A smectic liquid-crystal phase formed by achiral banana-shaped molecules (C-B2) is studied as one-dimensional piezoelectric crystals with fluidity of tilted-C-2V anisotropy in the other two spatial dimensions. Variation of the elastic-piezoelectric free energy of the system predicts the existence of racemic (R), left-, and right-handed chiral (H) antiferroelectric equilibrium structures, a result in agreement with the observation by Link [Science 278, 1924 (1997)]. The electric-field-induced ferroelectric state is shown to be a special C-2 symmetry, which confirms the measurements by optical second harmonic generation in a C-B2 by Macdonald [Phys. Rev. Lett. 81, 4408 (1998)]. The hysteresis loop of switching polarization is also discussed, and the molecular tilt angle dependence of spontaneous polarization is obtained as a function of the elastic and the piezoelectric constants.