﻿ 考虑不确定性的复合材料加筋壁板后屈曲分析模型验证方法
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Validation method for post-buckling analysis model of stiffened composite panels considering uncertainties
WANG Binwen, AI Sen, ZHANG Guofan, NIE Xiaohua, WU Cunli
Aircraft Strength Research Institute of China, Xi'an 710065, China
Abstract: In the lightweight structural design of stiffened composite panels, uncertainties in the geometric and material parameters lead to the uncertainty of ultimate load carrying capacity. Therefore, it is necessary to consider these uncertainties in model validation. A method to validate the post-buckling finite element model of stiffened composite panels considering the uncertain factors is proposed. Based on the orthogonal experimental design, the significance analysis of the uncertainty parameters was first carried out to obtain significance parameters, followed by the acquisition of a surrogate model by the Kriging model to represent the post-buckling characteristics. The probability distribution of the post-buckling loads for the stiffened composite panel was achieved by Monte Carlo simulations, and the accuracy of the post-buckling model verified by experimental data. This validation approach can be applied to similar engineering cases.
Keywords: composite materials    stiffened panels    post-buckling    uncertainties    model validation

1 考虑不确定性参数模型验证方法 1.1 参数显著性分析

1.2 代理模型

 图 1 两因素和三因素中心组合试验点分布 Fig. 1 Test point distributions of two-factor and three-factor central composite designs

Kriging模型是基于统计的插值模型[23]，包含回归部分与随机过程2部分：

 $\mathit{\boldsymbol{Y}} = {\mathit{\boldsymbol{F}}^{\rm{T}}}(\mathit{\boldsymbol{x}})\mathit{\boldsymbol{\beta }} + \mathit{\boldsymbol{Z}}(\mathit{\boldsymbol{x}})$ （1）

 ${E(\mathit{\boldsymbol{Z}}(\mathit{\boldsymbol{x}})) = 0}$ （2）
 ${ {\rm{var}} (\mathit{\boldsymbol{Z}}(\mathit{\boldsymbol{x}})) = {\sigma ^2}}$ （3）
 $\begin{array}{*{20}{c}} {{\rm{Cov}} (\mathit{\boldsymbol{Z}}({\mathit{\boldsymbol{x}}_i}), \mathit{\boldsymbol{Z}}({\mathit{\boldsymbol{x}}_j})) = {\sigma ^2}[{\mathit{\boldsymbol{R}}_{ij}}({\theta _k}, x_i^k, x_j^k)]}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i, j \in \{ 1, 2, \cdots , m\} ;k \in \{ 1, 2, \cdots , n\} } \end{array}$ （4）

 ${\mathit{\boldsymbol{R}}_{ij}}({\theta _k}, x_i^k, x_j^k) = {\rm{exp}}( - {\theta _k}|x_i^k - x_j^k{|^2})$ （5）

 ${{\sigma ^2} = \frac{{{{(\mathit{\boldsymbol{Y}} - {\mathit{\boldsymbol{F}}^{\rm{T}}}\mathit{\boldsymbol{\beta }})}^{\rm{T}}}{\mathit{\boldsymbol{R}}^{ - 1}}(\mathit{\boldsymbol{Y}} - {\mathit{\boldsymbol{F}}^{\rm{T}}}\mathit{\boldsymbol{\beta }})}}{m}}$ （6）
 ${{\mathit{\boldsymbol{\beta }}^*} = {{({\mathit{\boldsymbol{F}}^{\rm{T}}}{\mathit{\boldsymbol{R}}^{ - 1}}\mathit{\boldsymbol{F}})}^{ - 1}}({\mathit{\boldsymbol{F}}^{\rm{T}}}{\mathit{\boldsymbol{R}}^{ - 1}}\mathit{\boldsymbol{Y}})}$ （7）

 ${\mathit{\boldsymbol{\hat y}}({x_{{\rm{new}}}}) = {\mathit{\boldsymbol{f}}^{\rm{T}}}{\mathit{\boldsymbol{\beta }}^*} + {\mathit{\boldsymbol{r}}^{\rm{T}}}{\mathit{\boldsymbol{\gamma }}^*}}$ （8）
 ${{{\hat \sigma }^2}({x_{{\rm{new}}}}) = {\sigma ^2}[1 + {\mathit{\boldsymbol{D}}^{\rm{T}}}{{({\mathit{\boldsymbol{F}}^{\rm{T}}}{\mathit{\boldsymbol{R}}^{ - 1}}\mathit{\boldsymbol{F}})}^{ - 1}}\mathit{\boldsymbol{D}} - {\mathit{\boldsymbol{r}}^{\rm{T}}}{\mathit{\boldsymbol{R}}^{ - 1}}\mathit{\boldsymbol{r}}]}$ （9）

 ${{r_i}({\theta _k}, x_{{\rm{ new }}}^k, x_i^k) = {\rm{exp}}( - {\theta _k}|x_{{\rm{ new }}}^k - x_i^k{|^2})}$ （10）
 ${{\mathit{\boldsymbol{\gamma }}^*} = {\mathit{\boldsymbol{R}}^{ - 1}}(\mathit{\boldsymbol{Y}} - {\mathit{\boldsymbol{F}}^{\rm{T}}}{\mathit{\boldsymbol{\beta }}^*})}$ （11）
 ${\mathit{\boldsymbol{D}} = {\mathit{\boldsymbol{f}}^{\rm{T}}} - {\mathit{\boldsymbol{F}}^{\rm{T}}}{\mathit{\boldsymbol{R}}^{ - 1}}\mathit{\boldsymbol{r}}}$ （12）

 ${R^2} = 1 - \frac{{\sum\limits_{i = 1}^m {{{({y_i} - {{\hat y}_i})}^2}} }}{{\sum\limits_{i = 1}^m {{{({y_i} - \bar y)}^2}} }}$ （13）

1.3 蒙特卡罗模拟

1.4 基于试验数据的模型验证方法

 $Z = \left| {\frac{{E({Y_{{\rm{mod}}}}) - E({Y_{{\rm{exp}}}})}}{{E({Y_{{\rm{exp}}}})}}} \right| \times 100\%$ （14）

1.5 模型验证流程

 图 2 模型验证流程 Fig. 2 Model validation process
2 复合材料加筋壁板有限元模型验证

2.1 结构简介

 图 3 加筋壁板几何参数 Fig. 3 Geometric parameters of stiffened panel specimens

 组件 铺层信息 蒙皮 [45/0/-45/90/±45/02/452/0/-452/0/452/90/-452/0]s 下缘条 [45/03/-45/90]s 腹板 [45/02/45/02/-45/90/-45/02/-45/02/45]s 自由缘条 [45/02/-45/02/-45/02/-45/90]s

2.2 试验结果

 图 4 试验加载方式及支持 Fig. 4 Test loading mode and supporting

 试验件编号 1# 2# 3# 4# 5# 均值 破坏载荷/kN 1 148 1 226 1 196 1 184 1 188 1 188.4
2.3 有限元模型构建

 图 5 复合材料加筋壁板有限元模型边界条件 Fig. 5 Finite element model boundary condition of stiffened composite panel
2.4 模型验证 2.4.1 不确定性参数及其分布确定

 参数类型 参数意义 均值 标准差 几何参数 壁板单层厚度Tl/mm 0.12 0.001 2 筋条下缘条宽度Ws1/mm 55 0.55 筋条上缘条宽度Ws3/mm 20 0.2 筋条腹板高度HS/mm 40 0.4 筋条间距B/mm 146 1.46 翼肋间距Lrib/mm 630 6.3 材料参数 1方向弹性模量E11/MPa 129 000 6 450 2方向弹性模量E22/MPa 9 820 491 泊松比ν12 0.311 0.016 12方向剪切模量G12/MPa 5 290 264.5 13方向剪切模量G13/MPa 5 290 264.5 23方向剪切模量G23/MPa 3 430 171.5 纤维方向拉伸强度XT/MPa 1 720 86 纤维方向压缩强度XC/MPa 1 230 61.5 垂直于纤维方向拉伸强度YT/MPa 70 3.5 垂直于纤维方向压缩强度YC/MPa 220 11 剪切强度S/MPa 134 6.7 加载参数 加载点X向坐标Xdisp/mm 292 8.76 截面形心高度HRP/mm 9.805 0.294
2.4.2 参数显著性分析

 图 6 不确定性参数显著性分析 Fig. 6 Significance analysis of uncertain parameters
2.4.3 Kriging模型构建

 样本编号 后屈曲有限元结果/N Kriging模型计算结果/N 相对误差/% 1 1 352 430 1 326 512 -1.92 2 1 295 830 1 293 201 -0.2 3 1 261 520 1 259 545 -0.16 4 1 226 060 1 225 546 -0.04 5 1 191 520 1 191 236 -0.02 6 1 156 610 1 156 550 -0.01 7 1 120 850 1 121 487 0.06 8 1 084 740 1 086 187 0.13 9 1 113 880 1 050 470 -5.69 10 1 075 690 1 014 336 -5.7
2.4.4 验证分析

 图 7 试验与分析破坏状态对比 Fig. 7 Failure state comparison of test and analysis
 图 8 试验与分析的载荷-应变曲线 Fig. 8 Load-strain curves of test and analysis

 图 9 后屈曲载荷直方图和试验结果 Fig. 9 Histogram of post-buckling loads and test results
3 结论

1) 模型验证中充分考虑了材料组分和几何尺寸的不确定性，从而避免因随机因素干扰对模型可靠与否做出错误的判断。

2) 不确定参数的选择通过参数显著性分析来实现。为了保证计算精度和效率，选取对模型预测结果具有较大影响的参数，而去除影响小的参数。

3) 文中提出了考虑不确定因素的复合材料加筋壁板后屈曲分析有限元模型验证方法和流程，从模型本身和统计学2个层面对模型进行了验证，具有一定工程实用性，可以对复合材料结构线性和非线性有限元模型进行确认。

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http://dx.doi.org/10.7527/S1000-6893.2020.23987

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#### 文章信息

WANG Binwen, AI Sen, ZHANG Guofan, NIE Xiaohua, WU Cunli

Validation method for post-buckling analysis model of stiffened composite panels considering uncertainties

Acta Aeronautica et Astronautica Sinica, 2020, 41(8): 223987.
http://dx.doi.org/10.7527/S1000-6893.2020.23987