6. Target reliability in case of extreme effects
6.3. Target reliability estimation in case of seismic design situation
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curves (e.g. ISO standard fire curve) may lead inconsistent structural reliability. In case of the investigated example considering both ρ=0.4 and 0.9 correlation coefficients, reliability index β=2.9 – 3.3 may be achieved based on the time demand R45, R30 and R15, respectively, using EC3-1-2 conforming prescriptive design and ISO standard fire curve [BT10]. It means that design based on ISO standard curve is too conservative in case of structures with low failure consequences and it may be unsafe when a possible failure has considerable consequences. The target reliability index may be selected between 2.8 and 3.7 based on the possible failure consequences for industrial steel tapered portal frames with storage function. The presented values are calculated on the basis of Hungarian circumstances, considering the regulations of OTSZ 5.0 and TvMI 5.1.
The presented values may be used later in performance based design, however, further research work is needed in order to extend and validate the suggested numbers and in order to understand better the components of failure costs (Cf). The target reliability indices may be also influenced by the acceptance ability of the society and global economy of the country, so in some cases minimum limits may be used in order to ensure the minimum desired safety.
50 years service life: calculated
Relative cost of safety measure Minor consequences
Moderate consequences
Large consequences High – Severe fire 2.8 (2.8 – 2.9) 2.8 – 3.0 (2.8– 3.1) 2.8 – 3.2 (2.8 – 3.4) Moderate – Medium fire 2.8 (2.8 – 3.0) 2.8 – 3.2 (2.8 – 3.3) 2.8 – 3.5 (3.0 – 3.6) Low – Minor fire 2.8 (2.8 – 3.0) 2.8 – 3.3 (2.8 – 3.4) 3.2 – 3.5 (3.2 – 3.7) Table 6-4 – Calculated target reliability indices for industrial steel tapered portal frame considering ρ=0.4 and
ρ=0.9 correlation among the frames
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# hc1 - hc2 x tw,c + bc x tf,c hb1 - hb2 x tw,b + bb x tf,b db Equipment
load C0 C1
1 300-700x6+200x11 380-700x6+180x8 16 mm 0.2 kN/m2 47,750 € 860 € 2 300-750x6+220x14 380-770x6+180x12 24 mm 1.0 kN/m2 51,370 € 1,940 € 3 400-750x6+260x14 400-770x6+220x14 28 mm 2.0 kN/m2 55,260 € 2,640 €
Table 6-5 – Reference structural configurations
For sake of simplicity, the reliability indices are calculated only in longitudinal direction related to the failure of side diagonal braces without considering the sheeting system’s stiffness in the analysis. For dissipative design, the limit state function expresses a deformation check of tension braces where the allowable deformation is set 7Δy (Δy – axial deformation at expected tensile yielding load). The nonlinear structural response is obtained with pushover analysis considering 0.01Einit post yielding stiffness and considering second order (also known as P-Δ) effects of the gravity loads on the columns. N2 method [111] is invoked in order to find the target displacement.
In elastic design case, the limit state function expresses tension verification of braces in tension supposing full strength connection at the ends.
The calculated probabilities are not conditional values, they contain the occurrence of seismic action via the hazard curve. The diameter of side bracing is varied from 4 to 84 mm altogether in six investigations, since the reliability is calculated using both elastic and dissipative limit state functions (Fig. 6-6). Due the design criteria related to persistent design, tension-only braces with diameter db<16mm cannot be used since 16mm is minimum that is able to resist the resultant wind force.
Fig. 6-6 – Failure probabilities and reliability indices of investigated structures for Komárom site as a function of the bracings’ diameter
Regarding to the tension-only braces, db=16mm, 24mm and 28 mm diameters would be sufficient according to elastic design rules of EC8-1 in the first, second and third case, respectively.
The presented reliability indices in Fig. 6-6 imply that the EC8-1 conforming elastic design rules
4 12 20 28 36 44 52 60 68 76 84
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Diameter [mm]
β − reliability index
4 12 20 28 36 44 52 60 68 76 84
0 0.1 0.2 0.3 0.4 0.5 0.6
Diameter [mm]
Pf − failure probability
#1 dissipative
#2 dissipative
#3 dissipative
#1 elastic
#2 elastic
#3 elastic
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provided β ≈ 1.5 reliability index for ~1.0 D/C ratio in the investigated cases due to the high uncertainty in the seismic effects (Table 5-1 and Fig. 5-3). It does not necessarily mean that all of the structures designed according to the prescriptive design rules of EC8-1 standard have similarly low reliability. The presented values may be not generalized easily due to the following issues: a) the indices are valid for the investigated structural configuration considering that the dead loads are dominant; b) the structure is investigated in longitudinal direction thus its behaviour can be represented well with a simple SDOF system; c) the failure probabilities are calculated for an elastic structure neglecting the possibility of energy dissipation by plastic deformation except the fact that q=1.5 behaviour factor (also known as seismic response modification factor, R, in ASCE7-10 [115]) is used for the calculation of EC8-1 design spectra. In case of structures where higher vibration modes become dominant, higher reliability index may be calculated, however, the results imply the fact that the desired reliability level may not be achieved in some cases using the prescriptive rules of EC8-1. In case of brittle failure modes, dashed lines (Fig. 6-6) may describe better the achievable reliability level for seismic design situation. Reinforced concrete structures are not incorporated in this study, other researchers published low reliability values for shear failure of reinforced concrete sections, e.g. [113], thus this issue also needs to be further investigated.
In case of dissipative structural configurations where significant plastic deformations are allowed and the failure mechanism of the structure is controlled and designed, significantly higher reliability indices may be achieved using EC8-1 conforming design. For example, Zsarnóczay in [67] presented reliability indices between 2.9 and 3.6 for a large set of BRB frames (designed with q=7 behaviour factor). The investigated structure may also be capable to absorb energy during seismic excitation, tension failure of bracing elements with full strength connections can be considered as a dissipative failure mode. For this reason, the considerable structural reliability achieved with the help of prescriptive rules of EC8-1 is higher than β ≈ 1.5. Considering tension-only braces with db=16mm, 24mm and 28 mm diameters, respectively, the resulted reliability indices vary between 2.3 and 2.5. These reliability indices satisfy the criteria of JCSS (Table 1-1) suggested for seismic design (high relative cost of safety measure) of a structure having moderate consequences in case of failure, however, the target indices of EC0 (Table 1-1) may be hardly achieved.
The above discussed reliability index values (Fig. 6-6) and the following life cycle cost function, Eq. (42), are applied in order to investigate the optimal safety level in seismic design
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situation for Komárom site considering structures loaded with low, moderate and high gravitational and seismic loads (Table 6-5). The cost function is formulated similarly to Eq. (9) but here with only one variable that refers to a diameter for the braces:
( )
x C P( )
x C0 C1 x2CLC = f ⋅ f + + ⋅ , (42)
where C1 is the cost factor related to the braces. Different sites are not incorporated in this investigation, thus for sites having hazard curve characterized with different mean or different variation the calculated optimal reliability indices may be slightly different than the presented values in Section 7.4.2. The minimum of the presented life cycle function is found using direct search algorithm. Some increase or decrease in the seismicity have similar effect on failure probability than some increase or decrease in the gravitational loads. For this reason, the presented results may be extended for wider set of possible cases.
The calculated target reliability indices for seismic design are presented in Fig. 6-7 for 6 considered cases (Table 6-5) and with different Cf/Ctot ratios (this ratio for the reference structure is around 50 considering normal design conditions). It has to be noted that the presented values may be only valid for steel portal frame structures with similar configuration (tension-only braces for wind- and side bracing) and size analysed in longitudinal direction.
Fig. 6-7 – Target reliability indices for various consequences for seismic design of example portal frames
The results are summarized in Table 6-6; however, more accurate values can be read from Fig.
6-7 in a specific design situation. It can be concluded that the possible target reliability indices depend not only on the consequence but also on the seismicity or the severity of the seismic effect and on the required strengthening cost. In Fig. 6-5, from the point where the curves change to horizontal lines, they only indicate the values because the optimal diameter amplifiers are out of
1 1,5 2 2,5 3 3,5 4 4,5
1 10 100 1000 10000
β-target reliability index
Cf /Ctot
elastic 0.2kN/m2 elastic 1.0kN/m2 elastic 2.0kN/m2 dissipative 0.2kN/m2 dissipative 1.0kN/m2 dissipative 2.0kN/m2 JCSS (high relative cost) JCSS (moderate relative cost) JCSS (low relative cost) EC0 recommended limits
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the investigated range (4-84! mm). The presented indices in this section will be compared and evaluated later together with results of Section 7.4.
Table 6-6 – Calculated target reliability indices in longitudinal direction for industrial steel tapered portal frame situated next to Komárom - brittle and dissipative failure mechanism
The calculated reliability indices are lower than the suggested values in EC0 and they are closer to the recommendations of JCSS. Recommended values of Joint Committee on Structural Safety for low and moderate cost of safety measure seem to be appropriate for low and medium seismic mass, respectively. If significant concentrated loads act on the columns, higher seismic mass have to be considered during the design. In these cases target indices may be lower than the calculated ones and JCSS recommended values for high relative cost of safety measure may be used. Due to the fact that a wide range of possible consequences, different intensity of gravitational loads, elastic and dissipative structural configurations have been taken into account by the derivation of reliability index values; these conclusions may be extended to other type of steel structures, as well.
The EC8-1 standard defines importance classes in order to differentiate structures having different failure consequences (similarly to the consequence classes in EC0). EC8-1 uses the so-called importance factor (γI) multiplying the considered PGA in the analysis (it can be interpreted as the return period of considered seismic event changes). The examples given in the code are mainly related to structures with different functions, however, higher or lower consequence class may be selected during the design if the failure causes more or less significant economic losses, respectively. This differentiation (in better agreement with the discussion in the code) is made and possible target indices are presented in Table 6-6 based on assumptions made by the author related to Cf/Ctot boundaries. Importance classes III and IV represent extraordinary cases, importance classes I and II cover most of the cases in practice for industrial steel portal frames.
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The investigated configurations having tension-only braces with db=16mm, 24mm and 28 mm diameters, respectively, have been designed considered importance class II (ordinary structure, γI=1.0) and elastic design rules of EC8-1. In case of brittle failure modes, where no plastic behaviour may be developed, the resulted reliability indices varied between 1.2 and 1.8 considering all the importance classes for each configuration (marked as light red zone in Fig. 6-7). These values are rather low (lower than the lowest limit of JCSS) and may be not acceptable. In case of dissipative failure mode the indices varied from 2.3 to 2.7 (marked as light blue zone in Fig. 6-7).
Higher values (β>3) may be reached with design of DCH [28] structures (e.g. BRBF) [BT7]. These results point another advantage of dissipative design (it is well known that using dissipative design the cost of bracing system can be significantly reduced), namely that seismic structural reliability can be remarkably increased. Even in the case of a simple industrial portal frame designed elastically against seismic effects, it is favourable to give the opportunity to development of plastic deformations in order to avoid brittle failure modes (e.g. design full strength brace’s connections in order to let yield tension failure being the leading failure mode and avoid shear failure of bolts with some overstrength).
It seems that the target indices suggested by EC0 [52] are too high and not appropriate in seismic design situation in case of industrial portal frames having similar function and geometry to the investigated frame. Although this conclusion has been drawn based on results in longitudinal direction (for Komárom site); similar or lower eliability indices may be calculated in transversal direction since the cost of improving structural capacity is higher. Further investigation is necessary concentrating on target reliability indices for seismic design in order to extend the validity of conclusions for different kinds of structures, failure modes and sites. Parametric study results will help to extend the conclusions for wider range in Section 7.4.2. For the time being, β=2.0-3.5 seems to be economical and optimal for most of the practical cases. This target value may be satisfied without any problems if the designer makes an effort on overstrength brittle failure components and ensure dissipative failure modes.
The resulted reliability index considering elastic limit state (brittle) are too low for a structure designed elastically according to the recent regulation, namely EC8-1. It seems that the rules need to be revised in the future. At the moment, from the point of view of the seismic effects EC8-1 introduces a safety factor in the design procedure since the considerable peak ground acceleration has 10% probability of exceedance in 50 years (0.9 fractile of 50 years mean hazard curve, as it
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can be seen in Fig. 5-3). However, this 10% exceedance probability results only β ≈ 1.3 reliability index for 50 years and the fact that resistance is accounted on design value by the verification cannot increase the structural reliability significantly in case of the investigated steel structure since the limit state function is dominantly sensitive for seismic uncertainties (Table 5-1). In spite of the fact that the review of the rules in the code is not one of the scopes of this study, one possible issue may be the neglecting the uncertainty of hazard level (Fig. 6-8) in code conforming design (the input parameter related to seismic effect intensity is typically the 0.9 fractile of 50 years mean hazard curve). It is found that if the desired safety level were β ≈ 2.0 reliability index considering elastic limit state function, the 0.95 fractile hazard curve should be used for design purposes in order to cover the uncertainty in the intensity because there is a considerable uncertainty related to the usage of different attenuation and ground motion prediction models.
Fig. 6-8 –Hazard curves with different fractiles for Komárom site [106]
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