Uncertainty in shear resistance models of reinforced concrete beams according to fib MC2010

Transcription

1 Received: 23 August 2017 Revised: 15 November 2017 Accepted: 27 December 2017 DOI: /suco TECHNICAL PAPER Uncertainty in shear resistance models of reinforced concrete beams according to fib MC2010 Miroslav Sykora 1 Jan Krejsa 1 Jan Mlcoch 1 Miguel Prieto 2 Peter Tanner 3 1 Czech Technical University in Prague, Klokner Institute, Prague, Czech Republic 2 RISE CBI Cement and Concrete Research Institute, RISE Research Institutes of Sweden, RISE Built Environment, Borås, Sweden 3 Instituto de Ciencias de la Construcción Eduardo Torroja, Madrid, Spain Correspondence Miroslav Sykora, Department of Structural Reliability, Czech Technical University in Prague, Klokner Institute, Solinova 7, 16608, Prague 6, Czech Republic. miroslav.sykora@cvut.cz Funding information Spanish Ministry of Economy and Competitiveness, Grant/Award number: IPT ; Grantová Agentura České Republiky, Grant/Award number: S Load bearing capacity can be predicted by appropriate modeling of material properties, geometry variables, and uncertainties associated with an applied model for the failure mechanism under consideration. The submitted study investigates shear resistance model uncertainties for reinforced concrete beams with and without shear reinforcement, considering large test databases and various levels of approximation offered by fib Model Code Model uncertainty is treated as a random variable and its characteristics are obtained by comparing test and model outcomes. The sensitivity of model uncertainty with respect to basic variables is analyzed. For beams with stirrups, Level III is recommended for practical applications. Its predictions are shown to be independent of the amount of shear reinforcement and have reasonable bias and dispersion around test results. For beams without shear reinforcement, the use of Level II is advisable and a distinction between lightly reinforced and moderately to heavily reinforced beams should be made. KEYWORDS fib model code 2010, model uncertainty, partial factor, reinforced concrete, shear reinforcement, shear resistance 1 INTRODUCTION Adequate description of the uncertainties in resistance models was recognized as one of key issues in reliability investigations of new and existing reinforced concrete (RC) structures. 1 Particularly when description of load effects and structural responses is highly complex, and operational procedures are based on simplified approaches, model uncertainty may dominate structural reliability and its quantification and explicit inclusion in reliability analysis is required. 2 Such cases can be identified by the lack of consensus amongst experts; alternative models used in research studies and practical applications; and different levels of approximation (LoA) 3 introduced in design codes. This is the case with shear resistances of RC structures, recognizing that shear strengths predicted by different design codes for a Discussion on this paper must be submitted within two months of the print publication. The discussion will then be published in print, along with the authors' closure, if any, approximately nine months after the print publication. particular beam or section can vary by factors of more than two. 4 Shear resistances of beams have stimulated an extensive debate over recent decades, as there is not yet an international consensus on which variables are important, 5 which mechanisms govern 6,7 and how to codify existing understanding. 8 Features of several models for shear resistances of beams with and without special shear reinforcement (hereafter referred to as shear models for brevity) were investigated and improvements proposed The complexity of shear transfer actions and modeling was recognized and described in detail in References 16,17. Model uncertainty as an indicator of the difference between model and real structural resistances was explicitly addressed by Russo et al 10 who analyzed the uncertainty associated with the models in EN and American Concrete Institute (ACI) 318M Bentz et al 4 were focused on the ACI procedure and two approaches belonging to the Modified Compression Field Theory. The recent fib. International Federation for Structural Concrete wileyonlinelibrary.com/journal/suco Structural Concrete. 2018;19:

2 SYKORA ET AL. 285 study 1 discussed uncertainties in beam models and sectionoriented approaches according to EN ( EC2 ), covering non-deteriorated and corrosion-damaged beams. It was shown that for common structural members, models for resistances in bending or concentric axial compression are sufficiently accurate, with a mean of model uncertainty near to unity and a coefficient of variation ( CoV ) lower than 0.1. By contrast, shear models for beams with and without special shear reinforcement were found to suffer from many deficiencies. It was foreseen that great improvements could be achieved by using the shear models provided in the Model Code ( MC2010 ) and stimulating indications were obtained even for deep beams of large effective depths. 21 Sigristetal 8 briefly discussed model uncertainty for the shear resistance of beams according to MC2010. However, detailed analysis of model uncertainty and factors contributing to possible biases and scatter still appears to be needed. This is why uncertainties in the shear models provided by MC2010, based on simple engineering formulas beam models or section-oriented approaches are investigated here and their statistical characteristics are estimated on the basis of available experimental data. Beams with and without special shear reinforcement (hereafter shear reinforcement for brevity) are discussed. As shear reinforcement is commonly provided by vertical stirrups, 10 only this type of shear reinforcement is taken into account. It is noted that several recent studies discussed uncertainties in resistance models of concrete structural members for more complex failure modes, for instance: Rectangular columns subjected to axial force, bending moment, and shear force 22 Shear strength of slender RC beams 23 Corroded shear-critical beams 1,24,25 Slabs with plain bars and slabs on elastomeric bearings exposed to shear 26 Beams and columns under combination of axial and shear forces and bending moments analyzed by advanced numerical models Beams with corroded reinforcing bars exposed to bending. 32 These studies provided valuable information on model uncertainty for more complex situations. However, it is deemed essential to properly analyze features of the individual contributing failure modes before approaching combined failure modes or the effects of corrosion. 2 UNCERTAINTY IN RESISTANCE MODELS According to the Probabilistic Model Code of the Joint Committee on Structural Safety JCSS, 33 model uncertainty should describe random effects neglected in the model and simplifications in mathematical relations on which the model is based. Bulleit 34 viewed model uncertainty as a measure of how the model fits to test data and how the model predicts structural behaviour ; see also Reference 35. Based on these approaches, the concept of model uncertainty proposed in References 1,2 is adopted here. The uncertainties in resistance models are obtained from comparisons of physical tests and model results. Real structurespecific conditions need then to be taken into account when they significantly deviate from test conditions; scale effects are often important for shear tests. A general framework for the uncertainty assessment of shear models, with examples of influences affecting test and model results, is given in Figure 1. Computational options appear to be irrelevant in this study, since simple structural members and simple shear models are considered and all input data are available in test databases. Note that for advanced numerical models, more detailed classification of model results was proposed in Reference 36. The methodology for the treatment of test uncertainty in the model uncertainty assessment was proposed in Reference 2. In this study, the effect of test uncertainty is disregarded in the analysis of model uncertainty datasets. This is based on the results of the previous studies: The effects of test uncertainty and real boundary conditions were investigated in bulletin 224 of CEB (Comité Euro-International du Béton). 37 Test uncertainty was assessed by numerical simulations a mean around unity and CoV of 0.05 were found to characterize uncertainty of the complex test for buckling columns. For less demanding shear tests, a CoV less than 0.05 is expected. Assuming the CoV of test uncertainty 0.05, the study focused on the resistance models 1 concluded that test uncertainty could be ignored in the model uncertainty assessment when the CoV of model uncertainty exceeds 0.1. This is the case with the shear models under investigation as the CoV of their model uncertainty exceeds 0.1. If needed, appropriate modifications of the model uncertainty, such as increasing variability and/or adjustments of the mean value should be accepted to reflect the real conditions of a structure (Figure 1). In most cases expert judgements are essential and a general quantification of the effect of structure-specific conditions is hardly possible. Care should be taken to avoid dual considerations of some of the effects given in Figure 1. For instance, quality control can be reflected separately from model uncertainty; see EN Detailed discussion on structure-specific conditions is beyond the scope of this study. The model uncertainty θ is here treated as a random variable. The multiplicative relationship for θ can be assumed 33 :

3 286 SYKORA ET AL. FIGURE 1 General framework of the model uncertainty assessment and examples for analytical shear models (adapted from References 1,2) RðX,YÞ= θðx,yþr model ðxþ ð1þ where R = response of a structure real resistance obtained from test results; R model = model resistance estimate of the resistance based on a model; X = vector of basic (random) variables X i included in the model; and Y = vector of variables neglected in the model, but possibly affecting the resistance. Longitudinal reinforcement, aggregate size, and modulus of elasticity are examples of variables Y for some shear models. In addition to X and Y variables, it is beneficial to define test parameters, that is, variables observed during testing. Obviously the test parameters should ideally include all X and Y variables, but may also cover other variables that, for instance, affect outcomes of a test procedure. As an example, the ambient air humidity hardly affects the real compressive strength of concrete, but may affect the accuracy of sensors (uncertainty of test method according to Figure 1). In this study the model uncertainty is assessed using the following procedure 2 : 1. Compilation of a database of model uncertainty observations: Any design bias is excluded from the calculation of R model, for instance, real concrete strengths instead of characteristic values are applied in shear models in the model uncertainty assessment. Ranges of test parameters such as material strength classes and geometry including the amount of steel are made available to represent the sample space of experimental observations for which model uncertainty is investigated. When the theoretical failure mode predicted by the model differs from test failure mechanism, the test result should be eliminated from the database. However, further investigations may be useful. As an example, consider a specimen assumed to fail in shear by yielding stirrups, that is, with a lower model shear force which can be sustained by the yielding shear reinforcement than the maximum shear force limited by the crushing of compression struts. When the specimen fails due to crushing, this can be indicative of either inadequate execution of the specimen, overestimated model capacity with respect to crushing, or an overly conservative model for shear reinforcement resistance. Consequently, more detailed analysis focused on both models involved is inevitable. Note that it may be difficult to identify a real experimental failure mode, for example, in the case of interacting shear force and bending moment. 2. Statistical assessment of the dataset, including tests of unbiased sampling, outliers, and goodness of fit of the probability distribution; in this study Grubb s test of outliers is performed to identify test results possibly affected by errors, incorrect records, etc. considering a significance level of Despite the fact that this test is based on the assumption of normality, it is deemed to be reasonably accurate in model uncertainty assessments. When a lognormal distribution is accepted, model uncertainty values can be transformed to a normally distributed variable.

4 SYKORA ET AL Suitable probabilistic description of the model uncertainty; lognormal distribution with the origin at zero is commonly an appropriate probabilistic model. 2,33 When generalizing the model uncertainty beyond the scope of the database, trends in its mean and dispersion characteristics should be carefully considered. Extrapolation with respect to basic variables for which significant trends are observed may be dubious. In the extrapolation care should be taken not to move to another failure mode, material model or conditions; for example, from ordinary- to high-strength concrete or shear of ordinary to high beams with a possible arch action. When using large databases, the values of some variables, such as inner dimensions or material properties difficult to obtain experimentally, are often unknown. Mean values from several tests or even nominal values are then commonly used for the purpose of comparison, and thus the model outcomes may become biased. This influence may be particularly important for small-scale tests, as is often the case with shear experiments. 3 UNCERTAINTY IN SHEAR MODELS BEAMS WITH STIRRUPS 3.1 Shear models under consideration Three LoA are accepted for shear resistance of RC beams according to MC2010 (hereafter e.g., Level I, Level II, and Level III ). With an increasing level more input data are required, evaluation becomes slightly more complex and the accuracy is expected to increase. However, all the shear models are based on analytical relationships that are essentially easy to compute. Considering no axial compressive force and f c in MPa, resistances according to Levels I and II are obtained as follows: n h i R model ðxþ= min max k v min 8, f c b w z;ρ w b w zf yw cotξ, where: h i k ε min 1, ð30=f c Þ 1=3 f c b w zsinξcosξg ð2þ Level I (z in mm): k v = 180/(1, z); k ε = 0.55; ξ is confined to <30,45 >; Levels II and III: k v = max[0.4(1 V E /V R,max )/(1 + 1,500ε x );0], where V R,max (ξ) =k ε min[1, (30/f c ) 1/3 ] f c b w z sin ξ cos ξ; k ε = min[1/( (ε x +(ε x ) cot 2 ξ)), 0.65]; ξ is bounded as < ,000ε x,45 >. with ε x =(M E /z + V E )/(2E s ρ l b w d) where M E = V E z and z = 0.9d (d in mm). For Level III the shear resistance is estimated as: For V R V R,max (ξ min ) Level II should be used; For V R < V R,max (ξ min ): 1 R model ðxþ= ρ w b w zf yw cotξ + k v min 8;f =2 c b w z ð3þ where ξ is same as in Level II and ξ min denotes the lower limit for ξ. Notation of the basic variables affecting the shear resistance and input data for all models are provided in Table 1; Table 2 provides notation and ranges of auxiliary variables derived from the variables included in the database. Variables ε x, k v, k ε, ξ and limit shear resistance V R,max are determined iteratively for varying shear force at failure in order to find a value of V E that maximizes shear resistance V R. The strain in the core layer ε x is calculated at the cross section located at the distance d from the face of supports according to Section of MC2010. The symbol ξ for the angle between concrete compression struts and the main tension chord is introduced here instead of θ (as used in MC2010 and EC2) to avoid confusion with the symbol for the model uncertainty. In addition, a deterministic value is considered for the modulus of elasticity of reinforcement, E s = 200 GPa, in Levels II and III. Insignificant influence of its small variability for ordinary steels on obtained shear capacity justifies this simplification. 3.2 Test results A database of 459 tests of beams with stirrups 17,23,40 is used here to assess the uncertainty in the MC2010 models. Ranges of material and geometrical characteristics of tested beams are given in Table 1. The database covers a wide range of beams with low to high concrete strengths, shear reinforcement ratio and effective depths. Beams with light, moderate and heavy shear reinforcement are included. Grubb s test of outliers reveals two outlying observations for Levels I and III and three for Level II. However, with the test information available it is not possible to reveal the reasons for such extreme observations and these are not eliminated from the database. It is numerically verified that due to the large size of the database, the elimination of these observations has a negligible effect on mean and CoV of model uncertainty. The sensitivity of skewness is briefly discussed below. 3.3 Model uncertainty characteristics For each experiment the model resistances are assessed from Equations (2) and (3); the model uncertainty is evaluated from Equation (1). The sample moment characteristics of model uncertainty (mean μ, CoV V and skewness ω) for the whole database are given in Table 3. It appears that

5 288 SYKORA ET AL. TABLE 1 Notation and ranges of basic variables included in the database for beams with stirrups Symbol Description Min. Max. Applied in level a/d ( ) Shear span-to-depth ratio b w (mm) Smallest width of a cross section in the tensile area I III d (mm) Effective depth 95 1,890 I III f c (MPa) Concrete compressive strength I III f yw (MPa)* Yield strength of stirrups I-III s (mm)* Stirrup spacing V fail (kn) Shear force at failure ,185 - ρ 1 = A sl /(b w d) (%) Longitudinal reinforcement ratio (A sl area of longitudinal II, III reinforcement) ρ w = A sw /(b w s); in % *,** Shear reinforcement ratio (A sw area of shear reinforcement) I-III ρ w f yw (MPa)* Strength of shear reinforcement I-III *Parameters available only for the tests considered in References 17,40. **The MC2010 criterion for minimum shear reinforcement is not met for about 25% of beams see section 3.4. TABLE 2 Notation and ranges of auxiliary variables derived from the variables included in the database for beams with stirrups Level I II III Symbol Description Min. Max. Min. Max. Min. Max. ε x ( 0.001) Strain in the core layer k v ( ) Strength reduction factor for concrete cracked in shear k ε ( ) Strength reduction factor for concrete cracked in compression ξ ( ) Angle between concrete compression struts and the main tension chord Level III is the most appropriate model the mean of the model uncertainty is close to unity and the CoV is the smallest. For all three levels, the values of skewness are significantly affected by eliminating outliers. A comparison of the V- and ω-values that is often informative regarding the selection of an appropriate probabilistic distribution 41 fails to provide unambiguous indications in this case. The question of an appropriate probabilistic distribution is left open for further studies, as the detailed analysis is beyond the scope of this contribution which is focused on key issues of biases and scatters in model predictions. A simple sensitivity analysis 40 is conducted to verify the influence of the basic variables on model uncertainty. Trends in θ with a basic variable are assessed using the correlation coefficient r i (correlation between θ and X i ). Medium correlations between θ and ρ w or ρ w f yw (r 0.5 to 0.3) and weak correlations for the other shear parameters ( r < 0.3) are observed for Levels I and II. The influence of ρ w or ρ w f yw on θ is considerably reduced ( r 0.1) for Level III which is its key improvement. Medium correlations θ d and θ s are observed (r 0.4 to 0.3); detailed results of correlation analysis were provided in Reference 42. Figure 2 depicts variation of the model uncertainty with the strength of shear reinforcement, its linear trend and bounds adopted here for lightly, moderately, and heavily reinforced beams. Sample characteristics of θ for light to heavy reinforced beams are provided in Table 3. It follows that the mean of model uncertainty μ depends on the strength of shear reinforcement for Levels I and II, while the effect on CoV is TABLE 3 Sample characteristics of model uncertainty beams with stirrups Level of approximation I II III Description of sample n μ V ω μ V ω μ V ω Whole database (0.28) (0.32) (0.39) Lightly reinforced beams (ρ w f yw 1 MPa) (0.18) (0.12) (0.31) Moderately reinforced beams (1 MPa < ρ w f yw 2 MPa) Heavily reinforced beams (2 MPa < ρ w f yw ) The values in underlined are indicative only, since the database contains beams not complying with the MC2010 criterion for minimum shear reinforcement see section 3.4. The values of coefficient of skewness in brackets are the estimates obtained for samples without outlying observations.

6 SYKORA ET AL. 289 less significant for all the Levels. For Levels I and II, μ clearly decreases with increasing ρ w f yw and its differentiation with respect to this parameter is advised. Based on the results given in Table 3, μ 1.14 and V 0.23 may be accepted for Level III. For Levels I and II both the characteristics μ and V are larger. To summarize, it is recommended that preference be given to a model with the highest accuracy if data required for analysis are available. For most practical cases, the additional data for Levels II and III E s and ρ l are known. Computational demands for all the considered models are similar Levels I to III are based on analytical relationships which are easy to evaluate. Consequently, it is recommended that the Level III based on the sum of stirrups and concrete contributions model be used. 3.4 The effect of the criterion on minimum shear reinforcement All the results presented in section 3.3 are based on a database that includes 118 lightly reinforced beams that fail to comply with the MC2010 criterion assessed for each tested beam as: ρ w, min =0:08 f c=f yw ð4þ Figure 2 shows the histogram of ρ w,min f yw -values affected by different values of concrete strengths. Firstly, model uncertainty characteristics for the subdatabase of lightly reinforced beams (n = 318) are analyzed. It appears that the exclusion of the 118 beams have insignificant effect on the characteristics provided in Table 3 for Levels I III (change lower than 5%). This suggests that criterion (4) should be carefully verified and a lower ρ w,min -value could be proposed based on a more detailed analysis of a larger number of tests of lightly reinforced beams. It is noted that the exclusion of the 118 beams affects the model uncertainty characteristics for the whole database and Level I, which poorly covers the effect of changing reinforcement ratio. This can be judged already from the mean values in Table 3 that are, for Level I, sensitive to the amount of shear reinforcement. Obviously, the exclusion of the beams decreases the relative importance of the results for lightly reinforced beams. Level III captures well the effect of changing the reinforcement ratio and the exclusion has negligible effect on θ-characteristics. For Level II also this effect is of no practical relevance. Similar to the considerations related to ρ w,min, it might be argued that some other basic variables range out of the usual or recommended limits (Table 1) and that relevant tests should be omitted from the analysis. For instance the database contains 80 specimens with ρ l > 0.04 (considering the EC2 limit) and 9 beams with f c < 20 MPa. Additional numerical verifications confirm that elimination of these test results have negligible effect on the model uncertainty characteristics provided in Table 3. 4 UNCERTAINTY IN SHEAR MODELS BEAMS WITHOUT SHEAR REINFORCEMENT 4.1 Shear models under consideration MC2010 provides two LoA to estimate shear resistances for beams without shear reinforcement. Considering no axial compressive force and f c in MPa, shear resistance is estimated according to Levels I and II as follows: R model ðxþ= min f c,8 k v b w z ð5þ The levels differ in evaluation of the strength reduction factor k v (z and d g in mm): Level I: k v = 180/(1, z), Level II: k v = 0.4/(1 + 1,500ε x ) 1,300/(1,000 + k dg z). where k dg = 32/(16 + d g ) Notation of the basic variables affecting the shear resistance and input data for all the models are provided in Table 4; Table 5 provides notation and ranges of auxiliary variables derived from the variables included in the database. The auxiliary variables ε x, k v, and V E are determined iteratively as in the section for beams with stirrups. In addition the modulus of elasticity of steel reinforcement E s = 200 GPa is applied in Level II. FIGURE 2 Variation of θ with ρ w f yw for the whole database and Level III 4.2 Test results A database of 184 shear tests of beams without shear reinforcement was compiled at the University of Stellenbosch. 17,40 The ranges of the test parameters are reported in Table 4. The database covers a wide range of beams with low to medium concrete compressive strengths; and small,

7 290 SYKORA ET AL. TABLE 4 Notation and ranges of variables included in the database for beams without shear reinforcement Symbol Description Min. Max. Applied in level a/d ( ) Shear span-to-depth ratio b w (mm) Smallest width of a cross section in the tensile area 100 1,000 I, II d (mm) Effective depth I, II d g (mm) Maximum size of the aggregate II f c (MPa) Concrete compressive strength I, II f y (MPa) Strength of longitudinal reinforcement 276 1,779 V fail (kn) Shear force at failure ρ 1 = A sl /(b w d); in % Longitudinal reinforcement ratio (A sl area of longitudinal reinforcement) ρ 1 f y Strength of longitudinal reinforcement TABLE 5 Notation and ranges of auxiliary variables derived from the variables included in the database for beams without shear reinforcement Level I Level II Symbol Description Min. Max. Min. Max. ε x ( 10 3 ) Strain in the core layer k v ( ) Strength reduction factor for concrete cracked in shear CoV of model uncertainty while skewness is significantly affected (Table 6). As in the case with the beams with stirrups, some basic variables range out of the common limits (Table 4); for instance, 15 beams with ρ l > 4% are included in the database. It can be shown that elimination of these test results have negligible effect on the model uncertainty characteristics provided in Table 6. ordinary, and large effective depths. Lightly, moderately, and heavily reinforced beams are included. Whilst no limits regarding basic variables are provided for Level II, Level I is intended to be used when the following conditions are satisfied: f yk 600 MPa, f ck 70 MPa, d g 10 mm. Thirty-four samples failed to meet these conditions and were excluded from the database for Level I. However, this elimination has insignificant influence on μ and V. As is similar with the case of beams with stirrups, Grubb s test identifies an outlier for Level II, but the available test information does not make it possible to reveal reasons for the extreme observation, and this is not eliminated from the database. Again, additional numerical verification reveals that such elimination marginally affects mean and 4.3 Model uncertainty characteristics For each experiment, the model resistance is assessed from Equation (5) and the model uncertainty is evaluated from Equation (1). Sample characteristics of θ for the whole database are given in Table 6. It appears that Level II slightly overestimates shear strength, whilst Level I is overly conservative. The effects of basic variables on model uncertainty are determined by the same approach as in the case of beams with stirrups. Strong θ ρ l and θ ρ l f y correlations (r 0.7), medium θ b w and θ f y (r 0.5) and weak correlations for the other shear parameters ( r < 0.3) are observed for Level I. Weak correlations for all the shear parameters indicate that the Level II model adequately reflects changes in basic variables. Model uncertainty for Level I tends to increase with an increasing ρ 1 (Figure 3) and its differentiation with respect to this parameter is thus advisable. Sample characteristics of θ for light to heavy reinforced beams are provided in Table 6. Level I seems to provide TABLE 6 Sample characteristics of model uncertainty beams without shear reinforcement Level of approximation I II Description of sample n μ V ω n μ V ω Whole database (0.24) Beams with small amount of longitudinal reinforcement ρ l 1% Beams with medium ρ 1 1<ρ 1 2% Beams with high ρ (0.81) 2<ρ 1 The values of coefficient of skewness in brackets are the estimates obtained for samples without outlying observations.

8 SYKORA ET AL. 291 Differences in the databases used in the studies the model uncertainty characteristics obtained by Sigrist et al 8 would generally match those obtained in this study when lightly reinforced beams (Table 3) are disregarded in the present analysis. FIGURE 3 Variation of θ with ρ l for the whole database and Level I overly conservative estimates for moderately and heavily reinforced beams. The use of Level II is thus advisable; the distinction should be made between: Lightly reinforced beams for which shear resistance is systematically overestimated (μ 0.82, V 0.16). Moderately to heavily reinforced beams (μ 0.93, V 0.13). The coefficients of skewness around or below zero suggest that a normal distribution or distributions that can describe negative skewness be more appropriate rather than a conventional, positively skewed two-parameter lognormal distribution. This may have a significant bearing on structural reliability. As in the case of beams with stirrups, comparison of the V- and ω-values is non-informative, and a more detailed statistical analysis is needed to support the selection of an appropriate probabilistic distribution. 5 DISCUSSION 5.1 Other model uncertainty studies focused on MC2010 models Uncertainties related to the MC2010 models were briefly discussed in Reference 8. Analysis of a different database containing beams with stirrups, different geometry and exposed to normal force led to results slightly different from those presented here. For Levels I and II, Sigrist et al 8 obtained less conservative mean values (μ 1.5 and 1.35, respectively) and slightly smaller CoVs (V 0.2); for Level III larger mean (μ 1.2) and a smaller CoV (V 0.13) was reported without detailed analysis of the effects of basic variables. The differences particularly in the mean values obtained by Sigrist et al 8 and in the present study might be explained by: Inclusion of beams exposed to normal force, T- or I- beams and smooth bars that are excluded from the present study. Based on a large database of beams without shear reinforcement, Sigrist et al 8 derived μ 2.0 and V 0.2 for Level I and μ 1.15 and V 0.1 for Level II. The CoVs obtained by Sigrist et al 8 are in agreement with those reported in Table 6, but the mean values are larger by about 10 20%. While small discrepancies are always expected when different databases are utilized, the systematic difference between the obtained mean values needs further explanation. It is interesting to observe that for both Levels I and II, the difference between the mean values provided by Sigrist et al 8 and by Table 6 reduces to less than 9% when f ck (f c 8 MPa) is applied in Equation (5) instead of f c. With the reference to step 1 of the methodology in section 2 any design bias is excluded from the calculation of R model in the model uncertainty assessment, itis advocated here that model uncertainty characteristics should be investigated using mean values best (unbiased) estimates of contributing factors. Required reliability margin should then be achieved by appropriate combination of characteristic values and partial factors. This is in agreement with the findings obtained by Mancini et al 43 The comparison of results based on various databases is further complicated by the considerable sensitivity of Level II shear resistances to applied level loading through strain ε x. As an example consider the point loads applied to obtain the results in Table 6; these loads are deemed to correspond to mean resistances, replicating the tests in the database. However, at design phase the design values of resistance, lower than mean values by about 50% for CoV of 25% for shear resistance, are taken into account. The consideration of loads reduced by 50% increases Level II shear resistances by about 20% on average. Another study 44 confirmed the high bias and CoV for Level I and the acceptable scatter (V 0.15) for Level II. Again compared to the results in Table 6, a larger bias of Level II was indicated. 5.2 Uncertainties related to similar shear models Uncertainties associated with two similar shear models for beams with stirrups EC2 and the model proposed in Reference 45 were analyzed by Busse et al, 46 without providing detailed information about a test database. It can only be judged from the figures provided in the cited study that, regarding ρ w, the database was somewhat similar to that adopted here for beams with stirrups; that is, including most of the beams with a low shear reinforcement ratio (ρ w < 0.5%), some of them with 0.5% < ρ w < 1% and few with

9 292 SYKORA ET AL. ρ w > 1%. Busse et al 46 considered ρ w as the most important parameter instead of ρ w f yw. This makes a marginal difference, as these variables affect the uncertainty of the EC2 model in a similar way. 42 Busse et al obtained μ 1.35 and V 0.3 for the EC2 model; μ 1.12 and a very low V 0.07 were reported for the model in Reference 45. Level III, associated with a lower bias and with scatter, outperforms the EC2 model. This is deemed to be attributable to the facts that the Level III model considers shear strength by summing the contributions of concrete and stirrups and likely estimates the lower limit of the angle ξ in a more realistic way compared to the EC2 and Level I models. Recently, Yang et al 47 thoroughly analyzed shear resistances of beams without shear reinforcement and proposed improvements based on the critical shear displacement theory; the improved model surpassed the EC2 and MC2010 (probably Level II) models, having no bias and small scatter, V Another improvement with similar results was proposed by Mari et al. 14 However, the model uncertainty characteristics given in Table 6 suggest that Level II provides a good approximation for ordinary-strength concretes, with slight overestimation of shear resistance and reasonable scatter. Slobbe et al 44 obtained negligible bias and small CoV for the model adopted in Switzerland, based on the Critical Shear Crack Theory. 48 A substantial improvement seems to be achieved by the complex analytical model recently proposed by Mari et al, 49 applicable for beams with and without special shear reinforcement. Further studies should carefully verify the uncertainties of these models. 5.3 Implications for partial factor method verifications The model uncertainty factor for verifications using the partial factor method can be derived using the procedures provided in fib bulletin and related publications. 50,51 Assuming a lognormal distribution and unity characteristic (nominal) value of model uncertainty, the partial factor γ Rd can be obtained from the following relationship: γ Rd =1= ½μexpð α R βvþš ð6þ where α R = FORM sensitivity factor; and β = selected target reliability index Beams with stirrups Considering the model uncertainty characteristics given in Table 3 for Level III, the variation of model uncertainty factor with target reliability is displayed in Figure 4 for beams with stirrups and α R = 0.32, that is, non-dominant resistance variable according to ISO and fib bulletin The range of β represents the target lifetime levels commonly associated with the Ultimate Limit States for new and existing structures. 20,25,38,52 56 FIGURE 4 Variation of model uncertainty factor with target reliability for beams with stirrups, Level III and α R = 0.32 It follows from Figure 4 that the model uncertainty factor for Level III beams with stirrups attains values in a wide range from 1 to 1.3. For a common reference target level of β = 3.8, the model uncertainty factor varies between 1.05 and Considering the whole database, γ Rd = 1.16 is obtained from Equation (6); for a normal distribution the model uncertainty factor increases to It is interesting to note that regardless of the target reliability level, MC2010 indicates γ Rd = 1.08 for steel strength dominated failure modes such as yielding of stirrups in shear or yielding of longitudinal reinforcement in bending. Using the model uncertainty characteristics in Table 3, it can be easily demonstrated that the model uncertainty factor needs to be differentiated between Levels I and III. For instance, γ Rd 0.85 is obtained from Equation (6) for Level II, the whole database and β = Beams without shear reinforcement Considering the model uncertainty characteristics given in Table 6 for Level II, the variation of model uncertainty factor with target reliability is portrayed in Figure 5 for beams without shear reinforcement and α R = Figure 5 indicates that the model uncertainty factor for Level II beams without shear reinforcement should be larger than for Level III for beams with stirrups. For β = 3.8, the model uncertainty factor is around 1.25, except for lightly reinforced beams for which γ Rd The large γ Rd -values are attributable to the systematic overestimation of shear resistance at Level II as follows from the μ-values provided in Table 6. MC2010 indicates a low value, γ Rd = 1.1, for concrete strength dominated failure modes. Again, the distinction between model uncertainty factors for Levels I and II should be made using the model uncertainty characteristics in Table 6; for instance, γ Rd 0.7 is obtained for Level I, the whole database and β = 3.8.

10 SYKORA ET AL. 293 FIGURE 5 Variation of model uncertainty factor with target reliability for beams without shear reinforcement, Level II and α R = 0.32 It is noted that the model uncertainty factor is significantly affected by a selected α R -value. In the case of shear, model uncertainty may be dominant for, say, V > 0.15; see Reference 25 for detailed discussion. 6 CONCLUDING REMARKS 1. Quantification and explicit inclusion of resistance model uncertainty in reliability verification is essential when description of structural responses is excessively complex and operational procedures are based on simplifying approaches. This is the case with shear resistances of RC structures. 2. Model uncertainties should always be related to test uncertainties, the actual structural conditions and the computational model under consideration. However, it is argued that test uncertainty can be ignored in model uncertainty assessment for common cases of shear resistances. 3. Real resistance can often be expressed as a product of the model uncertainty and resistance obtained by the model. 4. Focusing on beams with stirrups and the three LoA offered by MC2010, it appears that: Model uncertainty is dominantly affected by the shear reinforcement ratio or the strength of shear reinforcement. For Levels I and II, medium correlation is observed, whilst this dependence vanishes at Level III, which is the key improvement for this level of approximation. The Level III model uncertainty can be characterized by mean μ 1.15 and CoV V 0.225; for the two lower levels both characteristics μ and V are larger as expected. In practical applications it is recommended that Level III be used, as data required for analysis are commonly available and no substantial computational demands are needed. It is important to emphasize that the main feature of Level III model is that it considers shear strength by summing the contributions of concrete and stirrups. This is judged to be the main reason why Level III outperforms Levels I and II as well as the EC2 model. It is also advised to carefully verify a limit on minimum shear reinforcement ratio, ρ w,min, using a large database of lightly reinforced beams; preliminary results suggest that shear resistance of beams with ρ w < ρ w,min can be predicted with similar accuracy as other lightly reinforced beams with ρ w > ρ w,min. 5. The analysis focused on beams without shear reinforcement and the two LoA provided in MC2010 reveals that: Level II: the distinction between lightly reinforced (a longitudinal reinforcement ratio lower than 1%) and moderately to heavily reinforced beams should be made. For the former, shear resistances tend to be systematically overestimated and such predictions should be treated with caution; for the latter, μ 0.93 and V can be taken into account as representative model uncertainty characteristics. Level I seems to provide excessively conservative estimates and the use of Level II is recommended, particularly for moderately to heavily reinforced beams. 6. The obtained model uncertainty characteristics can be directly used to establish partial factors for beams exposed to shear. However, uttermost care needs to be taken to adequately address the importance of model uncertainty and verify whether or not model uncertainty can be assumed to be a non-dominant resistance variable. Initial insights related to this issue were provided by fib Bulletin It is emphasized that all the presented results are based on comparisons of test results and shear resistance models from which any bias in values of basic variables is excluded, that is, no partial factors nor characteristic values are applied in shear resistance predictions. The selection of an appropriate probabilistic model for model uncertainty requires a more advanced statistical analysis and should be addressed within future activities. ACKNOWLEDGMENTS This study is an outcome of research project S, supported by the Czech Science Foundation, and of research project IPT , supported by the Spanish Ministry of Economy and Competitiveness.

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12 SYKORA ET AL Sykora M, Holicky M, Markova J. Verification of existing reinforced concrete bridges using the semi-probabilistic approach. Eng Struct. 2013;56: Caspeele R, Sykora M, Allaix DL, Steenbergen R. The design value method and adjusted partial factor approach for existing structures. Struct Eng Int. 2013;23(4): ISO 2394: General Principles on Reliability for Structures. 4th ed. Geneva, Switzerland: ISO; Steenbergen RDJM, Sykora M, Diamantidis D, et al. Economic and human safety reliability levels for existing structures. Struct Conc. 2015;16: Sykora M, Holicky M, Jung K, et al. Target reliability for existing structures considering economic and societal aspects. Struct Infrastruct Eng. 2016;13(1): Tanner P, Hingorani R. Acceptable risks to persons associated with building structures. Struct Concr. 2015;16(3): Bigaj-van Vliet A, Vrouwenvelder T. Reliability in the performance-based concept of fib model code Struct Concr. 2013;14(4): AUTHOR'S BIOGRAPHIES Miroslav Sykora (Lead author), Associate Professor Department of Structural Reliability Czech Technical University in Prague, Klokner Institute Solinova 7, 16608, Prague 6, Czech Republic Jan Krejsa, PhD Student Department of Structural Reliability Czech Technical University in Prague, Klokner Institute Solinova 7, Prague 6, Czech Republic Jan Mlcoch, PhD Student Department of Structural Reliability Czech Technical University in Prague, Klokner Institute Solinova 7, Prague 6, Czech Republic. Miguel Prieto, PhD, Researcher RISE Research Institutes of Sweden, RISE Built Environment, RISE CBI Cement and Concrete Research Institute, Brinellgatan 4, SE , Borås, Sweden Peter Tanner, PhD, Researcher Instituto de Ciencias de la Construcción Eduardo Torroja Serrano Galvache 4, Madrid, Spain How to cite this article: Sykora M, Krejsa J, Mlcoch J, Prieto M, Tanner P. Uncertainty in shear resistance models of reinforced concrete beams according to fib MC2010. Structural Concrete. 2018; 19: