Evaluation of calculation methods used for estimating the ultimate moment resistance of bridge decks reinforced with frp bars

A statistical research of the calculation methods for calculating ultimate moments of concrete elements reinforced with fiber-reinforced plastic is presented. For this purpose a database of experimental results has been collected. Calculations of the ultimate moment resistance were performed according to three design recommendations. Wilk-Shapiro test were used to determine the distribution of experimental and theoretical data. The statistical research to evaluate the calculation methods was performed by testing the statistical hypothesis on the differences between theoretical and experimental values. It is suggested to calculate the coefficient of confidence for assessing the accuracy of calculation methods.


Introduction
Concrete reinforced with steel bars is widely applied in modern construction. Corrosion of steel reinforcement in reinforced concrete structures in aggressive environments, however, causes serious problems during the service time of the constructions. Due to steel reinforcement corrosion, the service time of concrete constructions such as bridges shortens and the maintenance costs increase. Concrete bridge decks deteriorate more and faster than the other parts due to direct impact of the environment, moisture, freeze thaw cycles, de-icing salts, increasing traffic loads.
In order to avoid the problems caused by the corrosion, concrete constructions reinforced with steel reinforcement should be additionally protected from the factors causing corrosion or constructions can be reinforced with non-corroding reinforcement. As an alternative to steel reinforcement, FRP (fibre-reinforced plastic) reinforcement can be used (Bouguerra et al. 2011;Dang et al. 2014;Hassan et al. 2000;Matta, Nanni 2009).
Numerous studies on application and use of fibre reinforced polymers (FRP) in construction elements have been conducted. Additionally to reinforcement bars, wires and ropes, FRP can be used in the form of strips, sheets and lamellas. FRP reinforcement can also be applied not only for reinforcing new concrete constructions as internal reinforcement, it can also be used in strengthening the structures as external reinforcement (Aktas, Sumer 2014; Barris et al. 2012;Benzaid, Mesbah 2014;Daugevičius et al. 2012;Fayyadh, Razak 2014;Lale Arefi et al. 2014;Lapko, Urbanski 2015;Marčiukaitis et al. 2007;Meisami et al. 2013;Mostofinejad, Ilia 2014;Mostofinejad, Moghaddas 2014;Nelson, Fam 2014;Pakrastinsh et al. 2006;Serdjuks et al. 2003;Skuturna, Valivonis 2014a;Sprince et al. 2013;Sundarraja, Prabhu 2013;Szolomicki et al. 2015;Valivonis et al. 2014). FRP reinforcement, as compared to steel, is lighter, resistant to corrosion, has a higher tensile strength, and is transparent to magnetic fields and electrically nonconductive. FRP reinforcement bars are made from glass, aramid and carbon fibres are connected with epoxy resin, polyester and vinyl-ester. New generation FRP reinforcement in some countries is certified and meets the set standards. FRP reinforcement bars allow for handling the problems caused by reinforcement corrosion.
A bridge deck forms the driving surface and performs as a structural bending element. If FRP reinforcement bars are used in bridge decks, their service time, resistance to aggressive environment increase and the maintenance costs decrease. Published design recommendations and codes allow using FRP reinforcement in bridge decks and girders as the main reinforcement.
Long-term bridge studies, observations and inspections have been performed and results generated. The results show that deflections, cracking mode and deformations of concrete bridge decks reinforced with FRP reinforcement bars are almost the same as those of the decks reinforced with steel reinforcement. It has also been found out that decks reinforced with FRP work very well in an aggressive environment, freeze thaw cycles, sharp temperature fluctuations, de-icing salts, intensive traffic. No additional cracks or their development in concrete constructions have been noticed under such complicated conditions. It has neither been noticed that FRP reinforcement would be damaged by aggressive effects.
FRP reinforcement module is relatively smaller than that of the steel, therefore, it is expected that concrete bending elements reinforced with FRP bars will deflect more than the ones reinforced with steel reinforcement. However, field tests of functioning bridges show that with a proper design of constructions, their deflections do not exceed the permissible ones. Deformability is influenced by the strength of concrete, the height of cross-section, the percentage of reinforcement (Benmokrane et al. 2004;Benmokrane et al. 2006;El-Ragaby et al. 2007).
Bridge decks are exposed to repeated transport loads; therefore, their failure due to fatigue is possible. Bridge decks reinforced with FRP reinforcement bars have a better fatigue performance and a longer fatigue life than the decks reinforced with steel reinforcement bars. This can be explained by the fact that FRP reinforcement and concrete elasticity modules are closer as well as the fact that FRP reinforcement is linear-elastic up to failure.
Designing of bridge decks and load carrying capacity calculations can be done using different design norms. The essence of methods provided in design norms is to solve two balance equations. These equations are a sum of crosssection forces and moments. Ways of calculating the load carrying capacity differ in defining the reinforcement percentage and the failure mode, description of the stress-strain curve of concrete in compression, the ultimate strains of concrete, additional coefficients and calculation assumptions.

Analysis of design methods
The research is based on three calculation methods for ultimate moment resistance. These methods are provided in design codes (fib Bulletin No. 40 There are a few assumptions on which calculation methods are based: − sections of an element are plane before and after loading; − the ultimate compressive strain of concrete ε cu is 0.0035 according to FIB, ISIS and for ACI is equal to 0.003; − the linear behaviour of FRP reinforcement up to failure; − the bond between concrete and FRP reinforcement is perfect; − the tensile concrete performance is ignored. There are three modes of flexural failure of concrete elements reinforced with FRP. The first is balanced failure when FRP reinforcement ruptures and concrete crushes at the same time. The second is compression failure when concrete crushes before FRP reinforcement reaches its ultimate strain. And the last way of failure is governed by FRP rupture before concrete crushing. According to design codes, the ultimate moment resistance is calculated assessing the failure mode. It is possible to determine the failure mode when the reinforcement ratio ρ f of FRP reinforcement is compared to the balanced reinforcement ratio ρ fb . The balanced reinforcement ratio is based on the equilibrium of Design scheme: a -deck cross section; b -strain distribution at compression failure condition; c -design stress distribution at compression failure condition; d -strain distribution at tension failure condition; e -design stress distribution at tension failure condition for FIB, f -for ACI, g -for ISIS internal forces and can be expressed for FIB as Eq (1), for ACI as Eq (2), for ISIS as Eq (3).

,
( , , where f ck -the characteristic compressive strength of concrete, MPa; ε cu -ultimate concrete compressive strain; f fk -the characteristic tensile strength of FRP reinforcement, MPa; f fu -the ultimate tensile strength of FRP reinforcement, MPa; E f -elastic modulus for FRP, MPa; -the compressive strength of concrete MPa; α 1 and β 1 -stress block factors. The design scheme is presented in Fig. 1. The flexural concrete element will fail due to concrete crushing if ρ f > ρ fb . When, according to FIB, the ultimate resistance is calculated by equation: , where f cd -the design compressive strength of concrete, MPa; b -the width of the element, m; d -the effective depth of the cross-section, m; η and λ -stress block factors according to Eurocode 2; ξ -ratio of neutral axis depth to the effective depth. .
The strain in FRP reinforcement is calculated: where f f -the stress of FRP reinforcement, MPa; f fd -the design tensile strength of FRP reinforcement, MPa. According to ACI, the ultimate moment is calculated by Eq (8) when failure of concrete element is initiated by concrete crushing. , , (9) where f fu -the ultimate stress of FRP reinforcement, MPa. ISIS presents Eq (10) for ultimate moment calculation when failure is governed by concrete crushing. . (10) The depth of neutral axis is calculated as follows: , , where A f -the cross-sectional area of FRP reinforcement, m 2 . The flexural concrete element will fail due to FRP rupturing if ρ f > ρ fb . At this state calculation of the ultimate moment differs from the one mentioned above because the concrete compressive strain ε c is unknown.
When calculations are done according to FIB, the ultimate moment is calculated by Eq (13): . (13) The depth of neutral axis is solved by Eqs (14)- (15): , .
According to ACI, the calculation of ultimate moment is done in a simplified way when failure is due to FRP rupture: , , where ε fu -the ultimate strain of FRP reinforcement. According to ISIS, the calculation of the ultimate moment is based on the iteration process Eq (18) to assume the depth of the neutral axis x. The stress block parameters α and β can be found in the tables of code. The ultimate moment is calculated by Eq (19). . .

Database and statistical analysis
There were collected 127 experimental results of concrete elements reinforced with FRP bars. Also, a database was created for statistical analysis. All elements included into the database were of rectangular cross-section and tested in four-point bending. No additional preloading or damaging was applied to the elements before tests. The failure modes of tested elements were concrete crushing or FRP reinforcement rupturing. Table 1 provides the dimensions of the tested elements, properties of materials, experimental ultimate moment M exp and the ratios of experimental and theoretical results , , .
The results of the ultimate moment calculation show that the mean and variation coefficient of the ratio according to the FIB is 0.95 and 15.6%, according ACI is 1.07 and 14.9%, according ISIS is 1.05 and 14.7%. It is difficult to state the accuracy and relevance of calculation methods from results which are given in Table 1 and Fig. 3.  For this purpose a statistical analysis is performed to assess the accuracy and relevance of the calculation methods. Six data samples were made according to the ratio of FRP reinforcement. The size of samples n, the mean, standard deviations s, min and max values are provided in Table 2.
Wilk-Shapiro test for a small size sample was applied for determination of the distribution of the sample data (Shapiro, Wilk 1965). The data has normal distribution if the hypothesis is valid: where W -Shapiro-Wilk test value calculated according to the Eq (21); W a -critical value found in tables.
, ; a n-i+1 -coefficient taken from tables (Shapiro, Wilk 1965). Table 2 provides the results of Wilk Shapiro test when the significance level is α = 0.05. The test results show that the hypothesis is valid and all data have normal distribution.
The analysis of the results shows that the calculated and experimental ultimate moments are different. It may be stated that the calculated and theoretical results will be always different but this difference may be statistically negligible (Montgomery, Runger 2002).
The best case is if the means of the ratios equals to 1: .
A statistical hypothesis is formed to assess a statistical significance of the Eq (22): , Continued where are the means of the ratios , , of the experimental and theoretical results.
t test can be used when the data has normal distribution. Hypothesis H 0 is rejected and hypothesis H 1 is valid when: .
The confidence interval is calculated: . Table 3 presents statistical results of testing the hypothesis and confidence intervals when the significance is α = 0.05.
The analysis of statistical results shows that the ratio of experimental and theoretical results only in two samples of six, if calculated according to FIB, statistically significantly differ from the best case when the ratio of results is equal to 1. The differences are statistically negligible in the rest of the samples.
The received statistical results are worse when the ultimate moment is calculated according to ACI. The differences are statistically negligible in three samples. The results of the rest of the samples show that the ratios of the experimental and theoretical results statistically significantly differ from 1.
Also, a half of the results of testing the hypothesis show that the different is statistically significant when calculated according to ISIS.
The received results show that in the first sample the difference is negligible when the ultimate moments are calculated according to all three methods. In samples 5 and 6 the experimental and the theoretical results statistically significantly differ when calculated according to one of the three methods. In samples 2, 3 and 4 the different statistical negligible when calculations are made according to one of the three methods.
It is possible to determine which calculation method is more accurate and whether the difference between the theoretical results is statistically significant when statistics methods are used. For this purpose, a statistical hypothesis is created. Hypothesis H 0 is valid when the means and are equal. Alternative hypothesis H 1 is valid then the means are not equal. .
Hypothesis H 0 will be rejected if: .
The degrees of freedom v and t statistic are calculated according to Eqs (28)-(29) when the variances of two samples are unknown and not equal. . .
Three samples are analyzed and results are given in Tables 4, 5, 6. The analysis of the results shows that the difference between theoretical ultimate moments is statistical negligible in sample 1. There is no statistical difference between theoretical results in sample 6 when calculations of the ultimate moment are done according to ACI and ISIS. However, there is a statistical difference in sample 5 when FIB and ACI methods are used to calculate the ultimate moments.
In order to determine which calculation method is more accurate, it is suggested to calculate the coefficient of confidence (Eq (31)) in data samples 1 and 6 when the differences between theoretical results are statistically negligible (Skuturna, Valivonis 2014b). .
The coefficient of confidence sets the relation between the means of calculation results and the variation coefficients because it can be complicated to evaluate the design method if they are analyzed separately. The more accurate are the theoretical results of the ultimate moments, the closer they are to the experimental ones if the coefficient of confidence is closer to 1.
The values of the coefficients of confidence in sample 1 for FIB, ACI, ISIS methods are 1.36, 1.31 and 1.42 respectively. In sample 6 the coefficients of confidence for

Conclusions
1. The performed analysis shows that it is complicated to assess the calculation methods only according to the means and variation coefficients of calculation results. Statistical methods such as Wilk-Shapiro, t test are proposed to use for assessing the calculation methods of the ultimate moment resistance. 2. According to the experimental results of 127 concrete elements in flexure reinforced with fibre reinforced polymer bars, a statistical research of calculating methods of the ultimate moment was performed.
3. The statistical research has shown that in half of the samples the results significantly differ from the experimental ones when calculations are performed according to Guide for the Design