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1IntroductionReinforced concrete (RC) beams are essential components of civil engineering structures such as buildings and bridges. Affected by corrosive environmental effects such as concrete carbonisation and chloride ion erosion, RC beams often suffer from steel corrosion and concrete expansion and cracking, which reduce their shear capacity. This affects the safety of the structure [1]. Therefore, the study and establishment of a calculation model for the shear capacity of corroded RC beams are of great significance for the safety assessment and bearing capacity redesign of in-service RC structures.The shear mechanism of corroded RC beams is complex and has many influencing factors. At present, research done on it has a few shortcomings: first, the introduced steel corrosion correction coefficient is usually determined according to engineering experience or test data fitting analysis. Second, the research model usually takes the critical oblique crack inclination angle to be approximately 45°, ignoring the impacts of factors such as shear span ratio, augmentation ratio, counterbearing ratio, and steel corrosion on the critical oblique crack inclination angle. This leads to limited calculation accuracy [2]. Therefore, it is necessary to comprehensively consider the influence of steel corrosion on the yield augmentation of counterbearings, counterbearing ratio, augmentation ratio, effective shear cross-sectional area of RC beams, and other essential factors. At the same time, we also need to consider the impact of objective uncertainty and subjective uncertainty to establish a probability model for calculating the shear capacity of corroded RC beams.This paper first comprehensively considers the influence of steel corrosion on the yield augmentation of counterbearings, augmentation ratio, counterbearing ratio, critical oblique crack inclination, beam effective shear cross-sectional area, and other essential factors. At the same time, we established a deterministic model for calculating the shear capacity of corroded RC beams by combining the modified pressure field theory (MCFT) and the critical oblique crack inclination model, considering the influence of the shear span ratio [3].2Deterministic model for calculating the shear capacity of corroded RC beamsCorrosion of steel bars reduces the yield augmentation and effective cross-sectional area of counterbearings, leading to a decrease in the contribution of the counterbearing shear capacity and often causes concrete corrosion, expansion, cracking or cracking spalling [4]. The following comprehensive consideration of the influence of steel corrosion on the yield augmentation of counterbearings, augmentation ratio, counterbearing ratio, effective shear cross-sectional area of beams and other important factors will help to establish a deterministic model. Based on the MCFT, the shear capacity V of RC beams is mainly composed of the contribution value Vc of the concrete shear capacity and the contribution value Vs of the shear capacity of the counterbearings as:(1)V=Vc+VsV = {V_c} + {V_s}After the steel is corroded, the cross-section will be rusted, which will cause stress concentration in the weakest cross-section when the steel is stretched [5]. Scholars have established the relationship between the nominal yield augmentation fvyv of the corroded counterbearing and the rust loss rate ηsv of the counterbearing section as [6]:(2)fvyv=a1−a2ηsva3−ηsvfvy{f_{vyv}} = {{{a_1} - {a_2}{\eta _{sv}}} \over {{a_3} - {\eta _{sv}}}}{f_{vy}}where ηsv = (Av − Avc)/Av. Av is the cross-sectional area of the counterbearing before rusting and Avc is the cross-sectional area of the counterbearing after corrosion.(3)ρsc=Asc/(bh0); ρvc=Avc/(bs){\rho _{sc}} = {A_{sc}}/(b{h_0});\;{\rho _{vc}} = {A_{vc}}/(bs)where b is the section width of the beam and Asc is the cross-sectional area of the corroded longitudinal bars [7]. Under the action of external load P, when the oblique section of the beam RC undergoes shear failure, the typical crack distribution is as shown in Figure 1 [8]. The specific image is shown in Figure 2. According to MCFT, we can establish stress balance conditions as:(4)f1={Ecε1ε1≤εcr0.33fc′1+500ε1ε1>εcr{f_1} = \left\{ {\matrix{ {{E_c}{\varepsilon _1}} \hfill & {{\varepsilon _1} \le {\varepsilon _{cr}}} \hfill \cr {{{0.33\sqrt {f_c^\prime} } \over {1 + \sqrt {500{\varepsilon _1}} }}}\hfill & {{\varepsilon _1} > {\varepsilon _{cr}}} \hfill \cr } } \right.where ɛcr is the cracking strain of concrete, fc′{f_c^\prime}is the compressive augmentation of concrete and ɛ1 is the primary tensile strain [9]. The force situation of the beam RC is shown in Figure 3, which can be obtained from the vertical force balance condition as:(5)Avfvyc(f2sin2θ−f1cos2θ)bs{A_v}{f_{vyc}}\left( {{f_2}\mathop {\sin }\nolimits^2 \theta - {f_1}\mathop {\cos }\nolimits^2 \theta } \right)bsThe formula Av is the cross-sectional area of the counterbearing.Fig. 1RC beam with diagonal cracks. RC, reinforced concreteFig. 2The stress balance condition and stress Mohr circle of the tiny bodyFig. 3Schematic diagram of beam principal stress and force balanceCombining formulas (4) and (5) we obtain as follows:(6)V=vbhv=f1bhvtanθ+cotθ+bhvtanθ+cotθ(Avfvycbs+f1cos2θ)1sin2θ=f1bhvtanθ+cotθ(1+cot2θ)+Avfvycbhvbs(tanθ+cotθ)×sin2θ+cos2θsin2θ=f1bhvcotθ+Avfvychvscotθ\matrix{ V \hfill & { = vb{h_v} = {{{f_1}b{h_v}} \over {\tan \theta + \cot \theta }} + {{b{h_v}} \over {\tan \theta + \cot \theta }}\left( {{{{A_v}{f_{vyc}}} \over {bs}} + {f_1}\mathop {\cos }\nolimits^2 \theta } \right){1 \over {\mathop {\sin }\nolimits^2 \theta }}} \hfill \cr {} \hfill & { = {{{f_1}b{h_v}} \over {\tan \theta + \cot \theta }}\left( {1 + \mathop {\cot }\nolimits^2 \theta } \right) + {{{A_v}{f_{vyc}}b{h_v}} \over {bs\left( {\tan \theta + \cot \theta } \right)}} \times {{\mathop {\sin }\nolimits^2 \theta + \mathop {\cos }\nolimits^2 \theta } \over {\mathop {\sin }\nolimits^2 \theta }} = {f_1}b{h_v}\cot \theta + {{{A_v}{f_{vyc}}{h_v}} \over s}\cot \theta } \hfill \cr } According to formula (6), it can be seen that the shear capacity V of the beam RC is related to the diagonal tension stress f1. Based on Eq. (4), it can be seen that the diagonal tension stress f1 is related to the principal tension strain ɛ1. ɛ1 is related to the spacing of oblique cracks and the critical oblique crack dip angle θ. Therefore, the iterative calculation is needed to solve Eq. (6), and the calculation process is cumbersome as (7)ε1=a7εy=a7fvyc/Es{\varepsilon _1} = {a_7}{\varepsilon _y} = {a_7}{f_{vyc}}/{E_s}where a7 is the influence coefficient of counterbearings and ɛy = fvyc/Es is the yield strain of the counterbearing. Corrosion of steel bars often causes concrete expansion, cracking or spalling, which reduces the effective shear cross-sectional area of RC beams [10]. The corrosion rate of the counterbearing section is as follows:(8)bc={b,ηsv≤30%b−2(c+dsv)+s5.5,ηsv>30%, s≤5.5cb−5.5s(c+dsv)2,ηsv>30%, s>5.5c{b_c} = \left\{ {\matrix{ {b,} \hfill & {{\eta _{sv}} \le 30\% } \hfill \cr {b - 2(c + {d_{sv}}) + {s \over {5.5}},} \hfill & {{\eta _{sv}} > 30\% ,\;s \le 5.5c} \hfill \cr {b - {{5.5} \over s}{{(c + {d_{sv}})}^2},} \hfill & {{\eta _{sv}} > 30\% ,\;s > 5.5c} \hfill \cr } } \right.where dsv is the diameter of the counterbearing. Combining formulas (1), (4), (7) and (8) the following is obtained:(9)Vn=Vc+Vs=f1bchvcotθ+Avfvychvscotθ=0.33bchvfc′1+500a7fvycEscotθ+fvycAvshvcotθ{V_n} = {V_c} + {V_s} = {f_1}{b_c}{h_v}\cot \theta + {{{A_v}{f_{vyc}}{h_v}} \over s}\cot \theta = {{0.33{b_c}{h_v}\sqrt {f_c^\prime} } \over {1 + \sqrt {500{a_7}{{{f_{vyc}}} \over {{E_s}}}} }}\cot \theta + {f_{vyc}}{{{A_v}} \over s}{h_v}\cot \theta Equation (9) cannot consider the objective uncertainties of the RC beam geometric dimensions, material properties, boundary constraints and other factors, which will lead to a certain degree of discreteness in the calculation results. Therefore, it is necessary to comprehensively consider the effects of objective uncertainty and subjective uncertainty.3A probabilistic model for calculating the shear capacity of corroded RC beams3.1Establishment of probabilistic shear capacity modelThe values of the parameter a1, a2, a3 in formula (2), parameter a4, a5, a6 in formula (2), and parameter a7 in formula (7) are mainly determined based on the engineering experience or test data fitting analysis. This paper introduces the probability model parameter β1 ∼ β7 to consider the impact of objective uncertainty. The probability model parameter β8 is introduced to consider the impact of subjective uncertainty as (10)Vncp=0.33bhvfc′1+500β7fvycpEscotθp+Avfvycphvscotθp+β8+ξσ{V_{ncp}} = {{0.33b{h_v}\sqrt {f_c^\prime} } \over {1 + \sqrt {500{\beta _7}{{{f_{vycp}}} \over {{E_s}}}} }}\cot {\theta _p} + {{{A_v}{f_{vycp}}{h_v}} \over s}\cot {\theta _p} + {\beta _8} + \xi \sigma (11)fvycp=β1−β2ηsvβ3−ηsv{f_{vycp}} = {{{\beta _1} - {\beta _2}{\eta _{sv}}} \over {{\beta _3} - {\eta _{sv}}}}(12)θp=(β5λ+β6)⋅arctan(−β4knsc+a42knsc2+4(1−β4)knscknvc2(1−β4)knvc){\theta _p} = \left( {{\beta _5}\lambda + {\beta _6}} \right) \cdot {\rm{arctan}}\left( {\sqrt {{{ - {\beta _4}{k_{nsc}} + \sqrt {a_4^2k_{nsc}^2 + 4(1 - {\beta _4}){k_{nsc}}{k_{nvc}}} } \over {2(1 - {\beta _4}){k_{nvc}}}}} } \right)where Vncp is the probability value of the shear capacity of the corroded RC beam, ξ σ is the systematic error of the probability model, ξ is a standard normal distribution random variable and σ is the standard deviation of the systematic error.3.2Determination of probability model parametersAccording to the engineering experience or existing test data, the prior distribution of the probability model parameter β can be determined. Then the posterior distribution of β can be determined using Bayesian theory as:(13)P(β|Vncp)=P(β)P(Vncp|β)∫P(β)P(Vncp|β)dVncpP(\beta |{V_{ncp}}) = {{P(\beta )P({V_{ncp}}|\beta )} \over {\int P(\beta )P({V_{ncp}}|\beta )d{V_{ncp}}}}where P(β |Vncp) is the posterior distribution probability density function of β, P(β) is the primary distribution probability density function of β, P(Vncp|β) is the likelihood function of β and ∫P(β)P(Vncp|β)dVncp is the normalisation factor. For this reason, this paper uses the Markov chain Monte Carlo (MCMC) method to determine the posterior distribution information of β. The basic parameters are shown in Table 1.Table 1Basic parameters of corroded RC beams and test values of shear capacity testNumberingb/mmh/mmλρ1/(%)fvy/MpaVn1Vn2Vn3Vn411001751.51.940.4449.7152.8723.6827.0721001752.51.940.4450.7148.9321.473831001751.51.940.4459.9250.1622.7826.6941001752.51.940.4452.4448.9620.2137.9251001752.51.940.4454.0551.7920.7538.8461001751.51.940.4461.9158.1322.9928.2271001752.51.940.4453.2854.0721.0539.9181001752.51.940.4453.2549.4820.2238.791001752.51.940.4452.7948.9620.2138.21101001752.51.940.4453.6550.520.538.7111001751.51.940.4462.6565.8622.7429.84121001751.51.940.4463.5367.2822.9930.02131001751.51.940.4465.2974.6722.9931.97141001752.51.940.4453.7348.9620.2138.98151001751.51.940.4466.0676.0223.1932.12161001751.51.940.4466.7677.8623.3633.261712020021.920.3273.8452.8630.0139.34181001752.51.940.4458.1356.1821.6641.39191001751.51.940.4468.0677.5223.3633.532012020021.920.3277.7473.1230.3242.34Based on the experimental data in Table 1 combined with formula (13) and the MCMC method, we can determine the posterior distribution information of each probability model parameter. The paper first uses K − S, a test to determine the empirical probability distribution type of the probability model parameter β. When the significance level is 0.05, and the sample size is 1,000, the critical value of the K − S test is 0.043. The D value of K − S test of each probability model parameter is shown in Table 2.Table 2Probability model parameter K − S test D valueParameterβ1β2β3β4Lognormal distribution0.0320.0470.0490.05Weibull distribution0.0670.0530.0580.059index distribution0.0550.5060.5410.56Gamma distribution0.0310.0390.0450.048Normal distribution0.030.0370.0360.042Parameterβ5β6β7β8Lognormal distribution×0.0380.064×Weibull distribution×0.0680.048×Index distribution×0.5860.473×Gamma distribution×0.0370.053×Normal distribution0.0310.0350.0340.025The distribution type with the smallest D value is used as the posterior distribution of each probability model parameter. We found that none of the probability model parameters refused to obey the normal distribution. The average value of β4 is 0.46, which indicates that the contribution of concrete shear capacity accounts for about 46% of the RC beam shear capacity. The mean value of β5 is negative, which indicates that the critical oblique crack inclination angle of the RC beam shows a decreasing trend with the increase in the shear-span ratio.4Comparative analysis and verification4.1Validation of the probabilistic shear capacity modelBased on determining the posterior distribution information of the probability model parameter β, we can determine the probability statistical characteristic value of the shear bearing capacity of the RC beam and the confidence interval of different confidence levels by combining Eq. (13) with the MCS method. Taking the 50% and 95% confidence intervals as an example, the comparative analysis of the mean value (Vm) of the probability model and the test value (Vt) of the experiment is shown in Figure 4.Fig. 4The distribution of experimental test values in the confidence interval of the probability modelIt can be seen from Figure 4 that nearly 1/2 of the measured values are within the 50% confidence interval, and almost all of the measured values are within the 95% confidence interval. This shows that the established probability model can better describe the probability distribution characteristics of the beam's shear capacity.4.2Calibration analysis of deterministic shear capacity modelTaking the deterministic shear capacity calculation model as an example, the distribution of the calculated values of each model and the experimental test values within the confidence interval of the probability model is as shown in Figure 5.Fig. 5The distribution of the calculated value of the deterministic model in the confidence interval of the probability modelAlthough nearly half of the calculated value of the model Vn1 falls within the 95% confidence interval, the overall deviation is far from the measured value. This shows that the dispersion is large, and the calculation accuracy is limited. The calculation results of the Vn2 model are highly discrete, and most of the data points fall outside the 95% confidence interval. This shows that the calculation accuracy of the model is low.In addition, we can use the probability model of this paper to calibrate the calculation accuracy of the traditional deterministic model. Choose a corroded beam from Table 1 as an example. The mean value and standard deviation of the shear capacity determined by the probability model are 52.64 kN and 8.77 kN, respectively. After the K–S test, the shear bearing capacity does not refuse to obey the normal distribution, and its probability density distribution is as shown in Figure 6. The calculated values of models Vn1, Vn2, Vn3, and Vn4 are 58.13 kN, 56.18 kN, 21.66 kN, and 41.39 kN, respectively. The calculated values of the Vn1 and Vn2 models are both more significant than the average, and the calculated values of the Vn3 and Vn4 models are both less than the average. The calculated value of the Vn3 model is too small. It can be seen that the probability model established in this paper can be used to calibrate the calculation accuracy of the traditional deterministic shear capacity model.Fig. 6Calibration of the calculation accuracy of the deterministic model5ConclusionThis paper combined the modified pressure field theory, Bayesian theory and MCMC method to establish a probability model for calculating the shear capacity of corroded beams. The established probability model for calculating the shear capacity of corroded beams not only has a rigorous theoretical basis but also comprehensively considers the effects of subjective and objective uncertainties. It has good applicability and calculation accuracy, which can reasonably describe the probability distribution characteristics of the shear capacity of corroded beams.
Applied Mathematics and Nonlinear Sciences – de Gruyter
Published: Jan 1, 2022
Keywords: reinforced concrete beam; mathematical; statistical algorithm; fatigue load; bending performance; 03B48
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