Chapter 5: FORM Analysis of EN 1992 Crack Model: Methodology, Results and Discussion . 87
5.3 Results and Discussion
5.3.3 Influence of Section Thickness
Figure 5.9: Influence of Effective Tension Area (End Restraint)
Clearly, the choice of effective depth of tension area affects the overall reliability of the crack model for both restraint conditions– although this effect was slight for both restraint conditions.
During the course of the analysis, where the stochastic nature of the variables had been accounted for, the random variables (cover in particular) increased in magnitude. Meaning that essentially, the concrete cover value at the limit state/ failure point is greater than the initial cover value (40 mm). This increase in value would subsequently lead to an effective depth value that was greater than what would normally be prescribed for the hc,eff = 2.5(c + φ/2) variation of the crack model in a deterministic analysis. Thus, a comparably larger than usual effective tension area would be in effect. This would then result in an increase in the likelihood of the limit state being exceeded, and hence a decrease in the reliability indices obtained. A result that is somewhat counterintuitive and could be better explained through a closer examination of the limit state function. This was done by performing a sensitivity analysis of the EN 1992 restrained shrinkage crack model and the results of which are reported in chapter 6.
Covers beyond 50 mm make h/2 the limiting effective tension depth for a 250 mm thick section, whilst bar diameter contributes little to the effective depth (cover is more influential, as was uncovered in the parametric study). Therefore a small selection of section thicknesses were analysed; namely, 250, 300 and 350 mm for the edge restraint case whilst for end restraint section thicknesses analysed varied from 250 mm to 500mm since section thickness featured in the end restraint case.
5.3.3.1 Edge Restraint
As the section thickness was increased, so did the reliability index (Figure 5.10). The only real difference that section thickness made in the edge restraint case (where hc,eff = 2.5(c+ φ/2)) was that thicker sections require more reinforcement to meet a particular steel to gross concrete cross- sectional area ratio. For instance for a steel to gross cross-sectional concrete area of say 2% a 250 mm section would require 2500 mm 2/sectional face, while 3000 mm 2 and 3500 mm2/sectional face is required for 300 and 350 mm respectively. The increase in reinforcement area results in an increase in the effective steel content (ρp,eff) and subsequently a decrease in crack spacing and crack widths obtained (since the model contains the reciprocal of the effective steel content ratio).
Thus it appears as though the reliability index of the edge restraint crack model increases with an increase in the section thickness.
For the range of section thicknesses analysed, the amount of area required to maintain the target reliability index (βt = 1.5) varied slightly, going from 1.62 %As (giving a steel reinforcing area of 4050 mm 2 for a 250 mm thick section), 1.36 %As (4080 mm 2 for a 300 mm section thickness and 1.17%As (4095 mm 2 for a 350 mm thick concrete section) – an average decrease of 18%
with each 50 mm increase in section thickness. All in all, the steel content is believed to have had the most impact of the reliability performance of the edge restraint crack model rather than the inherent variabilities of the random variables within the model. Thus the physical model rather than the reliability model influenced the outcome of this particular reliability assessment (where section thickness was varied).
Figure 5.10: Influence of Section Thickness (Edge Restraint)
5.3.3.2 End Restraint
It can be clearly seen that an increase in section thickness results in an increase in the reliability index (Figure 5.11), as was found in the edge restraint case. For reinforcement ratio of say 2%, the reliability index is 1.64, 1.97, 2.3, 2.65, 2.93 and 3.19 for section thicknesses 250, 300, 350, 400, 450 and 500 mm sequentially. Comparing the reliability indices obtained for h = 250, 300 and 350 mm at 2% As in the end restraint crack model to those of edge restraint it is observed that the edge restraint crack model produced higher reliability indices (2, 2.38 to 2.69 corresponding to h = 250, 300 and 350mm respectively for the edge restraint case as compared to the 1.64, 1.97 and 2.33 of the end restraint case for h = 250, 300 and 350 mm).
A few factors come into effect in this result. As the section thickness varies, so did the k coefficient- this coefficient accounts for the presence of a non-linear stress distribution (varying between 1 for section thicknesses less than 300 mm and 0.65 for section thicknesses greater than 800 mm, values between these limits being interpolated). Table 5.2 gives those k values used and their corresponding section thickness. Being directly proportional to the restrained strain, the decrease in this coefficient with the increase in section thickness resulted in a decrease in the restrained strain. A decrease in the restrained strain means that a smaller crack width is attained,
0 0.5 1 1.5 2 2.5 3 3.5 4
0 0.5 1 1.5 2 2.5 3
Reliability Index (β)
%As
h=250mm h=300mm h=350mm
βt= 1.5
resulting in a decrease in the likelihood of the crack width limit being exceeded (increasing the model’s reliability).
Table 5.2: Change of k Coefficient with Increasing Section Thickness (by interpolation) h (mm) k
250 1
300 1
350 0.965
400 0.93
450 0.895
500 0.86
Also, much like in the case for edge restraint, the increase in amount of reinforcement area required to meet particular ratio of reinforcing steel to gross concrete cross-sectional area may have also influenced the results. Once again, larger section thicknesses require larger amounts of reinforcement to meet a certain steel to concrete ratio. This then decreases the likelihood of crack limit exceedance and increases the reliability performance of the crack model. The amount of reinforcement required for the reliability index to be met ranged from 1.94%As (giving a reinforcing steel area of 4859 mm 2) to 1.33%As (6655 mm 2) for the range of section thickness from 250 mm to 500 mm considered in this analysis.
To directly compare the results of the end restraint crack model with that of the edge restraint crack model, the amount of reinforcement required to meet the target reliability index for section thicknesses from 250 to 350 mm were examined. At βt = 1.5, a reinforcement to concrete percentage of 1.94, 1.80 and 1.65% are required for section thicknesses 250, 300 and 350 mm respectively (larger percentages of reinforcement are required here than the 1.62, 1.36 and 1.17%
As respectively found for edge restraint). An average decrease of about 8% was observed with every 50 mm increase in section thickness, almost half that experienced in the edge restraint case (Figure 5.11). For the remainder section thicknesses of 400, 450 and 500 mm the corresponding percentage of reinforcement required to meet the reliability index are 1.5, 1.41 and 1.33%As
respectively (Figure 5.11). None of the reinforcement requirements for the section thicknesses analysed exceeded the maximum feasible limit for reinforcement at the target reliability index.
Figure 5.11: Influence of Section Thickness (End Restraint)
Evidently, section thickness does influence the reliability of the crack model, more so in the end restraint crack model than for the edge restraint crack model in that more reinforcing is required to meet the target reliability index. Although, the effect held by section thickness in the edge restraint case has more so to do with the increase in reinforcing required maintaining a particular reinforcement ratio (thus being more a testament to the influence held by the steel reinforcing in the edge restraint crack model) - the variable itself has no role in the edge restraint crack model.
Hence, the section thickness actually appearing in the end restraint crack model by default has more of an effect on the end restraint model. A closer examination of the sensitivity factors obtained for a predetermined reliability index should expose to what extent section thickness influences the end restraint crack model.