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Journal of ZheJiang University (Engineering Science)  2025, Vol. 59 Issue (11): 2352-2360    DOI: 10.3785/j.issn.1008-973X.2025.11.014
    
Probabilistic model for shear capacity of corroded RC beams based on Gaussian process regression
Chong WANG1(),Lizhao DAI2,*(),Bin CHEN1
1. China Rail Bridge and Tunnel Technologies Limited Company, Nanjing 210061, China
2. School of Civil Engineering, Changsha University of Science and Technology, Changsha 410114, China
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Abstract  

A probability model for predicting the shear capacity of corroded RC beams based on hybrid tree-structured hierarchical Bayesian estimation and priority memory BFGS (TEP-L-BFGS) optimized Gaussian process regression (GPR) was proposed aiming at the problem that the objective uncertainty of the input parameters and the subjective uncertainty of the hyperparameter formulation were difficult to be considered in deterministic data-driven methods, resulting in low prediction accuracy. The influence of data dimensionality and scale on prediction accuracy was analyzed. The predictive accuracy and generalization performance of the proposed model were validated by comparing with traditional mechanism-driven methods and other data-driven methods. Results indicate that the proposed method can consider the objective uncertainties related to the geometric parameters and degradation levels of corroded RC beams, as well as the subjective uncertainties associated with the prior distributions of hyperparameters, demonstrating high generalization capability. The dimensionality and scale of input features significantly impact the prediction accuracy of model, making the preprocessing of input features essential. GPR quantifies the uncertainty of predicted values compared to deterministic models, significantly improving the prediction accuracy of shear capacity for corroded RC beams.



Key wordsreinforced concrete beam      corrosion      shear capacity      machine learning      feature selection     
Received: 04 November 2024      Published: 30 October 2025
CLC:  TU 375  
Fund:  国家自然科学基金资助项目(52278140);湖南省科技创新计划资助项目(2023RC3142).
Corresponding Authors: Lizhao DAI     E-mail: 1135128340@qq.com;lizhaod@csust.edu.cn
Cite this article:

Chong WANG,Lizhao DAI,Bin CHEN. Probabilistic model for shear capacity of corroded RC beams based on Gaussian process regression. Journal of ZheJiang University (Engineering Science), 2025, 59(11): 2352-2360.

URL:

https://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2025.11.014     OR     https://www.zjujournals.com/eng/Y2025/V59/I11/2352


基于高斯过程回归的锈蚀RC梁抗剪承载力概率模型

针对确定性数据驱动方法难以考虑输入参数的客观不确定性与超参数拟定的主观不确定性导致预测精度较低的问题,提出基于混合树形分层贝叶斯估计和优先内存BFGS(TEP-L-BFGS)优化高斯过程回归(GPR)的锈蚀RC梁抗剪承载力预测概率模型,探讨数据维度和尺度对预测精度的影响. 通过与传统机理驱动方法和其他数据驱动方法进行比较,验证了所提模型的预测精度和泛化性能. 结果表明,该方法可以同时考虑锈蚀RC梁几何参数、退化程度的客观不确定性和超参数先验分布的主观不确定性影响,具有较高的泛化能力. 输入特征的维度及尺度显著影响模型的预测精度,有必要对输入特征进行预处理. 相较于确定性模型,GPR可以量化预测值的不确定性,显著提高了锈蚀RC梁抗剪承载力的预测精度.


关键词: 钢筋混凝土梁,  锈蚀,  抗剪承载力,  机器学习,  特征选择 
Fig.1 Execution step of TEP-L-BFGS optimized GPR model
数据集极值${f_{\mathrm{c}}}$/MPa${h_0}$/mm$\lambda $${\eta _{\mathrm{w}}}$/%${\rho _{\mathrm{l}}}$/%${\rho _{\mathrm{v}}}$/%${A_{{\mathrm{sv}}}}$/mm2$b$/mm${\eta _{\mathrm{l}}}$/%${f_{{\mathrm{yv}}}}$/MPa$s$/mm${d_{\mathrm{v}}}$/mm${f_{\mathrm{y}}}$/MPa$h$/mm
训练集
(133组)
max26.855214.797.23.270.90265.46254.026.0626.035013.0706.0610.0
min12.091301.001.220.1456.55120.00.0275.0806.0210.0180.0
测试集
(57组)
max26.615214.793.82.792.90265.46254.030.24626.030513.0706.0610.0
min9.731301.001.220.1456.5595.00275.0806.0210.0180.0
验证集
(22组)
max14.571502.9469.43.600.4487.1315069.43601806.0490300
min14.471001.1702.400.2356.551000300706.0300200
Tab.1 Statistics table of input feature range
映射方法训练集测试集
R2RMSEMAER2RMSEMAE
原始特征0.97515.25010.6370.89623.26217.441
Johnson变换0.9965.8573.4970.19564.60129.330
标准化0.9919.1985.4060.91720.68912.404
Johnson变换标准化0.98910.1596.3780.93618.21812.661
Tab.2 Predictive performance metrics before and after feature mapping
Fig.2 Comparison of model prediction accuracy of different mapping feature methods
Fig.3 Feature correlation analysis
Fig.4 Relationship between number of feature and prediction accuracy
Fig.5 Predicted probability density of shear capacity of A3 specimen
Fig.6 Prediction result and confidence interval of GPR model
Fig.7 Analysis of feature sensitivity
模型模型公式
文献[2]模型${V_{\mathrm{p}}} = \xi (\lambda ,{\eta _{\mathrm{w}}})\dfrac{{0.08+4{\rho _{\mathrm{l}}}}}{{\lambda - 0.3}}{f_{\mathrm{c}}}b{h_0}+\alpha ({\eta _{\mathrm{w}}})\dfrac{{0.25+0.4\lambda }}{s}{A_{{\mathrm{vc}}}}{f_{{\mathrm{yv}}}}{h_0}.$ $\alpha ({\eta _{\mathrm{w}}}) = 1 - 0.077{\eta _{\mathrm{w}}},\;{A_{{\mathrm{vc}}}} = {A_{\mathrm{v}}}(1 - {\eta _{\mathrm{w}}}).$
$ \xi (\lambda ,{\eta _{\mathrm{w}}}) = \left\{ \begin{gathered} {\text{ }}1,\qquad\qquad\qquad\;\;{\eta _{\mathrm{w}}} \leqslant {\eta _{{\mathrm{cr}}}} ; \\ {({\eta _{\mathrm{w}}}/{\eta _{{\mathrm{cr}}}})^{0.069\lambda - 0.43}},{\eta _{\mathrm{w}}} > {\eta _{{\mathrm{cr}}}} . \\ \end{gathered} \right.\;\;{\eta _{{\mathrm{cr}}}} = 10.4(c/{d_{\mathrm{v}}}^2)+{f_{{\mathrm{cu}},{\mathrm{k}}}}/{d_{\mathrm{v}}}. $
文献[7]模型${V_{\mathrm{p}}} = 1.75\dfrac{{{f_{\mathrm{t}}}{b_{\mathrm{c}}}{h_0}_{\mathrm{c}}}}{{\lambda +1}}+\dfrac{{{f_{{\mathrm{yvc}}}}{A_{{\mathrm{vc}}}}{h_0}}}{s}.$${f_{{\mathrm{yvc}}}} = {f_{{\mathrm{yv}}}}\dfrac{{1 - 1.121\;9{\eta _{\mathrm{w}}}}}{{1 - {\eta _{\mathrm{w}}}}} \geqslant 0.$
文献[3]模型${V_{\mathrm{p}}} = \psi {f_{\mathrm{c}}}b{h_0}\left[\dfrac{{0.08}}{{\lambda - 0.3}}+\dfrac{{100{\rho _{\mathrm{l}}}}}{{\lambda {f_{\mathrm{c}}}}}\right]+\alpha \dfrac{{(0.4+0.3\lambda )}}{s}{A_{\mathrm{v}}}{f_{{\mathrm{yv}}}}{h_0}.$
$\psi = \left\{ \begin{gathered} {\text{ }}1,{\text{ }}{\eta _{\mathrm{l}}} \leqslant 5\text{%} ; \\ 1.098 - 1.96\eta_{\mathrm{l}},{\eta _{\mathrm{l}}} > 5\text{%} . \\ \end{gathered} \right.\;\;{\text{ }}\alpha = 1 - 1.059{\eta _{\mathrm{w}}} \geqslant 0.$
Tab.3 Empirical model for calculating shear capacity of RC beam
模型训练集测试集
R2RMSEMAER2RMSEMAE
RF0.97814.1018.3110.91620.81312.765
Xgboost0.9947.3594.8330.94416.95211.859
MLP0.98611.6587.0730.94417.08011.451
GPR0.98710.9387.0040.94716.63211.462
文献[2]模型0.78047.46024.5400.57747.89735.182
文献[3]模型0.87434.70028.4200.84428.48223.882
文献[7]模型0.64266.12153.2100.43062.02352.141
Tab.4 Comparison of predictive indicator of different model
Fig.8 Comparison of prediction accuracy of different model
[33]   叶见曙, 李国平. 结构设计原理[M]. 4版. 北京: 人民交通出版社, 2018: 95-97.
[34]   戴明江, 杨鸥, 肖岩 纵筋锈蚀对钢筋混凝土梁抗剪性能影响[J]. 工业建筑, 2016, 46 (11): 74- 79
DAI Mingjiang, YANG Ou, XIAO Yan Influence of longitudinal bar corrosion on shear behavior of RC beams[J]. Industrial Construction, 2016, 46 (11): 74- 79
[1]   JUAREZ C A, GUEVARA B, FAJARDO G, et al Ultimate and nominal shear strength in reinforced concrete beams deteriorated by corrosion[J]. Engineering Structures, 2011, 33 (12): 3189- 3196
doi: 10.1016/j.engstruct.2011.08.014
[2]   徐善华, 牛荻涛 锈蚀钢筋混凝土简支梁斜截面抗剪性能研究[J]. 建筑结构学报, 2004, 25 (5): 98- 104
XU Shanhua, NIU Ditao The shear behavior of corroded simply supported reinforced concrete beam[J]. Journal of Building Structures, 2004, 25 (5): 98- 104
doi: 10.3321/j.issn:1000-6869.2004.05.016
[3]   霍艳华. 锈蚀钢筋混凝土简支梁受剪承载力研究[D]. 南昌: 南昌大学, 2007.
HUO Yanhua. Research on shear capacity of simply supported concrete beam with corroded reinforcement [D]. Nanchang: Nanchang University, 2007.
[4]   ALASKAR A. Shear behaviour of slender RC beams with corroded web reinforcement [D]. Waterloo: University of Waterloo, 2013.
[5]   赵羽习, 金伟良 锈蚀箍筋混凝土梁的抗剪承载力分析[J]. 浙江大学学报: 工学版, 2008, 42 (1): 19- 24
ZHAO Yuxi, JIN Weiliang Analysis on shearing capacity of concrete beams with corroded stirrups[J]. Journal of Zhejiang University: Engineering Science, 2008, 42 (1): 19- 24
[6]   余波, 陈冰 锈蚀钢筋混凝土梁抗剪承载力计算的概率模型[J]. 工程力学, 2018, 35 (11): 115- 124
YU Bo, CHEN Bing Probabilistic model for shear strength of corroded reinforced concrete beams[J]. Engineering Mechanics, 2018, 35 (11): 115- 124
doi: 10.6052/j.issn.1000-4750.2017.06.0479
[7]   李士彬, 张鑫, 贾留东, 等 箍筋锈蚀钢筋混凝土梁的抗剪承载力分析[J]. 工程力学, 2011, 28 (Suppl.1): 60- 63
LI Shibin, ZHANG Xin, JIA Liudong, et al Analysis for shear capacity of reinforced concrete beams with corrosion stirrups[J]. Engineering Mechanics, 2011, 28 (Suppl.1): 60- 63
[8]   FENG D C, LIU Z T, WANG X D, et al Failure mode classification and bearing capacity prediction for reinforced concrete columns based on ensemble machine learning algorithm[J]. Advanced Engineering Informatics, 2020, 45: 101126
doi: 10.1016/j.aei.2020.101126
[9]   张玉平, 马超, 李传习, 等 基于改进BP算法的混凝土热学参数反演与预测[J]. 交通科学与工程, 2021, 37 (1): 42- 50
ZHANG Yuping, MA Chao, LI Chuanxi, et al Back analysis and prediction of thermal parameters of concrete based on improved BP algorithm[J]. Journal of Transport Science and Engineering, 2021, 37 (1): 42- 50
doi: 10.3969/j.issn.1674-599X.2021.01.007
[10]   KUMAR A, ARORA H C, KAPOOR N R, et al Machine learning intelligence to assess the shear capacity of corroded reinforced concrete beams[J]. Scientific Reports, 2023, 13 (1): 2857
doi: 10.1038/s41598-023-30037-9
[11]   FU B, FENG D C A machine learning-based time-dependent shear strength model for corroded reinforced concrete beams[J]. Journal of Building Engineering, 2021, 36: 102118
doi: 10.1016/j.jobe.2020.102118
[12]   NGUYEN T H, NGUYEN D T, NGUYEN D H, et al Evaluation of residual strength of corroded reinforced concrete beams using machine learning models[J]. Arabian Journal for Science and Engineering, 2022, 47 (8): 9985- 10002
doi: 10.1007/s13369-021-06493-8
[13]   戴理朝, 王冲, 袁平, 等 基于可解释机器学习的锈蚀RC构件抗剪承载力预测模型[J]. 吉林大学学报: 工学版, 2024, 54 (11): 3231- 3243
DAI Lizhao, WANG Chong, YUAN Ping, et al Prediction model for shear capacity of corroded RC beams based on interpretable machine learning[J]. Journal of Jilin University: Engineering and Technology Edition, 2024, 54 (11): 3231- 3243
[14]   于晓辉, 王猛, 宁超列 基于机器学习的钢筋混凝土柱失效模式两阶段判别方法[J]. 建筑结构学报, 2022, 43 (8): 220- 231
YU Xiaohui, WANG Meng, NING Chaolie A machine-leaning-based two-step method for failure mode classification of reinforced concrete columns[J]. Journal of Building Structures, 2022, 43 (8): 220- 231
[15]   GUAN X, SUN H, HOU R, et al A deep reinforcement learning method for structural dominant failure modes searching based on self-play strategy[J]. Reliability Engineering and System Safety, 2023, 233: 109093
doi: 10.1016/j.ress.2023.109093
[16]   LIU J, ALEXANDER J, GU Q, et al Gaussian process regression-based load-carrying capacity models of corroded prestressed concrete bridge girders for fast-screening and reliability-based evaluation[J]. Engineering Structures, 2023, 285: 116040
doi: 10.1016/j.engstruct.2023.116040
[17]   YU Z, XIE W, YU B, et al Probabilistic prediction of joint shear strength using Gaussian process regression with anisotropic compound kernel[J]. Engineering Structures, 2023, 277: 115413
doi: 10.1016/j.engstruct.2022.115413
[18]   SCHULZ E, SPEEKENBRINK M, KRAUSE A A tutorial on Gaussian process regression: modelling, exploring, and exploiting functions[J]. Journal of Mathematical Psychology, 2018, 85: 1- 16
doi: 10.1016/j.jmp.2018.03.001
[19]   HEAD J D, ZERNER M C A Broyden-Fletcher-Goldfarb-Shanno optimization procedure for molecular geometries[J]. Chemical Physics Letters, 1985, 122 (3): 264- 270
doi: 10.1016/0009-2614(85)80574-1
[20]   ZHU C, BYRD R H, LU P, et al Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization[J]. ACM Transactions on Mathematical Software, 1997, 23 (4): 550- 560
doi: 10.1145/279232.279236
[21]   BERGSTRA J, BARDENET R, BENGIO Y, et al. Algorithms for hyper-parameter optimization [C]//Proceedings of the 24th International Conference on Neural Information Processing Systems. Granada: Curran Associates Inc. 2011, 2546–2554.
[22]   李冰, 高向华, 王小惠, 等 局部区段锈蚀的钢筋混凝土梁抗剪承载力试验研究[J]. 混凝土与水泥制品, 2010, (6): 60- 65
LI Bing, GAO Xianghua, WANG Xiaohui, et al Experimental study on shear capacity of reinforced concrete beam with partial section corrosion[J]. China Concrete and Cement Products, 2010, (6): 60- 65
doi: 10.3969/j.issn.1000-4637.2010.06.016
[23]   杨晓明, 吴桐, 王耀耀 小剪跨比锈蚀钢筋混凝土梁受剪性能试验研究[J]. 建筑结构学报, 2019, 40 (12): 147- 154
YANG Xiaoming, WU Tong, WANG Yaoyao Experimental study on shear behavior of corroded reinforced concrete beams with low shear span ratio[J]. Journal of Building Structures, 2019, 40 (12): 147- 154
[24]   陈正. 钢筋锈蚀后RC梁抗剪性能的试验研究与分析[D]. 扬州: 扬州大学, 2022.
CHEN Zheng. Experimental study and analysis of shear strength of RC beams due to reinforcement corrosion [D]. Yangzhou: Yangzhou University, 2022.
[25]   FU C, HUANG J, DONG Z, et al Shear behavior of reinforced concrete beams subjected to accelerated non-uniform corrosion[J]. Engineering Structures, 2023, 286: 116081
doi: 10.1016/j.engstruct.2023.116081
[26]   中华人民共和国住房和城乡建设部. 混凝土结构设计规范: GB50010-2010 [S]. 北京: 中国建筑工业出版社, 2015.
[27]   国振喜. 简明钢筋混凝土结构计算手册[M]. 3版. 北京: 机械工业出版社, 2017: 41-58.
[28]   HE Y, ZHENG Y Short-term power load probability density forecasting based on Yeo-Johnson transformation quantile regression and Gaussian kernel function[J]. Energy, 2018, 154: 143- 156
doi: 10.1016/j.energy.2018.04.072
[29]   HAUKE J, KOSSOWSKI T Comparison of values of Pearson's and Spearman's correlation coefficients on the same sets of data[J]. Quaestiones Geographicae, 2011, 30 (2): 87- 93
doi: 10.2478/v10117-011-0021-1
[30]   赵丽, 胡翮. 统计学基础[M]. 南京: 南京大学出版社, 2017: 213-216.
[31]   CHEN X, JEONG J C. Enhanced recursive feature elimination [C]//Proceedings of the 6th International Conference on Machine Learning and Applications. Cincinnati: IEEE, 2007: 429−435.
[32]   柳世涛. 受腐蚀钢筋混凝土梁抗剪性能研究[D]. 长沙: 中南大学, 2013.
LIU Shitao. Research on shear behavior of corroded RC beams [D]. Changsha: Central South University, 2013.
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