Ising models, with quenched bond disorder along one direction on a square lattice, are studied by the transfer-matrix method. The exact criticality condition is rederived, and extended to an arbitrary variation, in magnitude and/or sign, of the interactions from layer to layer. With competing interactions, a phase diagram is obtained, with a zero-temperature spin-glass point. We consider
2n different nearest-neighbor couplings layered and repeated periodically along one direction. The phase boundaries, including zero-temperature bicritical points, the order parameters, and the ground-state degeneracies are illustrated by a detailed discussion for
n=2. Closed-form free energies are given for
n=2and3. The unique criticality condition is derived for any
n, the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ corresponding to quenched disorder. Our extended result is made possible by a similarity transformation in the free-fermion representation. A conjecture emerges for the criticality condition of randomly layered ferromagnetic
q-state Potts models.