Results show that lognormal probability distribution could better model wind ... The statistical data and probability distribution function (PDF) for normalized wind ...
Investigation of wind load probability models and their effect on structural reliability
M.A. Barkhordari, M.A. Shayanfar, H. Baji, H.R. Ronagh
Abstract Wind loads are very important in the design of buildings as the load combinations, including the wind load, often govern the design. Compared to other loads, such as gravity load, wind loads are more uncertain and therefore bring about a lower safety index as the ratio of wind to gravity loads increases. Wind load, which is evaluated in terms of wind pressure, is a product of several factors. Wind speed is the main parameter. Other factors depend on the structural shape and geometry. In order to perform a realistic and accurate reliability analysis, it is important to find a suitable model for wind pressure. Previous studies have used Extreme Type I for modeling the wind load. In this study, based on wind speed data for 105 stations located in non-hurricane regions, this wind pressure model is evaluated and then modified. The best-fit models investigated were Lognormal and Extreme Type I. For these, the probability distribution parameters were averaged and general models were then obtained. Results show that lognormal probability distribution could better model wind pressure, although the Extreme Type I model is also close to Lognormal. It is shown that using lognormal probabilistic distribution leads to more conservative reliability indices than those from Extreme Type I. Keyword: Wind load, Wind Speed, Load combination, Reliability, Safety index, Monte Carlo method
1. Introduction Modern structural design codes have adopted reliability-based approaches for code calibration and determination of design safety factors. Assessments of risk and reliability analysis of structures require statistical descriptions of loads and resistance. Loads on structures are one of the main sources of uncertainty in the design process. Load studies, in general, serve two main objectives, being i) estimation of statistical characteristics of the load and development of stochastic models, and ii) estimation of nominal design values which are usually calculated at an acceptable level of risk during the service lifetime (typically = 50 years) of the structure. The acceptable level of risk differs for each load. Modern reliability procedures require full definition of the statistical model for loads, which contains mean and standard deviation as well as probability distribution function. A general load, Q, in a reliability analysis is often defined as: Q=A×B×C
(1)
Where A, B and C represent the load itself, the variation due to load models (which transform the actual spatially and temporarily varying load into a statistically equivalent uniformly distributed load), and uncertainties that arise from analysis methods that transform the loads to load effects respectively. Apart from dead and live loads, which are classified as gravity loads, all other loads have environmental sources. Wind, snow and earthquake loads are representative of wind speed, snow thickness and ground acceleration respectively. These environmental loads are associated with a high degree of uncertainty. Evaluation of these loads is possible upon knowing some basic parameters, some of which relate to the environmental parameters, whilst the others depend on the specification of the structure. Statistical description of these loads is based on statistical data for each of the basic parameters that define the load. Wind load mainly depends on wind speed, which varies based on the type of terrain surrounding the structure, height above ground, shielding from other buildings, and location of the building. The basic equation for calculating wind pressure is defined as (2) Where ρ is the density of the air and V is the wind speed. The basic pressure, q, in the above equation is used to evaluate the wind force on the structure. When dealing with building structures, some other factors, such as wind gust factor, exposure and pressure coefficients, transform the basic wind pressure, calculated using above equation, to actual wind pressure on the structure. Compared with Equation 1, the basic wind pressure, q, corresponds to factor A. Wind speed is the more important parameter in the probabilistic study of wind load. The natural wind speed contains a high level of uncertainty in comparison with other parameters. The combination of gust, exposure and pressure coefficients corresponds to factor B of Equation 1. Factor C, however, is related to the approximation in applied analysis methods and model idealization. In order to define a statistical model for wind load, it is necessary to have statistical models for basic parameters that represent factors A, B and C. In the current study, a probabilistic analysis of wind load in the United States is presented. A wind load model for non-hurricane-prone sites was considered in this study. The statistical data for wind speed were derived from a recent study by Cheng and Yeung [2]. The wind speed data used by Cheng and Yeung were obtained from the US National Climatic Center database. Since their study only dealt with wind models for non-hurricane-prone sites, hurricane-prone sites are not considered here either. Statistical data for other random parameters other than wind speed were derived from a recent study by Ellingwood and Tekie [3], in which the revised uncertainty models for basic variables were evaluated using the Delphi model. Results of the Delphi model have also been used in other recent investigations [1 and 2]. The Monte Carlo simulation method is then used to develop a statistical model for wind load. The statistical data and probability distribution function (PDF) for normalized wind load
(with respect to nominal wind according to ASCE 7-05 [1]) in all sites were obtained, and then these data averaged over different sites.
2. Wind speed data Probability distributions, selected to model annual extreme wind speed as well as its mean and standard deviation, are important for the wind load definition. Several studies have been performed to find models that best fit a set of wind speed data. Three types of extreme value probability distributions, which are Type I (Gumble), Type II (Frechet) and Type III (Reverse Weibull), are available for wind speed data fit. The Type I distribution has a thinner upper tail than Type II, but is still not bounded from above. Previous probabilistic considerations as well as available empirical evidence suggest that the asymptotic probability distributions of the largest values with unlimited tail are appropriate for modeling the largest annual wind speed [3]. ASCE 7 building code [1] wind map is based on Extreme Type I distribution. Extreme Type I distribution is the most common distribution function and has been used worldwide in many design codes. A recent study by Simiu and Heckert [9], however, has shown that Type I is conservative, and that Type III’s extreme value distribution may provide a better fit to the annual extreme wind speed data. Their results, however, have not been confirmed by others. For example, Cheng and Yeung [2] later stated that, based on the conditional mean exceedance (CME) method, the annual extreme gust wind speeds are better predicted by the Extreme Type III model. However, the results that can be obtained from a simple statistical analysis and a graphic curve fitting approach can equally reveal that Type I is in fact a better model for the estimation of the wind probability function. In this study, it was assumed that wind speed data follow Extreme Type I function. The method of order statistics and the method of moment should both be examined to estimate the best-fit Extreme Type I probability distribution function for the wind speed data [8]. In this study, the method of moment has been applied to estimate the statistical data for wind speed probability distribution function. Because we are only concerned with the wind load model in non-hurricane regions in this research, only the 105 stations (of the total 143 stations in Cheng and Yeung’s study) that are not located in the eastern coastal hurricane regions were selected. These are shown in Table 1. The eastern states of US contiguous have been removed. According to the map provided by Petreka and Shahid [6], the nominal wind speeds for selected stations (which are also reflected in ASCE 7/98 and its following revised edition) are 90 or 85 mph. Petreka and Shahid used the superstations concept and the Extreme Type I distribution to estimate the 50-year normal target and many other recurrence periods corresponding to the wind speed maps.
Table 1. Wind speed statistical data in 105 stations in US contiguous (Cheng and Yeung [2]) Sta. No. 3860 3872 3927 3945 3947 13729 13741 13748 13866 13877 13882 13891 13893 13897 13962 13966 13967 13968 13985 13994 13995 13996 14820 14821 14826 14827 14836 14837 14839 14842 14847 14852 14860 14891 14898 14913 14914 14918 14922 14923 14925 14933 14936 14939 14943 14944 23034 23042 23044 23047
Sta. Abv. HTS BKW DFW COL MCI EKN ROA ILM CRW BRI CHA TYS MEM BNA ABI SPS OKC TUL DDC STL SGF TOP CLE CMH FNT FWA LAN MSN MKE PIA CIU YNG ERI MFD GRB DLH FAR INL MSP MLI RST DSM HUR LNK SUX FSD SJT LBB ELP AMA
m (years) 16 16 32 21 18 15 16 19 16 18 19 18 15 21 18 27 21 21 19 21 19 21 19 22 15 17 19 19 17 16 16 15 15 15 16 24 21 16 18 18 19 21 16 18 19 19 19 18 21 21
Mean m/sec 23.96 24.78 28.14 30.44 28.18 24.23 27.87 27.38 26.02 26.55 25.02 25.50 28.51 26.68 28.08 29.33 29.98 26.99 29.34 27.70 25.35 30.01 29.43 27.84 27.48 28.31 27.53 28.92 29.95 28.60 26.51 28.42 29.43 29.14 26.66 26.52 25.12 25.44 26.85 30.32 32.10 29.01 29.84 28.79 29.75 30.14 31.49 30.64 28.41 30.86
Cov 0.127 0.109 0.146 0.155 0.161 0.110 0.161 0.161 0.141 0.171 0.190 0.129 0.132 0.181 0.137 0.133 0.143 0.130 0.103 0.149 0.110 0.140 0.131 0.155 0.148 0.058 0.112 0.197 0.116 0.154 0.097 0.140 0.112 0.116 0.181 0.132 0.125 0.112 0.104 0.117 0.119 0.107 0.106 0.143 0.097 0.131 0.146 0.137 0.168 0.093
Vn m/sec 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23
Sta. No. 23065 23066 23154 23155 23157 23160 23169 23174 23185 23188 23234 24011 24029 24033 24089 24090 24127 24128 24131 24144 24146 24149 24153 24155 24156 24157 24221 24225 24227 24229 24232 24233 24243 93193 93814 93817 93819 93820 93821 93822 94008 94014 94224 94789 94814 94822 94830 94846 94847 94849
Sta. Abv. GLD GJT ELY BFL BIH TUS LAS LAX RNO SAN SFO BIS SHR BIL CPR RAP SLC WIN BOI HLN FCA LWS MSO PEN PIH GEG EUG MFR OLY PDX SAE SEA YKM FAT CVG EVV IND LEX SDF SPI GGW ISN AST JFK HTL RFD TOL ORD DTW APN
m (years) 16 16 15 20 16 21 20 40 21 21 40 21 16 21 15 17 36 17 21 17 17 16 17 21 21 21 19 21 16 32 19 18 21 21 17 17 23 18 20 19 21 16 20 15 16 19 16 24 24 16
Mean m/sec 31.40 29.06 27.42 21.23 27.82 27.80 30.35 21.85 31.65 20.46 28.62 27.57 29.67 28.88 29.82 31.71 29.29 27.51 25.60 27.55 25.11 25.57 26.14 28.29 29.39 25.88 23.97 23.09 22.55 26.96 24.53 22.73 25.91 20.01 28.78 27.17 31.00 25.35 28.25 27.11 29.65 27.31 31.67 27.71 23.49 28.21 26.54 27.22 25.49 23.63
Cov 0.083 0.121 0.058 0.141 0.123 0.149 0.159 0.130 0.131 0.158 0.187 0.123 0.067 0.095 0.059 0.077 0.140 0.166 0.121 0.109 0.132 0.109 0.122 0.115 0.122 0.090 0.128 0.124 0.145 0.189 0.142 0.135 0.112 0.131 0.145 0.145 0.262 0.128 0.191 0.117 0.131 0.099 0.112 0.093 0.099 0.138 0.103 0.147 0.131 0.104
Vn m/sec 40.23 40.23 40.23 38.00 38.00 40.23 40.23 38.00 40.23 38.00 38.00 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 38.00 40.23 38.00 38.00 38.00 38.00 38.00 38.00 38.00 38.00 38.00 40.23 40.23 40.23 40.23 40.23 40.23 40.23 40.23 38.00 40.23 40.23 40.23 40.23 40.23 40.23 40.23
23050 23061 23062
ABQ ALS DEN
18 16 35
29.83 27.11 25.54
0.053 0.108 0.077
40.23 40.23 40.23
94860 24143
GRR GTF
21 17
28.09 27.30
0.130 0.118
40.23 40.23
In Table 1, parameter m represents the number of years corresponding to the wind speed records. Cov, Mean and Vn are coefficient of variation, mean and nominal wind speed respectively. Data shown in Table 1 that are available in the Cheng and Yeung study were obtained from the US National Climatic Center database. Figure 1 shows the frequency of coefficient of variations among the selected stations. In almost 70 percent of the stations the coefficient of variation is between 0.10 and 0.15.
Figure 1. Frequency of coefficient of variations over selected stations
In all stations, the years of record-keeping are greater than 15. The overall mean of coefficient of variation in the 105 selected stations is about 0.131. The variation of extreme annual wind speed with the standard deviation in each station is shown in Figure 2. A linear regression line forced through the origin has a slope, which is the coefficient of variation, of 0.1283. The tendency for the standard deviation of the maximum annual wind speed to increase as the mean value of the maximum annual wind speed increases is not strong. The correlation factor between maximum annual wind speed standard deviation and mean is about 0.30.
Figure 2. Best fitted line for annual wind speed mean and standard deviation
Figure 3 shows the relationship between the reference wind speed for various return periods and the coefficient of variation in all selected stations. The relationship is based on Extreme Type I distribution. In Figure 3, only the variability of wind speed has been considered and the other uncertainties have been neglected. The wind pressure is proportional to the square of the wind speed. Figure 3 shows that in the range of 0.10 to 0.15, which includes the coefficient of variation of a majority of stations, the wind load factor that equals the 500-year to 50-year wind pressure ranged from almost 1.31 to 1.42 respectively. This factor is close to the value of (1.6)*(0.85) =1.36 that is used in ASCE 7-05. The factor of 0.85 represents the directionality effect. Return periods of 25 and 100 years have been used to estimate the importance factor for temporary and post-disaster buildings. For a 100-year return period and the most frequent range of coefficient of variation, the importance factor range between 1.09 and 1.12. The estimated importance factor for temporary buildings, based on a 25-year return period, also ranges between 0.88 and 0.92. Importance factors for temporary and post-disaster buildings according to ASCE 7-05 are 1.11 and 0.87 respectively.
Figure 3. Normalized wind pressure with respect to 50-year return period
Wind load factors are sensitive to the selected probability distribution function. Whalen and Simiu [12] showed that for long return periods, using the Extreme Type I model leads to more conservative load factors than Extreme Type III. Current ultimate strength design is based on a 500-year return period and for this return period, the results for Extreme Type III and I are close to each other.
3. Wind load analysis Main wind-force resisting systems (MWFRS) should be designed to resist gravity and wind loads. In moment resisting frames and other lateral resistance systems, load combinations containing the wind load are dominant combinations in design, especially in tall building structures. In these types of structures, ultimate strength limit states as well as serviceability limit states are considered. According to ASCE 7-05, design wind load pressure for MWFRS of buildings of all heights shall be determined by Equation 3. (3)
In Equation 3, and are velocity pressure at heights z and h. Parameter h is the height of the building, Cp is the external pressure coefficient for windward and leeward faces, and G is the peak gust wind factor. Equation 4 is used to determine the velocity pressure along the height. (4) , and are velocity pressure exposure coefficient, topographic factor and basic wind speed respectively. The factor is the importance factor. Basic design wind speed is defined as the wind speed measured at the standard height of 10 meters above ground in open country with a 50-year mean recurrence interval, which is equivalent to 2 percent annual probability of exceedance. Ultimate strength states are used to check the strength of load-bearing systems against applied loads. The 500-year return period is often used to check the structural members against ultimate limit states. The design load combinations for structures, components and foundations according to ASCE 7-05 are shown in Equations 5 and 6. In these load combinations, the wind load factors convert the 50-year wind pressure to almost 500-year wind pressure. It is assumed that wind directionality has been considered in evaluating the nominal or design wind load. (5) (6) The above load combinations are based on maximum wind load and average live load. Some other codes, such as the Australian code, make direct use of the 500-year wind map instead of the 50-year wind map. In this case, wind load appears with no factor in the load combinations like those shown in Equations 5 and 6. As discussed in the following sections, the upper tail of the probability distribution function is used to fit the best probability model. The fitted model is very sensitive to the selected upper tail region. For serviceability limit states that are used to provide building comfort ability and functionality, excessive deflection, vibration and deterioration are checked. Chapter C, Appendix C of ASCE 7-05 code commentary defines the load combination with a 5% annual probability of exceedance for checking the short-term effects, as shown in Equation 7. This load combination corresponds to a 20-year return period. (7) The above load combination is used to check the lateral inter-story drift of buildings. ASCE 7-05 stated that using factored wind load in checking serviceability is excessively conservative, so using the less conservative 20-year return period would seem to be more reasonable. A previous study by Galambos and Ellingwood [5] suggested the 8-year wind load for checking serviceability limit states.
In the next sections of this study, the statistical moment and probability distribution of annual and 50-year wind load pressure are estimated.
4. Statistical analysis of wind load Reliability assessment of various structural components requires statistical moments and probability distribution of wind load. Statistical information on actual wind load is estimated based on the wind speed and relevant statistical data relating to code factors. In Equations 3 and 4, the derivation of wind pressure is shown based on ASCE 7-05. Wind pressure is a product of wind speed and other code factors. The main random variable is the wind speed, but other code factors do affect the wind pressure statistical model. Ellingwood [4] statistically characterized wind loads on structures for developing load factors for the ANSI code. Statistical information for seven stations, which were located in nonhurricane regions, was used in that study. These data had been directly obtained from a previous study by Simiu [10]. Extreme Type I was used to model the annual wind speed data. As the extent of information available for the estimation of statistical data for other coderelated factors (such as pressure and gust factors) is minimal, previous typical data for these factors were used by Ellingwood. It was assumed that the means of these factors were equal to their code values. The probability distribution function for all of these factors was assumed to be a normal function. Two methods that could generally be applied to estimate the statistical moments and the probabilistic model for wind load are moment and order statistics methods. In the method of moment, the distribution parameters are obtained by replacing the expectation and standard deviation of wind pressure with the corresponding statistics of the sample. In the method of order statistics, the evaluation of parameters of cumulative distribution function is based on least squares fitting of straight line to the data on the probability paper. Since wind pressure is the product of a number of random variables, the probabilistic model for it tends towards the Lognormal distribution. However, Ellingwood [4] has proposed Extreme Type I (largest) distribution, based on the method of order statistics for wind load model. Recently, Rosowsky and Cheng [7] used the same procedure for three stations in hurricane-prone regions and proposed the Lognormal model for wind load. In this study, the same procedure as that applied in previous studies is used for estimating the annual and the 50-year wind pressure probabilistic models. As mentioned in the previous sections, statistical data for wind speed (shown in Table 1) are derived directly from the study by Cheng and Yueng [2]. Statistical information on other code-related factors are obtained from the recent study by Ellingwood and Tekei [3], in which the Delphi method was used to revise uncertainties among these code factors. The statistical data for basic random variables considered in this study are shown in Table 2 below.
Table 2. Statistical data for basic random variables Parameter Wind Speed Exposure factor Gust Factor Windward pressure coefficient Leeward pressure coefficient Wind direction factor Model error
Bias1 Table 1 0.96 0.96 0.86 0.92 1.01 1.00
COV2 Table 1 0.116 0.098 0.145 0.152 0.093 0.050
CDF Extreme Type I Normal Normal Normal Normal Normal Normal
1- Mean to nominal value 2- Coefficient of variation
All nominal values for basic variables were derived from ASCE 7-05 at 10 m height above the ground. All other code-related factors, such as topographic factor, are assumed to be deterministic. It is possible to include a directionality factor of 0.85 when calculating the velocity pressure. In the recent revision of the ASCE code, this factor has been applied directly in nominal wind load calculations and the wind load factor has been modified to account for this consideration. In order to maintain consistency with the ASCE load combinations, this factor has been incorporated into the statistical analysis of the current study. The model error is used to account for approximation and errors arising from the analysis methods and simplification. This factor reflects the uncertainty of converting the load to load effects. Assuming that the annual maximum gust wind speeds are statistically independent and identically distributed, the cumulative distribution function (CDF) of N year maximum wind speed can be derived from the CDF of the annual maximum gust wind speed. If V follows the Type I distribution, the lifetime design speed for N years, VN, also follows Type I distribution. The mean and coefficient of variation of VN are given by Equations 8 and 9 [8]. ̅
̅
√
(8) ̅ ̅
(9)
The method of ordered statistics and the curve fitting tool have been used in this research. For comparison, different regions of the probability paper were used to find the best-fit model. As mentioned previously, the probabilistic parameters of the estimated model are sensitive to the regions selected on the probability paper. In addition to the inherent uncertainty in the sample data, the errors of both sampling and observation have been considered in estimating the total uncertainties in predicting N year wind speed data from limited available data. Observation and sampling errors have been considered as described in the available literature [4, 8 and 11]. In the previous study by Ellingwood [1], the coefficient of variation due to sampling and
observation errors for a 50-year return period were taken as
̅ √
̅
and 0.02 respectively.
For other return periods, a similar formula could be derived using moment method.
5. Probabilistic analysis results Monte Carlo simulations along with systematic Latin hypercube technique have been applied to calculate the probability required for the curve-fitting process. Two probability distribution functions, i.e. Lognormal and Extreme Type I, have been used. The goodness of fit for these probability functions was checked by comparing the two. Firstly, at each station, the curvefitting procedure was conducted and the fitted curve parameters were then evaluated. The fitted curves’ parameters were averaged at all stations. The fitted curve parameters are sensitive to the probability region considered for the curve-fitting purpose. The range above the 90th percentile is the most important region of probability density function. In many of the past studies, such as the one that was used for ASCE code calibration [4], the 90th percentile area was used. For comparison purposes, the entire range of probabilities was used to find the best-fit curves. The wind pressure was normalized based on nominal wind pressure calculated according to ASCE 7-05. The main random variable considered is wind pressure over nominal wind pressure ratio. Wind pressure data were obtained for the annual and the 50-year return periods. Figure 4 contains the resulting fitted lines for several considered cases of the probability paper coordinates. The horizontal axis in Figure 4 is the inverse of cumulative density function. The vertical axis for Extreme Type I shows the normalized wind load while that for Lognormal shows the logarithm of normalized wind load. The location and scale parameters have been calculated based on linear fitting over wind load data at each station. The resulting parameters were then averaged over all the stations. For Extreme Type I, location and scale parameters were converted to probability function parameters and averaged over the stations. Equations 10 and 11 show the parameters for Extreme Type I and Lognormal distributions respectively. (
)
(10) (11) α and u are the scale and scale parameters for the Extreme Type I while and are the location and scale parameters for the Lognormal function. Parameter represents the cumulative normal probability function. The probability distribution parameters were determined at each station, and then the mean and coefficient of variation of fitted curve were calculated based on these parameters.
Figure 4. Normalized wind pressure in probability paper
Figure 4 shows that for the range above the 90th percentile, the normalized wind pressure is more linear than that in the entire range. Graphically, it seems that Lognormal probability function is more suitable for modeling the wind pressure, especially for the range above the 90th percentile. Based on the Central Limit Theorem, since wind pressure is a product of several parameters, it may tend towards the Lognormal distribution. However, as mentioned previously, Ellingwood [4] has proposed Extreme Type I (largest) distribution for wind pressure while recently, Rosowsky and Cheng [1] used Lognormal for modeling wind pressure. They stated that neither Type I nor Type II largest distribution was found to fit for the wind load data while Lognormal fitted for the wind data very well. Figure 5 shows the histogram of scale and location parameters for both Extreme Type I and Lognormal functions over different stations. The results are based on fitting above the 90 th percentile, and show a large fluctuation. In a previous study [4], location and scale parameters for seven stations were averaged. Then the mean and standard deviation of these averages were used to model the wind load. This model was the basis for the ANSI code calibration. Using the same procedure, the parameters of Lognormal and Extreme Type I were averaged among the 105 selected stations. The average parameters and the
corresponding mean and coefficient of variation are shown in Table 3. The effect of using average location and scale parameters on the structural reliability is considered in the next section. It is reasonable to use averaging of parameters over stations that are near populated areas like major cities. Using weighted averaging, in which at each station a weight is considered based on the population of surrounding areas, would seem reasonable. However, in this study, the results were averaged without considering any weight. As seen in Figure 5, a wide difference exists between the scale and location parameters.
Figure 5. Scale and location parameters histogram
It should be noted that the results shown in Table 3 include the directionality factor. The coefficient of variation calculated based on fitting over the entire range is less than that for the range above the 90th percentile. Both fitted Lognormal and Extreme Type I distributions have similar coefficients of variation and mean. The bias factor in Table 3 represents the mean to nominal for the wind pressure. Table 3. Statistical parameters for fitted functions of normalized wind pressure Above 0.90 Parameters Statistical Data
Function
1
2
Entire Rage Parameters Statistical Data 1
2
location
scale
Bias
COV
location
scale
Bias
COV
-0.4050
0.3827
0.7177
0.3972
-0.3184
0.3158
0.7645
0.3239
4.6125 0.5637 0.6889 0.4035 Extreme Type I 1– Mean to Nominal Value 2– Coefficient of Variation
5.2153
0.6646
0.7753
0.3171
Lognormal
The appropriateness of fit for both Lognormal and Extreme Type I is calculated based on Rsquare. Equation 12 shows the calculation of this parameter. ∑ ̂
̅
∑
̅
(12)
SSR is the sum of squares of the regression and SST is the sum of squares about the mean. Mathematically, the residual for a specific predictor value is the difference between the response value y and the predicted response value ̂. is the real response. R-square can take up any value between 0 and 1, with a value closer to 1 indicating that a greater proportion of variance is accounted for in the model. Figure 6 shows the variation in R-square at different stations.
Figure 6. R-square variation over stations
In comparison to the Extreme Type I function, the Lognormal function fits the data better. Results show that fitting in the range above the 90th percentile fits the wind pressure data better than that in the entire range. Using these results, the average bias and coefficient of variation for the Lognormal model are 0.72 and 0.40 respectively. For the Extreme Type I model, the estimated bias factor and coefficient of variation are 0.69 and 0.40 respectively. The bias factor of 0.69 in this case includes the directionality factor. Removing the 0.85 directionality factor will result in 0.81 for the bias factor. This could be compared with 0.78, which was estimated in the Ellingwood study [4]. The same procedure was carried out for annual wind pressure. Again, Lognormal distribution was found to be the best fit for wind pressure, and the estimated bias factor and coefficient of variation were 0.37 and 0.42 respectively. Using Extreme Type I for modeling wind pressure resulted in 0.35 for the bias factor and 0.46 for the coefficient of variation.
6. Effect of wind model on structural reliability The type and parameters of probabilistic models for loads have direct effects on structural reliability indices. In this section, the effect of using different models while averaging over stations is studied. For simplicity, one of the main ASCE load combinations is considered. The additive effect of dead, live and wind loads has been considered as in the load combination below: (13)
In Equation 13, DL, LL and WL represent dead load, live load and wind load respectively. In the above load combination, the maximum lifetime (50-year) wind load is added to arbitrary point in time live load. These data are constructed from the available literature. It is assumed that the resistance component is tensile strength with a bias factor of 1.05 and a coefficient of variation of 0.11. Table 4 shows the statistical information for dead and live loads and the tensile strength. Live to dead load ratio is assumed to be 0.50. Different wind to dead load ratios were considered. The so-called advanced first order-second moment method was utilized to evaluate the reliability indices. Table 4. Statistical parameters of loads and resistance Component Dead load Live Load Tensile Strength
Bias1 1.05 0.24 1.05
COV2 0.10 0.50 0.11
Model Normal Gamma Lognormal
1. Bias=Mean to Nominal2. COV=Coefficient of Variation
Using the statistical data shown in Table 4 and those obtained for the wind load for both Lognormal and Extreme Type I, reliability analyses were performed in order to calculate the reliability indices. Figure 7 shows the resulting reliability indices for different stations. These results are based on wind to dead loads ratio of 1.0. The calculated reliability indices for each station, based on Lognormal and Extreme Type I models, are very close to each other. This indicates that there is minimal difference between using Lognormal and Extreme Type I models in reliability analysis. The average reliability indices of all stations for Lognormal and Extreme Type I were 2.982 and 3.120 respectively. Generally, using the Lognormal model leads to more conservative results than the Extreme Type I model. Considering that the Lognormal model fits the wind data much better than Extreme Type I, and that it leads to more conservative results in comparison to Extreme Type I, it seems reasonable to use the Lognormal model instead of Extreme Type I in any wind load reliability analysis.
Figure 7. Variation in reliability indices over stations
Although the average reliability indices for the Lognormal model is less than that for Extreme Type I, in some of the stations investigated, the reliability indices resulting from the Lognormal model were greater than those from the Extreme Type I model.
In order to quantify the effect of wind to dead load ratios on reliability indices, several reliability analyses were conducted for different wind to dead load ratios. In these analyses, the averaged statistical models (see Table 3) were used. Figure 8 shows the reliability indices for both Lognormal and Extreme Type I models. Results show that the Lognormal model leads to lower reliability indices over all wind to dead load ratios in comparison to the Extreme Type I model. As expected, by increasing the wind to dead load ratio, the reliability indices decrease. The difference between reliability indices for Lognormal and Extreme Type I is low and in the range of 5 percent.
Figure 8. Reliability Indices variation over stations
Figure 8 shows that reliability indices for most of the stations lie between 2.0 and 4.0 for a wind to dead load ratio of 1.0. The coefficient of variation of reliability indices is around 0.25 for both Lognormal and Extreme Type I. This coefficient of variation is twice the coefficient of variation of annual wind speed, which is around 0.12.
7. Conclusion Wind pressure is a product of several factors such as wind speed, pressure coefficient, peak gust wind, velocity pressure coefficient, etc. In this study, nominal wind pressures were calculated, based firstly on ASCE 7-05, and from these the normalized wind pressure was obtained. Wind speed data for 105 stations in non-hurricane regions were extracted from data presented in Cheng and Yeung [2]. These data were obtained from the US National Climatic Center database. It was assumed that wind speed data followed Extreme Type I distribution. Statistical data for other factors were obtained from a recent study by Ellingwood and Tekei [3], which used the Delphi study to revise uncertainties among these code factors. The method of statistical order (by the means of Monte Carlo simulation) was used to evaluate the 50-year wind pressure model. In this method, the best-fitted linear equation was extracted from plotted wind pressure data on the probability axes coordinate system. Lognormal probability function, as well as Extreme Type I function, was used in this research. Data fitting was performed on the entire set of data and on the set corresponding to
those above the 90th percentile, in order to find the probability model parameters. The calculated model parameters for Lognormal and Extreme Type I were then averaged over the stations to get the overall model for wind pressure. Results showed that Lognormal is a more appropriate model for wind pressure than Extreme Type I. In order to quantify the effect of the wind pressure model on structural reliability, a set of reliability analyses were performed, based on the resultant wind pressure model, for the case of tensile strength and for one of the basic load combinations in the ASCE code. Reliability indices resulting from assumed Lognormal and Extreme Type I models were close to each other, with differences as low as 5 percent. However, using the Lognormal model leads to more conservative results in comparison to Extreme Type I. Considering that the Lognormal model fits the wind pressure data better than the Extreme Type I model, and that it leads to more conservative results, it can be confidently suggested that that the Lognormal model is the better probability distribution function to use to model wind pressure.
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