  Given a real closed field $R$, we identify exactly four proper reducts of $R$
which expand the underlying (unordered) $R$-vector space structure. Towards
this theorem we introduce a new notion, of strongly bounded reducts of linearly
ordered structures:
  A reduct $\mathcal M$ of a linearly ordered structure $\langle
R;<,\cdots\rangle $ is called \emph{strongly bounded} if every $\mathcal
M$-definable subset of $R$ is either bounded or co-bounded in $R$. We
investigate strongly bounded additive reducts of o-minimal structures and as a
corollary prove the above theorem on additive reducts of real closed fields.
