Chapter 6: Sensitivity Analysis of EN 1992 Crack Model: Methodology, Results and Discussion
6.2 Results and Discussion
6.2.2 Theoretical Partial Safety Factors
was h/2. The sensitivity factors obtained for section thickness where hc,eff = h/2 are greater in magnitude than those obtained where hc,eff = 2.5(c +φ/2), but still negligible. The sensitivity factors of concrete tensile strength are quite comparable between the two variations of the end restraint crack model, with the hc,eff = h/2 containing end restraint crack model slightly larger in magnitude (larger by factor 1.06) at wlim = 0.2 mm and model uncertainty CoV 0.3.
The biggest difference between the two variations of the end restraint crack model was in the influence of the concrete cover variable. The larger magnitude of the concrete cover sensitivity factor in the model containing hc,eff = 2.5(c +φ/2) will mean that larger crack widths are calculated with this reliability model. Thus the reliability indices produced under this model will be lower as compared to the end restraint crack model where the effective depth of tension was h/2.
Table 6.5: Theoretical Partial Factors of Random Variables for Edge Restraint Crack Model (βt = 1.5, hc, eff = 2.5(c +φ/2))
wlim
(mm)
Model
uncertainty CoV
%As required
γc
(concrete cover)
γθ
(model uncertainty) 0.3
0.1 0.756 1.18 1.09
0.15 0.797 1.14 1.17
0.2 0.847 1.12 1.26
0.25 0.902 1.09 1.35
0.3 0.963 1.08 1.45
0.2
0.1 1.229 1.18 1.09
0.15 1.302 1.15 1.17
0.2 1.390 1.12 1.26
0.25 1.491 1.10 1.35
0.3 1.600 1.08 1.45
0.1
0.1 3.294 1.18 1.09
0.15 3.553 1.15 1.17
0.2 3.884 1.12 1.26
0.25 4.280 1.10 1.35
0.3 4.743 1.09 1.44
For concrete cover the theoretical psf’s were relatively constant irrespective of the model uncertainty CoV and the crack width limit, Figure 6.16 illustrates this (ranging from approximately 1.1, up to a maximum value of 1.2). A larger range of variations in the theoretical psf’s of model uncertainty were found as the model uncertainty CoV increased (approximately from 1.1 to 1.4, as shown in Figure 6.17). An increase in the crack width limit resulted in a decrease in the influence of the concrete cover (as indicated in Figure 6.16) with an increase in influence being found for model uncertainty (referring to Figure 6.17).
Figure 6.16: Edge Restraint Theoretical Partial Safety Factors of Concrete Cover (c) for Varying Model Uncertainty Coefficient of Variance (hc, eff =2.5(c +φ/2))
Figure 6.17: Edge Restraint Theoretical Partial Safety Factors of Model Uncertainty (θ) for Varying Model Uncertainty Coefficient of Variance (hc, eff =2.5(c +φ/2))
6.2.2.2 Edge Restraint (hc, eff= h/2)
Even with where the effective depth of tension area was h/2, no real adjustment was required for section thickness to meet the desired reliability index. This was evident across all assessed crack
1 1.1 1.2 1.3 1.4 1.5
0.1 0.15 0.2 0.25 0.3
Partial Factor (γc)
Model Uncertainty CoV
wlim=0.3mm wlim=0.2mm wlim=0.1mm
1 1.1 1.2 1.3 1.4 1.5
0.1 0.15 0.2 0.25 0.3
Partial Factors (γθ)
Model Uncertainty CoV
wlim=0.3mm wlim=0.2mm wlim=0.1mm
width limits and model uncertainty CoV’s considered in the analysis. Model uncertainty required the most adjustments to meet the target reliability index, with theoretical psf’s from about 1.1 (model uncertainty CoV = 0.1) to 1.5 (model uncertainty CoV = 0.3) for the crack width limits considered. The concrete cover follows after model uncertainty with psf’s from 1.01 (wlim = 0.3 mm, model uncertainty CoV = 0.3). This was indicative of the slight influence the concrete cover had on the edge restraint crack model where h/2 was the effective depth of tension area. Presented in Table 6.6 are the theoretical psf’s obtained for the edge restraint crack model where hc,eff = h/2.
Table 6.6: Theoretical Partial Factors of Random Variables for Edge Restraint Crack Model (βt = 1.5, hc, eff =h/2)
wlim
(mm)
Model
uncertainty CoV
%As Required
γh
(section thickness)
γc
(concrete cover)
γθ
(model uncertainty) 0.3
0.1 0.690 1.00 1.03 1.15
0.15 0.745 1.00 1.02 1.24
0.2 0.804 1.00 1.01 1.32
0.25 0.866 1.00 1.01 1.41
0.3 0.930 1.00 1.01 1.50
0.2
0.1 1.114 1.00 1.05 1.15
0.15 1.210 1.00 1.03 1.23
0.2 1.316 1.00 1.02 1.32
0.25 1.424 1.00 1.02 1.41
0.3 1.541 1.00 1.02 1.49
0.1
0.1 2.935 1.00 1.11 1.13
0.15 3.251 1.00 1.08 1.22
0.2 3.617 1.00 1.06 1.30
0.25 4.0354 1.00 1.05 1.39
0.3 4.511 1.00 1.04 1.48
Figure 6.18 illustrates how the theoretical psf’s required for section thickness were generally unaffected by the increase in crack width limit. For concrete cover, increases in the crack width resulted in a decrease in the theoretical psf required to attain the target reliability index (observing from Figure 6.19). Model uncertainty psf’s increased with an increase in crack width limit (referring to Figure 6.20). Increases in the model uncertainty CoV resulted in decreases in the theoretical psf attained for concrete cover and an increase in those theoretical psf’s values obtained for model uncertainty (reading from Figure 6.19 and 6.20 respectively). In the case of section thickness, little variation was experienced across the range model uncertainty CoV’s considered in this analysis (as illustrated in Figure 6.18).
Figure 6.18: Edge Restraint Theoretical Partial Safety Factors of Section Thickness (h) for Varying Model Uncertainty Coefficient of Variance (hc, eff = h/2)
Figure 6.19: Edge Restraint Theoretical Partial Safety Factors of Concrete Cover (c) for Varying Model Uncertainty Coefficient of Variance (hc, eff = h/2)
1 1.1 1.2 1.3 1.4 1.5
0.1 0.15 0.2 0.25 0.3
Partial Factor (γh)
Model Uncertainty CoV
wlim=0.3mm wlim=0.2mm wlim=0.1mm
1 1.1 1.2 1.3 1.4 1.5
0.1 0.15 0.2 0.25 0.3
Partial Factor (γc)
Model Uncertainty CoV
wlim=0.3mm wlim=0.2mm wlim=0.1mm
Figure 6.20: Edge Restraint Theoretical Partial Safety Factors of Model Uncertainty (θ) for Varying Model Uncertainty Coefficient of Variance (hc, eff = h/2)
The adjustments required for concrete cover to meet the target reliability were greater by a factor of 1.07 where the effective depth of tension zone was 2.5(c +φ/2) –for wlim = 0.2 mm and model uncertainty CoV of 0.3 (a small increase in the psf required between the effective depth hc,eff = h/2 to where hc,eff is 2.5(c +φ/2). The theoretical psf’s calculated for model uncertainty where the effective depth is h/2 were 1.03 times greater than where the effective depth of the tension was 2.5(c +φ/2). This slight increase was found where the crack width limit was 0.2 mm and the variability of model uncertainty was set at 0.3.
6.2.2.3 End Restraint (hc, eff = 2.5(c +φ/2))
It may be observed from Table 6.7 that the section thickness had obtained negligible theoretical psf’s. Being the most influential random variable, model uncertainty had partial factors from about 1.1 to 1.4for the range of crack width limits considered (for corresponding model uncertainty CoV’s 0.1 and 0.3). This was followed by concrete cover with theoretical psf’s from 1.07 to 1.1 (at model uncertainty CoV of 0.3). The effective concrete tensile strength is a material property and thus a resistance variable, the theoretical partial safety factor for concrete tensile strength would be implemented in design codes as 1/γfct,eff to obtain the design value for this variable . For the effective concrete tensile strength theoretical psf’s were generally from 1.1 at model uncertainty CoV of 0.3 to about 1.2 for model uncertainty CoV of 0.1 – implemented as 0.91 to 0.83 respectively (referring to Table 6.7).
1 1.1 1.2 1.3 1.4 1.5
0.1 0.15 0.2 0.25 0.3
Partial Factor (γθ)
Model Uncertainty CoV
wlim=0.3mm wlim=0.2mm wlim=0.1mm
Table 6.7: Theoretical Partial Factors of Random Variables for βt = 1.5 (hc, eff =2.5(c +φ/2)) wlim
(mm)
Model
uncertainty CoV
%As required
γh
(section thickness)
γc
(concrete cover)
γfct,eff
(effective concrete tensile strength)
1/γfct,eff γθ
(model uncertainty)
0.3
0.1 1.380 1.00 1.11 1.22 0.82 1.06
0.15 1.404 1.00 1.10 1.20 0.84 1.12
0.2 1.435 1.00 1.09 1.17 0.85 1.19
0.25 1.471 1.00 1.08 1.15 0.87 1.28
0.3 1.510 1.00 1.07 1.13 0.88 1.37
0.2
0.1 1.762 1.00 1.11 1.22 0.82 1.06
0.15 1.794 1.00 1.01 1.20 0.84 1.12
0.2 1.835 1.00 1.09 1.17 0.85 1.19
0.25 1.883 1.00 1.08 1.15 0.87 1.28
0.3 1.935 1.00 1.07 1.13 0.88 1.37
0.1
0.1 2.751 1.00 1.11 1.22 0.82 1.06
0.15 2.807 1.00 1.10 1.20 0.84 1.12
0.2 2.877 1.00 1.09 1.17 0.85 1.19
0.25 2.959 1.00 1.08 1.15 0.87 1.28
0.3 3.049 1.00 1.07 1.13 0.88 1.37
The theoretical partial safety factors obtained for the random variables remained mostly steady across all crack width limits considered as may be deduced from the Figures 6.21 to 6.24, particularly for section thickness (as shown in Figure 6.21). Nonetheless, as the crack width limit decreased the theoretical psf’s of the effective concrete tensile strength and model uncertainty decreased (referring to Figures 6.23 and 6.24 respectively). The theoretical partial safety factors of concrete cover increased with a decrease in crack width limit (reading from Figure 6.22).
Increases in the variability of the model uncertainty decreased the theoretical partial safety factors required for section thickness (negligible decrease), concrete cover (decrease was also found to be marginal, but not as small as for section thickness) and for the effective concrete tensile strength for reliability compliance (as may be observed in Figures 6.21, 6.22 and 6.23 respectively). Model uncertainty’s theoretical partial safety factors increased considerably with an increase in model uncertainty variability (Figure 6.24).
Figure 6.21: End Restraint Theoretical Partial Safety Factors of Section Thickness (h) for Varying Model Uncertainty Coefficient of Variance (hc, eff = 2.5(c +φ/2))
Figure 6.22: End Restraint Partial Safety Factors of Concrete Cover (c) for Varying Model Uncertainty Coefficient of Variance (hc, eff = 2.5(c +φ/2))
1 1.1 1.2 1.3 1.4 1.5
0.1 0.15 0.2 0.25 0.3
Partial Factor (γh)
Model Uncertainty CoV
wlim=0.3mm wlim=0.2mm wlim=0.1mm
1 1.1 1.2 1.3 1.4 1.5
0.1 0.15 0.2 0.25 0.3
Partial Factors (γc)
Model Uncertainty CoV
wlim=0.3mm wlim=0.2mm wlim=0.1mm
Figure 6.23: End Restraint Theoretical Partial Safety Factors of the Effective Concrete Tensile Strength (fct,eff) for Varying Model Uncertainty Coefficient of Variance (hc, eff = 2.5(c +φ/2))
Figure 6.24: End Restraint Theoretical Partial Safety Factors of Model Uncertainty (θ) for Varying Model Uncertainty Coefficient of Variance (hc, eff = 2.5(c +φ/2))
6.2.2.4 End Restraint (hc, eff = h/2)
Model uncertainty, once again, required the largest theoretical partial safety factor with values from approximately 1.1 to 1.4 (at model uncertainty CoV’s of 0.1 and 0.3 respectively) for the
1 1.1 1.2 1.3 1.4 1.5
0.1 0.15 0.2 0.25 0.3
Partial Factor (γfct,eff)
Model Uncertainty CoV
wlim=0.3mm wlim=0.2mm wlim=0.1mm
1 1.1 1.2 1.3 1.4 1.5
0.1 0.15 0.2 0.25 0.3
Partial Factor (γθ)
Model Uncertainty CoV
wlim=0.3mm wlim=0.2mm wlim=0.1mm
crack width limits considered in this analysis. The theoretical psf’s for the effective concrete tensile strength follows after model uncertainty with theoretical psf’s from about 1.1. As before mentioned, when applying the calculated theoretical partial safety factor for the effective concrete tensile in a design code the factor 1/γfct,eff will be used rather than γfct,eff. since it is a material property (and thus a resistance variable). The theoretical psf’s obtained for concrete cover was generally around 1.02. Section thickness had a small influence on the end restraint crack model (where hc,eff = h/2) and hence obtained theoretical partial safety factors of about 1 for all crack widths limits and model uncertainty CoV’s considered. These results may be observed in Table 6.8.
Table 6.8: Theoretical Partial Factors of Random Variables for βt = 1.5 (hc, eff = h/2) wlim
(mm)
Model Uncertainty CoV
%As Required
γh
(section thickness)
γc
(concrete cover)
γfct,eff
(effective concrete tensile strength)
1/γfct,eff γθ
(model uncertainty
0.3
0.1 1.347 1.00 1.02 1.26 0.79 1.07
0.15 1.375 1.00 1.02 1.23 0.82 1.14
0.2 1.409 1.00 1.02 1.19 0.84 1.22
0.25 1.447 1.00 1.01 1.17 0.86 1.30
0.3 1.489 1.00 1.01 1.14 0.88 1.40
0.2
0.1 1.718 1.00 1.03 1.26 0.80 1.07
0.15 1.755 1.00 1.03 1.22 0.82 1.13
0.2 1.800 1.00 1.02 1.19 0.84 1.22
0.25 1.851 1.00 1.02 1.17 0.86 1.30
0.3 1.906 1.00 1.02 1.14 0.88 1.39
0.1
0.1 2.679 1.00 1.04 1.25 0.80 1.07
0.15 2.741 1.00 1.04 1.22 0.82 1.13
0.2 2.818 1.00 1.03 1.19 0.84 1.21
0.25 2.906 1.00 1.03 1.16 0.86 1.30
0.3 3.001 1.00 1.03 1.14 0.88 1.39
There were slight variations in the theoretical partial safety factors obtained across the crack width limits considered for all random variables (particularly for section thickness). Increases in the crack width limit meant an increases in the theoretical partial safety factors required for the effective concrete tensile strength and model uncertainty (as shown in Figures 6.27 and 6.28 respectively). The concrete cover had theoretical partial safety factors that decreased in value as
the crack width limit was increased (referring to Figure 6.26). Increases in the variability of the model uncertainty resulted in there being a decrease in values of the theoretical partial safety factors obtained for concrete cover and the effective concrete tensile strength (as shown in Figures 6.26 and 6.27). The theoretical partial factors obtained for model uncertainty increased as the variability in the model uncertainty was increased (as observed in Figure 6.28).
Figure 6.25: End Restraint Theoretical Partial Safety Factors of Section Thickness (h) for Varying Model Uncertainty Coefficient of Variance (hc, eff =h/2)
1 1.1 1.2 1.3 1.4 1.5
0.1 0.15 0.2 0.25 0.3
Partial Factor (γh)
Model Uncertainty CoV
wlim=0.3mm wlim=0.2mm wlim=0.1mm
Figure 6.26: End Restraint Theoretical Partial Safety Factors of Concrete Cover (c) for Varying Model Uncertainty Coefficient of Variance (hc, eff =h/2)
Figure 6.27: End Restraint Theoretical Partial Safety Factors of the Effective Concrete Tensile Strength (fct,eff) for Varying Model Uncertainty Coefficient of Variance (hc, eff = h/2)
1 1.1 1.2 1.3 1.4 1.5
0.1 0.15 0.2 0.25 0.3
Partial Factors (γc)
Model Uncertainty CoV
wlim=0.3mm wlim=0.2mm wlim=0.1mm
1 1.1 1.2 1.3 1.4 1.5
0.1 0.15 0.2 0.25 0.3
Partial Factor (γfct,eff)
Model Uncertainty CoV
wlim=0.3mm wlim=0.2mm wlim=0.1mm
Figure 6.28: End Restraint Theoretical Partial Safety Factors of Model Uncertainty (θ) for Varying Model Uncertainty Coefficient of Variance (hc, eff =h/2)
Comparing the theoretical psf’s obtained for the end restraint crack model where the effective depth was h/2 to the end restraint crack model where hc,eff = 2.5(c +φ/2), it may be found that the theoretical psf’s obtained for most variables were greater in value, but only slightly. For a 0.2 mm crack width limit and at a model uncertainty CoV of 0.3, factors of 1.02 and 1.01 were where the model uncertainty and the effective concrete tensile strength’s respective theoretical partial safety factors were greater in the case where the effective depth was hc,eff = h/2 as compared to the end restraint crack model where hc,eff = 2.5(c +ϕ/2). Considering concrete cover, the theoretical partial safety factors where hc,eff = 2.5(c +φ/2) was greater in magnitude by factor 1.06 than where the effective depth were h/2 (a larger difference in magnitude than those experienced by model uncertainty and the effective concrete tensile strength at the same crack limit of 0.2 mm and model uncertainty CoV of 0.3). Additionally, section thickness had theoretical psf’s amounting to 1 in either variations of the end restraint crack model. Overall, the theoretical partial safety factors obtained for the respective variables were quite comparable.