TiNi-based shape memory alloys have been widely used in the fields of engineering and medicine due to their shape memory effect and superelasticity, which are displayed during martensitic transformations [1]. However, the martensitic transformation in this system is least understood compared with other systems due to the complexity of the transformation and sensitivity to alloying addition. It is well known that most TiNi-based alloys have an ordered B2 phase at high temperatures and transform to a stable low temperature martensitic phase when the temperature decreases. For TiNi alloy, the parent B2 phase transforms into a monoclinic B19′ martensite. Moreover, the martensitic transformation of TiNi alloy is significantly affected by the addition of the third elements as regards the transformation temperature [2][3][4]. With Fe addition, the martensitic transformation temperature is decreased drastically [5]. With low addition of Pd and Cu, the martensitic transformation temperatures are also decreased [6]. Despite the importance of these transformations in applications of TiNi-based alloys, several fundamental aspects of these transformations have not been well understood. The most puzzling problem is the origin of the unique B19′ martensite, which found no analogy in other usual B2 alloys, such as CuZn and NiAl. The second puzzle is the mechanism of the effect of the third element on the martensitic transformation temperature of TiNi. Some previous works show that the martensitic transformation is related both to elastic properties and electronic structure [7][8][9][10]. Recently, Ren and Otsuka et al. [11] proposed that the B19′ martensite structure is a result of the coupling between a non-basalplane shear (the C 44 shear) and the basal plane shear (the C′ shear), and verified this argument by some experimental studies of elastic constants of TiNi-based alloys. But studies on the origin of the B19′ martensite from theoretical calculation of their elastic properties have not been reported, and the second puzzle has still remained unclear.In this paper, we report the first theoretical results on the elastic properties and electronic structure of B2 shape memory alloys by the first-principles method for understanding their martensitic transformation behavior. The calculations presented in this study are performed with the CASTEP code [12], based on density functional theory, using Vanderbilt-type ultrasoft pseudopotentials [13] and a plane-wave expansion of the wave functions. We use the local density approximation (LDA) to describe the exchange and correlation potential. The structure is optimized with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method, and convergence is assumed when the forces on atoms are less than 0.03 eV/Å. The plane-wave cutoff energy of 550 eV is employed. The calculations are done using a (14, 14, 14) Monkhorst-Pack grid. As for shape memory alloys are studied by the plane-wave psedudopotential method within the local density approximation. The elastic constants and density of states are...