  An orientation of an undirected graph $G$ is an assignment of exactly one
direction to each edge of $G$. The oriented diameter of a graph $G$ is the
smallest diameter among all the orientations of $G$. The maximum oriented
diameter of a family of graphs $\mathscr{F}$ is the maximum oriented diameter
among all the graphs in $\mathscr{F}$. Chv\'atal and Thomassen [JCTB, 1978]
gave a lower bound of $\frac{1}{2}d^2+d$ and an upper bound of $2d^2+2d$ for
the maximum oriented diameter of the family of $2$-edge connected graphs of
diameter $d$. We improve this upper bound to $ 1.373 d^2 + 6.971d-1 $, which
outperforms the former upper bound for all values of $d$ greater than or equal
to $8$. For the family of $2$-edge connected graphs of diameter $3$, Kwok, Liu
and West [JCTB, 2010] obtained improved lower and upper bounds of $9$ and $11$
respectively. For the family of $2$-edge connected graphs of diameter $4$, the
bounds provided by Chv\'atal and Thomassen are $12$ and $40$ and no better
bounds were known. By extending the method we used for diameter $d$ graphs,
along with an asymmetric extension of a technique used by Chv\'atal and
Thomassen, we have improved this upper bound to $21$.
