Chapter 5: FORM Analysis of EN 1992 Crack Model: Methodology, Results and Discussion . 87
5.3 Results and Discussion
5.3.1 Influence of cover and φ/ρ p, eff
Table 5.1: Feasible limit for reinforcement for select section thicknesses (minimum bar spacing 75 mm)
h (mm) Max.
Feasible As
(%)
250 3.35
300 2.79
350 2.39
400 2.09
450 1.86
500 1.68
It must be reiterated that comparisons of reliability assessments with those of similar past research findings are quite difficult in that unless the same limit state and statistical parameters (mean value and standard deviation of the random variables) are used, direct comparisons would prove to be somewhat inaccurate. Thus alternative measures of results verification may have to be employed. As a measure of assurance that the FORM analysis was correctly executed, hand calculations were conducted alongside those calculations done via Microsoft Excel (acting as a double check of the results obtained).
Reliability models used in the analysis are:
a) Edge restraint with depth of effective tension zone taken to be 2.5(c +φ/2) b) Edge restraint with depth of effective tension zone taken to be h/2
c) End restraint with depth of effective of tension zone taken to be 2.5(c +φ/2) d) End restraint with depth effective of tension zone taken to be h/2
The majority of the reliability analysis was conducted with 2.5(c +φ/2) being the depth of effective tension area as it is the limiting depth of effective tension area for most combinations of section thicknesses, concrete covers and bar diameters. However there were certain instance where models containing h/2 was more appropriate, these instances are mentioned where they apply.
crack model of EN 1992. The concrete cover affects the formulation of the limit state function by dictating the limiting effective tension depth. Thus the cover value was selected such that the same limit state function was used in the reliability analysis- making direct comparisons of results between concrete covers possible.
5.3.1.1 Edge Restraint
For a 250 mm thick section, the influence of concrete cover and the φ/ρp, eff was examined by varying the cover value against a selection of steel reinforcement ratios. Cover values of 50, 60 and 70 mm were used in this analysis, making h/2 the appropriate effective depth of tension area for the analysis. The reliability of the crack model increased with a decrease in concrete cover (as illustrated in Figure 5.4). For example for a steel ratio of say 2%, the reliability indices for 70, 60 and 50 mm are 1.48, 1.68, and 1.90 respectively. A decrease in cover would result in a decrease in the crack spacing attained, thus decreasing the overall crack width obtained. Crack widths beyond the crack width limit are then less likely to occur, hence the increase in reliability with the decrease in concrete cover.
Moreover, irrespective of the cover value selected, the reliability index would increase with an increase in the amount of reinforcement used. Undoubtedly, where more reinforcement is applied to a concrete section the more resistant the member will become against tensile stresses, and so less cracking occurs.
More reinforcement is required to meet the target reliability index as concrete cover increases.
The target reliability index (βt = 1.5) is met at steel to concrete ratios of about 1.69%, 1.85% and 2.02% for a cover value of 50 mm, 60 mm and 70 mm respectively. This amounts to about an average increase in reinforcement of 10% with every 20% relative increase in covers selected (or otherwise per 10 mm absolute increase in concrete cover value).
Figure 5.4: Influence of Cover and φ/ρp, eff (Edge Restraint)
On comparing the reliability index directly against the φ/ρp,eff ratio (as shown in Figure 5.5) the φ/ρp,eff ratios that ensure that the target reliability index is met for concrete covers 50, 60 and 70 mm are: 1.18, 1.08 and 0.99 m respectively (Figure 5.5). Since an increase in the amount of steel reinforcement used will result in a decrease in the φ/ρp,eff ratio, it may be concluded that the reliability increases with a decrease in the φ/ρp,eff ratio. Where the bar diameter had been found in the deterministic analysis to have little influence on the crack width model, the influence of the φ/ρp,eff ratio may be deduced to have come mostly from the steel reinforcing and partially from the section thickness’s stochastic nature within the limiting effective tension depth equation. The gradient of the graph of Figure 5.5 was close to -2 for the cover values considered indicating a strong relationship.
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
Reliability Index (β)
% As
c=50mm c=60mm c=70mm
βt= 1.5
Figure 5.5: Influence of φ/ρp, eff Ratio on Reliability Index (Edge Restraint)
5.3.1.2 End Restraint
For end restraint conditions, with the same selection of concrete cover values, as with edge restraint, it can clearly be deduced that a decrease in cover results in an increase in the reliability index calculated (Figure 5.6). For instance, the reliability indices for end restraint at 2% As would go from 1.17, 1.34, 1.52 for concrete values 70, 60 and 50 mm respectively (achieving lower reliability indices than edge restraint for the same %As value).
A steel reinforcement to concrete percentage of 1.99%, 2.09% and 2.19% is required for the selected concrete covers 50, 60 and 70 mm respectively to meet the target reliability index (βt = 1.5). This translates to about a 5% increase in reinforcement required per 20% relative increase in the concrete cover- half the value found for edge restraint. This is indicative of the greater influence had by concrete cover on the edge restraint crack model as compared to that of the end restraint. Comparing this result to that of the edge restraint crack model, it is evident that slightly more reinforcement is required to meet the target reliability index for end restraint. This makes end restraint the more conservative of the two variations of the restrained shrinkage crack model.
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
Reliability Index (β)
ϕ/ρp,eff (m)
c=50mm c=60mm c=70mm
βt = 1.5
Figure 5.6: Influence of Cover and φ/ρp, eff (End Restraint)
Much like for the edge restraint condition, the reliability index decreases as the φ/ρp, eff ratio increases. The target reliability index (βt = 1.5) is met where the φ/ρp, eff ratio is at 1, 0.96 and 0.92 m for covers 50, 60 and 70 mm respectively (Figure 5.7). The gradient of the φ/ρp, eff ratio to reliability indices graphs across the concrete cover values selected was about -4, having a strong impact on the reliability of the end restraint crack model (reading from Figure 5.7). This was a stronger relationship than in the edge restraint case.
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
Reliability Index (β)
%As
c=50mm c=60mm c=70mm
βt= 1.5
Figure 5.7: Influence of φ/ρp, eff Ratio on Reliability Index (End Restraint)
Clearly, the concrete cover selected has some bearing on the reliability of the crack model for both end and edge restraint. For both restraint conditions, increasing the cover decreases the reliability of the crack model. The amount of reinforcing also influences the reliability levels that may be achieved by the crack model and thus the contributions of the φ/ρp, eff ratio cannot be ignored. This is evident in the figures representing the change in reliability indices with respect to the φ/ρp, eff ratio for both end and edge restraint (Figures 5.5 and 5.7, respectively). So therefore, as the EN 1992 crack model stands, both terms (c and the φ/ρp, eff ratio) of the crack spacing have a noteworthy influence on the eventual reliability of the crack model.