In this paper, we study the $σ$-self-orthogonality of constacyclic codes of length $p^s$ over the finite commutative chain ring $\mathbb F_{p^m} + u \mathbb F_{p^m}$, where $u^2=0$ and $σ$ is a ring automorphism of $\mathbb F_{p^m} + u \mathbb F_{p^m}$. First, we obtain the structure of $σ$-dual code of a $λ$-constacyclic code of length $p^s$ over $\mathbb F_{p^m} + u \mathbb F_{p^m}$. Then, the necessary and sufficient conditions for a $λ$-constacyclic code to be $σ$-self-orthogonal are provided. In particular, we determine the $σ$-self-dual constacyclic codes of length $p^s$ over $\mathbb F_{p^m} + u \mathbb F_{p^m}$. Finally, we extend the results to constacyclic codes of length $2 p^s$.