拱梁接觸單向彈性基底的穩定性分析

Abstract

本論文探討受單向彈性基底拘束的拱梁在集中力作用下的平衡解及其穩定性。所謂的單向基底意指拱梁和基底間的接觸力只能是壓力，這大幅的增加了問題的困難性。我們利用模態展開法來分析拱梁接觸彈性基底後的變形。為此，先假設拱梁和彈性基底接觸段的長度和中心點的位置為已知，而接觸段間的分佈接觸力可近似為已知大小的離散集中力。這個情形下，我們可以求得對應的拱梁的變形。另一方面，拱梁的變形必需滿足接觸段的幾何約束和彈性基底的特性，由此，利用牛頓法遞迴修正先前猜測的接觸段的幾何參數和接觸力的大小，便可以得到正確的接觸力。我們另外應用打靶法驗證模態展開法所得解的正確性。當多重平衡解存在時，利用能量法可以判別各個平衡解的穩性。利用所開發的數值方法，我們詳細分析了單向彈性基底對拱梁平衡解和其穩定性的影響。

This thesis investigates the equilibrium solutions and their stability for an arch constrained by a unilateral elastic foundation under a concentrated load. The unilateral foundation can only support compressive reaction forces and hence significantly increases the complexity of the problem. We employ the modal expansion method to analyze the deformation of the arch when it comes into contact with the unilateral foundation. To this end, we first assume that the length and central position of the contact segment are known, and the distributed contact force within this segment can be approximated by discrete concentrated forces of known magnitude. Under these assumptions, the corresponding deformation of the arch can be obtained. On the other hand, the deformation must satisfy the geometric constraints of the contact segment as well as the constitutive law of the elastic foundation. By applying Newton’s method, we iteratively correct the initially assumed geometric parameters of the contact segment and the magnitude of the contact forces to arrive at the correct solutions. Additionally, we verify the accuracy of the modal expansion solution using the shooting method. When multiple equilibrium solutions exist, the energy method is used to determine their stability. With the developed numerical approach, we conduct a detailed study on the effects of the unilateral elastic foundation on the equilibrium solutions and stability of the arch.