基于概率密度演化的锈蚀混凝土梁时变可靠性分析
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摘要
建立了锈蚀钢筋混凝土梁的极限状态函数,其中引入了锈蚀不均匀系数考虑钢筋锈蚀不均匀性和随机性。根据概率守恒原理引出锈蚀钢筋混凝土梁极限状态函数的广义概率密度演化方程,并介绍了其TVD(total variation diminishing)差分格式的有限差分方法。运用"吸收边界条件"提出了基于概率密度演化理论的锈蚀钢筋混凝土梁时变可靠度计算方法。以三个锈蚀钢筋混凝土梁为例,展示广义概率密度演化方程以及时变可靠度的计算结果。通过100万次Monte Carlo模拟方法以及2范数误差指标考察了概率密度演化及时变可靠度计算结果的精准度。二阶矩信息和可靠度的对比结果显示:概率密度演化法能够以较小的计算代价,较为精确地捕捉锈蚀钢筋混凝土梁在服役期间的概率密度演化信息,较为准确地预测其时变可靠度。
The limit state function for corroded reinforced concrete beams was established, in which the non-uniformity coefficient of corrosion was introduced to consider the non-uniformity of corrosion and randomness of steel bars. Then, according to the principle of conservation of probability, the generalized probability density evolution equation of the limit state function for corroded reinforced concrete beams was derived. Meanwhile,the finite difference method of TVD(total variation diminishing) difference scheme was recommended. Next, the time-dependent reliability calculation method based on the probability density evolution theory was proposed for corroded reinforced concrete beams through ‘absorption boundary conditions'. Further, the generalized probability density evolution equation and the calculation results of time-varying reliability were shown by taking three corroded reinforced concrete beams as examples. Moreover, the accuracies of the probability density evolution and the time-dependent reliability calculation were investigated through one million-scale of Monte Carlo simulation and the two-norm error index. The comparison results of the second-order moment information and the reliability show that the probability density evolution method can accurately capture the probability density evolution information of corroded reinforced concrete beams during its service period with a relative low calculation cost, and then accurately predict the time-varying reliability.
The limit state function for corroded reinforced concrete beams was established, in which the non-uniformity coefficient of corrosion was introduced to consider the non-uniformity of corrosion and randomness of steel bars. Then, according to the principle of conservation of probability, the generalized probability density evolution equation of the limit state function for corroded reinforced concrete beams was derived. Meanwhile,the finite difference method of TVD(total variation diminishing) difference scheme was recommended. Next, the time-dependent reliability calculation method based on the probability density evolution theory was proposed for corroded reinforced concrete beams through ‘absorption boundary conditions'. Further, the generalized probability density evolution equation and the calculation results of time-varying reliability were shown by taking three corroded reinforced concrete beams as examples. Moreover, the accuracies of the probability density evolution and the time-dependent reliability calculation were investigated through one million-scale of Monte Carlo simulation and the two-norm error index. The comparison results of the second-order moment information and the reliability show that the probability density evolution method can accurately capture the probability density evolution information of corroded reinforced concrete beams during its service period with a relative low calculation cost, and then accurately predict the time-varying reliability.
引文
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[14] ZHU W, FRANCOIS R. Corrosion of the reinforcement and its influence on the residual structural performance of a 26-year-old corroded RC beam[J]. Construction and Building Materials, 2014, 51: 461- 472.
[15] GU X L, GUO H Y, ZHOU B B, et al. Corrosion non-uniformity of steel bars and reliability of corroded RC beams[J]. Engineering Structures, 2018, 167: 188-202.
[16] ALMUSALLAM A A. Effect of degree of corrosion on the properties of reinforcing steel bars[J]. Construction and Building Materials, 2001, 15(8): 361-368.
[17] FRANGOPOL D M, LIN K Y, ESTES A C. Reliability of reinforced concrete girders under corrosion attack[J]. Journal of Structural Engineering, 1997, 123(3): 286-297.
[18] FRANGOPOL D M, LIN K Y, ESTES A C. Life-cycle cost design of deteriorating structures[J]. Journal of Structural Engineering, 1997, 123(10): 1390-1401.
[19] APOSTOLOPOULOS C A,DEMIS S,PAPADAKIS V G. Chloride-induced corrosion of steel reinforcement-mechanical performance and pit depth analysis[J]. Construction and Building Materials, 2013, 38: 139-146.
[20] PAPADOPOULOS M P, APOSTOLOPOULOS C A, ZERVAKI A D, et al. Corrosion of exposed rebars, associated mechanical degradation and correlation with accelerated corrosion tests[J]. Construction and Building Materials, 2011, 25(8): 3367-3374.
[21] STEWART M G, AL-HARTHY A. Pitting corrosion and structural reliability of corroding RC structures: experimental data and probabilistic analysis[J]. Reliability Engineering & System Safety, 2008, 93(3): 373-382.
[22] ZHANG W P, CHEN H, GU X L. Tensile behaviour of corroded steel bars under different strain rates[J]. Magazine of Concrete Research, 2016, 68(3): 127-140.
[23] DU Y G, CLARK L A, CHAN A H C. Residual capacity of corroded reinforcing bars[J]. Magazine of Concrete Research, 2005, 57(3): 135-148.
[24] 陈建兵,张圣涵. 非均布随机参数结构非线性响应的概率密度演化[J]. 力学学报, 2014, 46(1): 136-144. (CHEN Jianbing, ZHANG Shenghan. Probability density evolution analysis of nonlinear response of structures with non-uniform random parameters[J]. Chinese Journal of Theoretical and Applied Mechanics, 2014, 46(1): 136-144. (in Chinese))
[25] LI J, CHEN J. The principle of preservation of probability and the generalized density evolution equation[J]. Structural Safety, 2008, 30(1): 65-77.
[26] LI J, CHEN J, SUN W, et al. Advances of the probability density evolution method for nonlinear stochastic systems[J]. Probabilistic Engineering Mechanics, 2012, 28: 132-142.
[27] CHEN J, ZHANG S. Improving point selection in cubature by a new discrepancy[J]. SIAM Journal on Scientific Computing, 2013, 35(5): 2121-2149.
[28] LI J. Probability density evolution method: Background, significance and recent developments[J]. Probabilistic Engineering Mechanics, 2016, 44: 111-117.
[29] 秦权,贺瑞,杨小刚. 在时变结构可靠度领域中有必要澄清一个错误概念[J]. 工程力学, 2009, 26(8): 201-204. (QIN Quan, HE Rui, YANG Xiaogang. Clarification of a wrong conception in the field of time-varying structural reliability[J]. Engineering Mechanics, 2009, 26(8): 201-204.(in Chinese))
[2] AMEY S L, JOHNSON D A, MILTENBERGER M A, et al. Predicting the service life of concrete marine structures: an environmental methodology[J]. ACI Structural Journal, 1998, 95(2): 205-214.
[3] VENEZIANO D, CORNELL C A, GRIGORIU M. Vector-process models for system reliability[J]. Journal of the Engineering Mechanics Division, 1977, 103(3): 441- 460.
[4] 赵国藩,贡金鑫,赵尚传. 工程结构生命全过程可靠度[M]. 北京:中国铁道出版社, 2004. (ZHAO Guofan, GONG Jinxin, ZHAO Shangchuan. Life cycle reliability of engineering structures[M]. Beijing: China Railway Publishing House, 2004. (in Chinese))
[5] 贡金鑫,赵国藩. 考虑抗力随时间变化的结构可靠度分析[J]. 建筑结构学报, 1998,19(5): 43-51.(GONG Jinxin, ZHAO Guofan. Reliability analysis for deteriorating structures[J].Journal of Building Structures, 1998, 19(5):43-51.(in Chinese))
[6] 张俊芝. 服役工程结构可靠性理论及其应用[M]. 北京:中国水利水电出版社, 2007. (ZHANG Junzhi. Reliability theory and its application of engineering structure in service[M]. Beijing: China Water & Power Press, 2007. (in Chinese))
[7] MOURELATOS Z P,MAJCHER M,PANDEY V, et al. Time-dependent reliability analysis using the total probability theorem[J]. Journal of Mechanical Design, 2015, 137(3):0314053.
[8] 李杰,范文亮. 钢筋混凝土框架结构体系可靠度分析[J]. 土木工程学报, 2008, 41(11): 7-12. (LI Jie, FAN Wenliang. System reliability analysis of RC frames[J]. China Civil Engineering Journal, 2008, 41(11):7-12. (in Chinese))
[9] LI J, CHEN J. Stochastic dynamics of structures[M]. Singapore: John Wiley & Sons, 2009.
[10] 李杰,陈建兵. 随机动力系统中的概率密度演化方程及其研究进展[J]. 力学进展, 2010,40(2): 170-188. (LI Jie, CHEN Jianbing. Advances in the research on probability density evolution equations of stochastic dynamical systems[J]. Advances in Mechanics, 2010, 40(2): 170-188. (in Chinese))
[11] 何涛,李杰. 基于概率密度演化方法的地下结构可靠度分析[J]. 力学季刊, 2009, 30(4): 530-536.(HE Tao, LI Jie. Reliability analysis of underground structures based on probability density evolution method[J]. Chinese Quarterly of Mechanics, 2009,30(4):530-536. (in Chinese))
[12] RODRIGUEZ J, ORTEGA L M, CASAL J. Load carrying capacity of concrete structures with corroded reinforcement[J]. Construction and Building Materials, 1997, 11(4): 239-248.
[13] ZHANG R, CASTEL A, FRANCOIS R. Serviceability limit state criteria based on steel: concrete bond loss for corroded reinforced concrete in chloride environment[J]. Materials and Structures, 2009, 42(10): 1407-1421.
[14] ZHU W, FRANCOIS R. Corrosion of the reinforcement and its influence on the residual structural performance of a 26-year-old corroded RC beam[J]. Construction and Building Materials, 2014, 51: 461- 472.
[15] GU X L, GUO H Y, ZHOU B B, et al. Corrosion non-uniformity of steel bars and reliability of corroded RC beams[J]. Engineering Structures, 2018, 167: 188-202.
[16] ALMUSALLAM A A. Effect of degree of corrosion on the properties of reinforcing steel bars[J]. Construction and Building Materials, 2001, 15(8): 361-368.
[17] FRANGOPOL D M, LIN K Y, ESTES A C. Reliability of reinforced concrete girders under corrosion attack[J]. Journal of Structural Engineering, 1997, 123(3): 286-297.
[18] FRANGOPOL D M, LIN K Y, ESTES A C. Life-cycle cost design of deteriorating structures[J]. Journal of Structural Engineering, 1997, 123(10): 1390-1401.
[19] APOSTOLOPOULOS C A,DEMIS S,PAPADAKIS V G. Chloride-induced corrosion of steel reinforcement-mechanical performance and pit depth analysis[J]. Construction and Building Materials, 2013, 38: 139-146.
[20] PAPADOPOULOS M P, APOSTOLOPOULOS C A, ZERVAKI A D, et al. Corrosion of exposed rebars, associated mechanical degradation and correlation with accelerated corrosion tests[J]. Construction and Building Materials, 2011, 25(8): 3367-3374.
[21] STEWART M G, AL-HARTHY A. Pitting corrosion and structural reliability of corroding RC structures: experimental data and probabilistic analysis[J]. Reliability Engineering & System Safety, 2008, 93(3): 373-382.
[22] ZHANG W P, CHEN H, GU X L. Tensile behaviour of corroded steel bars under different strain rates[J]. Magazine of Concrete Research, 2016, 68(3): 127-140.
[23] DU Y G, CLARK L A, CHAN A H C. Residual capacity of corroded reinforcing bars[J]. Magazine of Concrete Research, 2005, 57(3): 135-148.
[24] 陈建兵,张圣涵. 非均布随机参数结构非线性响应的概率密度演化[J]. 力学学报, 2014, 46(1): 136-144. (CHEN Jianbing, ZHANG Shenghan. Probability density evolution analysis of nonlinear response of structures with non-uniform random parameters[J]. Chinese Journal of Theoretical and Applied Mechanics, 2014, 46(1): 136-144. (in Chinese))
[25] LI J, CHEN J. The principle of preservation of probability and the generalized density evolution equation[J]. Structural Safety, 2008, 30(1): 65-77.
[26] LI J, CHEN J, SUN W, et al. Advances of the probability density evolution method for nonlinear stochastic systems[J]. Probabilistic Engineering Mechanics, 2012, 28: 132-142.
[27] CHEN J, ZHANG S. Improving point selection in cubature by a new discrepancy[J]. SIAM Journal on Scientific Computing, 2013, 35(5): 2121-2149.
[28] LI J. Probability density evolution method: Background, significance and recent developments[J]. Probabilistic Engineering Mechanics, 2016, 44: 111-117.
[29] 秦权,贺瑞,杨小刚. 在时变结构可靠度领域中有必要澄清一个错误概念[J]. 工程力学, 2009, 26(8): 201-204. (QIN Quan, HE Rui, YANG Xiaogang. Clarification of a wrong conception in the field of time-varying structural reliability[J]. Engineering Mechanics, 2009, 26(8): 201-204.(in Chinese))