﻿ 曲线片型加筋壁板的稳定性优化设计
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Design and optimization of improving stability of curvilinear blade-stiffened panels
XU Yuanming , WANG Dong
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:Stiffened panels are widely used in the aerospace field as a typical structure. To further expand the design space of straight stiffened panels, an optimization framework for curvilinear blade-stiffened panels was developed based on ModelCenter optimization environment, Catia and Abaqus. Stiffeners were generated by equally-splitm cubic B-spline interpolation function, which had no limit to the number of parameters, thus providing a sufficiently large optimization space for curvilinear stiffened panels. On the other hand, two methods were taken to load on stiffeners including collecting loaded edges of stiffeners and making stiffeners perpendicular to panel edges and some new configurations of stiffened panels were proposed. Optimization results show that for single-axle load, curvilinear stiffened panels in comparison with straight stiffened panels make very small stability improvement. But for multi-axial load, curvilinear stiffened panels have better stability than straight stiffened panels.

1 优化框架

 图 1 曲线片型加筋板优化框架Fig. 1 Optimization framework of curvilinear blade-stiffened panels
1.1 曲筋曲线生成方法

f(x)为定义在区间[a,b]上的函数,f(xi)=fi(i=0,1,…,n),求三次样条函数s(x)使得:

ci由式(5)求得.

1.2 优化数学模型

 图 2 等距剖分三次B样条曲线Fig. 2 Equally-split cubic B-spline curve

 图 3 曲筋布局形式(横向为板长、纵向为板宽)Fig. 3 Configurations of curvilinear blade-stiffened panels (horizontal edges represent plate length, longitudinal edges represent plate width)
1.2.1 设计变量

1.2.2 约束条件

1) 几何约束条件:

2) 考虑到工程需求:

3) 加筋板需要满足稳定性和强度要求.

① 第三强度准则:von Mises等效应力最大值σvm小于材料屈服载荷σy，即

② 稳定性条件:要求实际载荷小于临界屈曲载荷,在有限元特征分析里面,可表示为

1.2.3 目标函数

2 计算实例 2.1 筋端相聚形式的曲筋优化

 图 4 端点闭合的曲筋构型Fig. 4 Configurations of collecting loaded edges of stiffeners

 图 5 加筋板几何尺寸与边界条件Fig. 5 Geometries and boundary conditions
2.1.1 单轴载荷下的优化结果

 参数 数值 E/GPa 73.085 ν 0.33 σ/MPa 427.47 ρ/(kg·m-3) 2 700

 设计变量 下界 上界 f ′is -3 3 f ′ie -3 3 hsi/mm 5 50 tsi/mm 0.2 10

 变量 光板 1条直筋 2条曲筋 2条直筋 4条曲筋 m/kg 1.179 36 1.179 36 1.179 36 1.179 36 1.179 36 σy/N 11 466 37 165 40 503 56 169 60 388 σvm/kPa 13 900 52 167 84 152 69 755 126 532 σs/kPa 0 16 828 43 502 21 357 99 348 n 435 2 004 328 1 693 hsi/mm 30.76 30.82 36.26 35.29 tsi/mm 2.343 2 2.122 2 2.221 4 1.718 1 tp/mm 2.6 2.342 6 2.112 5 2.024 7 1.710 4 f ′1s 0 -0.603 8 0 0.425 5 f ′1e 0 0.426 9 0 -0.390 4

 图 6 单轴载荷下光板和单直筋的一阶模态Fig. 6 First step mode of bare board and one straight stiffener board for single-axial load
 图 7 单轴载荷下双曲筋和双直筋的一阶模态Fig. 7 First step mode of two curvilinear stiffeners board and two straight stiffeners board for single-axial load
 图 8 单轴载荷下4条曲筋的一阶模态Fig. 8 First step mode of four curvilinear stiffeners board for single-axial load
2.1.2 组合载荷下的优化结果

 变量 光板 1条直筋 2条曲筋 2条直筋 4条曲筋 m/kg 1.179 36 1.179 36 1.179 36 1.179 36 1.179 36 σy/N 5 021 12 995 15 635 17 109 21 935 σvm/kPa 13 793 41 041 82 583 63 069 114 560 σs/kPa 0 2 459 36 799 5 403 61 168 n 407 1 021 997 1 594 hsi/mm 32.27 27.91 43.82 26.39 tsi/mm 2.332 1 2.148 1.980 4 1.858 6 tp/mm 2.6 2.331 3 2.148 1.980 1 1.856 4 f ′1s 0 -0.593 6 0 -0.621 8 f ′1e 0 0.593 6 0 -0.621 8 y11/mm 150 150 205.54 116.67

 图 9 组合载荷下光板和单直筋的一阶模态Fig. 9 First step mode of bare board and one straight stiffener board for multi-axial load
 图 10 组合载荷下双曲筋和双直筋的一阶模态Fig. 10 First step mode of two curvilinear stiffeners board and two straight stiffeners board for multi-axial load
 图 11 组合载荷下4条曲筋的一阶模态Fig. 11 First step mode of four curvilinear stiffeners board for multi-axial load
2.2 筋条垂直于板边情况下的曲筋板优化

2.2.1 单轴载荷下的优化结果

 变量 光板 双直筋 双曲筋 m/kg 1.179 36 1.179 36 1.179 36 σy/N 11 466 62 327 62 899 σvm/kPa 13 900 88 130 92 446 σs/kPa 0 1 478 16 546 n 1 146 1 854 hsi/mm 38.08 38.22 tsi/mm 2.044 8 2.043 1 tp/mm 2.6 2.043 8 2.042 2 f ′1s 0 0 f ′1e 0 0 y11/mm 93.88 94.49 y12/mm 93.88 97.00

 图 12 单轴载荷下光板和双直筋的一阶模态Fig. 12 First step mode of bare board and two straight stiffeners board for single-axial load
 图 13 单轴载荷下双曲筋的一阶模态Fig. 13 First step mode of two curvilinear stiffeners board for single-axial load
2.2.2 组合载荷下的优化结果

 变量 光板 双直筋 S1,2={1,1}+中间一个插值点 S1,2={3,5} S1,2={3,2} S1,2={1,1}+中间无插值点 m/kg 1.179 36 1.179 36 1.179 36 1.179 36 1.179 36 1.179 36 σy/N 5 021 17 109 19 795 15 211 18 066 16 716 σvm/kPa 13 793 63 069 102 650 65 622 301 447 81 850 σs/kPa 0 5 403 72 738 32 403 313 266 71 830 n 997 2 391 1 467 2 444 1 343 hsi/mm 43.82 20.76 22.71 28.38 35.09 tsi/mm 2.332 1 2.476 0 2.354 2 2.492 7 2.070 5 tp/mm 2.6 1.980 4 2.111 1 2.344 1 2.276 4 2.070 4 f ′1s 0 0 0 f ′1e 0 0 0 y11/mm 43.73 0 134.24 16.40 150 y12/mm 264.89 251.98 142.62 124.17 y21/mm 192.52 67.35

 图 14 组合载荷下光板和双直筋的一阶模态Fig. 14 First step mode of bare board and two straight stiffeners board for multi-axial load
 图 15 组合载荷下S1,2={1,1}+中间一个插值点 布局和S1,2={3,5}布局的一阶模态Fig. 15 First step mode of S1,2={1,1} with one interpolation point and S1,2={3,5} for multi-axial load
 图 16 组合载荷下S1,2={3,2}布局和 S1,2={1,1}+中间无插值点布局的一阶模态Fig. 16 First step mode of S1,2={3,2} and S1,2= {1,1} with no interpolation point for multi-axial load
3 结 论

1) 在端点闭合的曲筋布局中,证明了2条端点闭合的曲筋比一根直筋更好地提高了板的稳定性,应用这种结构形式,可以将加筋板上的任何一根直筋替换为端点闭合的曲筋.

2) 在筋条垂直于板边的布局中,研究了若干种曲筋布局方案,证明了在相同筋条数目情况下,有多种曲筋布局加筋板比直筋加筋板更具优越性.说明了曲线片型加筋板可以获得比传统直筋加筋板更优的稳定性性能,为曲筋加筋板设计的深入研究提供了基础.

3) 除稳定性之外,曲筋加筋板的后屈曲特性、疲劳特性、损伤容限和振动特性等性能也需要优化研究.这些特性也同样影响着曲筋加筋板的发展.

4) 曲筋加筋板存在很大的发展潜力,但如果将直筋加筋板合理布局,也能获得很好的稳定性性能.由于曲筋加筋板计算量大,加工工艺复杂,所以只有制造工艺和计算机水平发展到一定水平,曲筋加筋板才能体现出它的优势.

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#### 文章信息

XU Yuanming, WANG Dong

Design and optimization of improving stability of curvilinear blade-stiffened panels

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(4): 567-573.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0257