We explore BiFeO3 under tensile strain using first-principles calculations. We find that the actual structures are more complex than what had been previously thought, and that there is a strong shear deformation type structural instability which modifies the properties. Specifically, we find that normal tensile strain leads to structural instabilities with a large induced shear deformation in (001) BiFeO3 thin films. These induced shear deformations in (001) BiFeO3 thin films under tension stabilize the (001) BiFeO3 thin films and lead to Cc and Ima2 phases that are more stable than the Pmc21 phase at high tensile strain. The induced shear deformation shifts the Cc to Ima2 phase transition towards lower tensile strain region (~1% less), prevents monoclinic tilt and oxygen octahedral tilts, and increases the ferroelectric polarization. The induced shear deformation also strongly affects the electronic structure. The results are discussed in relation to growth of BiFeO3 thin films on cubic and tetragonal substrates involving high levels of tensile strain.