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The main purpose of this paper is to investigate the exact solutions of cylindrically symmetric spacetime in the context of f( R, T) gravity (Harko et al. in Phys. Rev. D 84:024020, ), where f( R, T) is an arbitrary function of Ricci scalar R and trace of the energy momentum tensor T. We explore the exact solutions for two different classes of f( R, T) models. The first class f( R, T)= R+2 f( T) yields a solution which corresponds to an exterior metric of cosmic string while the second class f( R, T)= f( R)+ f( T) provides an additional solution representing a non-null electromagnetic field. The energy densities and corresponding functions for f( R, T) models are evaluated in each case. [ABSTRACT FROM AUTHOR]