FUNCIONARIOS EJECUTIVOS DEL REGISTRANTE
ÍTEM 15. ANEXOS, CUADROS DE LOS ESTADOS FINANCIEROS
= statistical mean of variable R.
At the optimal design limit, the mean of R becomes:
max
The coefficient of variation for the strength is Cov R.
Load
The total applied nominal moment at the ASCE/SEI 7-10 700-year wind speed is:
1 700
MT = MD+M where:
M T 1 = total nominal moment at ASCE/SEI 7-10 700-year wind speed,
To standardize the comparisons between ASD and LRFD, and for any specified year of wind, all analyses and compari-sons are based on the total nominal moment for the LRFD 700-year total applied moment equal to 1.0:
1.0
1 700
MT = MD + M =
and, the dead load moment can be represented by:
1 700
MD = −M
The calibration and comparison varies M 700 from 1.0 to 0.0, while M D varies from 0.0 to 1.0 so that the total applied nomi-nal moment at the ASCE/SEI 7-10 700-year load remains 1.0.
The total applied nominal moments for ASD and other LRFD year wind speeds are adjusted to be equivalent to the ASCE/
SEI 7-10 700-year wind speed load case.
Given that the nominal moment from wind for any year wind can be determined by:
700
V T = wind speed for any year T wind speed, and M WT = nominal wind moment at any year T , and the total applied nominal moment becomes:
1 700
M T z = total applied nominal moment at any year T wind speed.
To determine the mean wind moment for the reliability analyses, the mean moment at the 50-year wind speed is determined from the ASCE/SEI 7-10 wind speed relation:
0.36 0.10 ln 12 50
and the nominal wind moment at the 50-year wind speed becomes:
50 2
M = λ V M 700
The nominal moment at the 50-year wind speed is propor-tional to V 2 by:
50 502
M ∝K K GC V d z d
where:
K d =directionality coefficient, K z = elevation coefficient,
G = gust factor, and C d = drag coefficient.
The mean wind moment for the reliability analyses is:
50 502
M ∝K K GC V d z d
where the variables are the means. Assuming that K d does not vary, the other non-wind speed variables’ nominal values are related to the means by the bias factors. Combining them into a single bias factorλ P gives:
K GCz d = λ λ λ Kz G Cd K GCz d = λ PK GCz d
Considering that the map design values may differ from the statistical mean of the 50-year wind speed, the mean 50-year wind speed can be represented by:
50 50 700
X V = bias for the 50-year wind speed, µ50 = mean 50-year wind speed, and V 50 = map design 50-year wind speed.
The mean wind moment for the reliability analyses becomes:
50 2 2
M = λ λ λ P V X M 700
where:
700 7002
M ∝K K GC V d z d
Referring back to the basis that all compariso
Referring back to the basis that all comparisons are equatedns are equated with a total ASCE/SEI 7-10 applied nominal moment of:
with a total ASCE/SEI 7-10 applied nominal moment of:
11
λ DD== bias factor for dead load moment. bias factor for dead load moment.
The mean load effect
The mean load effect on the structure becomes:on the structure becomes:
11
whereQQ== the mean moment. the mean moment.
T
To find the coefficient of vao find the coefficient of variation forriation forQQ, first the coefficient, first the coefficient of variation for the mean wind moment is determined from:
of variation for the mean wind moment is determined from:
22 22 22 22 22
50
C 50
Coovv M M == ( ( CoovvC V V )) ++CoovvC K K zz ++CCoovv +GG +CCoov v C C d d
Noting that
Noting that V V in the in the V V 22 term is 100% correlated, and the term is 100% correlated, and the coefficient of
coefficient ofV V 22 (Cov (Cov V V 22) is two times the coefficient of varia-) is two times the coefficient of varia-tion of
tion ofV V (Cov (Cov V V ).).
The combination of the statistical properties for
The combination of the statistical properties for the deadthe dead and wind moments to determine the coefficient of variation and wind moments to determine the coefficient of variation for the total mean moment
for the total mean momentQQ results in: results in:
11 700700
AssumingQQ and andRR are lognormal and are lognormal and independent:independent:
ln
whereσσ is the standard deviation of is the standard deviation of the variable indicated.the variable indicated.
The reliability index The reliability indexββ is: is:
ln
The LRFD reliability analysis was coded into a
The LRFD reliability analysis was coded into a spreadsheetspreadsheet to study four different regions in the United States:
to study four different regions in the United States:
•
• Florida Coastal Region,Florida Coastal Region,
•
• Midwest and Western Region,Midwest and Western Region,
•
• Western Coastal Region, andWestern Coastal Region, and
•
• Southern Alaska Region.Southern Alaska Region.
Inputs for LRFD
Inputs for LRFD reliability analyses spreadsheet:reliability analyses spreadsheet:
V
V 300300,,V V 700700,,V V 17001700 per ASCE/SEI 7-10 design wind speeds per ASCE/SEI 7-10 design wind speeds µ
µ5050,,V V µµ5050,,V V 5050 per ASCE/SEI 7-05 design wind speeds per ASCE/SEI 7-05 design wind speeds LRFD reliability analyses inputs are in Table 7-1.
LRFD reliability analyses inputs are in Table 7-1.
Global inputs (for all regions):
Global inputs (for all regions):
λ
Table 7-2 shows the global inputs (inputs are highlighted).
Table 7-2 shows the global inputs (inputs are highlighted).
The results for the Midwest and Western Region ASCE/SEI The results for the Midwest and Western Region ASCE/SEI 7-10 700-year wind speed are shown in the Table 7-3 (other 7-10 700-year wind speed are shown in the Table 7-3 (other regions are similar). For the 300-year wind speed, the results regions are similar). For the 300-year wind speed, the results are in Table 7-4. Notice that the total nominal moment, are in Table 7-4. Notice that the total nominal moment, M M T T 22,, is less than 1.0 since the wind moment,
is less than 1.0 since the wind moment, M M 300300, is less than, is less than M M 700700.. Likewise, for the 1,700-year wind speed,
Likewise, for the 1,700-year wind speed, M M T T 22 is larger than 1.0 is larger than 1.0 since
since M M 17001700 is greater than is greater than M M 700700, as shown in Table 7-5 for the, as shown in Table 7-5 for the Midwest and Western Region.
Midwest and Western Region.
Using the 300-year wind speed requires less nominal Using the 300-year wind speed requires less nominal resis-tance; conversely, using the 1,700-year wind speed increases the tance; conversely, using the 1,700-year wind speed increases the required nominal resistance. Because the mean load
required nominal resistance. Because the mean load QQ and its and its variation do not change, this difference in required nominal variation do not change, this difference in required nominal resistance changes the reliability indices
resistance changes the reliability indices ββ accordingly. accordingly.
V
Table 7-1. able 7-1. LRFD reliability analyses inputs.LRFD reliability analyses inputs.
T
Table 7-2. able 7-2. Global inputs.Global inputs.
C
Table 7-3. able 7-3. Results for the Midwest and WResults for the Midwest and Western Region, 700-year wind speed.estern Region, 700-year wind speed.
700 Year Wind
Table 7-4. able 7-4. Results for the Midwest and WResults for the Midwest and Western Region, 300-year wind speed.estern Region, 300-year wind speed.
300 Year Wind