## 1 Introduction
Until now, only a fraction of force identification methods for nonlinear structures have been concerned with estimating the force identification or simultaneous identification problems for nonlinear structures based on nonlinear recursive estimation tools. In the force identification process, a dynamic analysis with a finite element model of the structure is usually required to obtain numerical structural responses. The time integration method is an efficient numerical tool for the response simulation of a large-scale structural system and structural dyanmic testing [26; 27; 28; 29; 30; 31]. The step-by-step integration method can also be divided into two classes: explicit and implicit methods. For instance, central differentiation method is a commonly used explicit integration method while it is conditionally stable. Implicit step-by-step integration method is an alternative tool to solve the equation of motion, which is unconditionally stable. It should be noted that the stability performance mentioned above is related to linear, but not nonlinear structures. The convergence performance of the implicit nonlinear dynamic analysis is totally different and it is difficult to guarantee convergence for the nonlinear dynamic analysis. Therefore, a forward and inverse dynamic analysis may be inaccurate or even unstable due to the nonlinearities, including hysteresis and geometric nonlinearity which exist in structures. Selecting a time integration method for the inverse dynamic analysis of a nonlinear structural system will also influence the accuracy and stability of the analysis. The energy-conserving integration method (ECIM), has been proposed [26; 27; 28; 29], especially for geometric nonlinearity, but, until now, it has never been concerned about the inverse dynamic problem which renders the inverse analysis inaccurate. In this paper, a force identification method is proposed for nonlinear structures. A general version of ECIM was developed and applied to the inverse dynamic analysis when considering structural nonlinearity. In the dynamic analysis of structures with geometric nonlinearity, the physical matrices of the structures are time variant, because the small deformation hypothesis cannot work in this condition. Without the optimization methods in this study, the convergence of the dynamic analysis would influence the inverse analysis adversely. ECIM was selected to solve the problem in this study to ensure convergence of the dynamic analysis. In the process of dynamic load identification, it is recognized that there are a large number of unknowns in force time history which could have an adverse effect on identification results. A Chebyshev polynomial decomposition method was utilized for the external force identification of nonlinear structures. The order of Chebyshev polynomials was determined using a practical method based on frequency analysis of the structural response, from which a formulation was obtained with regression to determine the decomposition order. Numerical simulation was conducted with a two-dimension truss structure and a base isolated building. The load identification method has also been validated by experimental studies of a seven-story plane frame, a series of quasi-static tests of full-scale rubber isolation and hybrid simulation of a nonlinear isolated structure. Structural uncertainties in materials, unexpected ground motion, and measurement noise are considered to validate the robustness of the proposed method.