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利用结合纠偏百分位思想的Bootstrap方法对疲劳分散系数进行置信区间分析的可信性已经在上节中得到了验证，下面对文献[1]中选取的GC4、30CrMnSiNi2A钢合金的5种状态共140个试件的疲劳寿命试验数据和LY12、LC4、LC9铝合金的10种状态共295个试件的疲劳寿命试验数据进行疲劳分散系数置信区间分析。为了研究使用载荷谱中的高中低载荷对试验数据标准差及疲劳分散系数的影响，文献[1]在数据处理时，每组标准差
数据除
和
的试验数据单独统计外，其他二者间的试验数据按
各数据取平均值。由于置信区间不能像点估计那样取平均值，因此表9 4和表9 5中列出了不同应力水平下的标准差
的置信区间的详细结果。
表9 4 140个钢合金试件的标准差
和疲劳分散系数
置信区间计算表
|
材料 |
|
试件
个数 |
原始样本
Si估计值 |
原始样本
Lf估计值 |
S纠偏Boot
置信区间 |
Lf纠偏Boot
置信区间 |
||
|
GC-4 |
~0.60 |
6 |
0.0941 |
|
2.1 |
[0.0390, 0.1278] |
|
[1.4, 2.8] |
|
Kt=1 |
~0.45 |
9 |
0.2504 |
7.2 |
12.5 |
[0.1455, 0.3417] |
[3.2, 15.0] |
|
|
11 |
0.3691 |
18.0 |
[0.2591, 0.4576] |
[7.6, 35.9] |
||||
|
R=-1 |
~0.30 |
10 |
0.3673 |
|
18.0 |
[0.3233, 0.4200] |
|
[12.7, 27.2] |
|
30CrMnSiNi2A |
~0.60 |
4 |
0.1148 |
|
2.6 |
[0.0313, 0.1482] |
|
[1.3, 3.5] |
|
Kt=2.9 |
~0.45 |
4 |
0.1069 |
2.5 |
2.3 |
[0.0448, 0.1348] |
[1.5, 3.1] |
|
|
5 |
0.0920 |
2.1 |
[0.0789, 0.1029] |
[1.9, 2.3] |
||||
|
R=0.1 |
~0.30 |
6 |
0.1485 |
|
3.3 |
[0.0971, 0.1946] |
|
[2.2, 4.9] |
|
材*2 |
~0.60 |
3 |
0.0224 |
|
1.2 |
[0.0229, 0.0256] |
|
[1.2, 1.2] |
|
Kt=2.9 |
~0.45 |
3 |
0.0460 |
1.5 |
3.2 |
[0.0385, 0.0508] |
[1.4, 1.6] |
|
|
6 |
0.2028 |
5.2 |
[0.1365, 0.3001] |
[3.0, 11.5] |
||||
|
6 |
0.1358 |
3.0 |
[0.1153, 0.1729] |
[2.6, 4.1] |
||||
|
R=0.5 |
~0.30 |
4 |
0.1332 |
|
3.1 |
[0.0083, 0.1564] |
|
[1.1, 3.7] |
|
材*2 |
~0.60 |
3 |
0.0643 |
|
1.7 |
[0.0630, 0.0655] |
|
[1.7, 1.8] |
|
Kt=4.1 |
~0.45 |
4 |
0.1172 |
2.7 |
2.4 |
[0.0261, 0.1482] |
[1.2, 3.5] |
|
|
3 |
0.0838 |
2.1 |
[0.0575, 0.0962] |
[1.6, 2.3] |
||||
|
4 |
0.1065 |
2.4 |
[0.0417, 0.1372] |
[1.4, 3.2] |
||||
|
R=0.5 |
~0.30 |
4 |
0.1981 |
|
5.3 |
[0.1096, 0.2607] |
|
[2.5, 9.0] |
|
材*2 |
~0.60 |
6 |
0.0487 |
|
1.5 |
[0.0140, 0.0665] |
|
[1.1, 1.7] |
|
Kt=3 |
~0.45 |
11 |
0.1113 |
2.4 |
2.2 |
[0.0353, 0.1448] |
[1.3, 3.1] |
|
|
10 |
0.1041 |
2.3 |
[0.0743, 0.1407] |
[1.8, 3.0] |
|
|||
|
6 |
0.0766 |
1.9 |
[0.0491, 0.0993] |
[1.5, 2.2] |
|
|||
|
R=0.445 |
~0.30 |
12 |
0.1359 |
|
2.9 |
[0.0942, 0.1972] |
|
[2.1, 4.6] |
|
平均 |
~0.60 |
|
|
|
1.8 |
|
|
|
|
~0.45 |
|
|
|
4.5 |
|
|
|
|
|
~0.30 |
|
|
|
6.5 |
|
|
|
|
|
|
|
140 |
23.3 |
|
|
|
|
|
表9 5 295个铝合金试件的标准差
和疲劳分散系数
置信区间计算表
|
材料 |
|
试件
个数 |
原始样本
Si估计值 |
原始样本
Lf估计值 |
Si纠偏Boot
置信区间 |
Lf纠偏Boot
置信区间 |
||
|
LY12CZ |
~0.60 |
11 |
0.1362 |
|
2.9 |
[0.1030, 0.1788] |
|
[2.2, 4.1] |
|
Kt=1 |
~0.45 |
10 |
0.1475 |
3.2 |
2.9 |
[0.0836, 0.2098] |
[1.9, 5.2] |
|
|
12 |
0.1157 |
2.5 |
[0.0769, 0.1635] |
[1.8, 3.6] |
||||
|
R=0.02 |
~0.30 |
11 |
0.2089 |
|
5.1 |
[0.1567, 0.2801] |
|
[3.4, 8.9] |
|
LY12B |
~0.60 |
3 |
0.0329 |
|
1.3 |
[0.0233, 0.0376] |
|
[1.2, 1.4] |
|
Kt=3 |
~0.45 |
4 |
0.1418 |
|
3.4 |
[0.0204, 0.1733] |
|
[1.2, 4.3] |
|
R=0.1 |
~0.30 |
7 |
0.1893 |
|
4.6 |
[0.1365, 0.2540] |
|
[3.0, 7.7] |
|
LY12B |
~0.60 |
3 |
0.0320 |
|
1.3 |
[0.0114, 0.0359] |
|
[1.1, 1.4] |
|
Kt=1 |
~0.45 |
5 |
0.0523 |
1.5 |
1.9 |
[0.0444, 0.0632] |
[1.4, 1.7] |
|
|
3 |
0.0956 |
2.3 |
[0.0252, 0.1057] |
[1.2, 2.5] |
||||
|
R=0.5 |
~0.30 |
3 |
0.2391 |
|
7.9 |
[0.2060, 0.2622] |
|
[5.9, 9.6] |
|
LY12B |
~0.60 |
7 |
0.0717 |
|
1.8 |
[0.0617, 0.0863] |
|
[1.6, 2.0] |
|
Kt=5 |
~0.45 |
7 |
0.0942 |
|
2.1 |
[0.0745, 0.1222] |
|
[1.8, 2.7] |
|
R=0.5 |
~0.30 |
8 |
0.2003 |
|
4.9 |
[0.1530, 0.2728] |
|
[3.4, 8.8] |
|
LY12B |
~0.60 |
8 |
0.0659 |
|
1.7 |
[0.0485, 0.0812] |
|
[1.5, 1.9] |
|
Kt=3 |
~0.45 |
10 |
0.0810 |
|
1.9 |
[0.0555, 0.1063] |
|
[1.5, 2.3] |
|
R=0.1人工 |
~0.30 |
14 |
0.3447 |
|
14.3 |
[0.2753, 0.4381] |
|
[8.4, 29.5] |
|
LY12CZ |
~0.60 |
10 |
0.0579 |
|
1.6 |
[0.0456, 0.0737] |
|
[1.4, 1.8] |
|
Kt=1 |
~0.45 |
10 |
0.1259 |
2.7 |
2.8 |
[0.0888, 0.1645] |
[2.0, 3.6] |
|
|
10 |
0.1373 |
2.9 |
[0.1169, 0.1644] |
[2.5, 3.6] |
||||
|
|
~0.30 |
10 |
0.2406 |
|
6.6 |
[0.1954, 0.2954] |
|
[4.7, 10.2] |
|
LY12CZ |
~0.60 |
10 |
0.0630 |
|
1.6 |
[0.0465, 0.0815] |
|
[1.4, 1.9] |
|
Kt=2.5 |
~0.45 |
10 |
0.1147 |
2.5 |
2.4 |
[0.0862, 0.1469] |
[2.0, 3.2] |
|
|
10 |
0.0916 |
2.1 |
[0.0561, 0.1218] |
[1.6, 2.6] |
|
|||
|
10 |
0.1241 |
2.7 |
[0.0817, 0.1717] |
[1.9, 3.9] |
|
|||
|
|
~0.30 |
10 |
0.1784 |
|
4.1 |
[0.0660, 0.2630] |
|
[1.7, 7.9] |
|
LC9 |
~0.60 |
5 |
0.0544 |
|
1.6 |
[0.0403, 0.0684] |
|
[1.4, 1.8] |
|
Kt=1 |
~0.45 |
5 |
0.0411 |
|
1.4 |
[0.0290, 0.0579] |
|
[1.3, 1.6] |
|
R=0.1 |
~0.30 |
7 |
0.1373 |
|
3.0 |
[0.0933, 0.1818] |
|
[2.1, 4.3] |
|
LC4 |
~0.60 |
5 |
0.0537 |
|
1.6 |
[0.0402, 0.0675] |
|
[1.4, 1.7] |
|
Kt=1 |
~0.45 |
6 |
0.0364 |
1.3 |
2.0 |
[0.0212, 0.0521] |
[1.2, 1.5] |
|
|
6 |
0.0520 |
1.5 |
[0.0312, 0.0698] |
[1.3, 1.8] |
||||
|
5 |
0.1524 |
3.5 |
[0.1015, 0.2323] |
[2.3, 6.8] |
||||
|
4 |
0.0836 |
2.0 |
[0.0797, 0.0952] |
[2.0, 2.2] |
||||
|
4 |
0.0844 |
2.0 |
[0.0409, 0.1035] |
[1.4, 2.4] |
||||
|
4 |
0.0587 |
1.6 |
[0.0275, 0.0716] |
[1.3, 1.8] |
||||
|
|
~0.30 |
4 |
0.1169 |
|
2.7 |
[0.0167, 0.1443] |
|
[1.2, 3.4] |
|
LC4 |
~0.60 |
3 |
0.0707 |
|
1.8 |
[0.0456, 0.0815] |
|
[1.5, 2.0] |
|
Kt=1 |
~0.45 |
3 |
0.0596 |
1.7 |
2.3 |
[0.0596, 0.0596] |
[1.7, 1.7] |
|
|
4 |
0.1091 |
2.5 |
[0.0270, 0.1329] |
[1.3, 3.1] |
||||
|
3 |
0.0708 |
1.8 |
[0.0620, 0.0771] |
[1.7, 1.9] |
||||
|
5 |
0.1457 |
3.3 |
[0.0611, 0.1998] |
[1.7, 5.2] |
||||
|
|
~0.30 |
6 |
0.1252 |
|
2.8 |
[0.0684, 0.1653] |
|
[1.7, 3.8] |
|
平均 |
~0.60 |
|
|
|
1.7 |
|
|
|
|
~0.45 |
|
|
|
2.3 |
|
|
|
|
|
~0.30 |
|
|
|
5.6 |
|
|
|
|
|
|
|
295 |
43 |
|
|
|
|
|
由表9 4和表9 5的结果可以看出，载荷高低对疲劳分散系数置信上下限有影响。采用文中所述的方法得到的疲劳分散系数的置信区间随载荷高低的变化规律同文献[1]相同，即在其它条件不变的情况下，疲劳分散系数随应力的提高而降低。这进一步说明采用本文方法对疲劳分散系数进行置信区间分析的思路是可行的，也是更可信的。而且通过该方法所得的分散系数的置信上限来估算飞机的安全疲劳寿命能够得到更偏安全的结果，更适用于工程应用。
本章针对疲劳寿命小子样统计分析问题，首先采用Bootstrap方法模拟母体标准差的抽样分布，然后结合纠偏的百分位法估算母体标准差的置信区间，进而对疲劳分散系数进行置信区间分析。通过对140个钢合金试件和295个铝合金试件的疲劳试验数据进行疲劳分散系数置信区间分析发现，疲劳分散系数置信限随应力的提高而降低，由疲劳安全寿命分散系数置信上限可以得到更偏安全的飞机安全疲劳寿命。为疲劳寿命试验数据的工程应用提供了依据，同时也探索了一种试验数据分析的参考方法。
