Methyl-ammonium lead iodide perovskite (CH3NH3PbI3) has drawn great attention due to its excellent photovoltaic properties. Because of its loosely compacted structure, the structural, electronic and optical properties of CH3NH3PbI3 are sensitive to external modulations. Strain effects on CH3NH3PbI3 are fully investigated by the first principles calculations. The results indicate that the inorganic framework deforms under compression or stretch and the embedded organic CH3NH3+ molecules rotate correspondingly. A band gap oscillation and a new structural phase in response to the external strain were observed for the first time. These phenomena are explained with the nonlinear structural deformation and phase transition under the external strains. The semi-quantitative relationship between the band gap variation and geometry change under the external strain is obtained. We found that the shift of valence band maximum under the external strain is mostly determined by the most stretched or compressed Pb-I bond of CH3NH3PbI3, and the shift of the conduction band minimum under the external strain is likely to be determined by the largest Pb-I-Pb bond angle in the system. These results are important for understanding of strain effects on semiconductors and guiding the experiments to improve the performance of the perovskite solar cells.