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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