Nonlinear vibrations of earth structures

Abstract

This article offers a detailed analysis of the current state of plane structure dynamics, addressing the complexities posed by non-homogeneous materials exhibiting nonlinear elastic and viscoelastic properties during real-life operations. The study proposes a comprehensive mathematical model and algorithm to investigate the dynamic behavior of such structures, employing the non-linear hereditary Boltzmann-Volterra theory to describe viscoelastic material properties accurately. Nonlinear oscillatory systems are analyzed using Lagrange equations based on the d’Alembert principle. The problem is approached through a multi-step process. Initially, the linear elastic problem of the structure’s natural oscillations is solved to determine its natural frequencies and modes of oscillations. Subsequently, these eigenmodes are employed as coordinate functions to address forced nonlinear oscillations in viscoelastic non-homogeneous systems. The complexity of the problem necessitates solving a Cauchy problem comprising a system of nonlinear integrodifferential equations. To illustrate the methodology, the study examines the Gissarak earth dam, considering real operational conditions and nonlinear, viscoelastic material properties near resonant modes of vibrations. Utilizing numerical methods, the dynamic behavior of the structure is analyzed, assessing displacements and stress components at different time points under non-stationary kinematic action. Stress concentration regions within the structure are identified for resonant vibrations, allowing the evaluation of its strength. Furthermore, the impact of nonlinear elasticity and viscoelasticity on the structural dynamics is quantified. This research provides valuable insights into the behavior of plane non-homogeneous structures, considering real-world scenarios and material complexities, ultimately contributing to an improved understanding of structural dynamics and facilitating the identification and mitigation of potential structural challenges.