Starting from different characters of failure models of lightweight aggregate concrete and normal concrete, the problems of current failure models of concrete were analyzed. It is pointed out that the failure surface of lightweight aggregate concrete should be continuum, smooth and protruding curved surfaces in principal stress space. The failure model of lightweight aggregate concrete described by using quadratic functions in principal stress space was brought forward based on isotropic mechanical characters of concrete materials and was quantified with experimental results of representative stress points. The envelope curves of any cross sections of the quantified elliptical failure model, including tensile and compressive meridians and failure envelope curves in deviated plane, are shown to be elliptical curves. The problems of current failure models of concrete were solved, in which the tensile and compressive meridians and failure envelope curves in deviated plane were described by using different types of functions. The disadvantage of tensile and compressive meridians with sharp corner due to using of quadratic parabola is overcome. The precision of the suggested elliptical failure model of lightweight aggregate concrete is verified by comparing with experimental data.