含不確定性的複合材料結構之強健最佳設計研究

Abstract

纖維強化疊層板是由許多單層組成，其最佳設計是在選取使用單層的數目與各層的角度，以獲得最佳的強度、勁度或其它力學性質，受限於製作工藝水準及使用機具的精度，複合材料疊層板的實際製作厚度與各層角度與設計值會有誤差，材料性質的變異性也大於金屬材料，使得實際的複合材料結構特性與設計預期的理想狀況會有一些差異。機率法及反最佳設計法為常用於處理複合材料厚度、角度及材料性質不確定性的強健最佳設計方法，但機率法須有很多原始資料，反最佳設計法需要大量的數值運算而只運用於設計變數較少的強健最佳設計問題。 本論文提出新的強健最佳設計方法，所提方法藉著含設計變數及非設計變數的不確定量及原限制條件對不確定量的靈敏度之修正限制條件，將不確定性的影響直接以近似解處理，使靈敏度除作為設計改良方向的參考依據外，並作為以近似解求解反最佳設計次問題的媒介，所提出的方法因只有最佳設計次問題須以數值法求解，可降低數值運算時間，而運用於較複雜問題的強健最佳設計上。文中提出僅處理設計變數不確定性，與同時考慮設計變數及非設計變數不確定性的強健最佳設計方法，並以具解析解的複合材料樑結構確認所提方法的正確性。再將其運用於結構較複雜及設計變數較多的複合材料結構的強健最佳設計，包括含圓孔及不含圓孔疊層板的強度與勁度之最佳設計、疊層板的熱挫曲最佳設計與具金屬內襯的複合材料圓筒結構之強健最佳設計。 強健最佳設計結果顯示，單層厚度與角度不確定性及材料性質不確定性對複合材料的最佳設計的影響量隨考慮問題而有顯著差異，材料性質不確定性對複合材料疊層板開孔附近的應力之影響特別顯著，而單層厚度不確定性對複合材料疊層板的熱挫曲之影響很大。對使用次數較少而採用塑性設計的具金屬內襯複合材料圓筒結構而言，金屬內襯的厚度愈小，複合材料圓筒結構的最佳重量愈小，故在製作工藝允許的情形下，金屬內襯的厚度愈小愈好；若使用次數較多而採用彈性設計時，金屬內襯的最佳厚度不是發生在0的狀況。

Fiber reinforced composite material is composed of many plies. The problem of optimal design of composite structure is to select the proper ply arrangements so as to achieve the highest performance for the specified requirement of strength, stiffness or other mechanical properties. Due to the uncertainties in material properties and the variations of ply thickness and ply orientation in manufacturing, the practical design properties can be different from what the designers predict. Robust optimal design techniques such as anti-optimization method and probabilistic optimal method are frequently used to deal with the optimal design problems of composite structures with uncertainties, however the traditional anti-optimization method is time consuming, and the accurate probability distribution needed for probabilistic optimal method is not easy to obtain. An innovative method of optimization considering design-variable uncertainties, such as ply thickness and orientation uncertainties, and non-design-variable uncertainties such as material property uncertainties is proposed. By including the sensitivities and uncertainties in the modified constraints, a robust optimum design problem is formulated. Besides being used to determine the most appropriate direction in the optimization algorithm, the sensitivities in the modified constraints are also served as the media to evaluate the effects of manufacturing uncertainties. In the proposed approach the numerical method is still needed for the optimization sub-problem, however the anti-optimization sub-problem is analytically rather than numerically solved. It is therefore more efficient in computing time than traditional anti-optimization technique where both optimization sub-problem and anti-optimization sub-problem are numerically solved. A beam-like composite laminate with analytical solution was used to verify the accuracy of the proposed method. The proposed method was then used to perform the robust optimal design of complex laminate structures including holed and non-holed laminate with stress and stiffness constraints, laminate with thermal buckling constraint, and fiber reinforcement composite cylinder with metallic liner subjected to uniform pressure and local loads. The influences of ply thickness and orientation uncertainties and material property uncertainties on optimal weight were surveyed. The most significant effect of material uncertainty is found for the holed laminate plate, and the most significant effect of ply uncertainty is occurred for laminate plate with thermal buckling constraint. Thickness and allowable strain of metallic liner are the other two factors affecting the optimal weight of the fiber reinforcement composite cylinder with metallic liner. If the allowable strain of composite layer is less than that of the metallic liner, the metallic liner should be kept as small as possible to obtain an optimal design, and the optimal thickness of metallic liner occurs at a particular value other than very close to zero if the allowable strain of composite layer is larger than that of the metallic liner.