Motor stand Base plate
F.2 Strength Factor
Using the calculated loads from the design verification analysis, the bending moment and shear calculated at the motor to stand interface were applied to the corrected dimensions and the resulting bending plus axial stress was computed to be about 7,000 psi. The material was A-53 Grade B, with 35 ksi specified yield strength. The median yield strength for low carbon steels is about 1.20 times the specified yield, resulting in a median yield strength of 42 ksi. The median factor of safety relative to yield is then 42/7 = 6.0.
It was assumed that any appreciable plasticity in the motor stand would result in misalignment between the motor and pump shafts and damage the coupling between the motor and pump assembly thus, the function would be lost. The plastic hinge shape factor in this case is greater than 1.5 but a full plastic hinge in a support assembly that assure alignment of rotating members is likely beyond a median capacity. Failure was assumed to occur at a factor of 1.5 beyond yield.
The onset of yielding was considered to be about a 95% confidence value. The resulting capacity factor is:
There are two sources of uncertainty in the capacity factor—the material yield strength and the capacity beyond yield. The specified yield is a 95% confidence value, so the uncertainty on yield strength is:
The uncertainty in the failure threshold is estimated based on the assumptions that the yield point is a 95% confidence capacity. The resulting β is: U
Example Problem for Service Water Pump
Combining βuy and βuf by SRSS results in a βuc on capacity of:
28 . 0 ) 245 . 0 14 . 0
( 2 2 1/2
UC = + =
β
F.3 Equipment Response Factor
Equipment response variables are Qualification Method, Damping, Modeling, Mode Combination and Earthquake Component Combination.
F.3.1 Qualification Method
The analysis was a response spectrum finite element analysis which is considered to be unbiased.
However, the modal response of the motor stand was computed at 3.9 Hz. The motor stand was modeled as being more flexible than the actual geometry. The moment of inertia calculated from the field dimensions taken by the IPEEE team showed that the stiffness in the weak axis was actually 6.5 times the stiffness calculated in the model. So the actual frequency of the motor on its stand would be greater by the square root of 6.5, or about 10 Hz vs 3.9 Hz from the computer model. This frequency is at the peak of the broadened spectrum shown in Figure F-3. The spectral acceleration defined at 2% damping used in the design analysis was 0.42g. 5% damping is considered to be a median damping value. As shown in Figure F-3, if the 5% damped design spectral peak is unbroadened, and the best estimate of frequency is 10 Hz, the qualification method factor including damping is the ratio of 2% damped spectral acceleration at 3.9 Hz vs 5%
damped spectral acceleration at 10 Hz.
86 . 50 0 . 0
43 . RQM =0 =
Example Problem for Service Water Pump
Assumed Unbroadened
Spectrum
Figure F-3
Demand Response Spectrum, 5% Damping
The uncertainty associated with the unbroadening of the design basis response spectrum can be estimated from the ratio of the broadened to unbroadened spectral acceleration at 10 Hz. Since the peak of the broadened spectrum is an upper bound (at least a 99% non-exceedance
probability value, i.e., 2.33 β value) the qualifications method uncertainty is U
0.50 0.04 ln0.55 2.33
1
UQM = =
β
F.3.2 Damping
The calculated dominant frequency of the motor stand was about 3.9 Hz and the associated two percent damped spectral acceleration used in the analysis was 0.42 g. Five percent damping is considered to be median centered. However, when computing the damping factor and its uncertainty on response, the frequency should first be corrected to reflect the actual motor stand geometry vs. the softer geometry used in the model. For this case the damping factor was incorporated in the qualification method factor. The response at 10 Hz is near the peak of the spectrum shown in Figure F-3. In the amplified response regime, the ratio of spectral
accelerations at different damping levels can be approximated as the square root of the damping rates. If 2% damping is considered a -2 β value, the uncertainty in damping is: U
2 0.23 ln 5 2 1
UD = =
β
Example Problem for Service Water Pump
F.3.3 Modeling
At the estimated 10 Hz elastic frequency of the motor stand, and considering the unbroadened spectrum in Figure F-3, the spectrum is very steep, so a small error in frequency results in a large change in spectral acceleration. Note that we have already accounted for the conservatism in broadening in the qualification method factor.
For fragility calculations, it should be considered that as the response goes beyond the linear region, the effective frequency reduces and in the case for the steep slope of the demand
spectrum and the corrected pump motor stand frequency, the response could significantly reduce.
It is also a common to slightly over estimate stiffness by assuming idealized inflexible bounding conditions. It was judged that a reasonable estimate of the effective spectral acceleration would be about half way between the 5% damped peak of 0.55g and the valley of 0.31g, or about 0.43g.
The elastic spectral acceleration used in developing the qualification method factor was 0.5g.
The modeling factor can then be computed as: The modeling factor, including damping, is then:
16 . 43 1 . 0
50 . FM=0 =
The range of response from the peak to the valley is considered to be plus or minus 2.33 β and U the β for modeling is computed as: U
0.12 (.55/.31) 2(2.33) ln
1
UM =
= β
F.3.4 Mode Combination
The analysis was multimode dynamic analysis. The first mode was predominantly the pump column and second mode was the pump motor and stand. The next modes are above 13 Hz and are not particularly influential to the pump stand response. There is no bias in the mode
combination method so the mode combination factor is unity:
0 . 1 FMC =
The randomness on mode combination when the response is predominantly in one mode is estimated to be:
05 .
MC 0
R =
β
F.3.5 Earthquake Component Combination
The analysis was in accordance with the licensing criteria wherein the worst horizontal
directional response was combined with the vertical response by absolute sum. Median centered response is considered to be a combination of all three directional responses by the 100, 40, 40
Example Problem for Service Water Pump
rule. The critical response is all horizontal and then 100, 40, 40 horizontal vector is 1.08 times the single direction response. The earthquake component combination factor is then:
FECC = 1/1.08 = 0.93
Assuming that both horizontal components being in phase is a 3σ extreme, the βR for earthquake component combination is:
1.08 0.09 ln 2
RECC =
= β
F.3.6 Equipment Response Factor
The equipment response factor is then:
FRE = FQM (FD)(FM)(FMC)(FECC)
FRE = 0.86(1.0)(1.16)(1.0)(0.93) = 0.93
βs are combined by SRSS to yield:
βR = (0 + 0 + 0 + 0.052 + 0.092)1/2 = 0.10 βU = (0.042 + 0.232 + 0.122 + 0 + 0)1/2 = 0.26
F.4 Structural Response Factor
The pump structure is founded on rock and has a fundamental frequency of about 10 Hz as indicated by the top of roof response spectrum in Figure F-3. Design damping was 4% as indicated by Regulatory Guide 1.61 for structures at less than half yield. The pump structure is simple and torsional effects are minimal. Spectra were developed from a fixed base 2
dimensional model.
Variables considered are spectral shape, damping, modeling, mode combination and ground motion incoherence.
F.4.1 Spectral Shape
The UHS spectrum for the site, when anchored to the DBE pga, exceed the DBE spectrum at the 10 Hz fundamental frequency of the structure by 45%. The resulting spectral shape factor is:
69 . 45 0 . 1 FSS = 1 =
From Reference F2, the randomness is about:
Example Problem for Service Water Pump
For fragilities anchored to pga, the uncertainty at 10 Hz is about:
16 .
SS 0
U =
β
In addition, the earthquake ground motion is specified as the average of two horizontal
components. The peak response in one direction will govern the structural response of the 2D model. From Tables 3-2 and 3-3 of Reference F2,
92
The 4% design damping utilized in development of response spectra is considered median for the low stress level in the structure.
FD = 1.0
UHS spectra are only provided at 5% damping. At 10 Hz the UHS spectra are rising as opposed to the decayed peak of the DBE spectrum. In this case, the uncertainty in response due to damping is estimated as:
15
The model is simple, is fixed base and dominant response is primarily in a single mode as indicated by the spectrum in Figure F-3. Code properties were used for concrete thus, according to the guidance in Reference F2, the calculated frequency is considered to be median centered.
FM = 1.0
Example Problem for Service Water Pump
The UHS spectra slope is not very steep at 10 Hz and the uncertainty due to variability in structural frequency is estimated as:
10 .
f 0
U = β
The mode shape for the short stiff fixed base structure is considered to not vary significantly from variations in modeling parameters.
05
The response is primarily in a single mode.
FMC = 1.0
F.4.5 Ground Motion Incoherence
The structure is 58” x 124’ in plan dimension. At 10 Hz, the GMI factor was calculated to be:
FGMI = 1.07 βU = 0.03
F.4.6 Structural Response Factor
The resulting structural response factor is:
67