## 1 Introduction  
Natural foundations often exhibit layering and nonlinearity, making their consolidation characteristics highly intricate. Consequently, accurately predicting consolidation settlement in engineering practice is challenging. This challenge primarily arises from numerous assumptions inherent in existing theories, such as the assumptions of either fully permeable or impermeable boundaries, and the assumption that soil compressibility or permeability remains constant during the consolidation process. These assumptions fail to accurately depict the consolidation characteristics of soil. In light of this, extensive research efforts have been undertaken by numerous scholars. Since a dual-layered foundation represents the simplest form of layered foundation, [PERSON] [1] formulated the analytical solution for the one-dimensional consolidation of double-layered soil subjected to instantaneous loading. Building upon [PERSON]'s [1] seminal groundwork, [PERSON] and [PERSON] [2] further extended the analytical framework to address consolidation challenges within layered soil, encompassing a range of boundary conditions and complex stress histories. In subsequent developments, [PERSON] [3] and [PERSON] [4] independently established analytical solutions for one-dimensional consolidation of double-layeredsoil and layered soil, respectively, considering scenarios involving variable loading. Thereafter, different scholars conducted in-depth research on one-dimensional consolidation of layered soils from various aspects, including rheology, nonlinearity, structural properties, non-Darcy flow, etc. [5; 6; 7; 8]. On the other hand, the nonlinear characteristics of soils evidently influenced the consolidation of soils. Employing the \(e\)-log\(\sigma^{\prime}\) relationship as a foundation, [PERSON] and [PERSON] [9] postulated the synchronous evolution of the permeability coefficient (\(k_{\
u}\)) and compression coefficient (\(m_{\
u}\)) during consolidation, yielding the inaugural analytical solution for one-dimensional nonlinear consolidation. Based on this, [PERSON] et al. [10] developed an analytical solution for one-dimensional nonlinear consolidation of single-layer soil under cyclic loading. [PERSON] et al. [11] developed an analytical solution for nonlinear consolidation of double-layer soil under cyclic loading. Due to the assumption that the permeability coefficient and volume compression coefficient change synchronously, the consolidation coefficient remains unchanged. Subsequently, [PERSON] et al. [12] solved the one-dimensional nonlinear consolidation problem by using the finite difference method with the well-recognized \(e\)-log\(\sigma^{\prime}\) and \(e\)-log\(k_{\
u}\) relationship, which has a wider scope. [PERSON] et al. [13] derived the nonlinear radial consolidation of vertical drains under a general time-variable loading based on [PERSON]'s assumption [11]. [PERSON] [14] and [PERSON] [15] deduced one-dimensional analytical solutions for nonlinear consolidation under instantaneous loading, adhering to a simplified version of [PERSON]'s assumption [12]. Notably, none of the aforementioned one-dimensional nonlinear consolidation inquiries have contemplated the stratification of soil. Driven by [PERSON]'s [11] assumption, [PERSON] et al. [6] achieved an analytical solution for one-dimensional nonlinear consolidation of double-layered soil, a contribution subsequently expanded upon by [PERSON] et al. [16] for the analytical treatment of one-dimensional nonlinear consolidation within layered soil. [PERSON] [17] addressed the one-dimensional nonlinear consolidation challenge in layered soil under variable loading using the DQM. While the nonlinear relationship between \(e\)-log\(\sigma^{\prime}\) and \(e\)-log\(k_{\
u}\) proposed by [PERSON] [12] was more applicable, the consolidation equation thus derived also assumed a nonlinear character, often yielding semi-analytical or numerical solutions. Building upon [PERSON]'s paradigm [12], [PERSON] et al.