In this paper, equivalent quadratic-representative volume elements (EQ-RVEs) in the direct FE² framework are proposed to reduce the computational cost for the multiscale analysis of nonlinear problems. In the offline stage, training loads are applied to fully discretized RVEs to obtain boundary displacement-force snapshots of EQ-RVEs for the construction of a radial basis function (RBF) surrogate model. The RBF surrogate model provides an explicit relation to define the internal force vector and tangent stiffness matrix of EQ-RVEs. The trained EQ-RVEs are implemented as a user element (UEL) to link the reduced micro-scale elements to macro-scale elements in a commercial software Abaqus. Benchmark tests indicate that the proposed EQ-RVE direct FE² method preserves the macroscopic accuracy of FE² simulations and significantly reduces computation time, offering an efficient tool for the multiscale analysis of composite materials.