For transverse loading of the same composite the components are now in an iso-stress situation and the total transverse strain is just the sum of the fiber strain and matrix strain.

For the iso-stress case, the strains are linearly distributed between the components so that:
e_C = e_FV_F + e_M(1 - V_F)

Combining this result with Hooke's Law gives the transverse Young's modulus of the composite:

E_C = (E_F E_M) / E_F (1 - V_F) + E_MV_F

From: McMahon and Graham, :"The Bicycle and the Walkman," Merion (1992)

The lower diagram shows how the longitudinal and transverse values of the Composite Young's modulus depends on the fiber volume fraction. In a realistic material only volume fractions between 5% and 80% are of use.