1. bookAHEAD OF PRINT
Zeitschriftendaten
License
Format
Zeitschrift
eISSN
2444-8656
Erstveröffentlichung
01 Jan 2016
Erscheinungsweise
2 Hefte pro Jahr
Sprachen
Englisch
access type Uneingeschränkter Zugang

Mathematical statistics algorithm in the bending performance test of corroded reinforced concrete beams under fatigue load

Online veröffentlicht: 30 Dec 2021
Volumen & Heft: AHEAD OF PRINT
Seitenbereich: -
Eingereicht: 17 Jun 2021
Akzeptiert: 24 Sep 2021
Zeitschriftendaten
License
Format
Zeitschrift
eISSN
2444-8656
Erstveröffentlichung
01 Jan 2016
Erscheinungsweise
2 Hefte pro Jahr
Sprachen
Englisch
Abstract

Harsh environments such as alternating wet and dry conditions and cyclic loading cause erosion in steel bars, leading to severe damage to structures. Steel bars are also prone to bending under long-term fatigue loads. This paper establishes the bending performance test of corroded reinforced concrete (RC) beam under fatigue load based on the mathematical–statistical algorithm. It investigates the influence of heavy load on the fatigue performance of damaged RC beams, the influence of corrosive environment on the fatigue performance of RC beams, the degree of corrosion of steel bars in concrete beams under fatigue loading, and the distribution of chloride ions. The study results found that the stress ratio has a significant effect on the maximum crack width of the specimen beam. The greater is the stress ratio, the longer the fatigue life. This directly affects the performance of the specimen beam under fatigue loading. For this reason, we should pay special attention to the impact of corrosion on the fatigue performance of a structure during the design of steel bars.

MSC 2010

Introduction

Reinforced 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].

Deterministic model for calculating the shear capacity of corroded RC beams

Corrosion 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: V=Vc+Vs V = {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]: fvyv=a1a2ηsva3ηsvfvy {f_{vyv}} = {{{a_1} - {a_2}{\eta _{sv}}} \over {{a_3} - {\eta _{sv}}}}{f_{vy}} where ηsv = (AvAvc)/Av. Av is the cross-sectional area of the counterbearing before rusting and Avc is the cross-sectional area of the counterbearing after corrosion. 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: f1={Ecε1ε1εcr0.33fc1+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: Avfvyc(f2sin2θf1cos2θ)bs {A_v}{f_{vyc}}\left( {{f_2}\mathop {\sin }\nolimits^2 \theta - {f_1}\mathop {\cos }\nolimits^2 \theta } \right)bs The formula Av is the cross-sectional area of the counterbearing.

Fig. 1

RC beam with diagonal cracks. RC, reinforced concrete

Fig. 2

The stress balance condition and stress Mohr circle of the tiny body

Fig. 3

Schematic diagram of beam principal stress and force balance

Combining formulas (4) and (5) we obtain as follows: 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 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: bc={b,ηsv30%b2(c+dsv)+s5.5,ηsv>30%,s5.5cb5.5s(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: Vn=Vc+Vs=f1bchvcotθ+Avfvychvscotθ=0.33bchvfc1+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.

A probabilistic model for calculating the shear capacity of corroded RC beams
Establishment of probabilistic shear capacity model

The 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 β1β7 to consider the impact of objective uncertainty. The probability model parameter β8 is introduced to consider the impact of subjective uncertainty as Vncp=0.33bhvfc1+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 fvycp=β1β2ηsvβ3ηsv {f_{vycp}} = {{{\beta _1} - {\beta _2}{\eta _{sv}}} \over {{\beta _3} - {\eta _{sv}}}} θp=(β5λ+β6)arctan(β4knsc+a42knsc2+4(1β4)knscknvc2(1β4)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.

Determination of probability model parameters

According 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: P(β|Vncp)=P(β)P(Vncp|β)P(β)P(Vncp|β)dVncp P(\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) is the likelihood function of β and ∫P(β)P(Vncp)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.

Basic parameters of corroded RC beams and test values of shear capacity test

Numbering b/mm h/mm λ ρ1/(%) fvy/Mpa Vn1 Vn2 Vn3 Vn4

1 100 175 1.5 1.94 0.44 49.71 52.87 23.68 27.07
2 100 175 2.5 1.94 0.44 50.71 48.93 21.47 38
3 100 175 1.5 1.94 0.44 59.92 50.16 22.78 26.69
4 100 175 2.5 1.94 0.44 52.44 48.96 20.21 37.92
5 100 175 2.5 1.94 0.44 54.05 51.79 20.75 38.84
6 100 175 1.5 1.94 0.44 61.91 58.13 22.99 28.22
7 100 175 2.5 1.94 0.44 53.28 54.07 21.05 39.91
8 100 175 2.5 1.94 0.44 53.25 49.48 20.22 38.7
9 100 175 2.5 1.94 0.44 52.79 48.96 20.21 38.21
10 100 175 2.5 1.94 0.44 53.65 50.5 20.5 38.7
11 100 175 1.5 1.94 0.44 62.65 65.86 22.74 29.84
12 100 175 1.5 1.94 0.44 63.53 67.28 22.99 30.02
13 100 175 1.5 1.94 0.44 65.29 74.67 22.99 31.97
14 100 175 2.5 1.94 0.44 53.73 48.96 20.21 38.98
15 100 175 1.5 1.94 0.44 66.06 76.02 23.19 32.12
16 100 175 1.5 1.94 0.44 66.76 77.86 23.36 33.26
17 120 200 2 1.92 0.32 73.84 52.86 30.01 39.34
18 100 175 2.5 1.94 0.44 58.13 56.18 21.66 41.39
19 100 175 1.5 1.94 0.44 68.06 77.52 23.36 33.53
20 120 200 2 1.92 0.32 77.74 73.12 30.32 42.34

Based 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 KS, 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 KS test of each probability model parameter is shown in Table 2.

Probability model parameter KS test D value

Parameter β1 β2 β3 β4

Lognormal distribution 0.032 0.047 0.049 0.05
Weibull distribution 0.067 0.053 0.058 0.059
index distribution 0.055 0.506 0.541 0.56
Gamma distribution 0.031 0.039 0.045 0.048
Normal distribution 0.03 0.037 0.036 0.042
Parameter β5 β6 β7 β8
Lognormal distribution × 0.038 0.064 ×
Weibull distribution × 0.068 0.048 ×
Index distribution × 0.586 0.473 ×
Gamma distribution × 0.037 0.053 ×
Normal distribution 0.031 0.035 0.034 0.025

The 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.

Comparative analysis and verification
Validation of the probabilistic shear capacity model

Based 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. 4

The distribution of experimental test values in the confidence interval of the probability model

It 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.

Calibration analysis of deterministic shear capacity model

Taking 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. 5

The distribution of the calculated value of the deterministic model in the confidence interval of the probability model

Although 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. 6

Calibration of the calculation accuracy of the deterministic model

Conclusion

This 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.

Fig. 1

RC beam with diagonal cracks. RC, reinforced concrete
RC beam with diagonal cracks. RC, reinforced concrete

Fig. 2

The stress balance condition and stress Mohr circle of the tiny body
The stress balance condition and stress Mohr circle of the tiny body

Fig. 3

Schematic diagram of beam principal stress and force balance
Schematic diagram of beam principal stress and force balance

Fig. 4

The distribution of experimental test values in the confidence interval of the probability model
The distribution of experimental test values in the confidence interval of the probability model

Fig. 5

The distribution of the calculated value of the deterministic model in the confidence interval of the probability model
The distribution of the calculated value of the deterministic model in the confidence interval of the probability model

Fig. 6

Calibration of the calculation accuracy of the deterministic model
Calibration of the calculation accuracy of the deterministic model

Basic parameters of corroded RC beams and test values of shear capacity test

Numbering b/mm h/mm λ ρ1/(%) fvy/Mpa Vn1 Vn2 Vn3 Vn4

1 100 175 1.5 1.94 0.44 49.71 52.87 23.68 27.07
2 100 175 2.5 1.94 0.44 50.71 48.93 21.47 38
3 100 175 1.5 1.94 0.44 59.92 50.16 22.78 26.69
4 100 175 2.5 1.94 0.44 52.44 48.96 20.21 37.92
5 100 175 2.5 1.94 0.44 54.05 51.79 20.75 38.84
6 100 175 1.5 1.94 0.44 61.91 58.13 22.99 28.22
7 100 175 2.5 1.94 0.44 53.28 54.07 21.05 39.91
8 100 175 2.5 1.94 0.44 53.25 49.48 20.22 38.7
9 100 175 2.5 1.94 0.44 52.79 48.96 20.21 38.21
10 100 175 2.5 1.94 0.44 53.65 50.5 20.5 38.7
11 100 175 1.5 1.94 0.44 62.65 65.86 22.74 29.84
12 100 175 1.5 1.94 0.44 63.53 67.28 22.99 30.02
13 100 175 1.5 1.94 0.44 65.29 74.67 22.99 31.97
14 100 175 2.5 1.94 0.44 53.73 48.96 20.21 38.98
15 100 175 1.5 1.94 0.44 66.06 76.02 23.19 32.12
16 100 175 1.5 1.94 0.44 66.76 77.86 23.36 33.26
17 120 200 2 1.92 0.32 73.84 52.86 30.01 39.34
18 100 175 2.5 1.94 0.44 58.13 56.18 21.66 41.39
19 100 175 1.5 1.94 0.44 68.06 77.52 23.36 33.53
20 120 200 2 1.92 0.32 77.74 73.12 30.32 42.34

Probability model parameter K − S test D value

Parameter β1 β2 β3 β4

Lognormal distribution 0.032 0.047 0.049 0.05
Weibull distribution 0.067 0.053 0.058 0.059
index distribution 0.055 0.506 0.541 0.56
Gamma distribution 0.031 0.039 0.045 0.048
Normal distribution 0.03 0.037 0.036 0.042
Parameter β5 β6 β7 β8
Lognormal distribution × 0.038 0.064 ×
Weibull distribution × 0.068 0.048 ×
Index distribution × 0.586 0.473 ×
Gamma distribution × 0.037 0.053 ×
Normal distribution 0.031 0.035 0.034 0.025

Song, L., Fan, Z., & Hou, J. Experimental and analytical investigation of the fatigue flexural behavior of corroded reinforced concrete beams. International Journal of Concrete Structures and Materials., 2019; 13(1): 1–14 SongL. FanZ. HouJ. Experimental and analytical investigation of the fatigue flexural behavior of corroded reinforced concrete beams International Journal of Concrete Structures and Materials 2019 13 1 1 14 10.1186/s40069-019-0340-5 Search in Google Scholar

Huang, Y., Wei, J., & Dong, R. Stiffness of corroded partially prestressed concrete T-beams under fatigue loading. Magazine of Concrete Research., 2020; 72(7): 325–338 HuangY. WeiJ. DongR. Stiffness of corroded partially prestressed concrete T-beams under fatigue loading Magazine of Concrete Research 2020 72 7 325 338 10.1680/jmacr.18.00187 Search in Google Scholar

Akid, A. S. M., Wasiew, Q. A., Sobuz, M. H. R., Rahman, T., & Tam, V. W. Flexural behavior of corroded reinforced concrete beam augmentationened with jute fiber reinforced polymer. Advances in Structural Engineering., 2021; 24(7): 1269–1282 AkidA. S. M. WasiewQ. A. SobuzM. H. R. RahmanT. TamV. W. Flexural behavior of corroded reinforced concrete beam augmentationened with jute fiber reinforced polymer Advances in Structural Engineering 2021 24 7 1269 1282 10.1177/1369433220974783 Search in Google Scholar

Banu, S. T., Chitra, G., Awoyera, P. O., & Gobinath, R. Structural retrofitting of corroded fly ash based concrete beams with fibres to improve bending characteristics. Australian Journal of Structural Engineering., 2019; 20(3): 198–203 BanuS. T. ChitraG. AwoyeraP. O. GobinathR. Structural retrofitting of corroded fly ash based concrete beams with fibres to improve bending characteristics Australian Journal of Structural Engineering 2019 20 3 198 203 10.1080/13287982.2019.1622490 Search in Google Scholar

Garhwal, S., Sharma, S., & Sharma, S. K. Monitoring the flexural performance of GFRP repaired corroded reinforced concrete beams using passive acoustic emission technique. Structural Concrete., 2021; 22(1): 198–214 GarhwalS. SharmaS. SharmaS. K. Monitoring the flexural performance of GFRP repaired corroded reinforced concrete beams using passive acoustic emission technique Structural Concrete 2021 22 1 198 214 10.1002/suco.202000247 Search in Google Scholar

Zhao, Y., Huang, Y., Du, H., & Ma, G. Flexural behaviour of reinforced concrete beams augmentationened with pre-stressed and near surface mounted steel–basalt-fibre composite bars. Advances in Structural Engineering., 2020; 23(6): 1154–1167 ZhaoY. HuangY. DuH. MaG. Flexural behaviour of reinforced concrete beams augmentationened with pre-stressed and near surface mounted steel–basalt-fibre composite bars Advances in Structural Engineering 2020 23 6 1154 1167 10.1177/1369433219891595 Search in Google Scholar

Yu, Y. L., Yin, S. P., & Na, M. W. Bending performance of TRC-augmentationened RC beams with secondary load under chloride erosion. Journal of Central South University., 2019; 26(1): 196–206 YuY. L. YinS. P. NaM. W. Bending performance of TRC-augmentationened RC beams with secondary load under chloride erosion Journal of Central South University 2019 26 1 196 206 10.1007/s11771-019-3993-y Search in Google Scholar

Huang, Y., Wei, J., & Dong, R. Stiffness of corroded partially prestressed concrete T-beams under fatigue loading. Magazine of Concrete Research., 2020; 72(7): 325–338 HuangY. WeiJ. DongR. Stiffness of corroded partially prestressed concrete T-beams under fatigue loading Magazine of Concrete Research 2020 72 7 325 338 10.1680/jmacr.18.00187 Search in Google Scholar

Gençoğlu, M. & Agarwal, P. Use of Quantum Differential Equations in Sonic Processes. Applied Mathematics and Nonlinear Sciences., 2020; 6(1): 21–2. GençoğluM. AgarwalP. Use of Quantum Differential Equations in Sonic Processes Applied Mathematics and Nonlinear Sciences 2020 6 1 21 2 10.2478/amns.2020.2.00003 Search in Google Scholar

Hu, X., Li, J. & Aram, Research on style control in planning and designing small towns. Applied Mathematics and Nonlinear Sciences., 2020; 6(1): 57–64 HuX. LiJ. Aram Research on style control in planning and designing small towns Applied Mathematics and Nonlinear Sciences 2020 6 1 57 64 10.2478/amns.2020.2.00077 Search in Google Scholar

Empfohlene Artikel von Trend MD

Planen Sie Ihre Fernkonferenz mit Scienceendo