Shape memory alloy (SMA) wire jackets for concrete are distinct from conventional jackets of steel or fiber reinforced polymer (FRP) since they provide active confinement which can be easily achieved due to the shape memory effect of SMAs. This study uses NiTiNb SMA wires of 1.0 mm diameter to confine concrete cylinders with the dimensions of 300 mm × 150 mm (L × D). The NiTiNb SMAs have a relatively wider temperature hysteresis than NiTi SMAs; thus, they are more suitable for the severe temperature-variation environments to which civil structures are exposed. Steel jackets of passive confinement are also prepared in order to compare the cyclic behavior of actively and passively confined concrete cylinders. For this purpose, monotonic and cyclic compressive loading tests are conducted to obtain axial and circumferential strain. Both strains are used to estimate the volumetric strains of concrete cylinders. Plastic strains from cyclic behavior are also estimated. For the cylinders jacketed by NiTiNb SMA wires, the monotonic axial behavior differs from the envelope of cyclic behavior. The plastic strains of the actively confined concrete show a similar trend to those of passive confinement. This study proposed plastic strain models for concrete confined by SMA wire or steel jackets. For the volumetric strain, the active jackets of NiTiNb SMA wires provide more energy dissipation than the passive jacket of steel.