Analytical solutions for mechanical responses of cast-in-place lattice anchoring frame girders based on transfer matrix method

This investigation presents the analytic solution for the mechanical responses of cast-inplace lattice anchoring frame girders based on the transfer matrix method, facilitating the frame girder design during slope support. The underlying elastic foundation model, which idealizes the supporting soil as a series of independent springs, is adopted to describe the soil-structure interaction. Firstly, the lattice-type beams are divided into longitudinal and horizontal beams, which are considered as Winkler-type elastic foundation beams under multiple concentrated forces, respectively. The corresponding anchorage force distribution formula is proposed based on the force balance and deformation coordination principles of the longitudinal beams and horizontal beams at the anchor points. Then, the longitudinal beam or horizontal beam after splitting is considered to be composed of multiple beam elements, where each beam element is subjected to only one concentrated force. The general solution for any beam element is obtained, and subsequently, the transfer matrix of the considered whole beam is established according to the continuity conditions between the beam elements. By applying the boundary conditions, the closed-form solution for the mechanical responses of the frame girders is obtained. Finally, the rationality of this solution is verified by comparing it with the outcomes of finite element calculations and published centrifugal model tests. Numerical cases are also conducted to explore the effects of key design parameters. Notably, an anchor spacing of 3.0-3.5 m reduces the maximum bending moment by approximately 28%, and a cantilever length of 1.0 m further optimizes the internal force distribution and reduces the maximum bending moment by about 33%, which provides practical design recommendations for the corresponding engineering application.