We compare two approaches to better understand the electromechanical properties of BaTiOs under finite electric fields. In the first approach Linearized Augmented Plane Wave (LAPW) computations of total energies and polarization are performed under zero field, and appropriate for zero temperature, arid finite field behavior is modeled by considering the free energy changes E - P induced by the field. In the second approach the temperature arid electric field dependencies of polarization and strain are obtained for BaTiO3 by Monte Carlo simulations using an effective Hamiltonian with parameterization based on ultrasoft pseudopotential calculations. The LAPW calculation results in an initial strain response larger than the effective Hamiltonian simulation. The effective Hamiltonian shows a similar but smaller response at low temperatures, but also shows a field-induced intermediate disordered monoclinic phase at higher temperatures. The strain-mode coupling in the effective Hamiltonian is smaller than obtained with LAPW. The overall picture of polarization rotation induced by an applied field is found with the effective Hamiltonian. similar to the LAPW results.