  A strong orientation of a graph $G$ is an assignment of a direction to each
edge such that $G$ is strongly connected. The oriented diameter of $G$ is the
smallest diameter among all strong orientations of $G$. A block of $G$ is a
maximal connected subgraph of $G$ that has no cut vertex. A block graph is a
graph in which every block is a clique. We show that every bridgeless graph of
order $n$ containing $p$ blocks has an oriented diameter of at most $n-\lfloor
\frac{p}{2} \rfloor$. This bound is sharp for all $n$ and $p$ with $p \geq 2$.
As a corollary, we obtain a sharp upper bound on the oriented diameter in terms
of order and number of cut vertices. We also show that the oriented diameter of
a bridgeless block graph of order $n$ is bounded above by $\lfloor \frac{3n}{4}
\rfloor$ if $n$ is even and $\lfloor \frac{3(n+1)}{4} \rfloor$ if $n$ is odd.
