P–$\mathrm{\Delta }$ Effect Analysis of Tall Slender Structures Subjected to Arbitrary Time-Variant Axial Forces: A Differential Quadrature-Based Approach

Traditional methods for analyzing the P–$\mathrm{\Delta }$ effect in tall structures often fail to fully account for time-varying axial forces, potentially underestimating the impact of the P–$\mathrm{\Delta }$ effect on structural safety. To address this limitation, this paper introduces a high-precision method based on the Differential Quadrature Method (DQM) for advanced analysis of the P–$\mathrm{\Delta }$ effect, applicable to both distributed mass, and concentrated mass structural systems. This method discretizes the partial differential equations governing the motion of structures subjected to arbitrary lateral and axial dynamic loads using Differential Quadrature (DQ) principles, applying DQ weighting matrix corrections method to handle boundary conditions. Additionally, it incorporates the unconditionally stable Newmark average acceleration method to solve for dynamic responses inclusive of the P–$\mathrm{\Delta }$ effect. The method is validated through case studies of high-rise buildings and tall bridge piers, demonstrating strong agreement with results from general finite element software. Results indicate that the proposed method not only effectively captures the influence of time-invariant and time-varying axial forces on lateral vibration characteristics but also provides high-precision dynamic response analysis, offering a highly efficient, and practical tool for analyzing dynamic responses in tall structures.