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This article presents an experimental study to determine the tensile properties of the envelope fabric Uretek3216A under mono-uniaxial, uniaxial cyclic, and biaxial cyclic loading. First, the mono-uniaxial, uniaxial cyclic, and biaxial cyclic tests were carefully carried out on the envelope fabric, and the corresponding stress–strain behaviors were discussed. Then, the elastic constants were calculated from the experimental data of these tests and the influences of the uniaxial loading cycle and the determination options with different stress ratios were discussed. For the biaxial tests, the elastic constants were determined with and without the constraint of the reciprocal relationship to investigate its significance. Finally, a comparison of the elastic moduli between uniaxial and biaxial tensile tests was presented. Results show that the nonlinearity and orthotropy of the envelope fabric could be attributed to the mechanical properties and unbalanced crimp of their constitutive yarns, respectively. The elastic constants vary noticeably with the experimental protocols, as well as the determination options for biaxial tests, and then in the real design practice, elastic constants should be determined for specific loading conditions and stress distributions depending on the project’s needs.