Supplemental Material S1: Finite-Difference Discretization and Joint-Node Derivations for Longitudinal Time-dependent Response of Existing Pipelines Triggered by Tunnel Down-crossing Considering Pipeline-soil Interface Gap Evolution

This supplemental material provides detailed finite-difference discretization formulas, joint-node derivations, virtual-node treatments, and global matrix definitions supporting the time-dependent pipeline response analysis presented in the associated manuscript entitled “Longitudinal Time-dependent Response of Existing Pipelines Triggered by Tunnel Down-crossing Considering Pipeline-soil Interface Gap Evolution.” The material is intended to supplement the theoretical formulations in the main manuscript and to improve the reproducibility of the proposed analytical framework.