Load factor calibration for the proposed 2005 edition of the National ...

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432 Can. J. Civ. Eng. Vol. 30, 2003Table 2. Survey of concrete floor thickness variability from 2000.Thickness difference (mm)TypicalWorstCase Min. Max. Min. Max. Bias CoV Note50 mm topping on precast –5 +10 –10 +20 1.00–1.10 0.140–0.400–10 +10 –40 +50 1.00–1.23 0.077–0.176 Depends on camber of precast member65–75 mm cover slab, –5 +5 — —steel deck–5 +15 –10 +50 1.02–1.06 0.049–0.078 Extreme values are for unshoredconstruction150 mm one-way slabbetween beams–5 +5 –5 +25 Slopes tough, especially if sloped in twodirections–5 +10 –25 +25200 mm two-way slab –10 +20 — — 1.03–1.06 0.062–0.065–10 +15 –15 +40 Always have more around the columns;there is a buildup there300 mm two-way slab –10 +10 — — 1.00–1.06 0.033–0.056–10 +20 –20 +50expected value of the 50 year maximum equivalent uniformlydistributed live load, Lmax (in kN/m 2 ), isFig. 1. Fifty year maximum live load bias coefficient versus influencearea.[2] Lmax = 0. 895 + 76 . / AIwhere A I is the influence area (in m 2 ). The correspondingnominal live load, L (in kN/m 2 ), from the 1995 NBCC is[3] L = 24 . ( 03 . + 98 . / B)where 2.4 is the specified live load (in kN/m 2 ) and B is thetributary area (in m 2 ). According to ASCE7-98 (ASCE2000), the influence area equals the tributary area for twowayslabs, twice the tributary area for interior beams, andfour times the tributary area for columns. Using these relationships,the bias is simply the ratio of eq. [2] to eq. [3] andso depends on the type of element and its influence area.The 50 year maximum live load bias is shown in Fig. 1for realistic ranges of influence areas of two-way slabs,beams, and columns. The ASCE (2000) curve in Fig. 1 indicatesthe bias if the nominal live load is reduced with increasinginfluence area in accordance with ASCE7-98instead of eq. [2]. The 30 year extreme live load model usedfor previous Canadian design code calibration (Allen 1975)has a bias of 0.70 and a CoV of 0.30; the equivalent 50 yearmaximum live load has a bias of 0.78 and a CoV of 0.27 andis also shown in Fig. 1.The present study adopted a bias for the 50 year maximumlive load effect of 0.90, based on several considerations.Two-way slabs and other components with smallinfluence areas have greater bias, but the consequences offailure of these elements are less severe than those for elementssupporting larger tributary areas. Provided they arenot weak in shear, two-way slab systems possess considerablestrength reserves that are not usually considered by thedesigner. The average bias for beams, as computed for the12 values shown by the markers in Fig. 1 and so weightedtowards the smaller influence areas, is 0.94. The averagebias for columns, as computed for the 16 values shown bythe markers in Fig. 1 and so weighted towards the smallerinfluence areas, is 0.87. Although larger bias occurs forlarger influence areas, the large dead load that the columnmust sustain in this case makes the design relatively insensitiveto the magnitude of the live load. Figure 1 also indicatesthat the live load reduction specified in the NBCC is conservativewith respect to that in ASCE7-98 for beams and forcolumns with influence areas less than 1000 m 2 , and so thelive load bias is less than the value of 1.00 used to calibrateASCE7-98 (Ellingwood 1999). The average ratios of NBCCbias to ASCE7-98 bias, as determined by averaging thepoints shown by markers in Fig. 1, are 0.95 for beams and0.84 for columns. Lastly, the bias of 0.90 for use and occupancyload is greater than the value 0.78, which is equivalentto that used to calibrate the 1975 edition of the NBCC(Allen 1975).The CoV of the maximum equivalent uniformly distributedlive load for office floors during a 50 year referenceperiod is also based on the simulation results for officebuildings (Ellingwood and Culver 1977). This study reported2the variance of the 50 year maximum live load, σ L max (inkN/m 2 ), is© 2003 NRC Canada
Bartlett et al. 4332[4] σ L max = 0. 033 + 4. 025/A IThus the CoV of the live load is obtained by dividing thesquare root of eq. [4] by eq. [2]. The resulting relationshipbetween the CoV and the influence area is represented bythe curve marked “load only” in Fig. 2. The CoV values arerelatively insensitive to the influence area considered, rangingfrom 0.162 to 0.182.The transformation of the live load into a live load effectintroduces modelling and analysis factors. The modellingfactor addresses the idealization of the actual live load as anequivalent uniformly distributed load and has been assumedunbiased with a CoV of 0.20 (Ellingwood et al. 1980). Kariyawasam(1996) assumed these values pertain to columns,and a reduced CoV of 0.10 pertains to beams. The analysisfactor reflects the transformation of load to load effect andhas been assumed unbiased with a CoV of 0.05 (Ellingwoodet al. 1980). Using these values, the variation of CoV for the50 year maximum equivalent uniformly distributed load withinfluence area is plotted in Fig. 2. The CoV values forbeams proposed by Kariyawasam are consistently smallerthan those for columns. All CoV values are relatively insensitiveto the influence area. The CoV values for columns aresimilar to that assumed by Allen (1975) for calibration ofprevious editions of the NBCC, if the difference between the30 year and 50 year reference periods is taken into accounted.In past investigations, a Gumbel distribution has beenused to represent the total live load effect, including modellingand analysis factors (Ellingwood et al. 1980;Kariyawasam 1996). In the present investigation, the liveload was assumed to be a Gumbel variate with a bias of 0.9and a CoV of 0.17. The combined effect of the modellingand analysis factors has a bias of 1.0 and a CoV of [0.20 2 +0.05 2 ] 1/2 = 0.206 and was assumed to be time-independent. Itis further assumed that the overall effect of modelling andanalysis is normally distributed, because its uncertainty isdominated by a single factor, the modelling uncertainty, andbecause in the absence of data or studies concerning themodelling uncertainty, a normal distribution seems as reasonableas any other.Point-in-time live loadsStatistical analyses of point-in-time live load have beencarried out by others (Ellingwood and Culver 1977; Chalkand Corotis 1980; Ellingwood et al. 1980). The load may bemodelled as a sustained component, with a duration from 2to 8 years, and an extraordinary live load that occurs once ortwice per year. For the present study, it is assumed that theduration of the point-in-time live load, which includes boththe sustained component and the extraordinary live loadevent, is 6 months.The point-in-time live load is assumed to have a Weibulldistribution, with parameters derived from those of the50 year maximum live load distribution following the procedurespresented by Wen (1990). The bias is 0.273 and theCoV is 0.674. The transformation factors used to convertload into load effect were assumed identical for the 50 yearmaximum live load and the point-in-time live load.Fig. 2. Fifty year maximum live load coefficient of variation(CoV) versus influence area.Wind loadThe simplified procedure in the 1995 NBCC adopts a simple“gust factor” approach to estimate wind load on buildings.The wind pressure, p, against the surface of a buildingis[5] p = qC e C p C gwhere q is the reference velocity pressure, C e is the exposurefactor, C p is the external pressure coefficient, and C g is thegust factor. Although values for each of these variables arespecified in the NBCC, all are uncertain. As described in thenext section, the reference velocity pressure depends on theclimatic conditions at the site and, assuming these conditionsdo not change, can be considered as a stochastic process.Thus the distribution of the maximum pressure that will occurover a given design life can be derived from the annualmaximum pressure distribution. The various coefficients thattransform the reference velocity pressure to a pressure on thesurface of a building will be considered time-independentrandom variables.Reference wind pressuresIn Appendix C of the 1995 NBCC, the 1 in T year referencewind pressure, q T , is determined from the 1 in T yearwind velocity, V T ,as2[6] qT= 1 ρVT2where ρ is the density of air (1.2929 kg/m 3 for dry air at 0°Cand standard atmospheric pressure). The density of air wasconsidered deterministic because its CoV, roughly 2.5–5.0%(Ellingwood et al. 1980; MacGregor et al. 1997), is considerablyless than the CoV of V 2 T .Reference velocity pressures presented in Appendix C ofthe 1995 NBCC are derived from 1 h wind velocity datafrom over 100 sites, typically airports, with 10–22 years ofrecord (NBCC 1995). Maps were prepared showing contoursof the 1 in 30 year wind velocity and dispersion parametersfor Gumbel distributions fit to the data. For the locations© 2003 NRC Canada
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