A secret sharing scheme is a cryptographic technique to protect a secret from loss and leakage by dividing it into shares. A ramp secret sharing scheme can improve efficiency in terms of the information ratio by allowing partial information about the secret which is composed of several sub-secrets to leak out. The notion of strong security has been introduced to control the amount of information on every subset of the sub-secrets that unauthorized sets can obtain. However, there have been proposed few methods to construct strongly secure schemes for general access structures. Furthermore, all existing methods need strong assumptions, which lead to a loss of efficiency and a limitation of access structures. In this paper, we show that any ramp secret sharing scheme which is linear over a sufficiently large field can be transformed into a strongly secure scheme with the same access structure preserving the information ratio. Since our method only requires linearity, our strongly secure schemes can realize arbitrary access structures and achieve smaller information ratio than the previous schemes.