<p>In engineering systems, local connections often induce significant nonlinear dynamic behavior. Although the Sparse Identification of Nonlinear Dynamics (SINDy) framework has recently emerged as a pivotal tool for understanding such dynamics, supporting the system design, analysis, and control, it faces notable challenges in real-world continuous structural systems. In this paper, a reliable sparse identification of continuous structural systems is proposed. The proposed method first uses a state-space model to solve the approximation and estimation problem of high-dimensional linear system. It then performs output feature mapping for sparse nonlinear candidate feedback forces. A Bayesian sparse regression with continuous spike-and-slab sparsity-promoting priors is introduced to reliably determine the type and coefficients of the nonlinearities. Both simulation and experiment demonstrate the robustness and feasibility. In a numerical example, the proposed method accurately selects models at noise levels up to 30%, outperforming other sparse linear regression techniques. Experimental validation on a cantilever beam with asymmetric clearances further showcases its applicability to complex nonlinear stiffness scenarios.</p>