具有奇異應力矩形Mindlin板

Abstract

幾何不連續的現象，在板構件中往往衍生了應力奇異點(stress singularity)之問題。如何正確模擬具應力奇異之行為，明顯影響數值分析之正確性。本論文提出有限元素法配合二階形狀函數，並於包含奇異點附近的元素引入應力奇異之漸近解，即角函數(corner function)，確保能得到準確解。經以具有V型缺口之扇形板及具有裂縫矩形板之收斂性分析，證明了本研究發方法之可行性及正確性。於有限元素法中加入角函數的確能有效加速數值解之收斂。最後，應用本研究方法於探討不同裂縫長度、位置與方向對矩形板振動頻率及模態之影響。

A sharp corner often occurs in a plate component and causes stress singularity at the corner. The accuracy of numerical analysis for such plate often depends on the correctness of modeling the stress singularity behavior. This work proposes a finite-element-based technique to analyze Mindlin plates with stress singularities. The proposed technique imposes the corner functions, which are confined in a small area covering a corner and describe the stress singularity behavior at the corner, into the conventional finite element approach. The validity and correctness of the approach is verified by convergence studies for the vibrations of a sectorial plate and a rectangular plate with a crack. The distribution of stress resultants near the crack is also investigated for a rectangular plate with a crack under static loading. Adding corner functions to the conventional finite element approach can effectively accelerate the convergence of the numerical results. Finally, the proposed technique is applied to investigate the effects of the length, position, and orientation of a crack on the vibration frequencies of a cracked rectangular plate.