requirements are needed to guarantee correct application. Underlying requirements in this case are that both $$R_m$$ and $$\Delta \sigma_R$$ are in MPa and that you should use just numbers from these quantities in the equation.

So the question is: How to make the equation (or any other equation with similar problems) consistent?

This case of a problem is not a general one, because the result, $$N$$, is dimensionless. If the result of an equation needs to have a unit, solution to the inconsistency problem like striping $$R_m$$ and $$\Delta \sigma_R$$ of their units needs also to add specific unit to the resulting number.

This question is related also to physics and academia, but academia seems to lack tex support and physics does not have relevant tags like "unit". I have found only 2 somewhat related questions in academia, this one with a link to this one.

Since both $$R_m$$ and $$\Delta \sigma_R$$ are supposed to be in MPa for the equation to work as it is, the constant 89.72 needs to be in MPa as well. Best general solution to this problem seems to be defining this constant outside of the equation along with the unit:

$$N = \left(\frac{2.69R_m + A}{\Delta \sigma_R}\right)^{10}$$

where:

• $$N$$ is number of cycles (dimensionless)
• $$R_m$$ is ultimate tensile strength (usually in MPa, any pressure unit will work)
• $$\Delta \sigma_R$$ is stress range from fatigue design curve (usually in MPa, any pressure unit will work)
• $$A = 89.72 \;\mathrm{MPa}$$ is a constant

In this way, the equation is consistent and specific unit for either $$R_m$$ or $$\Delta \sigma_R$$ is no longer required.

• If someone wants to use psi (pounds per square inch) the 2.69 would also have to change, but he might not consider that and get a wrong answer. So for this equation, I would make the 2.69 a B which would be defined as 2.69 MPa^-1. The original poster was asking in general, for all equations with unit dependent numbers. So, I would say put constants for all unit dependent numbers with definitions. Commented May 5 at 21:26
• @WHG I don't really think so, because 2.69 is dimensionless, so it cannot be unit dependent. The equation could be rewritten as $N = \left(2.69\frac{R_m}{\Delta\sigma_R} + \frac{A}{\Delta\sigma_R}\right)^{10}$, where units of $R_m$ and $\Delta\sigma_R$ will cancel out. This is why $A$ has to have a unit. Commented May 6 at 8:18