Flexible equipment and distributive systems that are designed by analysis usually have ductile failure modes but for purposes of screening it is assumed that the failure mode is non-ductile such as for failure of anchor bolts or welds or buckling. Fillet welds are shown in Reference E1 to have as much larger margin than implied by the design codes so fillet welds should not be the basis for the screening computation. The following assumptions are made for the screening calculation:
• Probable frequency range is 3-10 Hz (flexible equipment).
• 2% damping was used for the design whereas 5% is considered median with 2% defined as a -2 β case. U
• The screening is only applicable to locations outside of the primary containment where the effects of hydrodynamic loads are minimal.
• Code margins are those inherent in the ASME code where the allowable stress may be as high as 70% of the specified ultimate strength, resulting in a nominal safety factor of 1.43.
Development of Generic Fragility Descriptions for Purposes of Screening Based Upon Design Criteria
The safety factors of expansion anchors and welds are greater so the ASME criteria governs for the screening calculation.
The methodology used follows that described in Reference E2.
E.4.1 Strength Factor The Strength Factor, Fs, is:
DBE N Fs FU −
=
FU is the ultimate strength (stress/load), N is the normal load or stress and DBE is the seismic load or stress. Since the 0.5g UHS structure spectra are essentially equal to the 0.25g DBE in-structure spectra in the 3-10 Hz frequency range, the basis for DBE stress is considered to be anchored to 0.5g pga. Conservatively, consider that the only normal load effect is from weight which usually is a low contribution to the ultimate load capacity. The median normal load is assumed to be 10% of the ultimate capacity with a + 1β value equal to 20% of the ultimate capacity. The median ultimate strength is about 1.1 times the code specified value. The code specified value is set at the 95% confidence level, which is a -1.65 β value. It is assumed that U the average demand is 70% of the code allowable with 100% assumed as a 95% probability value (+ 1.65 ). The code allowable can be as high as 70% of the code specified ultimate strength, therefore the median load/stress is (.7)(.7) = 0.49 of the ultimate capacity based on code specified strength. The DBE load/stress is then (0.49-0.1) = 0.39 times the ultimate capacity.
The strength factor is then:
βU
56 . 39 2 . 0
1 . 0 1 .
Fs=1 − =
Using the approximate second moment method from Reference E2 for calculating β, the β is U computed to be 0.30.
E.4.2 Equipment Response Factor
The equipment response factor consists of the product of the individual factors for the variables of Qualification Method, Damping, Modeling, Mode Combination and Earthquake Component Combination.
Qualification Method
It is assumed that a dynamic response spectrum analysis was conducted as opposed to a more conservative static coefficient method. There is no particular bias in the response spectrum method but considerable conservatism can accumulate from several variables. In the case of piping, often the envelope response spectra were used which is very conservative. However, if we consider a single flexible component or a subsystem supported from the same elevation, this
Development of Generic Fragility Descriptions for Purposes of Screening Based Upon Design Criteria
conservatism cannot be counted. The practice of peak broadening and smoothing introduces conservatism. A prior study of single degree of freedom systems revealed that the degree of conservatism for spectra soothing in the frequency range of interest is about a factor of 1.2 with a
of 0.09.
The factor of conservatism that results from using 2% damping in design vs 5% median is quantified by:
The design spectra were used to find an average value of 1.17 for the spectral acceleration ratios between 2% and 5% damping in the 3-10 Hz frequency range. If 2% damping is a -2 value,
Modeling error can arise from frequency error and mode shape difference between the model and the actual response. The model would normally be median centered so the modeling factor would be unity. The spectra peak at very low frequency, are fairly flat between 5 and 10 Hz and have a factor of 2 difference from 3-5 Hz (Figure E-1). For the steepest slope between 3 and 4 Hz, a ±1 βf frequency shift results in a 25% difference in response. Thus:
0.22 ln(1.25)
Uf = =
β
The response β due to mode shape error is estimated to be about 0.15. Combining βs by SRSS, the β for modeling is:
Development of Generic Fragility Descriptions for Purposes of Screening Based Upon Design Criteria
Mode Combination
Mode combination was by SRSS which is median centered. The response variability due to mode combination is estimated as a βR of 0.15 for multimode response of distribution systems.
Earthquake Component Combination
Earthquake components were combined by SRSS. Response is usually dominated by the two horizontal directions. Considering the two horizontal components to be in phase as a 3 βR case:
0.12 2
3ln 1
R = =
β
Equipment Response Factor Results
Combining the response factors as the product of the individual factors and the βs by the SRSS rule, the equipment response factor and its variability are represented by:
4 . 1 FRE =
19 .
R =0 β
29 .
U =0 β
E.4.3 Structural Response Factor
Spectra were developed by probabilistic methods using a Latin Hypercube simulation process in which all important variables associated with structural response are included. The median results were used to derive the strength and response factors so the structural response factor is unity. The difference between the 50th and 84th percentile spectral accelerations in the 3-10 Hz frequency range defines the composite variability, β . This ratio averages about 1.25 for the C reactor building so the β is 0.22. There is approximately equal variability from random and C uncertainty variables and the corresponding βR and β are 0.22/U 2 = 0.16 each.
E.4.4 Fragility Description for Flexible Components Designed by Analysis
The median peak ground acceleration capacity is the product of the strength, equipment response and structural response factors times the reference 0.5g peak ground acceleration for the UHS.
As previously shown the 0.5g in-structure response spectra for the UHS are equivalent to the 0.25g DBE spectra, hence 0.5g is used as the reference earthquake pga to represent the DBE demand.
Am = 2.56 (1.4)(1.0)(0.5g) = 1.79g
Development of Generic Fragility Descriptions for Purposes of Screening Based Upon Design Criteria
The random and uncertainty variability are the SRSS of the βRs and β for the three variables Us and are computed to be:
25
R =. β
45 .
U =0 β
The HCLPF is computed as:
HCLPF = 1.79g exp(-1.65) (βR+βU) = 0.56g
This exceeds the 0.5g HCLPF and 1.50g median target set as the screening threshold in Section E.1 and, considering the conservative assumptions used in the derivation, this class of component and distribution system can be comfortably screened out subject to a walkdown verification that there are no vulnerable looking details. This calculation was used to screen out piping, cable trays and valves as well as the passive parts of instrument racks and electrical distribution cabinets.
Valves are rigid but the piping systems in which they are mounted are flexible and the piping response dictates the demand for the valves. Valve qualification data were reviewed and the valves usually had a large design margin above the specified demand so the above derivation is also considered applicable to valves with the exception of those that may be identified during the walkdowns that appeared to be outside of the seismic experience database.