主應力線應用於結構最佳化設計

Application of Principal Stress Lines on Structural Optimization Design

簡孟笙(Meng-Sheng Chien)

指導教授 ： 呂良正

國立臺灣大學/工學院/土木工程學研究所/碩士(2019年)

https://doi.org/10.6342/NTU201902476

摘要

隨著多樣化且具有流線形之大跨度薄殼結構在結構工程案例中日趨增加，本研究除了透過自由曲面建模之技術模擬薄殼結構，也導入最佳化理論於設計之過程，並結合本研究開發之主應力線生成演算法，以提供設計者兼具力學性質與美學外型之設計結果。
本研究將簡介何謂主應力線，詳述主應力線生成之概念及作法，從後處理計算、離散向量場內插到結合數值積分的應用，並透過多種載重及邊界條件的比對來探討主應力線之性質，且搭配實際樓版案例，討論使用主應力梁版系統之優異。
本研究利用NURBS作為主要的自由曲面建模方法，透過控制點、節點向量和基底函數等進行參數化控制和建模，其中利用最佳化方法之循序二次規劃法(Sequential Quadratic Programming, SQP)做最佳化設計，並結合主應力線生成演算法，探討最佳化薄殼結構在主應力線方向佈置加勁肋梁之影響。
真實世界中存在有眾多且多樣的實際網柵薄殼結構，本研究將參考部分案例，針對實際尺寸之網柵薄殼結構做最佳化設計，且結合主應力線之佈置方式，將其作為網柵主結構比較優劣，並考慮活載重、自重及風載重對結構的影響，其中，風載重將以計算流體力學(CFD)模擬結構所受之風壓分佈，在結合我國的設計規範下，定義出最佳化問題之限制式，使得實際網柵薄殼結構之模擬結果更具參考價值。
本研究分析工具以Python語言進行程式撰寫，結合有限元素軟體ABAQUS進行分析，程式內容將包含上述提到之NURBS曲面建模方法以及本研究所開發之主應力線生成演算法，透過ABAQUS進行最佳化之自動化建模與分析，以獲得最終設計結果。

關鍵字

主應力線 ； 自由曲面 ； 結構最佳化 ； 形狀最佳化 ； 尺寸最佳化 ； 計算流體力學 ； 薄殼網柵結構 ； 有限元素套裝軟體

並列摘要

Shell structures are widely used in structure design due to the benefit of their load-carrying capacity. However, considering the large-span shells failure region, the issues of designing reinforcement for the thin shell or supporting structure optimally for the grid shell are coming to be essential to be sorted out. Therefore, this thesis focuses on using principal lines generation techniques to stiffen the large span shell structures, model the grid shell structures and integrates with free-form surface based on optimization theory to obtain a final design considering both aesthetics and mechanical behaviors.
In this thesis, we suggest a computational algorithm to design and optimize ribs layout on given design domain to enhance the global structural and mechanical performance. The core concept of our method is to place ribs along the principal stress lines which demonstrate the paths of stress flow. Based on certain surface and external loads, Finite Element Analysis can be performed and solve the physical field numerically. With the post-processing from FEM results, directional vector field of principal stress on nodes can be obtained by eigenvalue calculation. We use the ideal of particle trajectories to generate the principal stress lines by numerical integration. Discussion of sensitivity, orthogonality, accuracy and other property of principal stress lines are all considered in this research.
Besides ribs layouts, shape optimization and sizing optimization are also the major parts and typical types of structural optimization. Our thesis using NURBS, which is a parametric method to approach the free-form surface, to construct and control our design domain. This method contains geometric tuning factors, such as control points, knots factors, etc. that gives user a great flexibility to handle the shape analytically. In the both shape and sizing optimization problems, sequential quadratic programming(SQP) is chosen as the optimization method to find the final design.
In our implementation, Demonstration of principal stress lines on shell structures are separated into two parts. Firstly, ribs play a reinforced role in the shell structure which can improve the overall structural performance and deflection. Secondly, some famous real grid shell structures are taken as examples and utilized as the initial models of optimization problems. After acquiring the optimal results, switch the grid with ribs along the stress flow to achieve better static performances. Mentioning about designing the real grid shell structure, some factors that may influence the shell structure must be considered, such as load cases, structural strength, maximum displacement, etc. Self-weight, live load and wind load are often considered as the essential load cases. Computational fluid dynamic (CFD) analysis will be used to attain the wind pressure among the complex shape of shell surface. After solving by CFD analysis, and then mapping as a distribution of wind pressure to the surface of shell structures. Furthermore, the constraints of optimization problems are defined by Taiwan construction specifications and limit conditions specified in the codes, such as limitation of strength and displacement. Through these considerations to real grid shell structure design, engineers and architects can get more practical optimal result as reference.
This research develops Python program to control the finite element analysis commercial software ABAQUS. With the analysis results from ABAQUS, the program can carry out the optimization analysis and find the final optimal results. The Python program allows users to create free-form surface by previous parametric NURBS method and generate the principal stress lines by our numerical method.

並列關鍵字

Principal stress lines ； Ribs layout ； Particle trajectories ； Free-form surface ； Structural optimization ； Shape optimization ； Sizing optimization ； Computational fluid dynamic ； Thin shell structure ； Grid shell structure

參考文獻

Abel, J. F., & Chilton, J. C. (2011). Heinz Isler – 50 years of “new shapes for shells”.Journal of The International Association for Shell and Spatial Structures, 52(3), 131-134.

Arora, J. S. (2011). Introduction to Optimum Design. London, Academic Press.

Basso, P., Del Grosso, A. E., Pugnale, A., & Sassone, M. (2009). Shape optimization of free form grid‐shells. Innovative Design & Construction Technologies – Building complex shapes and beyond Conference Paper.

Butcher, J. C. (1963). Coefficients for the study of Runge-Kutta integration processes.Journal of the Australian Mathematical Society, 3(2), 185-201.

Cheng, C. M., & Fu, C. L. (2010). Characteristic of wind loads on a hemispherical dome in smooth flow and turbulent boundary layer flow. Journal of Wind Engineering and Industrial Aerodynamics, 98(6-7), 328-344.