Chapter 3: Structural Reliability
3.6 Statistical Parameters of the EN 1992 Restrained Cracking Serviceability Limit State
3.6.2 Model uncertainty (θ)
The model uncertainty may be determined by comparing experimental data to those values obtained through the existing prediction model (JCSS, 2000). There are instances in which not
much data is available on the model uncertainty and experience and professional judgement is depended upon (Holický, 2009). Considering a sensitivity analysis conducted of the EN 1992 load induced crack model with respect to variations in model uncertainty conducted by McLeod (2013), model uncertainty had been found to bear the most influence on the tension load case and was found to be the second most influential random variable of the flexural loading case. The above-mentioned tension load case may be indicative of how influential model uncertainty might be on the restrained shrinkage crack model of EN 1992. ISO 2394:1998 (reproduced as SANS 2394:2004) includes model uncertainty as a random variable, θ, to be used in reliability assessments of performance functions accounting for a) inherent variability within the analysed model, b) inadequate knowledge and c) statistical uncertainty. Moreover, d) mathematical simplifications and assumptions made in developing the prediction model generates a certain degree of uncertainty (McLeod, Viljoen & Retief, 2016).
Looking more carefully into these above-mentioned sources of uncertainty with respect to the restrained shrinkage crack model it may be gathered that:
a) Cracking is a naturally random phenomenon with inherent variability.
b) The knowledge base regarding the stochastic nature of the restrained thermal and shrinkage cracking case is limited, meaning that there must be a heavy reliance on experience and professional judgement in this regard. Increased research in this area would result in a more accurate depiction of restrained cracking’s statistical parameters and thus increased accuracy in the reliability assessment of its model. Most knowledge in the area of reliability-based assessments of the cracking serviceability limit state veered towards those cracks resulting from load (be it a concrete member under flexure or tension).
The Eurocode 2 crack model along with other crack models have been tested against experimental data several times in previous research. One such comparison of the experimental crack widths to those predicted by EN 1992-3 was found in the investigations of Kamali et al. (2013) on the crack width control of a concrete slab bridge under restrained cracking (particularly for tensile forces in the transversal direction). It was determined in the course of this study that for 90% of all the observed crack widths, the EN 1992-3 crack model overestimated the crack widths (more crack widths were found to fall below where the measured crack widths equalled those estimated by EN 1992-3, as denoted by the broken red line of Figure 3.2). This experiment was done for crack widths greater than 0.2 mm. This fact is made clear in Figure 3.2 where the majority of the estimated crack widths are either comparable to the measured crack widths (at lower crack widths) or greater than the measured crack widths where the crack widths are
larger than ±0.4 mm. This finding reinforces the notion that the EN 1992-3 is conservative in its estimation of the crack width due to restrained strains.
Figure 3.2: Comparison of Measure Crack Widths against the EN 1992-3 Predicted Crack Widths for a Concrete Member Restrained Along its Base (Kamali, Svedholm and Johansson, 2013).
In another comparison of the EN 1992-3 and BS 8007 crack prediction models to observe cracks, both models were found to under-predict the observed crack widths – this is presented in Figure 3.3 (Bamforth, Shave & Denton, 2011). In some instances, this underestimation of observed cracks would be by as much as 50%. This is contrary to what was found in the previous case by Kamali et al. (2013), alluding to the considerable amount of scatter in model uncertainty of the EN 1992-3 crack model for restrained shrinkage. Both Kamali et al. (2013) and Bamforth et al.’s (2011) comparisons were done so against data obtained for research on the control of cracking resulting from restrained contraction.
Figure 3.3: Comparison of Measured Crack Widths to Predicted Crack Widths of BS 8007:1987 and EN 1992-3:2006 (Bamforth, Shave & Denton, 2011)
Although the determination of model uncertainty depends on the formulation of the prediction model used (McLeod, 2016), the findings of model uncertainty related to the load induced cracking model were included in the subsequent text. Given the scarcity of probability based investigations done on the restrained thermal and shrinkage strain cracking, the load induced crack case should give indications as to how the EN 1992 may be described in terms of its statistical parameters. Quan and Gengwei (2002) found the model uncertainty for the crack widths of reinforced concrete beams to have a coefficient of variance of 0.298 (or otherwise 0.3) and an estimated mean of 1.05. These results came after a statistical study of 116 beams with varying configurations, strengths and applied loads (Quan and Gengwei, 2002). The model uncertainty was found to follow a lognormal probability distribution model. Thus subsequently, a mean of 1 and a maximum coefficient of variance of 0.3 will be adopted for the reliability analysis in this thesis.
c) Statistical uncertainty results from there being some uncertainty in the ways in which statistical parameters are estimated. Increases in the data base and sample size of the cracking from restrained shrinkage through testing and recording of observations should increase the accuracy of reliability assessments.
d) Examples of mathematical simplifications or assumptions made in modelling cracking include, for instance, the crack spacing formula of EN 1992 which contains some empirical fixed-value coefficients (McLeod, Viljoen, Retief, 2016). Such as the coefficient k1, accounting for bond properties in EN 1992. A value of 0.8 is stipulated in EN 1992 for instances of good bond. This coefficient is the equivalent of BS 8007’s fct/fb
(taking on a value of 0.67 for type 2 deformed bars for class C35A concrete. Previous research has also indicated that 0.67 could be safely applied to all strength classes of concrete (Bamforth, 2007)). Even though the concrete tensile strength to reinforcing bond strength ratio (fct/fb) was found in past research to decrease with an increase in concrete strength class, EN 1992 gives a constant value (0.8 for good bond) that is to be applied across all strength classes. This would then mean that at higher concrete strength classes, the k1 coefficient provides an added margin of safety (or otherwise an added degree of conservatism).
Additionally, creep is accounted for particularly in the restraint factor since it has the effect of reducing restraint over time. However, where creep test methods are not given in the South African and British standards, most creep test methods involve loading concrete cylinders hydraulically and then measuring the deformation that results over time (Owens, 2013). This would mean that the creep value obtained would be based on compression rather than tension in the concrete (particularly tension arising from restrained contraction in the concrete). Thus in applying this same creep factor to tension cases (such as where there is restrained shrinkage) there could be a margin of error that arises since the creep prediction model does not necessarily represent the tension case.
Furthermore, it had been found in past research that the tensile creep of concrete is lower under restrained shrinkage as opposed to where the concrete is under constant stress (Sajedi et al., 2011).
Bearing all of these sources of uncertainty in mind, the EN 1992-3 restrained shrinkage crack model’s coefficient of variance value will be varied in the reliability assessment to gauge what influence it has on the reliability performance of the crack model. This would be a particularly relevant assessment given model uncertainty’s observed dominance in previous research (Retief, 2015).