## 6 Conclusions  
The mechanism of the FASSI with different embedded depths and slab lengths was investigated by using experimental tests, finite element simulations, parametric analyses, and a proposed simplified calculation formula. The main conclusions that can be drawn are as follows:  
* The load-displacement curves of FASSI under push or pull displacement can be divided into three stages: the elastic stage, the elastoplastic stage, and the failure stage. In the elastic stage, the longitudinal displacement of a FAS can be absorbed by the sand and no obvious cracks or deformations can be observed on the sand surface. In the elastoplastic stage, part of the sand enters the plastic deformation stage, and bumps and voids can be observed on the sand surface. In the failure stage, a through-going shear slip surface is formed, leading to an overall shear failure of the sand. * Under push displacement, the sand at the distal end of the approach slab is in a passive state and uplifted to form a bump due to compression; the sand at the proximal end of the approach slab is in an active state and subsides to form a void due to tension. Under pull displacement, the deformations of the sand at the distal and proximal ends of the approach slab are opposite to those under pull displacement. The pavement  
Figure 24: Comparison of \(k(H,L)\) and \(F_{\text{u}}(H,L)\) obtained from simplified calculation formula, tests, and FEM: (**a**) \(k(H,L)\); (**b**) \(F_{\text{u}}(H,L)\). deformation can be reduced with an increase in the embedded depth, which could reduce the risk of pavement settlement and cracking. * With an increase in the embedded depth or slab length, the stiffness and ultimate load of the load-displacement curves of FASSI increase under push displacement, the stiffness increases and the ultimate load decreases under pull displacement. * The longitudinal displacement transfer mode and vertical deformation distribution mode of FASSI are affected by the embedded depth and slab length. With an increase in the embedded depth or a decrease in the slab length, the sand deformation on its surface decreases, which is beneficial for avoiding pavement crack risks and improving pavement evenness. * A simplified calculation formula that can be used to predict the load-displacement curves of FASSI based on a bilinear function model was proposed by considering the embedded depth and slab length as the parameters. The functions for the embedded depth and slab length were obtained through three-dimensional surface fitting. This research forms the basis for more in-depth investigations on FASSI under cyclic load in jointless bridges. Based on the authors' experience of systems at different depths under cyclic load, with an increase in the embedded depth, the ductility and energy dissipation capacity of FASSI may be improved. In future research, the influence of the cyclic load and the different soil properties on FASSI will be investigated in detail through experimental tests. Conceptualization, B.B., J.X. and C.N. ; methodology, Y.T., B.B., J.X. and C.N. ; software, Y.T. ; validation, Y.T., B.B., J.X. and C.N. ; formal analysis, Y.T. and J.X. ; investigation, Y.T., resources, F.H. ; data curation, Y.T. and J.X. ; writing--original draft preparation, Y.T. ; writing--review and editing, Y.T., B.B., J.X., C.N. and F.H. ; visualization, Y.T. ; supervision, B.B., J.X. and C.N. ; project administration, J.X. ; funding acquisition, B.B. and J.X. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (51508103); the National Natural Science Foundation of China (51778148); the Recruitment Program of the Global Experts Foundation (TM2012-27); the 2023 Fujian Provincial Transportation Science and Technology Demonstration Project (SF20230202); the Open Project Fund of the Sustainable and Innovative Bridge Engineering Research Center of Fujian Province University (Grant No.