My title https://doi.org/10.14311/APP.2022.36.0127 Acta Polytechnica CTU Proceedings 36:127–134, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague ESTABLISHING MODEL UNCERTAINTY OF SLS REINFORCED CONCRETE CRACK MODELS APPLIED TO LOAD-INDUCED CRACKING Christina Helen McLeod University of KwaZulu-Natal, School of Engineering, Discipline of Civil Engineering, Mazisi Kunene Road, Glenwood, Durban 4041, South Africa correspondence: mcleodc@ukzn.ac.za Abstract. Although some research has been performed, serviceability limit state (SLS) concrete crack models have yet to be calibrated fully in probabilistic terms in structural design standards. This is partly due to the fact that the SLS is generally not the critical limit state in structural design. However, in some specialist structures such as water retaining structures, the SLS such as cracking is the limiting design criterion, specifically the crack width required to control leakage and thus requires a proper probabilistic analysis. In probabilistic crack models, the reliability of the crack model is determined by the performance function whereby the design limiting crack width is greater than the estimated crack width calculated using the appropriate design crack model. As cracking in concrete is a random mechanism with a high degree of variability and crack models tend to be at least semi-empirical with inherent uncertainty, model uncertainty of the crack model is significant and is applied in the reliability model as a random variable. A database was established of both short and long term cracking experimental data for the tension load case to quantify model uncertainty. However, the data for long term cracking is limited which meant that the model uncertainty in this case was not definitively established. This paper discusses the determination of model uncertainty for tension cracking, which could be extrapolated to other models where data is limited. Keywords: concrete crack models, model uncertainty, reliability. 1. Introduction Serviceability limit state (SLS) concrete crack models are treated nominally in probabilistic terms in struc- tural design standards. This is partly due to the fact that SLSs are generally not the critical limit states. However, in some specialist structures such as water retaining structures, SLS cracking is the limiting de- sign criterion, specifically the crack width required to control leakage. In addition, crack models are mostly empirically or semi-empirically derived so have not been assessed to any extent in reliability terms. How- ever, cracking in concrete is a random mechanism with a high degree of variability indicative of a significant level of model uncertainty which, in turn, influences the level of treatment of model uncertainty in prob- abilistic analyses. Retief (2015) [1] and Holický et al (2016) [2] outlined a framework to assess model uncertainty. The relative influence of model uncer- tainty can be divided into model uncertainty classes [1] classified by an increasing level of treatment, as having Nominal effect, Significant effect or Dominat- ing effect, with, respectively. When model uncertainty is at least significant, it is deemed to be justified to treat model uncertainty as a random variable in the probabilistic model. This contribution discusses the quantification of model uncertainty for short- and long-term tension cracking, which could be extrapo- lated to other models where data is limited and where SLS governs. The selection of the theoretical model for the general probabilistic model (GPM) is also im- portant as model uncertainty is specific to the chosen model. Load-induced tension cracking is controlled in structural concrete design standards by limiting the crack width for a given loading, section and reinforce- ment configuration. The maximum crack width is considered at the tension surface of the cross section. In probabilistic assessments of SLS crack models, the reliability of the crack model is determined by the performance function g, expressed as g = wlim − θ. wpredict where wlim is the design limiting crack width, θ is the model uncertainty as a random variable and wpredict is the estimated crack width calculated using the appropriate design crack prediction model. As model uncertainty is specific to the prediction model concerned, suitable load-induced crack mod- els were investigated, applicable to structures where serviceability cracking governs the design, such as re- inforced concrete water retaining structures (WRS). The South African industry utilises the now with- drawn British standard BS 8007 (1987) [3] and is at present updating the standard SANS 10100-1 (2004) [4] for the design of reinforced concrete structures by adopting the corresponding Eurocode [5], concurrently with the development of SANS 10100-3 (Draft) (2015) [6] for the design of WRS. McLeod (2013) [7] in a deterministic analysis of typical WRS wall configu- rations, showed that the Eurocode crack model was 127 https://doi.org/10.14311/APP.2022.36.0127 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Christina Helen McLeod Acta Polytechnica CTU Proceedings conservative considering the reinforcement ratio com- pared to BS 8007’ especially for the tension loading case (> 38 % increase), for a limiting crack width of 0,2 mm. This has significant economic consequences if EN 1992 adopted in South Africa. The release of the fib Model Code MC 2010 [8] included an update of the fundamental crack model of fib MC 90 on which the Eurocode crack model is based and allows for long term effects. The load-induced crack models for pure tension of BS 8007 (1987) [3], BS EN 1992-1 (2004) [5], fib MC 2010 [8] were thus considered and compared to the results obtained for flexural cracking, as reported in [9] and [10]. The results of this research are also of relevance to researchers developing Eurocode-related crack models and are applicable to structures other than WRS where serviceability cracking has a signif- icant level of importance in the design thereof, such as bridges. The results of this research will also be utilised in the probabilistic assessment of SLS cracking in existing structures such as WRS and bridges. 2. Model Uncertainty Treatment Uncertainty exists in all models and requires quan- tification as far as possible for proper reliability as- sessment and may be divided into two main types of uncertainty [1]. Inherent random variability (aleatory) exists in model parameters such as material properties and loads. As these parameters are treated as random variables in the reliability model, this inherent vari- ability is measured by their coefficients of variation. Epistemic uncertainty (that due to incomplete knowl- edge and statistical uncertainty) may be measured by a model factor treated as a random variable in the reliability model. This model factor is defined as the ratio of measured to predicted values, in this case, crack widths. This ratio, or model factor (MF), is then quantified in terms of its statistical parameters of mean, coefficient of variation (CoV) and probability distribution function (PDF). Model uncertainty is a measure of the performance of the prediction model where the mean of model factors indicates bias in the model. Thus, a MF mean of 1 indicates an unbiased model. A MF mean greater than 1 represents under- prediction; conversely, a mean less than 1 indicates an overprediction by the model. The variation of model factors is indicative of the level of uncertainty in the model - a higher variation signifying a higher uncer- tainty. For a model to be suitable over a reasonable range of design applications, it should behave consis- tently in terms of model factor bias and uncertainty over the range of application. Interdependence be- tween model uncertainty and parameters results in drift in the MF mean and an inconsistent model. With a large degree of randomness in concrete crack mechanisms and multiple influencing variables, crack models have high degree of variability (CoV well in excess of 0,1 which is the reference value for general structures). Model uncertainty (θ) is thus a significant variable in the reliability crack model, as shown by McLeod (2019) [9], requiring full probabilistic treat- ment. Some research (e.g. [11]) has been done in establishing model uncertainty as a model factor but mostly for flexural short term cracking. The mean and CoV of parameters were reported but not the PDF of model uncertainty necessary for reliability assessment. 3. Deterministic analysis of predicted crack width In determining the predicted crack width for the MF, typical configurations of WRS and thus critical load cases for cracking were investigated. Load-induced pure tension cracking occurs in the wall of circular water retaining structures (WRS) due to hoop stresses in the horizontal plane induced by water load, which is considered as a quasi-permanent load by both Eu- rocode and SANS design standards. The main rein- forcement is thus placed in the horizontal direction with any tension cracks running perpendicular to this reinforcement. As tension cracks in this case tend to be through the full cross section, this is a critical load case for water tightness. A brief summary of the chosen design standard for- mulations used to calculate the predicted crack width for the critical pure tension load case in a WRS follows. A more comprehensive discussion may be found in [9]. Crack mechanism philosophies differ in the mod- elling of the development and transfer of the stresses between the steel and concrete around a crack in the section, and the resulting strain incompatibility. As cracking is a serviceability limit state, linear elastic stress-strain theory applies. The predicted maximum crack width corresponds to a 5 % probability of ex- ceedance by most design code formulations [9],[10]. This may not probabilistically match the measured maximum crack width, and therefore contributes to model uncertainty. 3.1. BS 8007 crack model The BS 8007 crack model [3] is a no bond-slip empirical model. The maximum surface crack width, w, for tension is calculated using: w = 3 acr εm (1) Crack spacing is assumed to be a function of acr , as the distance from the crack considered to the nearest longitudinal reinforcing bar. The mean strain at the surface, εm, is calculated from: εm = ε1 − ε2 (2) where ε1 is apparent steel strain (εs) at the surface and ε2 is the concrete tension-stiffening effect. The equations to calculate tension stiffening strain for tension were derived empirically, calibrated to the specified maximum crack width limit (wlim) as follows: 128 vol. 36/2022 SLS reinforced concrete crack models uncertainty ε2 = 2 bt h 3 EsAs f or wlim = 0, 2 mm (3) ε2 = bt h Es As f or wlim = 0, 1 mm (4) Es is the steel modulus of elasticity, h is section depth, bt is the width of the section in tension and As is the area of the tension reinforcement. Interpolation for other crack widths is not possible which limits the application of this crack model. 3.2. EN 1992 crack model The EN 1992 crack width design equation [5] for pure tension cracking is a semi-analytical bond-slip model, developed from the compatibility relationship for cracking and assuming the stabilised cracking phase has been reached, namely: wm = Srm · εm (5) where wm is the mean crack width, Srm is average crack spacing and εm the mean strain. This crack width prediction model is based on the more funda- mental model of fib MC 90. The mean strain is: εm = εsm − εcm (6) where εsm is the mean strain in the reinforcement under loading calculated using linear elastic theory. The mean concrete strain, εcm, also known as tension stiffening, is calculated using: εcm = kt fct,ef f ρct,ef f (1 − αe ρpef f ) / Es (7) where αe is the modular ratio Es/Ec, ρp,ef f is the effective reinforcement ratio between the reinforce- ment area and the effective area of concrete in tension, fct,ef f is the mean tensile strength of the concrete at the time of cracking, Ec is the concrete modulus at the time of cracking, and kt is a factor dependent on the duration of load. A minimum limit of 0, 6 σs/Es (where σs is the steel stress) is placed on the mean strain. The design (or "maximum") crack width is required rather than the mean width. This maximum crack width is related to the mean crack spacing by the equation: wk = (βw Srm) εm (8) where (βw Srm) is the maximum crack spacing (Sr,max) and the factor βw (the ratio of Sr,max to Srm) has a value of 1,7 corresponding to 1, 64× stan- dard deviations from the mean (normal distribution of crack widths). The EN 1992 maximum crack spacing is: Sr,max = k3 c + k1 k2 k4 φ/ρp,ef f (9) where φ is the bar diameter, c is the concrete cover to the longitudinal reinforcement and k1 is a coeffi- cient taking into account of the reinforcement bond properties, having a value of 0,8. The values of the factors k3 and k4 are determined by European Union individual member countries’ National Annexes, with recommended values of 3,4 and 0,425 given for k3 and k4, respectively. The factor k2 allows for stress distribution and has a value of 1 for pure tension load- induced cracking. All k-values are empirical factors, therefore would contribute to model uncertainty. 3.3. MC 2010 crack model The fib MC 2010 crack model [8] is an update of the fib MC 1990 model. The design crack width, wd, as a maximum (assumed to be the 95th percentile, as EN 1992) is determined from: wd = 2 ls,max(εsm − εcm − η εsh) (10) where ls,max is the transfer length over which slip occurs (equal to half the crack spacing) and εsh is the free shrinkage strain over time. The factor η is zero for short-term cracking, and 1,0 for long-term cracking. The minimum limit on mean strain of EN 1992 is also specified by MC 2010. The transfer length is determined using: ls,max = k · c + 0.25 fctm τcms · φs ρp,ef f (11) where k is an empirical parameter to account for the influence of the concrete cover (k = 1, 0 can be assumed), τcms is the mean bond strength between steel and concrete (considered to be evenly distributed between two cracks) and φs is the nominal diameter of reinforcing bars. The ratio between the concrete tensile strength and mean bond strength (fctm/τcms) is 1/1,8 for stabilised cracking for both short and long-term loads. MC 2010 allows for crack width determination over both the crack formation stage and the stabilised cracking phase. The MC 2010 crack model is valid for c ≤ 75 mm. 4. Quantification of crack model uncertainty The model factor for the chosen concrete crack models was defined as the ratio between measured crack width and predicted crack width, wexp/wpred. A database of experimental values for load-induced tension cracking was thus compiled to establish the stochastic parame- ters of model uncertainty for the crack models of EN 1992-1-1, MC 2010 and BS 8007. The database was that from McLeod (2019) [9] updated to include more recent testing by Gribniak et al (2020) [13]. The statistical parameters and probability distribu- tion necessary for probabilistic analyses were deter- mined for the model factors of each crack model using standard statistical test methods to a 95 % confidence level. The same procedure of analysis was followed 129 Christina Helen McLeod Acta Polytechnica CTU Proceedings Researcher Element Type TestDuration No. of samples** Farra & Jaccoud [12] Ties with single reinforcing bar, square cross section. Short-term 71 Gribniak [13] Ties with 4 no. reinforcing bars, cover varied. 10 samples total, repeat samples. Short-term 4 Hartl (1977), UPM data [14], [15] Ties with single or 2 No. reinforcing bars, squarecross section. Short-term 48 Hwang [16] Slab elements reinforced in both directions, axial tension in one direction. Variation of cover and rein- forcement Short-term 34 Wu [17] Ties with single reinforcing bar, square cross section. Short (7) &long-term (4) 7 + 4 Eckfeldt [15] Ties with 1or 2 No. reinforcing bars, square cross section. Repeats 2 × 4 No. ties & 3 × 1 ties Long-term 11 *Final load steps considered only, EN 1992 minimum strain complied with. Table 1. Sources of experimental data - direct tension load-induced cracking. as detailed in [9] and [10]. Non-parametric normality tests such as Kolmogorov-Smirnov (using Lilliefors sig- nificance correction) and Shapiro-Wilks (corrected), as appropriate, were performed to a significance p of 0,05 to establish the estimated probability distribu- tion of the model factor. Graphical methods such as probability plots and box plots were used to validate the estimated probability distribution. Through the statistical analysis of model factors, a first assessment of the crack models’ performances was done. Pearson’s correlations were used to investigate the relationship between significant parameters and model uncertainty, and thus assess the consistency of each crack model. 4.1. Experimental database With data for existing WRS lacking and challenging to obtain, experimental data was used to establish model uncertainty of the crack models of EN 1992, BS8007 and MC2010. Both short- and long-term data was considered, given the quasi-permanent nature of the water load in WRS. However, the data for long term tension cracking [9] is limited. Sources for the database are summarised in Table 1. Given that the first filling of a WRS only occurs once concrete has reached at least its 28-day strength and considering the quasi-permanent nature of the wa- ter load, only data from the stabilised cracking stage was considered. Where steel stress is small, crack widths may be underpredicted resulting in an overes- timation of model uncertainty, unduly increasing the upper tail of the model uncertainty distribution. In addition, the applied loads and section geometry of water retaining structures typically result in a mean strain well above the minimum strain limit. Therefore, data was selected where the calculated mean strain was at least the specified minimum limit of 0, 6σs of EN 1992 and MC 2010 where σs is the steel stress determined using linear elastic theory. Crack widths measured on the final load step only were considered to ensure independence of samples. Results for any re- peat samples were averaged to prevent undue sample bias. Analyses were performed for EN 1992 and MC 2010 using the updated database, whilst the results for BS 8007 were taken from [9]. 5. Discussion on Model Uncertainty Quantification 5.1. Short term Tension The final sample size for short-term tension was 86 after repeats were averaged, which is sufficient to obtain a reasonable estimate of model uncertainty. The statistical parameters for model uncertainty are summarized in Table 2. The values obtained for the statistical parameters are similar to those reported on in [9]. There were some small differences on including the research of [13] which consisted of ties reinforced with multiple bars rather than the mostly single bars or uniform samples of the database of [9]. The statistical analysis showed that the EN 1992 crack model tends to be conservative, overestimating the predicted short-term tension crack widths, with a MF mean of 0,75. The MF mean of around 1 obtained for the MC 2010 crack model shows little bias in this model. The MC 2010 MF CoV of 0,32, however, is higher than that of the EN 1992 model (CoV of 0,25). The BS 8007 crack model underpredicts crack widths with MF means over 1,27. This bias increases signif- icantly when tension stiffening is determined using the smaller limiting crack width, which is problematic when developing a GPM as the model is not consistent over crack widths away from the given limiting crack width (either 0,1 or 0,2 mm). There is also some un- certainty in the determination of acr in ties reinforced with a single bar as the formulation for acr results in underprediction of the crack spacing and width. 130 vol. 36/2022 SLS reinforced concrete crack models uncertainty Statistical Parameter EN 1992 MC 2010 BS 8007 wlim 0.2 mm [9] BS 8007 wlim 0.1 mm [9] Mean 0.747 0.996 1.271 1.430 Standard Error 0.020 0.034 0.032 0.041 Median 0.722 0.956 1.225 1.398 Standard Deviation 0.183 0.319 0.290 0.369 COV 0.245 0.321 0.228 0.258 Sample Variance 0.033 0.102 0.084 0.136 Kurtosis 0.051 -0.581 -0.056 1.234 Skewness 0.540 0.427 0.441 0.777 Range 0.927 1.408 1.516 2.139 Minimum 0.374 0.416 0.582 0.657 Maximum 1.301 1.824 2.097 2.796 PDF LN LN N N Count 86 86 82 82 Table 2. Model uncertainty statistical parameters - short-term tension cracking. Figure 1. Probability Plots for EN 1992 and MC 2010 - short-term tension cracking. Referring to Table 2, both the EN 1992 and MC 2010 crack models exhibit a positive skewness which suggests that the distribution is not normal. For the EN 1992 model uncertainty, the Shapiro-Wilks non- parametric test rejected the null hypothesis, whilst for the MC 2010 crack model, the Shapiro-Wilks sig- nificance factor was just greater than 0,05. Thus, it is estimated that both models have a non-normal distri- bution for model uncertainty. Curve-fitting was done for both crack models, considering normal and lognor- mal distributions. This indicated that both EN 1992 and MC 2010 tends towards a lognormal distribution, as shown in Figure 1. Considering that a lognormal distribution produces lower reliability estimates than a normal distribution (so more conservative), and that the distribution of the MC 2010 crack model has a positive skewness and is not clearly normal, a lognormal distribution is assumed for both models. BS 8007 model uncertainty appears to have a normal distribution. 5.2. Long term tension Statistical analyses for long-term tension cracking are summarised in Table 3, and as reported in [9]. The sample size given is the final one after all repeats were averaged. Comparing the long-term model uncertainty sta- tistical parameters [9] to the short-term case, crack models that do not take long term shrinkage into account (BS 8007 and EN 1992) do not have a con- sistent model uncertainty as load duration increases. The EN 1992 model uncertainty mean increases from 0,75 to 0,90 for short to long-term tension loading. However, when considering the MC 2010 crack model, the MF means for short and long-term loading are very similar. The CoV for all models for long-term loading is lower than that of short-term tension, but still demonstrate the significant influence that model uncertainty has on the crack models. However, some restraint is required on interpreting the data given the small sample size. The sample size is too small to give a good indication of skewness, nevertheless, 131 Christina Helen McLeod Acta Polytechnica CTU Proceedings Statistical Parameter EN 1992 MC 2010 BS 8007 w 0,2 mm BS 8007 w 0,1 mm Mean 0.895 0.988 1.318 1.603 Standard Error 0.078 0.076 0.211 0.280 Median 0.860 0.946 1.089 1.353 Standard Deviation 0.220 0.214 0.597 0.793 COV 0.246 0.216 0.453 0.495 Sample Variance 0.048 0.046 0.356 0.629 Kurtosis -1.539 -0.059 3.779 2.151 Skewness 0.442 0.806 1.882 1.447 Range 0.571 0.618 1.807 2.328 Minimum 0.656 0.764 0.836 0.935 Maximum 1.227 1.382 2.643 3.262 PDF (estimated) LN LN N N Count 8 8 8 8 Table 3. Model uncertainty statistical parameters for long-term tension cracking [9]. a lognormal rather than normal distribution is sug- gested from the positive skewness values obtained for both the EN 1992 and MC 2010 crack models. As found for short-term tension, the BS 8007 model un- certainty statistical parameters are not consistent for the different tension stiffening models. 5.3. Evaluation of crack models - Pearson’s correlations The interdependence between selected parameters and model uncertainty for each model were evaluated, to- gether with further assessment of the performance of the EN 1992, MC 2010 and BS 8007 crack models by means of scatter plots, regression analyses and Pearson’s correlations. The selected parameters were reinforcing ratio (as % As), steel stress, concrete ten- sile strength (fctm), section depth (h), concrete cover (c) and section width (b). The Pearson’s correlation factors between the model factor and selected param- eters are summarised in Table 4 for each load case and crack model, respectively. The small sample size means that the long-term tension cracking correla- tions of [9] need to be viewed with caution. Little variation in the sample configurations for long-term tension loading [9] also results in correlations relating to section geometry being overestimated for all mod- els. This applies to some extent to short-term tension loading, as although the sample size is larger, many samples were limited to ties with a single reinforcing bar or have little variation. Adding the experimental programme of [13] (which consisted of repeat samples with multiple bars as opposed to much of the more uniform database of [9] with ties reinforced with single bars) resulted in differences to the short-term tension values obtained for the Pearson’s analyses reported in [9] for MC 2010 and EN 1992 even though the sample size increase was small. Further research using different configurations/real structures is required for tension cracking for a more accurate evaluation of interdependence between these parameters and the model factor. Referring to Table 4 and short-term tension, with the exception of the section depth to width ratio (h/b) for MC 2010, model uncertainty for the EN 1992 crack model has low to moderate correlations with all parameters. As would be expected, model uncer- tainty has a negative correlation with concrete tensile strength (fctm) for both EN 1992 and MC 2010. The inconsistency in the BS8007 crack model is clearly demonstrated as the Pearson’s correlations vary sub- stantially between the tension stiffening formulations. A larger sample size is required for a true reflection of the Pearson correlations for the long-term case [9], as can be evidenced by the extremely varied values between short- and long-term loading. 5.4. Comparison to Flexure Model Uncertainty As the database for long-term tension cracking was very limited, comparisons were made to analyses to quantify model uncertainty as reported in [9] and [10] for short- and long-term flexure loading applied to the EN 1992, MC 2010 and BS 8007 crack models. The statistical parameters for the MF for flexural cracking are summarised in Table ??. Further detail on the flexural crack models analyses can be found in [9, 10]. Comparing the short-term model uncertainty for tension to that of flexure, the means for the MC 2010 crack model are similar at approximately 1, although the variation for the tension case is less than that of flexure. This is probably partly due to the greater uniformity of sample configurations of the tension database compared to flexure. Lognormal distribu- tions are indicated for the EN 1992 and MC 2010 crack models for both flexural and tension cracking. However, model uncertainty parameters are similar enough to conclude that it is not necessary to distin- guish between the flexural and tension load cases if the MC 2010 crack model is considered for the GPM. Although the dataset for long term flexural cracking 132 vol. 36/2022 SLS reinforced concrete crack models uncertainty Load Case Parameter Correlation with Model Factor EN 1992 MC 2010 BS8007 w = 0, 2 mm [9] BS8007 w = 0, 1 mm [9] Short-term tension Steel stress 0.052 0.204 -0.284 -0.364 h/b 0.579 0.722 0.049 0.206 c 0.286 0.420 0.332 0.230 Bar dia, φ 0.518 0.570 -0.068 -0.241 fctm -0.294 -0.503 -0.030 0.080 % As 0.352 0.390 -0.036 -0.019 Long-term tension [9] Steel stress 0.825 0.641 0.840 0.777 h/b 0.490 0.335 0.602 0.479 c -0.825 -0.641 -0.840 -0.777 Bar dia, φ -0.414 -0.209 -0.544 -0.407 fctm -0.826 -0.724 -0.792 -0.773 % As -0.349 -0.156 -0.426 -0.334 Table 4. Pearson’s correlation matrix between model uncertainty & model parameters. Load Case Statistical parameter EN 1992 MC 2010 BS 8007 w = 0.2 mm BS 8007 w = 0.1 mm Short term Mean 1.107 1.052 1.185 1.112 CoV 0.397 0.376 0.380 0.459 PDF LN LN LN LN Count 164 164 164 164 Long term Mean 1.443 1.127 1.502 1.514 CoV 0.331 0.380 0.336 0.357 PDF LN LN LN LN Count 30 30 30 30 Table 5. Summary of statistical parameters of MF for flexural cracking [10]. [9, 10] is larger than that of the tension load case, further research for long-term loading in general is recommended, including expanding the database and application to real conditions, for a more accurate estimation of the statistical parameters of model un- certainty. Pearson’s correlations were performed for flexural cracking, as reported by [10]. Low to moderate cor- relations were found for all parameters except for % As for long-term flexure (r of 0,58). It is noted that the dataset for flexural cracking had more variation in terms of section and reinforcement configurations than that of tension cracking, which supports the sug- gestion that the uniformity of the dataset for tension cracking results in apparent dependence between these parameters and the MF. 6. Summary and concluding remarks In quantifying model uncertainty for pure tension crack models, the significant degree of uncertainty was confirmed, which justifies a full probabilistic treat- ment of model uncertainty and SLS crack models. The model uncertainty statistical parameters were determined for short and long-term tension cracking, although the latter case does require further investiga- tion to confirm values obtained. However, considering the MF statistical parameters determined for short and long-term flexural cracking ([9, 10]), the short- term MF values for tension cracking may be used for long-term MF’s when considering crack models such as MC 2010 which take long-term shrinkage into ac- count. Indications are also that for the MF, it is not necessary to distinguish between flexural and tension load-induced cracking. This simplifies the reliability assessment of load-induced cracking and the develop- ment of the GPM. In terms of crack model performance, the BS 8007 crack model is shown to be inconsistent as its tension stiffening model is dependent on the specified limiting crack width. Any variation in crack width away from this limit results in a high variation in the MF, and thus an inconsistent model. This is a point to note for any empirically-developed model which may be consid- ered for a GPM. It was also noted during the analyses that the point on the cross section where the crack was measured was generally not given in the experimental 133 Christina Helen McLeod Acta Polytechnica CTU Proceedings records, introducing some further uncertainty into the predicted crack widths due to potential mismatches between measured and estimated crack positions. Us- ing acr based on the geometry of the cross section, rather than a crack spacing formulation, for ties with single bars resulted in underprediction of the measured crack widths. This effect would be mitigated in struc- tures under real conditions. Crack models that omit shrinkage strain (EN 1992 and BS 8007) underpredict crack widths in the long-term loading case. MC 2010 was the most consistent crack prediction model and will thus be the basis for the GPM. Further to this research, data will be compiled for existing structures such as bridges and water retaining structures for use in the probabilistic analysis of the GPM. The experimental database was compiled and re- fined with WRS in mind, where SLS cracking is dom- inant in design, so focused on small crack widths. For long-term tension cracking in particular, it must be noted that as many of the elements tested were mostly ties reinforced with single reinforcing bars, the crack widths recorded may not necessarily be representative of real conditions in structures such as WRS where there are obviously multiple reinforcing bars in both the main and transverse directions. This was noted in [9] and [10], borne out in the further analyses. This requires further research to properly assess the influ- ence of section and reinforcement configurations on tension cracking, and which will include data from existing structures. References [1] J. V. Retief. Contributions to the implementation of the principles of reliability to the standardised basis of structural design. Dr of Eng Thesis, Stellenbosch University, South Africa. Supervisor GAP van Zijl, Co-supervisor C Viljoen, 2015. [2] M. Holický, J. V. Retief, M. Sýkora. Assessment of model uncertainties for structural resistance. Probabilistic Engineering Mechanics 45:188-97, 2016. https: //doi.org/10.1016/j.probengmech.2015.09.008. [3] BS 8007. Design of concrete structures for retaining aqueous liquids. British Standards Institute, 1987. [4] SANS 10100 -1 (2004): The structural use of concrete: Part 1: Design. South African Standards Division. Pretoria, 2004. [5] BS EN 1992-1-1: 2004. Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings. British Standards Institute, 2004. [6] SANS 10100 Part 3 (Draft): Design of concrete liquid retaining structures. SABS-TC98-SC2 Working group on implementation of South African National Standard for liquid retaining structures. South African Standards Division, Pretoria., 2015. [7] C. H. McLeod. Investigation into cracking in reinforced concrete water-retaining structures. 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Supervisor Prof RI Gilbert, 2010. 134 https://doi.org/10.1016/j.probengmech.2015.09.008 https://doi.org/10.1016/j.probengmech.2015.09.008 https://doi.org/10.1002/suco.201200060 https://doi.org/10.1002/suco.201900036 https://doi.org/10.1002/suco.201700248 https://doi.org/10.1016/j.engstruct.2020.110979 https://doi.org/10.1016/j.engstruct.2020.110979 Acta Polytechnica CTU Proceedings 36:127–134, 2022 1 Introduction 2 Model Uncertainty Treatment 3 Deterministic analysis of predicted crack width 3.1 BS 8007 crack model 3.2 EN 1992 crack model 3.3 MC 2010 crack model 4 Quantification of crack model uncertainty 4.1 Experimental database 5 Discussion on Model Uncertainty Quantification 5.1 Short term Tension 5.2 Long term tension 5.3 Evaluation of crack models - Pearson's correlations 5.4 Comparison to Flexure Model Uncertainty 6 Summary and concluding remarks References