Stochastic Analysis of Effective Moment of Inertia of Cracked In-Service Reinforced Concrete Beams
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摘要: 由于混凝土材料的不确定性和非线性特性，开裂钢筋混凝土梁的有效惯性矩很难准确地预测，往往影响了对结构正常使用极限状态的准确估计.按我国规范，推导了在正常使用极限状态范围内，钢筋混凝土梁的有效惯性矩无量纲表达，并利用蒙特卡洛抽样进行了随机分析；对应不同的配筋率，研究了有效惯性矩随机分析和确定性分之间的差异及其产生的机理，利用偏相关系数表达各随机变量与有效惯性矩之间的敏感性.分析结果表明：由于混凝土的开裂非线性，采用模型参数的均值进行确定分析的结果与采用模型随机参数进行随机分析结果的均值不一致，这种不一致是由混凝土截面开裂发生的随机性与开裂前后刚度的差异共同引起；通过随机分析结果回归，给出了钢筋混凝土梁有效惯性矩的预测均值与95%保证率的预测范围列表；混凝土抗压强度对有效惯性矩几乎没有影响，而混凝土抗拉强度的敏感性最大.Abstract: For cracked reinforced concrete structures in working service life, calculation of the stiffness of the cracked reinforced concrete beam is important for the serviceability design of reinforced concrete structures. However, owing to the uncertainty and nonlinearity of concrete, it is difficult to precisely predict the effective moment of inertia of cracked concrete beams. In this study, the dimensionless equation for the effective moment of inertia recommended in specification GB50010-2010 for in-service reinforced concrete beams was derived, and a Monte Carlo method was employed to analyse its stochastic properties. The dimensionless effective moment of inertia was calculated using both a stochastic analysis with random variables and a deterministic analysis with mean values of the parameters. The results indicate that the mean values obtained with the stochastic analysis are not in agreement with those calculated with the deterministic analysis owing to the cracking nonlinearity of concrete. In addition, through regression of the stochastic analysis results, mean values of the effective moment of inertia are estimated at the 95% confidence interval. Finally, a sensitivity coefficient reflecting the correlation between each random variable and the effective moment of inertia was calculated using a partial correlation coefficient. The results demonstrate that the tensile strength of concrete is the most sensitive variable, while the compressive strength of concrete has nearly no effect on the effective moment of inertia.
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图 1 无量纲截面惯性矩均值面
Figure 1. Surface of the mean values of dimensionless moment of inertia
图 2 不同配筋率的无量纲截面惯性矩均值曲线对比
Figure 2. Comparison of mean values of the dimensionless moment of inertia at varying reinforcement ratios
图 3 无量纲截面惯性矩的累积函数分布
Figure 3. Cumulative distribution function of the dimensionless moment of inertia
表 1 随机变量的随机特性
Table 1. Statistical properties of random variables
随机变量 均值 变异系数 分布形式 βfc=fc=βfcfcm 1 0.15 正态分布 βEc=Ec=βEc(fc/10)1/3 21 500 0.08 正态分布 βft=ft=βft(ft)2/3 0.3 0.15 正态分布 βEs=Es=βEsEsm 1 0.033 正态分布 
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表 2 公式(9)的拟合系数
Table 2. Fitting coefficients for Eq. (9)
系数 fcm/MPa 30 40 50 60 a0 -0.297 2 -0.060 0 -0.046 1 -0.043 3 a1 0.915 4 0.844 7 0.816 6 0.803 3 a2 1.383 0 0.864 9 0.810 9 0.788 5 a3 -0.345 8 -0.324 2 -0.309 5 -0.298 9 a4 -0.157 4 -0.110 9 -0.110 1 -0.120 5 a5 -1.272 0 -0.921 2 -0.879 2 -0.859 9 a6 0.048 8 0.045 1 0.042 4 -0.040 4 a7 0.063 5 0.061 4 0.060 1 0.060 5 a8 -0.018 4 -0.033 5 -0.034 1 -0.031 0 a9 0.330 4 0.254 6 0.245 3 0.240 8
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表 3 Csup的拟合系数(0.7Mcr＜M≤1.3Mcr)
Table 3. Fitting coefficients for Csup (0.7Mcr < M≤1.3Mcr)
系数 fcm/MPa 30 40 50 60 b0 3.459 3.768 3.565 4.160 b1 0.243 0.087 -0.065 -0.067 b2 -12.340 -13.000 -12.260 -14.060 b3 -0.891 -0.690 -0.628 -0.662 b4 3.150 3.143 3.275 3.339 b5 10.040 10.490 9.600 11.280 b6 0.057 -0.009 -0.009 -0.010 b7 0.550 0.563 0.519 0.577 b8 -2.697 -2.711 -2.717 -2.814 b9 -1.507 -1.552 -1.21 -1.681
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表 4 Csup的拟合系数(1.3Mcr＜M≤2Mcr)
Table 4. Fittingcoefficients for Csup(1.3Mcr < M≤2Mcr)
系数 fcm/MPa 30 40 50 60 b0 11.190 11.850 12.520 12.020 b1 -3.742 -3.418 -3.376 -3.522 b2 -15.160 -16.470 -17.590 -16.590 b3 0.814 0.775 0.792 0.968 b4 2.911 2.611 2.506 2.443 b5 7.419 8.251 8.925 8.348 b6 -0.100 -0.098 -0.094 -0.117 b7 -0.222 -0.209 -0.227 -0.276 b8 -0.647 -0.569 -0.523 -0.473 b9 -1.240 -1.413 -1.553 -1.448
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表 5 Cinf的拟合系数(0.7Mcr＜M≤1.3Mcr)
Table 5. Fittingcoefficients for Cinf (0.7Mcr < M≤1.3Mcr)
系数 fcm/MPa 30 40 50 60 b0 5.850 6.078 6.266 6.332 b1 0.390 0.456 0.329 0.387 b2 -17.960 -18.730 -19.170 -19.420 b3 -0.459 -0.445 -0.396 -0.396 b4 1.320 1.134 1.230 1.092 b5 14.870 15.640 15.980 16.240 b6 0.065 0.062 0.065 0.067 b7 0.117 0.118 0.068 0.064 b8 -0.909 0.820 -0.804 -0.730 b9 -3.715 -3.959 -4.058 -4.146
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表 6 Cinf的拟合系数(1.3Mcr＜M≤2Mcr)
Table 6. Fittingcoefficients for Cinf(1.3Mcr < M≤2Mcr)
系数 fcm/MPa 30 40 50 60 b0 -6.724 -6.985 -7.292 -7.615 b1 1.696 1.638 1.651 1.721 b2 9.641 10.080 10.580 11.080 b3 -0.151 -0.125 -0.127 -0.168 b4 -1.624 -1.609 -1.622 -1.657 b5 -4.673 -4.909 -5.180 -5.440 b6 0.003 0.003 0.002 0.013 b7 0.078 0.066 0.067 0.071 b8 0.389 0.395 0.398 0.406 b9 0.759 0.799 0.849 0.894
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BISCHOFF P H. Reevaluation of deflection prediction for concrete beams reinforced with steel and fiber-reinforced polymer bars[J]. Journal of Structural Engineering, 2005, 114(7):1499-66. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=5eb4eb09b58e59dc22467eacd2b49d7e 徐腾飞, 白雪濛, 赵人达.加载过程中钢筋混凝土梁弯曲型变的随机性分析[J].西南交通大学学报, 2015, 50(4):630-634. doi: 10.3969/j.issn.0258-2724.2015.04.009XU Tengfei, BAI Xuemeng, ZHAO Renda. Stochastic analysis of bending deflection for reinforced concrete beam in loading process[J]. Journal of Southwest Jiaotong University, 2015, 50(4):630-634. doi: 10.3969/j.issn.0258-2724.2015.04.009 GILBERT R I. Tension stiffening in lightly reinforced concrete slabs[J]. Journal of the Structural Division ASCE, 2007, 133(6):899-903. http://cn.bing.com/academic/profile?id=9ff00ac19894721356adbe553e227e57&encoded=0&v=paper_preview&mkt=zh-cn 丁大钧.钢筋混凝土构件抗裂度、裂缝和刚度[M].南京:南京工学院出版社, 1986:17-26. 过镇海, 时旭东.钢筋混凝土原理和分析[M].北京:清华大学出版社, 2007:257-267. 李志华, 苏小卒.钢筋混凝土受弯构件挠度计算方法综述分析[J].四川建筑科学研究, 2011, 37(2):30-34. doi: 10.3969/j.issn.1008-1933.2011.02.008LI Zhihua, SU Xiaozu. Review of study on the methods for computing deflections of reinforced concrete flexural members[J]. Sichuan Building Science, 2011, 37(2):30-34. doi: 10.3969/j.issn.1008-1933.2011.02.008 XU Tengfei, XIANG Tianyu, ZHAO Renda, et al. Stochastic analysis on flexural behavior of reinforced concrete beams based on piecewise response surface scheme[J]. Engineering Failure Analysis, 2016, 59:211-222. doi: 10.1016/j.engfailanal.2015.10.004 GILBERT R I, WARNER R F. Tension stiffening in reinforced concrete slabs[J]. Journal of the Structural Division, 1978, 104(2):1885-2900. http://cn.bing.com/academic/profile?id=bad026accbc3a0e2904b30472290761e&encoded=0&v=paper_preview&mkt=zh-cn BALAKRISHAN S, MURRAY D W. Concrete constitutive model for NLFE analysis of structures[J]. Journal of Structural Engineering, 1988, 114(7):1449-1466. doi: 10.1061/(ASCE)0733-9445(1988)114:7(1449) COLLINS M P, VECCHIO F J. The modified compression-field theory for reinforced concrete elements subjected to shear[J]. ACI Journal, 1986, 83(2):219-231. doi: 10.1021-nn1019972/ PARKHYA G K., MORLEY C T. Tension stiffening and moment-curvature relation for reinforced concrete elements[J]. ACI Journal, 1990, 87(5):597-605. https://www.concrete.org/publications/internationalconcreteabstractsportal/m/details/id/2680 FLOEGL H, MANG H A. Tension stiffening concept based on bond slip[J]. Journal of the Structural Division ASCE, 1982, 108(12):2681-2701. http://cn.bing.com/academic/profile?id=5a80f08c13e009794a567315b83588d0&encoded=0&v=paper_preview&mkt=zh-cn CHOIL C K, CHEUNG S H. Tension stiffening model for planar reinforced concrete members[J]. Computers and Structures, 1996, 59(1):179-190. doi: 10.1016/0045-7949(95)00146-8 BRANSON D E. Instantaneous and time-dependent deflections of simple and continuous reinforced concrete beams[R]. Alabama: Alabama Highway Department, 1963. ACI Committee 318. ACI 318-05 building code requirements for structural concrete[S]. Washington D. C.: American Concrete Institute, 2005. BISCHOFF P H, ANDREW S. Effective moment of inertia for calculating defections of concrete moment containing steel reinforcement and fiber-Reinforced polymer reinforcement[J]. ACI Structural Journal, 2007, 104(1):68-5. http://cn.bing.com/academic/profile?id=3c5770e47671fef7188c3c3e5b8f34d6&encoded=0&v=paper_preview&mkt=zh-cn 中华人民共和国住房与建设保障部. GB50010——2010混凝土结构设计规范[S].北京: 中国建筑工业出版社, 2010. 姜磊, 姚继涛, 信任, 等.验研究钢筋混凝土薄板受拉刚化效应[J].混凝土, 2011(1):62-64. http://d.old.wanfangdata.com.cn/Periodical/hnt201101016JIANG Lei, YAO Jitao, XIN Ren, et al. Tension stiffening in reinforced concrete slabs and test research[J]. Concrete, 2011(1):62-64. http://d.old.wanfangdata.com.cn/Periodical/hnt201101016 JUNG J K, MAHMOUD M R T, HYUK-CHUN N, et al. Reliability analysis to resolve difficulty in choosing from alternative deflection models of RC beams[J]. Mechanical Systems and Signal Processing, 2013, 37:240-252. doi: 10.1016/j.ymssp.2012.06.024 周婧, 陈秦, 王慧英.钢筋混凝土框架结构竖向不规则参数的概率评估[J].建筑结构学报, 2014, 35(3):39-45. http://d.old.wanfangdata.com.cn/Periodical/jzjgxb201403006ZHOU Jing, CHEN Qin, WANG Huiying. Probability evaluation of vertical regularity parameters for reinforced concrete frame structures[J]. Journal of Building Structures, 2014, 35(3):39-45. http://d.old.wanfangdata.com.cn/Periodical/jzjgxb201403006 褚松涛, 曹少卫, 高夕良.成都东客站承轨层桥建合一结构设计施工综合技术[J].建筑施工, 2010, 32(6):520-524. doi: 10.3969/j.issn.1004-1001.2010.06.016CHU Songtao, CAO Shaowei, GAO Xiliang. Comprehensive techonogy of structure design and construction for bridge and building combined rail bearing floor of cheng du east railway station[J]. Building Construction, 2010, 32(6):520-524. doi: 10.3969/j.issn.1004-1001.2010.06.016 徐腾飞, 向天宇, 赵人达.偏心钢筋混凝土受压柱长期变形分析[J].西南交通大学学报, 2014, 49(4):26-630. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=xnjtdxxb201404010XU Tengfei, XIANG Tianyu, ZHAO Renda. Long-term random deflection of eccentrically load RC column[J]. Journal of Southwest Jiaotong University, 2014, 49(4):26-630. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=xnjtdxxb201404010 徐腾飞, 向天宇, 白雪濛, 等.基于分片响应面的钢筋混凝土梁变形随机模拟[J].工程力学, 2014, 31(11):170-174. http://www.cnki.com.cn/Article/CJFDTOTAL-GCLX201411023.htmXU Tengfei, XIANG Tianyu, BAI Xuemeng, et al. Stochastic simulation of reinforced concrete beam with piecewise response surface[J]. Engineering Mechanics, 2014, 31(11):170-174. http://www.cnki.com.cn/Article/CJFDTOTAL-GCLX201411023.htm XU Tengfei, CASTEL A, GILBERT R I, et al. Modeling the tensile steel reinforcement strain in RC-beams subjected to cycles of loading and unloading[J]. Engineering Structures, 2016, 126:92-105. doi: 10.1016/j.engstruct.2016.07.043 XU Tengfei, CASTEL A. Modeling the dynamic stiffness of cracked reinforced concrete beams under low-amplitude vibration loads[J]. Journal of Sound and Vibration, 2016, 368:135-147. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=365141718e2d5c6cd5d6c8882cb8c065 VAL D V, STEWART M G, MELCHERS R E. Effect of reinforcement corrosion reliability of highway bridges[J]. Engineering Structure, 1998, 20(97):1010-1019. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=ad1fc36b75edcbc78ebdf06c9f3a7df9 YANG I H. Prediction of time-dependent effects in concrete structures using early measurement data[J]. Engineering Structures, 2007, 29:2701-2710. doi: 10.1016/j.engstruct.2007.01.015 YANG I H.. Uncertainty and sensitivity analysis of time-dependent effects in concrete structures[J]. Engineering Structures, 2007, 29:1366-1374. doi: 10.1016/j.engstruct.2006.07.015 徐腾飞, 吴涤, 汪军, 等.预应力混凝土简支足尺试验梁变形随机分析[J].中国公路学报, 2015, 28(9):67-72. doi: 10.3969/j.issn.1001-7372.2015.09.009XU Tengfei, WU Di, WANG Jun, et al. Stochastic analysis of deflection of prestressed concrete simply supported full-scale test beam[J]. China Journal of Highway and Transport, 2015, 28(9):67-72. doi: 10.3969/j.issn.1001-7372.2015.09.009 -
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