Suppose X is a pure dimensional projective complex scheme, reducible and non-reduced but without embedded components of lower dimension. Let $X=\cup X_i$ be the decomposition such that $X_i$ is set-theoretically irreducible and $\dim(X_i\cap X_j)<\dim(X_i)$.

Let $F_X$ be a torsion free sheaf of $\mathcal{O}_X$-modules. If $F$ is generically locally free on each $X_i$ then one can define its multi-rank. What to do if $F|_{X_i}$ is not generically locally free?

For example let $X:=(x^p=0)\subset\mathbf P^2$ be the multiple line. Let $F_j$ be the $\mathcal{O}_X$ module generated by $x^j$. For $0 < j < p$ it is not locally free at any point.

My guess: let $pt\in X_i$ be generic enough closed point. Let $p_i$ be the generic multiplicity of $X_i$. At the point $pt\in X$ consider the intersection of $X_i$ with the generic plane of the complementary dimension. This gives the sub-scheme $Y\subset X $ supported on $pt$. Take the fraction $\frac{length(F\otimes\mathcal{O}_Y)}{p_i}$ as the rank on $X_i$.

In the previous example this gives the rank of $F_j$: $\frac{p-j}{p}$.

Is this the commonly accepted definition? References?