This paper presents a unified calculation method and its application in determining the uniaxial mechanical properties of concrete with concrete strengths ranging from 10 to 140 MPa. By analyzing a large collection of test results of the uniaxial mechanical properties of normal-strength, high-strength and super high-strength concrete in China and performing a regression analysis, unified calculation formulas for the mechanical indexes of concrete are proposed that can be applied to various grades of concrete for determining the size coefficient, uniaxial compressive strength, uniaxial tensile strength, elastic modulus, and strain at peak uniaxial compression and tension. Optimized mathematical equations for the nonlinear stress-strain relationship of concrete, including the ascending and descending branches under uniaxial stress, are also established. The elastic modulus is almost constant throughout the elastic stage for the ascending branches of the stress-strain relationship for concrete. The proposed stress-strain relationship of concrete was applied to the nonlinear finite element analysis of both a steel-concrete composite beam and a concrete-filled steel tubular stub column. The analytical results are in good agreement with the experiment results, indicating that the proposed stress-strain relationship of concrete is applicable. The achievements presented in this paper can be used as references for the design and nonlinear finite element analysis of concrete structures.