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

Additive reducts of real closed fields and strongly bounded structures