利用MLS-Ritz法分析預載重下具內部裂縫方形3D彈性板之挫屈載重與自然頻率

Abstract

板在工程應用中，由於在製造過程中品質不良或工程實務上造成裂縫的產生；當板受載重時，裂縫會造成初始應力重新分配，導致與完整板之動態行為與穩定性不同。本研究以三維彈性理論為基礎，利用MLS-Ritz 法分析具內部裂縫方形板受面內載重下之穩定性與振動行為。Ritz法所需之允許函數是由z向(板厚方向)多項式配合由移動最小平方差法建構之x-y平面允許函數。在MLS建構允許函數時，於基底函數中導入裂縫函數來準確描述具裂縫板內應力之奇異性以及滿足跨越裂縫之板內位移不連續性，以提升傳統Ritz 法描述局部的物理行為的能力。 本研究對具有內部裂縫方形板之挫屈載重以及自然振動頻率進行收斂分析，並與文獻比較。根據此收斂性之結果，進一步探討不同裂縫長度、角度和位置、不同幾何邊界條件與荷重型式下對內部裂縫板之挫屈載重與振動行為的影響。此外，亦列出對所分析結果之挫屈模態與受載重下自然振動頻率模態圖供參考。

A plate is commonly used in engineering structures, and often suffers damage resulting from cracks, which may be caused by manufacturing defects, fatigue, and unexpectedly high loads, among other factors. A crack yields the reduction in the stiffness of a plate and redistributuions of the internal stresses in a loaded plate. Consequently, a crack can significantly change the stability and dynamic characteristics of a plate. The buckling loads and vibration frequencies of an internally cracked square plate under in-plane loads are determined by the MLS-Ritz method along with the three-dimensional elasticity theory. The admissible functions are constructed by polynomial functions in the thickness direction multiplied wiith the in-plane admissible functions established by the moving least-squares method with a set of enriched basis functions, which consists of crack functions showing the stress singularities at the fronts of a crack and the displacement discontinuity across the crack. In this study, convergence studies on the buckling loads and the vibration frequencies of square plates with internal cracks are carried out. The convergent results agree well with the published results based on thin plate theory. Then, the proposed approach is applied to investigate the effects of cracks’ lengths, orientations, and locations on the buckling loads and vibration frequencies of internally cracked square plates under different in-plane loading conditions and boundary conditions.