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Mechanical models of residually stressed fibre-reinforced solids, which do not resist bending, have been developed in the literature. However, in some residually stressed fibre-reinforced elastic solids, resistance to fibre bending is significant, and the mechanical behavior of such solids should be investigated. Hence, in this paper, we model the mechanical aspect of residually stressed elastic solids with bending stiffness due to fibre curvature, which up to the authors' knowledge has not been mechanically modeled in the past. The proposed constitutive equation involves a nonsymmetric stress and a couple-stress tensor. Spectral invariants are used in the constitutive equation, where each spectral invariant has an intelligible physical meaning, and hence they are useful in experiment and analysis. A prototype strain energy function is proposed. Moreover, we use this prototype to give results for some cylindrical boundary value problems. [ABSTRACT FROM AUTHOR]