利用MLS-Ritz法於預載重下具邊緣裂縫或V型缺口矩形板之挫屈載重與自然頻率振動分析

Abstract

板在工程中，由於使用性設計或是製造過程而產生裂縫或缺口，並因裂縫尖端造成應力集中，使裂縫或缺口將持續增長，其靜態加載與動態振動行為將顯著不同於完整板；本研究基於古典板理論下利用著名的Ritz法，並搭配以移動最小平方差法(MLS)引入增益型基底函數建構之允許函數，分析具邊緣裂縫或Ｖ型缺口矩形板受面內預載重下之穩定性與自然振動頻率。增益型基底函數為多項式函數與裂縫角函數，有別於傳統Ritz法其建構之允許函數能有效描述具邊緣裂縫或V型缺口板之幾何不連續與裂縫尖端之應力奇異性；為了穩定數值結果，本研究針對面內位移採取了多項式權函數與指數型搭配繞射權函數。本研究針對受線性預載重下具邊緣裂縫或Ｖ形缺口矩形板的應力強度因子、挫屈載重以及自然振動頻率進行精確度及效率之收斂分析，並與表列文獻值比較。最終本研究所列之收斂分析可觀測具邊緣裂縫或Ｖ形缺口矩形板之裂縫開口位置、裂縫長度、裂縫角度以及Ｖ形缺口開口角度對挫屈載重、振動頻率以及模態形狀之影響。面內位移之邊界條件為兩對邊滿足法向曳引力(normal traction)，其他為自由邊界。面外位移考慮了四種邊界條件，分別為SSSS、CSCS、CFCF以及SFSF；其中S、C與F分別代表了簡單支撐(simply supported)、全束制(clamped)以及全自由(free)邊界。

This study deals with vibration and stability analyses of a rectangular plate with a side crack or a v-notch based on classical plate theory using the famous Ritz method with the admissible functions that are constructed by the moving least square (MLS) method with enriched basis functions. The enriched basis functions consist of regular polynomial functions and crack functions that appropriately describe the stress singularities at the tip of a crack or a v-notch and show the discontinuities of displacement or slope crossing the crack. Comprehensive convergence studies on the stress intensity factor, buckling loads and vibration frequencies of a cracked or a v-notched rectangular plate under linear loading at its two opposite edges are carried out and demonstrate the accuracy and efficiency of the presented approach by comparing the present results with the published ones. Finally, the present approach is applied to investigate the effects of location, length and orientation of side cracks or v-notches on the buckling loads, vibration frequencies and mode shapes of cracked rectangular plates. The boundary conditions for in-plane are normal traction prescribed along two opposite edges and free along the other edges. Four boundary condition combinations are considered for out-of-plane, and they are SSSS, CSCS, CFCF, SFSF, where S, C, and F donate simply-supported, clamped, and free boundary conditions, respectively.