The effective transverse shear modulus of a transversely isotropic composite with columnar microstructure and with two cylindrically orthotropic phases is estimated by a one-parameter generalized self-consistent model. The equation characterizing the effective transverse shear modulus is a fourth-order polynomial equation instead of the quadratic one determined by the classical generalized self-consistent model. The result obtained in Part 11 of this work extends the relevant one of Hashin [Z. Hashin, Thermoelastic properties and conductivity of carbon/carbon fiber composites, Mech. Mater. 8 (1990) 293-308]. (C) 2007 Elsevier Ltd. All rights reserved.