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Analytical solutions for transverse matrix strain magnification and transverse fibre strain reduction in uniaxially aligned continuous fibre reinforced composites developed here for a Representative Volume Element (RVE), based on the Reuss rule and the principle of constant strain energy. The results obtained show respective matrix and fibre transverse strain magnification and reduction that increase with increasing volume fraction of the reinforcing fibre for both square and hexagonal fibre packing geometries. The square arrays register higher values of matrix and fibre transverse strain magnification and reduction, respectively, than hexagonal arrays. The values of transverse stress ratio between the fibre and matrix in the RVE central sub-region, and between the two and the RVE central sub-region composite are seen to vary with the volume fraction of reinforcing fibre. The results obtained here denote a dependency of respective matrix and fibre transverse strain magnification and reduction as well a stress ratio on the reinforcing fibre packing geometry and volume fraction.