基于基因表达式编程的FRP约束混凝土极限轴向应变预测

邓楚兵; 薛新华

doi:10.13801/j.cnki.fhclxb.20220125.002

基于基因表达式编程的FRP约束混凝土极限轴向应变预测

邓楚兵,
薛新华^,

四川大学　水利水电学院，成都 610065

详细信息

通讯作者:
薛新华，博士，教授，博士生导师，研究方向为岩土工程　E-mail: xuexinhua@scu.edu.cn

中图分类号: TU37
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出版历程
- 收稿日期: 2021-11-21
- 修回日期: 2022-01-12
- 录用日期: 2022-01-14
- 网络出版日期: 2022-01-26
- 刊出日期: 2023-01-14

Prediction of ultimate axial strain of FRP-confined concrete based on gene expression programming

DENG Chubing,
XUE Xinhua^,

College of Water Resource and Hydropower, Sichuan University, Chengdu 610065, China

摘要

摘要: 纤维增强树脂复合材料(FRP)以其质量轻、强度高、耐腐蚀和施工方便等优势被广泛应用于混凝土结构性能提升和受损构件加固中。FRP约束混凝土的极限条件是选择FRP种类、选择FRP厚度及确定包裹层数等必须要考虑的因素，现有极限应力模型的预测结果能够较好反地映真实情况，而现有极限轴向应变模型的预测精度偏低，故本文对极限轴向应变进行了研究。由于影响FRP约束混凝土极限轴向应变的因素较多，许多研究人员提出的模型在输入参数的选择上存在较大差异，故本文在通过基因表达式编程建立极限轴向应变模型的同时还探讨了不同输入形式对模型预测精度的影响。采用决定系数及平均绝对误差等5种统计指标对模型预测结果进行评价，并将其与现有模型进行对比分析。研究结果表明：原始数据和新数据组合的输入形式对应的模型具有最高的预测精度，因此在模型输入参数的选择上不能仅考虑原始数据或者新数据；与其他研究人员所提模型相比，本文所提模型预测精度更高，其决定系数为0.893，平均绝对误差等指标均在0.35以下。
- 基因表达式编程 /
- 纤维增强树脂复合材料 /
- 混凝土柱 /
- FRP约束混凝土 /
- 极限轴向应变
Abstract: Fiber reinforced polymer (FRP) is widely used in enhancing the performance of concrete structures and strengthening damaged components due to its advantages of light weight, high strength, corrosion resistance and convenient construction. The ultimate conditions of FRP-confined concrete are the important factors that must be considered in the selection of FRP types, FRP thickness and the number of covering layer. The prediction results of the existing ultimate stress model can better reflect the real situation, while the prediction accuracy of the existing ultimate axial strain model is low, so the ultimate axial strain was studied. Since there are many factors that affect the ultimate axial strain of FRP-confined concrete, the models proposed by many researchers have large differences in the choice of input parameters. Therefore, the influence of different input forms on the prediction accuracy of ultimate axial strain model was discussed while the ultimate axial strain model was established by gene expression programming. Five statistical indicators such as coefficient of determination and mean absolute error were used to evaluate the prediction results of model, which was compared with the existing prediction models. The research results show that the model corresponding to the input form of the combination of original data and new data has the highest prediction accuracy, so the selection of model input parameters should not only consider the original data or new data. Compared with the models proposed by other researchers, the prediction accuracy of the model proposed in this article is the highest. The coefficient of determination is 0.893, and the mean absolute error and other indicators are all below 0.35.
- gene expression programming /
- fiber reinforced polymer /
- concrete cylinder /
- FRP-confined concrete /
- ultimate axial strain

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图 1 基因表达式编程(GEP)语言示意图

Figure 1. Diagram of gene expression programming (GEP) language

d₀—Ultimate axial strain of unconfined concrete; d₁—Stiffness ratio; d₂—Strain ratio; d₃—Ultimate axial strain of FRP-confined concrete; 3Rt—Function symbol; c₂—Constant

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图 2 FRP约束混凝土极限轴向应变模型最优参数确定

Figure 2. Determination of optimal parameters of ultimate axial strain model of FRP-confined concrete

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图 3 不同输入形式下的FRP约束混凝土极限轴向应变模型的预测结果与试验结果对比

Figure 3. Comparison between experimental and prediction results of ultimate axial strain models of FRP-confined concrete under different input forms

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图 4 模型C (GEP模型)的表达式树(Sub-ET)

Figure 4. Expression trees (Sub-ET) of model C (GEP model)

Inv—Inverse; c₀, c₃, c₇—Constant

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图 5 FRP约束混凝土GEP模型的参数敏感性分析结果

Figure 5. Parameter sensitivity analysis results of GEP model for FRP-confined concrete

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图 6 各FRP约束混凝土极限轴向应变模型的预测结果与试验结果对比

Figure 6. Comparison between experimental and prediction results of ultimate axial strain models of FRP-confined concrete

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表 1 FRP约束混凝土的极限轴向应变模型

Table 1 Ultimate axial strain models of FRP-confined concrete

Model and year	Ultimate axial strain
Ahmad et al ^[23] (2020)	${\varepsilon _{ { { {\rm{cu} } } } } } = (1.85 + 7.46\rho _{ {\varepsilon } }^{ { {1} }{ {.171} } }\rho _{ {{\rm{k}}} }^{ { {0} }{ {.71} } }){\varepsilon _{ { { {\rm{co} } } } } }$
Yu et al^[24] (2011)	${\varepsilon _{ { { {\rm{cu} } } } } } = 0.0033 + 0.6{\left( \dfrac{ { {E_l} } }{ {f_{ { {{\rm{co}}} } }^{ {'} } } }\right)^{0.8} }{({\varepsilon _{ { { {\rm{h} },{\rm{rup} } } } } })^{1.45} }$
Benzaid et al^[25] (2010)	${\varepsilon _{ { {{\rm{cu}}} } } } = 2{\varepsilon _{ { {{\rm{co}}} } } } + 7.6\dfrac{ { {f_l} } }{ {f_{ { {{\rm{co}}} } }^{ {'} } } }{\varepsilon _{ { {{\rm{co}}} } } }$
Teng et al^[26] (2009)	${\varepsilon _{ { {{\rm{cu}}} } } } = 1.75{\varepsilon _{ { {{\rm{co}}} } } } + 6.5{\varepsilon _{ { {{\rm{co}}} } } }\rho _{ {\varepsilon } }^{ { {1} }{ {.45} } }\rho _{ {{\rm{k}}} }^{ { {0} }{ {.8} } }$
Al-Tersawy et al^[27] (2007)	${\varepsilon _{ { {{\rm{cu}}} } } } = 2{\varepsilon _{ { {{\rm{co}}} } } } + 8.16{\left( \dfrac{ { {f_l} } }{ {f_{ { {{\rm{co}}} } }^{ {'} } } }\right)^{0.34} }{\varepsilon _{ { {{\rm{co}}} } } }$
Ilki et al^[28] (2004)	${\varepsilon _{ { {{\rm{cu}}} } } } = {\varepsilon _{ { {{\rm{co}}} } } } + 20{\left( \frac{ { {f_l} } }{ {f_{ { {{\rm{co}}} } }^{ {'} } } }\right)^{0.5} }{\varepsilon _{ { {{\rm{co}}} } } }$
Xiao et al^[29] (2000)	${\varepsilon _{ { { {\rm{cu} } } } } } = \dfrac{ { {\varepsilon _{ { { {\rm{h} },{\rm{rup} } } } } } - 0.0005} }{ {7{ {\left( \dfrac{ { {E_l} } }{ {f_{ { {{\rm{co}}} } }^{ {'} } } }\right)}^{ - 0.8} } } }$
Samaan et al ^[30] (1998)	${\varepsilon _{ { {{\rm{cu}}} } } } = \dfrac{ {f_{ { {{\rm{cc}}} } }^{ {'} } - 0.872f_{ { {{\rm{co}}} } }^{ {'} } - 0.371{f_l} - 6.258} }{ {245.61f{ {_{ { {{\rm{co}}} } }^{ {'} } }^{0.2} } + 1.3456\dfrac{ { {E_{ { {{\rm{FRP}}} } } }T} }{D} } }$
Mander et al ^[31] (1988)	${\varepsilon _{ { { {\rm{cu} } } } } } = \left( 1 + 5\left( \dfrac{ {f_{ { { {\rm{cc} } } } }^{ {'} } } }{ {f_{ { {{\rm{co}}} } }^{ {'} } } } - 1\right)\right){\varepsilon _{ { { {\rm{co} } } } } }$
Notes: $f_{ { {{\rm{cc}}} } }'$ —Peak strength of FRP-confined concrete; ${\varepsilon _{ { {{\rm{cu}}} } } }$ —Ultimate axial strain of FRP-confined concrete; $f_{ { {{\rm{co}}} } }^{ {'} }$ —Peak strength of unconfined concrete; ${\varepsilon _{ { {{\rm{co}}} } } }$ —Peak strain of unconfined concrete; $D$ —Diameter of the concrete core; $T$ —Total thickness of the FRP jacket; ${E_{ { {{\rm{FRP}}} } } }$ —Elastic modulus of the FRP jacket; ${\varepsilon _{ { {{\rm{h,rup}}} } } }$ —Hoop rapture strain of the FRP jacket; ${f_l}$ —Confining stress; ${\rho _{ {{\rm{k}}} } }$ —Stiffness ratio; ${\rho _{{\varepsilon }}}$ —Strain ratio; E_l—Restraint stiffness; See Table 2 for variable units.

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表 2 FRP约束普通混凝土圆柱体试验数据的统计参数

Table 2 Statistical parameters of test data of FRP-confined normal concrete cylinder

Parameter	Min	Max	Median	Average	Standard deviation
$D$ /mm	100	200	152	146.080	17.816
$H$ /mm	200	788	305	314.106	94.341
$f_{ { {{\rm{co}}} } }^{ {'} }$ /MPa	26.2	55.2	42	40.577	7.100
${\varepsilon _{ { {{\rm{co}}} } } }$ /%	0.16	0.42	0.239	0.249	0.047
${E_{\rm{c}}}$ /GPa	24.21	35.14	29.77	29.692	2.731
$T$ /mm	0.11	5.21	0.495	0.794	0.882
${E_{ { {{\rm{FRP}}} } } }$ /GPa	13.6	629.6	105	154.604	113.36
${\varepsilon _{ { {{\rm{h}},{\rm{rup}}} } } }$ /%	0.19	3.09	1.08	1.208	0.504
${\rho _{ {{\rm{k}}} } }$	2.555	38.114	9.324	10.683	7.153
${\rho _{{\varepsilon }}}$	0.010	0.369	0.049	6.430	5.659
${f_l}$ /MPa	0.905	13.435	4.333	4.922	2.154
${\varepsilon _{ { {{\rm{cu}}} } } }$ /%	0.4	5.55	1.61	1.830	1.005
Notes: H—Height of the concrete core; E_c—Elastic modulus of the concrete core.

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表 3 FRP约束普通混凝土圆柱体试验数据的皮尔逊相关性分析结果

Table 3 Results of Pearson correlation analysis of test data of FRP-confined normal concrete cylinder

	${\rho _{ {{\rm{k}}} } }$	${\rho _{{\varepsilon }}}$	${f_l}$	${\varepsilon _{ { {{\rm{cu}}} } } }$
$D$	−0.223**	−0.064	−0.017	−0.121
$H$	−0.041	−0.131	−0.108	−0.127
$f_{ { {{\rm{co}}} } }^{ {'} }$	−0.211**	0.039	0.136	−0.029
${\varepsilon _{ { {{\rm{co}}} } } }$	−0.163*	−0.181*	−0.180*	0.236**
${E_{\rm{c}}}$	−0.198**	0.089	0.151	−0.003
$T$	0.161*	0.356**	0.553**	0.297**
${E_{ { {{\rm{FRP}}} } } }$	0.509**	−0.544**	−0.031	−0.165*
${\varepsilon _{ { {{\rm{h}},{\rm{rup}}} } } }$	−0.395**	0.926**	0.086	0.377**
${\rho _{ {{\rm{k}}} } }$	1	−0.334**	0.669**	0.379**
${\rho _{{\varepsilon }}}$	−0.334*	1	0.165*	0.301**
${f_l}$	0.669**	0.165*	1	0.653**
${\varepsilon _{ { {{\rm{cu}}} } } }$	0.379**	0.301*	0.653**	1
Notes: *—Correlation is significant at the 0.01 level (2-tailed); —Correlation is significant at the 0.05 level (2-tailed).

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表 4 FRP约束混凝土极限轴向应变模型的参数设置

Table 4 Parameters setting of ultimate axial strain model of FRP-confined concrete

Parameter types	Setting	Parameter types	Setting
Population size	50	Gene transposition rate	0.3
Head length	12	Gene recombination rate	0.3
Gene number	3	One-point recombination rate	0.4
Chromosome length	45	Two-point recombination rate	0.4
Connection function	Addition (+)	Insertion sequence transposition rate	0.3
Mutation rate	0.044	Root insertion sequence transposition rate	0.3

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表 5 FRP约束混凝土极限轴向应变模型的输入形式

Table 5 Input informs of ultimate axial strain model of FRP-confined concrete

No.	Model	Ultimate axial strain ${\varepsilon _{ { {{\rm{cu}}} } } }$
1	A	${\varepsilon _{ { {{\rm{cu}}} } } } = {f}(D,H,f_{ { {{\rm{co}}} } }^{ {'} },{\varepsilon _{ { {{\rm{co}}} } } },{E_{ {{\rm{c}}} } },T,{E_{ { {{\rm{FRP}}} } } },{\varepsilon _{ { {{\rm{h}},{\rm{rup}}} } } })$
2	B	${\varepsilon _{ { {{\rm{cu}}} } } } = {f}({f_l},{\rho _{ {\varepsilon } } },{\rho _{ {{\rm{k}}} } })$
3	C	${\varepsilon _{ { {{\rm{cu}}} } } } = {f}({\varepsilon _{ { {{\rm{co}}} } } },{\rho _{ {\varepsilon } } },{\rho _{ {{\rm{k}}} } })$
4	D	${\varepsilon _{ { {{\rm{cu}}} } } } = {f}(f_{ { {{\rm{co}}} } }^{ {'} },{\varepsilon _{ { {{\rm{co}}} } } },{f_l})$
5	E	${\varepsilon _{ { {{\rm{cu}}} } } } = {f}({\varepsilon _{ { {{\rm{co}}} } } },T,{E_{ { {{\rm{FRP}}} } } },{\varepsilon _{ { {{\rm{h}},{\rm{rup}}} } } },{f_l},{\rho _{ {\varepsilon } } },{\rho _{ {{\rm{k}}} } })$

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表 6 不同输入形式下的FRP约束混凝土极限轴向应变模型的预测结果

Table 6 Prediction results of ultimate axial strain models of FRP-confined concrete under different input forms

Model	${R^{\text{2}}}$	M_AE	R_RSE	R_MSE	M_APE
A	0.487	0.414	0.721	0.682	0.257
B	0.561	0.533	0.841	0.796	0.299
C	0.893	0.228	0.334	0.316	0.160
D	0.705	0.362	0.543	0.514	0.260
E	0.740	0.543	0.762	0.721	0.353
Notes: R²—Determination coefficient; M_AE—Mean absolute error; R_RSE—Relative square root error; R_MSE—Root mean square error; M_APE—Mean absolute percentage error.

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表 7 FRP约束混凝土GEP模型的参数重要性分析结果

Table 7 Parameter importance analysis results of GEP model for FRP-confined concrete

Model	Parameter	${R^{{2}}}$	Conclusion
GEP	Without ${\varepsilon _{ { {{\rm{co}}} } } }$	0.496	${\rho _{ {{\rm{k}}} } } > {\rho _{ {\varepsilon } } } > {\varepsilon _{ { {{\rm{co}}} } } }$
	Without ${\rho _{ {{\rm{k}}} } }$	0.112
	Without ${\rho _{{\varepsilon }}}$	0.397
	Without ${\varepsilon _{ { {{\rm{co}}} } } }$	0.522
Teng et al ^[26]	Without ${\rho _{ {{\rm{k}}} } }$	0.118	${\rho _{ {{\rm{k}}} } } > {\rho _{ {\varepsilon } } } > {\varepsilon _{ { {{\rm{co}}} } } }$
	Without ${\rho _{{\varepsilon }}}$	0.245

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表 8 各FRP约束混凝土极限轴向应变模型的预测结果

Table 8 Prediction results of ultimate axial strain models of FRP-confined concrete

Model	${R^{\text{2}}}$	M_AE	R_RSE	R_MSE	M_APE
GEP	0.893	0.228	0.334	0.316	0.160
Ahmad et al^[23]	0.743	0.327	0.524	0.496	0.206
Yu et al^[24]	0.611	0.394	0.721	0.683	0.199
Benzaid et al^[25]	0.664	0.814	1.129	1.068	0.410
Teng et al^[26]	0.701	0.357	0.563	0.533	0.208
Al-Tersawy et al^[27]	0.550	0.513	0.767	0.726	0.348
Ilki et al^[28]	0.682	0.881	1.029	0.974	0.671
Xiao et al^[29]	0.572	0.412	0.742	0.702	0.213
Samaan et al^[30]	0.478	1.713	2.061	1.950	0.971
Mander et al^[31]	0.681	0.379	0.544	0.613	0.204

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施引文献(33)

期刊类型引用(12)

1.	龙国文，曾开华，谢帮华，田海，邱智坚. 相变混凝土复合材料的性能研究综述及展望. 功能材料. 2024(10): 10038-10046 . 百度学术
2.	陈春鸣，张燕，彭建山. 轻质相变混凝土物理力学性能研究进展. 化工新型材料. 2024(S2): 294-298 . 百度学术
3.	肖桐，刘庆祎，张家豪，赵佳腾，刘昌会. 热固性树脂基复合相变材料的制备及其储能强化研究进展. 复合材料学报. 2023(03): 1311-1327 . 本站查看
4.	张宇，恽定康，邱菊. 冻融循环下相变混凝土剪力墙抗震性能研究. 徐州工程学院学报(自然科学版). 2023(01): 42-49 . 百度学术
5.	张家玮，黄玮，黄大建，李旭辉，马建虎，郑永. 基于微孔漂珠的相变微胶囊制备及其对砂浆力学和热性能影响. 复合材料学报. 2023(08): 4703-4719 . 本站查看
6.	于文艳，殷凯. 复合相变蓄热涂层的蓄放热调温特性. 科学技术与工程. 2023(31): 13492-13498 . 百度学术
7.	蔡昕辰，刘志彬，张云，白梅，刘锋. 相变材料在道路工程中的应用研究进展. 功能材料. 2021(12): 12013-12021 . 百度学术
8.	周建庭，聂志新，郭增伟，杨娟，郑忠. 相变混凝土的制备与性能研究综述. 江苏大学学报(自然科学版). 2020(05): 588-595 . 百度学术
9.	刘朋，缪正坤，田国华，刘伟，王东. 不同工况下陶粒吸附相变石蜡试验研究. 江苏科技信息. 2019(04): 44-46 . 百度学术
10.	司亚余，马芹永，罗小宝，顾皖庆，白梅. 癸酸-正辛酸与活性炭复合相变材料的制备. 安徽理工大学学报(自然科学版). 2019(03): 66-72 . 百度学术
11.	马芹永，白梅. 珍珠岩基相变骨料混凝土断裂特性试验与分析. 建筑材料学报. 2018(03): 365-369 . 百度学术
12.	顾皖庆，马芹永，白梅，罗小宝，司亚余. 月桂醇膨胀珍珠岩复合相变材料吸附试验与分析. 科学技术与工程. 2018(25): 230-235 . 百度学术