Chapter 6: Sensitivity Analysis of EN 1992 Crack Model: Methodology, Results and Discussion
6.2 Results and Discussion
6.2.4 Influence of the Choice of Reliability Index (β)
6.2.4.1 Edge Restraint
To determine the impact a change in the target reliability index on the EN 1992 edge restraint crack model with hc,eff = 2.5(c +ϕ/2) and where hc,eff = h/2 (referring to Table 6.9 and Table 6.10 respectively), the reliability index was changed from 0.5 to 2 whilst maintaining the crack width limit (wlim = 0.2 mm) and model uncertainty CoV (0.3). In the case where the effective depth of the tension zone was given by hc,eff = 2.5(c + ϕ/2), the relative influence held by both concrete cover and model uncertainty remained effectively constant through the changes in reliability index. Both the theoretical psf’s for concrete cover and model uncertainty increased with an increase in the reliability index. Regarding β = 0.5 as a base, theoretical partial safety factors for concrete cover increased by 6% and 10% with an increase in reliability index to 1.5 and 2 respectively. And for model uncertainty, an increase in theoretical partial factors of 32% and 51% where corresponding reliability indices β = 1.5 and 2 were compared against selected base reliability index 0.5. Clearly, a change in the reliability index has the most effect on the model uncertainty of the EN 1992 edge restraint model. Section thickness, having no part in the edge restraint model where hc,eff = 2.5(c + ϕ/2), has no relative influence with a sensitivity factor of 0 and theoretical partial safety factor of 1.
A 50% and 89% respective increase in reinforcement was obtained for reliability index 1.5 and 2 to be met with respect to reliability index 0.5– a substantial increase. A 25% increase in reinforcement was required where the reliability index was increased from 1.5 to 2. Thus a change in reliability index could have a considerable financial effect on the design of liquid retaining structures with elements restrained along their edge.
Table 6.9: Influence of Reliability Index on the Basic Variables of the EN 1992 Edge Restraint Crack Model (wlim = 0.2 mm, model uncertainty CoV = 0.3, hc,eff = 2.5(c + ϕ/2))
Sensitivity Factors Partial Factors
β %As required αh αc αθ γh γc γθ
0.5 1.060 0 -0.386 -0.922 1.000 1.018 1.098
1.5 1.600 0 -0.394 -0.919 1.000 1.081 1.446
2 2.000 0 -0.399 -0.917 1.000 1.115 1.657
The same exercise was extended to where the effective depth of tension zone, hc,eff,, was h/2 (results of which were presented in Table 6.10). The sensitivity factors for section thickness and model uncertainty were slightly influenced by the change in reliability index. Concrete cover, on the other hand, increased by 37% and 62% for reliability indices 1.5 and 1.2 when compared against the sensitivity factor when the reliability index was set at 0.5. Section thickness and
concrete cover obtained theoretical psf’s that varied only slightly as the reliability index was increased. Model uncertainty’s theoretical partial safety factor increased by 35% and 56% for reliability indices 1.5 and 2 respectively when compared against the theoretical partial factor obtained for where the reliability index was set at 0.5. A comparable result to those obtained for when the effective depth of the tension zone was 2.5(c + ϕ/2).
Reinforcements required to meet a reliability index of 1.5 and 2 as compared to those required for β = 0.5 are 46% and 79% respectively. A considerable increase in reinforcement, which would have a proportional impact on the cost of design where the reliability index is changed. An amount of 23% more reinforcement was required where the reliability index was changed from 1.5 to 2.
These were overall smaller increases as compared to the results for the case where the effective depth of tension was 2.5(c + ϕ/2).
Table 6.10: Influence of Reliability Index on the Basic Variables of the EN 1992 Edge Restraint Crack Model (wlim = 0.2 mm, model uncertainty CoV = 0.3, hc,eff = h/2)
Sensitivity Factors Partial Factors
β %As required αh αc αθ γh γc γθ
0.5 1.054 -0.027 -0.086 -0.996 1.000 0.995 1.110
1.5 1.541 -0.025 -0.118 -0.993 1.000 1.015 1.495
2 1.889 -0.024 -0.139 -0.990 1.000 1.031 1.732
6.2.4.2 End Restraint
For end restraint, the sensitivity factors for all basic variables remained relatively constant (referring to Table 6.11 where the effect depth of tension was 2.5(c + φ/2)). As the reliability index was increased from 1.5 to 2 the theoretical partial safety factors of concrete cover increases by 5% and 8% correspondingly. For the effective concrete tensile strength this increase was about 10% and 15% for reliability index 1.5 and 2 respectively as compared to the corresponding theoretical partial safety factor for reliability index 0.5. Considering the theoretical partial safety factors of model uncertainty, an increase of 27% and 43% was experienced for reliability indices 1.5 and 2 respectively as compared against the theoretical partial safety factor obtained where β was 0.5. Again, a change in the reliability index had the largest effect on the model uncertainty.
Increases of 26% and 42% in steel reinforcement would be required to meet a reliability index of 1.5 and 2 as compared against β = 0.5. An increase of the reliability index from 1.5 to 2 resulted in an increase in reinforcement of 13%. These increases were smaller than those required for edge restraint, although there are still significant. Evidently, the target reliability index set for the
EN 1992 cracking serviceability limit state has a considerable impact on the cost of the design of liquid retaining structures.
Table 6.11: Influence of Reliability Index on the Basic Variables of the EN 1992 End Restraint Crack Model (wlim = 0.2 mm, model uncertainty CoV = 0.3, hc,eff = 2.5(c + ϕ/2))
Sensitivity Factors Partial Factors
β %As required αh αc αfct,eff αθ γh γc γfct,eff 1/γfct,eff γθ
0.5 1.540 -0.024 -0.337 -0.504 -0.795 1.000 1.020 1.030 0.971 1.077 1.5 1.935 -0.023 -0.343 -0.503 -0.793 1.000 1.068 1.133 0.882 1.366
2 2.181 -0.023 -0.345 -0.502 -0.793 1.000 1.097 1.189 0.841 1.538
In the case where the effective depth of tension zone was h/2, the sensitivity factors for section thickness, concrete tensile strength and model uncertainty were only slightly affected by the change in reliability index (as evident in Table 6.12). A similar trend may be found for the theoretical partial safety factors where a small variation was experienced as the reliability index was increased. Concrete cover obtains sensitivity factors that increased in value by 19% and 31%
for reliability indices 1.5 and 2 correspondingly as compared to the sensitivity factor obtained for a reliability index of 0.5. The same comparison being applied to the theoretical partial safety factors (with β = 0.5 as the base) of concrete cover showed an increase in value of 2% and 3% for β = 1.5 and 2 respectively. However, model uncertainty had increases in value of 29% and 47%
for reliability indices 1.5 and 2 respectively, where β = 0.5 was the base of comparison – a comparable finding to where the hc,eff = 2.5(c + φ/2).
The demand in reinforcement increased by 24% and 39% for reliability indices 1.5 and 2 as compared to that which was required for a 0.5 reliability index. An increase in reliability index from 1.5 to 2 results in a 12% steel reinforcement. Once again, a comparable result to where hc,eff
= 2.5(c + φ/2).
Table 6.12: Influence of Reliability Index on the Basic Variables of the EN 1992 End Restraint Crack Model (wlim = 0.2 mm, model uncertainty CoV = 0.3, hc,eff = h/2)
Sensitivity Factors Partial Factors
β %As required
αh αc αfct,eff αθ γh γc γfct,eff 1/γfct,eff γθ
0.5 1.532 -0.047 -0.098 -0.532 -0.840 1.000 0.996 1.033 0.968 1.077 1.5 1.906 -0.045 -0.117 -0.531 -0.838 1.001 1.015 1.143 0.875 1.394 2 2.135 -0.044 -0.128 -0.530 -0.837 1.001 1.027 1.201 0.832 1.580
Overall, a large increase in reinforcement was observed where the reliability index was increased from 0.5 to 2. This was particularly evident for the edge restraint case. Results for where the effective depth of the tension zone was either 2.5(c + ϕ/2) or h/2 were comparable for both the edge and end restraint case. The above-mentioned observations were much greater than the EN 1992 load-induced cracking case (Retief, 2015) in which the amount of tension steel increased by 10% and 15% for β = 1.5 and 2 respectively (where β = 0.5 was set as a default value). Clearly, a change in the choice of reliability index of the EN 1992 restrained shrinkage crack model may be deduced to have a considerable effect on the cost of design. However, these increases in cost may be minor when compared to those required for structural failure where this serviceability limit state is not met. Further research into the cost of failure for the serviceability limit state is required;
this should give clearer insight into what the target reliability index should be for liquid retaining structures.