載重反應關聯法（Load response correlation method，簡記為LRC法)是將結構承受之外力載重歷時等值為其極限靜力載重分佈，對應於此載重分佈之靜力位移場域，即為結構系統之動力尖峰變形反應。因此，由LRC力學理論，結構系統承載其外力載重歷時之特定等值極限靜力載重分佈下，以靜力結構分析結果即可得系統之動力尖峰位移反應，因而能較為便捷地求得結構系統於動力載重下的極值反應行為。
	本論文研究成功地融合了LRC法與結構最佳化設計演算法，藉由本研究成果，靜力結構最佳化問題與設計方法，可通用地推廣至結構承受動力荷載的情況。另一方面，本論文之各結構最佳化演示例題中，除以靜力環境下的情況進行最佳化設計，且與其相對應文獻之分析與設計結果一一比較，以檢核本研究所提出融合LRC理論之最佳計算力學方法外，並使用有限元素套裝軟體ABAQUS對於結構設計進行動力分析，以驗證系統最佳設計結果之動力尖峰變形反應均符合設計限制條件。
	綜合而論，本研究提出一創新之結構最佳設計方法，期能將計算力學於結構動力最佳化設計問題推展至新的層面。

Load response correlation (LRC) method is used to transform dynamic load distributions applied on structures into equivalent extreme static loads. Structural responses correlated to such equivalent static load distribution are called the dynamic peak responses of the structures. Hence, through LRC method, dynamic peak responses of structural system under the distribution of equivalent extreme static loads, which are transformed from loading history, are thus simply obtained by static analysis.
This study successfully integrated LRC method with structural optimization algorithm. With the proposed design scheme, static structural optimization problems can be extended to another scenario which structures are subjected to dynamic loads. In this thesis, for each optimization example, optimization was not only performed under static loading condition, but the result was compared with references to examine the effectiveness of the proposed method for integrating LRC method and optimization algorithm. In addition, a finite element analysis package- ABAQUS was applied to execute dynamic analysis on optimized structures in order to identify that the peak dynamic responses of the optimized structures meet the design constraints.
To sum up, this research proposed an innovative structural optimization method, anticipating it could provide computational mechanics with a new avenue toward engineering design problems.

目錄
第一章	緒論	1
1-1	研究動機	1
1-2	研究目的	2
1-3	研究方法	3
1-4	論文章節及架構	5
第二章	理論與方法	6
2-1	結構最佳化設計簡介	6
2-2	結構最佳化設計演算法	7
2-3	最佳化分析方法與工具介紹	10
2-3-1	數學規劃法	11
2-3-2	有限元素套裝軟體	18
2-4	桁架結構分析	19
2-4-1	靜力分析	19
2-4-2	自然振動頻率	24
2-4-3	動力分析	26
第三章	載重反應關聯法	31
3-1	文獻回顧	31
3-2	理論推導	39
3-3	演算步驟	41
第四章	對應系統動力尖峰反應之結構最佳設計實例	44
4-1	尺寸最佳化設計	45
4-1-1	平面18根桿件桁架結構之尺寸最佳化設計	45
4-1-2	平面10根桿件桁架結構之尺寸最佳化設計	63
4-1-3	平面17根桿件桁架結構之尺寸最佳化設計	78
4-2	形狀最佳化設計	94
4-2-1	平面Michell桁架結構之形狀最佳化設計	94
4-2-2	平面18根桿件桁架結構之尺寸與形狀何最佳化設計	107
4-3	載重情況最佳化設計	123
4-3-1	空間22根桿件桁架結構之載重情況尺寸最佳化	123
4-3-2	空間25根桿件桁架結構之載重情況尺寸最佳化	146
4-3-3	平面37根桿件拱橋結構之載重情況形狀最佳化	168
第五章	結論與建議	181
5-1	結論	181
5-2	未來展望	182
參考文獻	183
附錄 A MATLAB fmincon函式示範例	195

 
圖目錄
圖 1-3-1 研究流程圖	3
圖 2-3-1 結構最佳化設計流程	11
圖 2-3-1 1 線性不等式限制條件下的最佳解	15
圖 2-3-1 2 線性等式和不等式限制條件下的最佳解	16
圖 2-3-1 3 邊界限制條件下的最佳解	17
圖 2-3-1 4 非線性限制條件下的最佳解	18
圖 2-4-1 1 區域座標系統之桿件節點力與位移示意圖	20
圖 2-4-1 2 桁架桿件與全域座標系統座標軸夾角示意圖	21
圖 2-4-3 1 Newmark 數值積分流程圖	30
圖 2-4-3 1 LRC法演算流程圖	43
圖 4-1-1 1 平面18根桿件桁架結構	46
圖 4-1-1 2 平面18根桿件桁架結構最佳化設計之迭代過程	49
圖 4-1-1 3最佳化平面18根桿件桁架結構比較圖	50
圖 4-1-1 4 18根桿件桁架結構之外力歷時	53
圖 4-1-1 5 平面18根桿件桁架結構節點11y方向位移歷時	56
圖 4-1-1 6 平面18根桿件桁架結構對於設計限制條件為正向極值反應之最佳化迭代過程	57
圖 4-1-1 7 平面18根桿件桁架結構對於設計限制條件為負向極值反應之最佳化迭代過程	57
圖 4-1-1 8 動力最佳化平面18根桿件桁架結構比較圖	58
圖 4-1-1 9 最佳化平面18根桿件桁架結構位移歷時之比較	59
圖 4-1-1 10 初始與最佳化平面18根桿件桁架結構 水平方向等值靜載重之比較	60
圖 4-1-1 11 初始與最佳化平面18根桿件桁架結構 垂直方向等值靜載重之比較	60
圖 4-1-2 1 平面10根桿件桁架結構	63
圖 4-1-2 2 平面10根桿件桁架結構迭代過程	66
圖 4-1-2 3 最佳化平面10根桿件桁架結構比較圖	67
圖 4-1-2 4 10根桿件桁架結構P1之外力歷時	70
圖 4-1-2 5 平面10根桿件桁架結構節點4位移歷時	73
圖 4-1-2 6 平面10根桿件桁架結構節點2位移歷時	73
圖 4-1-2 7 平面10根桿件桁架結構迭代過程	74
圖 4-1-2 8 最佳化平面10根桿件桁架結構比較圖	75
圖 4-1-3 1 平面17根桿件桁架結構	78
圖 4-1-3 2 平面17根桿件桁架結構迭代過程	81
圖 4-1-3 3 最佳化平面17根件桁架結構比較圖	82
圖 4-1-3 4 17根桿件桁架結構之外力歷時	86
圖 4-1-3 5 平面17根桿件桁架結構節點9位移歷時	89
圖 4-1-3 6 平面17根桿件桁架結構迭代過程	90
圖 4-1-3 7 最佳化平面17根桿件桁架結構比較圖	90
圖 4-2-1 1 平面Michell桁架結構	95
圖 4-2-1 2 平面Michell桁架結構迭代過程	97
圖 4-2-1 3 最佳化平面Michell桁架結構比較圖	98
圖 4-2-1 4 Michell桁架結構承受之外力歷時	101
圖 4-2-1 5 Michell桁架結構節點1y方向位移歷時	103
圖 4-2-1 6 Michell桁架結構正向極值反應迭代過程	104
圖 4-2-1 7 Michell桁架結構負向極值反應迭代過程	104
圖 4-2-1 8 動力最佳化平面Michell件桁架結構比較圖	105
圖 4-2-2 1 平面18根桿件桁架結構	107
圖 4-2-2 2 平面18根桿件桁架結構迭代過程	109
圖 4-2-2 3 最佳化平面18根桿件桁架結構比較圖	110
圖 4-2-2 4 18根桿件桁架結構之外力歷時	114
圖 4-2-2 5 平面18根桿件桁架結構節點1位移歷時	117
圖 4-2-2 6 平面18根桿件桁架結構正向極值反應迭代過程	118
圖 4-2-2 7 平面18根桿件桁架結構正向極值反應迭代過程	118
圖 4-2-2 8 動力最佳化平面22根桿件桁架結構比較圖	119
圖 4-3-1 1 空間22根桿件桁架結構	124
圖 4-3-1 2 空間22根桿件桁架結構迭代過程	127
圖 4-3-1 3 最佳化空間22根桿件桁架結構比較圖	128
圖 4-3-1 4 22根桿件桁架結構載重情況1之外力歷時	133
圖 4-3-1 5 22根桿件桁架結構載重情況2之外力歷時	134
圖 4-3-1 6 22根桿件桁架結構載重情況3之外力歷時	134
圖 4-3-1 7 空間22根桿件桁架結構載重情況1作用下節點1位移歷時	137
圖 4-3-1 8 空間22根桿件桁架結構載重情況2作用下節點1位移歷時	137
圖 4-3-1 9 空間22根桿件桁架結構載重情況3作用下節點1位移歷時	138
圖 4-3-1 10 空間22根桿件桁架結構正向極值反應迭代過程	139
圖 4-3-1 11 空間22根桿件桁架結構負向極值反應迭代過程	139
圖 4-3-1 12 動力最佳化空間22根桿件桁架結構比較圖	140
圖 4-3-2 1 空間25根桿件桁架結構	146
圖4-3-2 2 空間25根桿件桁結構架迭代過程	149
圖 4-3-2 3 最佳化空間25根桿件桁架結構比較圖	150
圖 4-3-2 4 25根桿件桁架結構載重情況1之外力歷時	156
圖 4-3-2 5 22根桿件桁架結構載重情況2之外力歷時	156
圖 4-3-2 6空間25根桿件桁架結構載重情況1作用下節點1位移歷時	159
圖 4-3-2 7 空間25根桿件桁架結構載重情況2作用下節點1位移歷時	160
圖 4-3-2 8 空間25根桿件桁架結構正向極值反應迭代過程	161
圖 4-3-2 9 空間25根桿件桁架結構負向極值反應迭代過程	161
圖 4-3-2 10 動力最佳化空間25根桿件桁架結構比較圖	162
圖 4-3-3 1 平面37根桿件拱橋結構	168
圖 4-3-3 2 平面37根桿件拱橋結構迭代過程	170
圖 4-3-3 3 最佳化平面37根桿件拱橋結構比較圖	171
圖 4-3-3 4 37根桿件桁架結構載重情況1之外力歷時	173
圖 4-3-3 5 37根桿件桁架結構載重情況2之外力歷時	174
圖 4-3-3 6 平面37根桿件桁架結構載重情況1作用下節點10位移歷時	176
圖 4-3-3 7平面37根桿件桁架結構載重情況2作用下節點10位移歷時	177
圖 4-3-3 8 平面37根桿件拱橋結構正向極值反應迭代過程	178
圖 4-3-3 9 平面37根桿件拱橋結構負向極值反應迭代過程	178
圖 4-3-3 10 動力最佳化平面37根桿件拱橋結構比較圖	179

表目錄
表 4-1-1 1 平面18根桿件桁架結構設計變數	46
表 4-1-1 2 平面18根桿件桁架結構設計條件	47
表 4-1-1 3 最佳化平面18根桿件桁架結構位移	50
表 4-1-1 4 最佳化平面18根桿件桁架結構	51
表 4-1-1 5 平面18根桿件桁架結構承載外力	52
表 4-1-1 6 平面18根桿件桁架結構外力歷時作用下正向和負向極值反應	54
表 4-1-1 7 平面18根桿件桁架結構模態	55
表 4-1-1 8 平面18根桿件桁架結構位移尖峰反應比較	56
表 4-1-1 9 正向極值反應限制條件之最佳化平面18根桿件桁架結構位移	59
表 4-1-1 10 負向極值反應限制條件之最佳化平面18根桿件桁架結構位移	59
表 4-1-1 11 初始與最佳化平面18根桿件桁架結構極值反應之比較	61
表 4-1-1 12 動力最佳化平面18根桿件桁架結構斷面尺寸	62
表 4-1-2 1 平面10根桿件桁架結構設計變數	64
表 4-1-2 2 平面10根桿件桁架結構設計條件	64
表 4-1-2 3 最佳化平面10根桿件桁架結構位移	67
表 4-1-2 4 最佳化平面10根桿件桁架結構應力	68
表 4-1-2 5 最佳化平面10根桿件桁架結構與文獻之比較	69
表 4-1-2 6 平面10根桿件桁架結構承載外力	70
表 4-1-2 7 平面10根桿件桁架結構各節點自由度之正向和負向極值反應	71
表 4-1-2 8 平面10根桿件桁架結構模態	72
表 4-1-2 9 平面10根桿件桁架結構由直接積分法與LRC法求得位移尖峰反應之比較	74
表 4-1-2 10 最佳化平面10根桿件桁架結構位移極值反應	75
表 4-1-2 11 最佳化平面10根桿件桁架結構應力	76
表 4-1-2 12 最佳化平面10根桿件桁架結構斷面資訊	77
表 4-1-3 1 平面17根桿件桁架結構設計變數	78
表 4-1-3 2 平面17根桿件桁架結構設計條件	79
表 4-1-3 3 最佳化平面17根桿件桁架結構位移	83
表 4-1-3 4 最佳化平面17根桿件桁架結構應力	84
表 4-1-3 5 最佳化平面17根桿件桁架結構與文獻之比較	85
表 4-1-3 6 平面17根桿件桁架結構承載外力資訊	86
表 4-1-3 7 平面17根桿件桁架結構外力作用下正向和負向極值反應	87
表 4-1-3 8 平面17根桿件桁架結構模態	88
表 4-1-3 9 平面17根桿件桁架結構位移尖峰反應比較	89
表 4-1-3 10 最佳化平面17根桿件桁架結構位移	91
表 4-1-3 11 最佳化平面17根桿件桁架結構應力	92
表 4-1-3 12 最佳化平面17根桿件桁架結構	93
表 4-2-1 1 Michell桁架結構座標設計變數	95
表 4-2-1 2 平面Michell桁架結構設計條件	96
表 4-2-1 3 最佳化平面Michell桁架結構位移	98
表 4-2-1 4 最佳化平面Michell桁架結構與文獻之比較	99
表 4-2-1 5 Michell桁架結構外力資訊	100
表 4-2-1 6 平面17根桿件桁架結構外力作用下正向和負向極值反應	101
表 4-2-1 7 Michell桁架結構模態	102
表 4-2-1 8 Michell桁架結構位移尖峰反應比較	103
表 4-2-1 9 最佳化平面Michell桁架結構位移	106
表 4-2-1 10 最佳化平面Michell桁架結構	106
表 4-2-2 1 平面18根桿件桁架結構設計條件	108
表 4-2-2 2 最佳化平面18根桿件桁架結構應力	111
表 4-2-2 3 最佳化平面18根桿件桁架結構位移	112
表 4-2-2 4 最佳化平面18根桿件桁架結構與文獻之比較	113
表 4-2-2 5 平面18根桿件桁架結構外力資訊	114
表 4-2-2 6 平面18根桿件桁架結構外力作用下正向和負向極值反應	115
表 4-2-2 7 平面18根桿件桁架結構模態	116
表 4-2-2 8 平面18桿件桁架結構位移尖峰反應比較	117
表 4-2-2 9 正向極值反應最佳化平面18根桿件桁架結構應力	120
表 4-2-2 10 負向極值反應最佳化平面18根桿件桁架結構應力	121
表 4-2-2 11 動力最佳化平面18根桿件桁架結構	122
表 4-3-1 1 空間22根桿件桁結構架節點座標及設計變數組合	125
表 4-3-1 2 空間22根桿件桁架結構設計條件	126
表 4-3-1 3 最佳化空間22根桿件桁架結構位移	129
表 4-3-1 4 最佳化空間22根桿件桁架結構應力	130
表 4-3-1 5 最佳化空間22根桿件桁架結構與文獻之比較	131
表 4-3-1 6 空間22根桿件桁架結構外力資訊	132
表 4-3-1 7 空間22根桿件桁架結構外力作用下正向和負向極值反應	135
表 4-3-1 8 空間22根桿件桁架結構模態	136
表 4-3-1 9 空間22根桿件桁架結構載重情況1作用下位移尖峰反應比較	137
表 4-3-1 10 空間22根桿件桁架結構載重情況2作用下位移尖峰反應比較	138
表 4-3-1 11 空間22根桿件桁架結構載重情況3作用下位移尖峰反應比較	138
表 4-3-1 12 正向極值反應最佳化空間22根桿件桁架結構位移	141
表 4-3-1 13 負向極值反應最佳化空間22根桿件桁架結構位移	142
表 4-3-1 14 正向極值反應最佳化空間22根桿件桁架結構應力	143
表 4-3-1 15 負向極值反應最佳化空間22根桿件桁架結構應力	144
表 4-3-1 16 動力最佳化空間22根桿件桁架結構	145
表 4-3-2 1 空間25根桿件桁架結構節點座標及設計變數組合	147
表 4-3-2 2 空間25根桿件桁架結構設計條件	148
表 4-3-2 3 最佳化空間25根桿件桁架結構位移	151
表 4-3-2 4 最佳化空間25根桿件桁架結構應力	152
表 4-3-2 5 最佳化空間25根桿件桁架結構與文獻之比較	154
表 4-3-2 6 空間25根桿件桁架結構外力資訊	155
表 4-3-2 7 空間25根桿件桁架結構外力歷時作用下正向和負向極值反應	157
表 4-3-2 8 空間25根桿件桁架結構模態	158
表 4-3-2 9 空間25根桿件桁架結構載重情況1作用下位移尖峰反應比較	159
表 4-3-2 10 空間25根桿件桁架結構載重情況2作用下位移尖峰反應比較	160
表 4-3-2 11 正向極值反應最佳化空間25根桿件桁架結構位移	163
表 4-3-2 12 負向極值反應最佳化空間25根桿件桁架結構位移	164
表 4-3-2 13 正向極值反應最佳化空間25根桿件桁架結構應力	165
表 4-3-2 14 負向極值反應最佳化空間25根桿件桁架結構應力	166
表 4-3-2 15 動力最佳化空間25根桿件桁架結構	167
表 4-3-3 1 平面37根桿件拱橋結構設計變數組合	168
表 4-3-3 2 平面37根桿件拱橋結構設計條件	169
表 4-3-3 3 最佳化平面37根桿件拱橋結構位移	172
表 4-3-3 4 最佳化平面37根桿件拱橋結構與文獻之比較	172
表 4-3-3 5 平面37根桿件拱橋結構外力資訊	173
表 4-3-3 6 平面37根桿件拱橋結構外力作用下正向和負向極值反應	174
表 4-3-3 7 平面37根桿件拱橋結構模態	175
表 4-3-3 8 平面37根桿件拱橋結構載重情況1作用下位移尖峰反應比較	176
表 4-3-3 9 面37根桿件拱橋結構載重情況2作用下位移尖峰反應比較	177
表 4-3-3 10 正向極值反應最佳化平面37根桿件拱橋結構位移	180
表 4-3-3 11 負向極值反應最佳化平面37根桿件拱橋結構位移	180
表 4-3-3 12 動力最佳化平面37根桿件桁架結構	180

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