In present work, we investigated the thermoelastic B2-B19' martensitic transformations in thermocycles under constant stress, sigma app, (epsilon(T) curves) and in loading/unloading cycles (sigma(epsilon) curves) in quenched [001]oriented single crystals of Ti-50.7 at%Ni and Ti-51.7 at%Ni alloys. Single crystals undergo either thermalinduced B2-B19' martensitic transformation (Ti-50.7 at%Ni) or strain glass transition (Ti-51.7 at%Ni), depending on their chemical composition. The transformation type defines the morphology of a martensite during cooling under stress. In Ti-50.7 at%Ni alloys during stress-assisted cooling the morphology is the mix of self-accommodation martensite and oriented martensite. In opposite, in Ti-51.7 at%Ni alloys the morphology consists of oriented martensite only. The following parameters in stress-assisted cooling/heating cycles are determined by transformation type and morphology of a martensite: the minimal stress, sigma min, for oriented martensite formation; the coefficient d epsilon for/d sigma that defines the strain dependence on stress, epsilon for (sigma app), and the presence of additional stage in the dependence Ms (sigma app). Similar in loading/unloading cycles the transformation type defines the energy barrier that should be overcome in order to start the stressinduced martensitic transformation, and, so, the stress level, sigma cr(Ms0).