- Email: [email protected]
Power spectra of wind pressures on low building roofs
Journal of Wind Engineering and Industrial Aerodynamics 74—76 (1998) 665—674
Power spectra of wind pressures on low building roofs K. Suresh Kumar1, T. Stathopoulos* School for Building, Concordia University, Montreal, Quebec, Canada H3G 1M8
Abstract Wind pressure spectra were measured at various tap locations on the roofs of several low building models placed in two types of terrain in order to determine their characteristic shape and derive a suitable analytical model for their representation. The results show that though spectra vary under different conditions, appropriately normalized spectra are found to be identical in many situations. On this basis, a suitable empirical form has been derived for the synthetic generation of normalized spectra. An attempt has also been made to classify complex spectral patterns, and suggest standard spectral shapes associated with various zones of each roof and their parameters. The synthetically generated pressure spectra can be utilized for the simulation of wind pressure fluctuations on roofs not only for the evaluation of extreme pressures in a risk-consistent way but also for fatigue design purposes. ( 1998 Published by Elsevier Science Ltd. All rights reserved. Keywords: Low building roofs; Power spectra; Wind pressures
1. Introduction Past experience reveals that low building roofs are severely affected by repeated high-pressure fluctuations during wind storms. More recently, the ASCE manual of practice for wind tunnel studies [1] refers to the growing concern in low-rise building industry regarding the local failure of fasteners and cladding due to the stress reversals caused by the fluctuating pressures acting on low building roofs. The simplification of pressure fluctuations to traditional pressure loading cycles is not appropriate in many cases. On the other hand, collection of long time histories using wind tunnels is time-consuming and expensive. Within this context, a new simulation scheme which can generate both Gaussian and non-Gaussian pressure fluctuations, based on the
* Corresponding author. 1 Present address: Faculty of Building, Architecture and Planning, FAGO, Eindhoven University of Technology, Postbus 513, 5600 MB Eindhoven, The Netherlands. 0167-6105/98/$19.00 ( 1998 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 6 0 - 9
666
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
characteristics of local pressure data obtained from low building roofs, has been suggested recently [2]. Spectra of wind pressures required for this simulation is the subject of this paper. In contrast with many studies that have been conducted with respect to the wind velocity spectrum (identification and analytical description), wind pressure spectra have received only limited attention perhaps due to their complex nature. Wind pressure spectra representing energy content in pressure fluctuations receive contributions not only from mechanical but also from building-generated turbulence, which varies depending upon the building configuration, exposure condition and architectural details. As a result, most of the previous studies focused only on isolated measurements for specific cases; however, the classification of spectra of spaceaveraged pressures over flat roof panels [3] and the more recent attempt to describe the basic shapes of spectra [4] are notable exceptions. In order to acquire systematic information on spectra of wind pressures acting on low building roofs, an extensive investigation has been carried out in this area. This paper presents the results of a study with the main objective to establish the overall characteristics of wind pressure spectra acting on low building roofs and provide an empirical model for their description. Thereafter, following normalization and classification of complex spectral patterns, standard spectral shapes in terms of their parameters are established for various roof geometries.
2. Measurement details In the present study, experimental measurements were carried out in the boundary layer wind tunnel of the Centre for Building Studies (CBS) of Concordia University. Appropriately scaled (1 : 400) plexiglass models of flat-roof, monoslope roof and gable-roof buildings 12—15 m high were tested for several wind angles in open and suburban terrain conditions. The wind speed at gradient height was approximately 11 m/s. Roof pressure fluctuations were measured by using SETRA 237 pressure transducers (0.1 psd range) placed in a scanivalve. A conventional restricted tube system with a flat frequency response up to 100 Hz was used as the transfer medium between the tap and the Setra transducer. Frequencies of pressure fluctuations at most tappings are expected to be well below this value. Pressure data were acquired in blocks of 8192 samples each at a sampling rate of 500 Hz using a waveform analyzer (DATA 6000) after each signal passed through a low-pass filter with a cut-off at Nyquist frequency of 250 Hz. Simultaneously, the pressure spectrum was also evaluated with the help of DATA 6000 analyzer. Smoothening of the spectra has been carried out by ensemble averaging for 16 records.
3. Characteristics of pressure spectra Typical samples of measured pressure spectra are shown in Figs. 1 and 2. The form of the spectrum adopted in this study is normalized (S( f )/p2) which is shown as
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
667
Fig. 1. Sample wind pressure spectra on flat roof.
Fig. 2. Evolution of spectral shapes on monoslope roof.
a function of the reduced frequency, F"f h/», where, f is the frequency, h is a typical building dimension (mean height of the building is chosen in this study), » is the mean velocity at the height of the building and p is the standard deviation of fluctuating pressure. As expected, wind pressure spectra measured on low building roofs vary with wind direction, roof geometry and tap location. At higher frequencies, the amplitudes of spectra seem to be similar; however, they are more different at lower frequencies in which most of the energy lies. Measurement results show that, in general, the amplitude of pressure spectrum decreases as the frequency increases.
668
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
However, a small growth in spectral amplitudes at dimensionless frequencies (F) between 0.1 and 0.2 ( f+20—40 Hz) has been observed at taps located in farwind (i.e., the windward edge region away from the windward corner in case of oblique wind angle) as well as leeward regions of the roof. These observed humps on spectra are possibly associated with shedding of dominant eddies in those regions. Typical examples are shown in Figs. 1 and 2 for flat (sample S2) and monoslope roofs (sample 3), respectively. Further, Fig. 2 presents the evolution of spectral shapes and the transition from one to the other for a monoslope roof building. It is also noted from experimental results that the pressure spectra normalized with respect to the variance (p2) corresponding to the same tap at open and suburban terrain conditions seem to be similar. Further, it can be seen from roof pressure spectra that significant energy is distributed over a wide frequency range indicating the broadband nature of the corresponding pressure fluctuations. In summary, the results show that although the spectra change for different roof geometries, terrain conditions, wind attack angle and tap location, they can be classified into two categories: (1) spectra whose amplitude decreases as the frequency increases and (2) spectra whose amplitude decreases up to a certain frequency and then grow to have another hump. Since the variance of pressure fluctuations also varies for different configurations, normalization of power spectra by the variance contributes to their similarity and assists in the generalization process.
4. Empirical model for pressure spectra Based on the similarities found in normalized spectra for various conditions, it may be appropriate to suggest a suitable analytical model for pressure spectra. Within this context, several well-known curve-fitting techniques have been employed to extract a suitable empirical equation for spectra. However, a different approach by using trial and error proved to be successful. The code is written in MATLAB environment with the help of available built-in functions. The independent variable ( f ), the dependent variable (S( f )/p2), initial values of parameters and the function to be fitted are inputs to the program. Traditional optimization methods have been used in the program to estimate the optimum parametric values iteratively by minimizing the sum of the squared residuals (i.e., the sum of the squares of the differences between the fitted spectral ordinates and the measured spectral ordinates). In this study, the Levenberg—Marquardt algorithm known for its robustness along with cubic polynomial interpolation line search method has been used to carry out least-squares optimization [5]. Several functions representing approximately the same shape as the roof pressure spectra have been tried and the following exponential function was found to be more appropriate: S( f )/p2"a e~c1f#a e~c2f, (1) 1 2 where S( f ) is the spectral ordinate, p2 is the variance, f is the frequency, a and a are 1 2 the position constants and c and c are the shape constants. The position constants 1 2
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
669
Fig. 3. Measured and fitted wind pressure spectra.
control the location of the spectra, whilst the shape constants control their shape. An example of the measured and fitted wind pressure spectra is shown in Fig. 3. The evolution of spectral curve is shown by plotting a e~c1f and a e~c2f separately. It is 1 2 clear that the term a e~c1f controls the position and shape of spectra at higher1 frequency region, while the term a e~c2f controls the position and shape of spectra at 2 lower-frequency region. The spectral fit appears satisfactory; for instance, the spectral fit has zero up-crossing or down-crossing rate (N )"57, positive or negative peak 0 rate (N )"150, irregularity factor (e)"0.38 and bandwidth parameter (b)"0.92 1 which are close to the corresponding target values of 55, 150, 0.37 and 0.93, respectively (N , N , e and b are estimated from corresponding spectra using the expressions 0 1 provided in Ref. [6]). Adequate performance of the proposed empirical expression has been established for the various cases examined. Though the fit is based on minimizing the squared residuals, the level of accuracy of the fit is quantified based on four spectral statistics (N , N , e and b); the fit which represents each of the target spectral 0 1 statistics within 10% is considered reasonable for practical applications.
5. Towards generalization It may be possible to categorize normalized spectra on a roof based on their similarities which is a gross simplification of the actual complex spectral patterns. It was decided, first, to classify the zones of Gaussian as well as non-Gaussian pressure fluctuations due to the difference in simulation methodologies used for each case. The statistics of pressure fluctuations was carefully checked and the corresponding zones have been identified for various ranges of wind direction. A particular zone is
670
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
Fig. 4. Gaussian and non-Gaussian zones for flat roofs.
considered non-Gaussian if the absolute values of skewness and kurtosis of pressure fluctuations at various taps are greater than 0.5 and 3.5, respectively. Fig. 4 shows the approximate Gaussian and non-Gaussian zones for a flat roof. Variable z is assumed to be 10% of least horizontal dimension or 40% of lower eave height whichever is less [7]. The non-Gaussian region is divided into two zones since, in general, two types of spectra have been found there. The first type observed on windward locations has generally continuous decreasing amplitudes while the second, observed on farwind or leeward locations, has another hump — see Fig. 1 (samples S1 and S2). A group of normalized spectra (measured) corresponding to farwind non-Gaussian region of a flat roof building under open and suburban exposures is shown in Fig. 5. Since spectra appear similar, they have been averaged and fitted by Eq. (1). The statistics of the fitted and observed spectra presented in Table 1 indicates that the fitted spectrum is a good representative of the observed spectra. In the case of Gaussian region, the normalized spectra measured at several locations on a flat roof at different conditions appear also similar. Therefore, they are averaged and fitted to obtain the most suitable spectra for that zone. A similar exercise has been carried out for various roof geometries with successful results. The final outcome of this exercise is a set of standard spectral shapes and their parameters associated with different regions of each roof. The standard spectral shapes of a flat roof building are presented in Fig. 6; each spectrum is assigned a subscript where the first letter stands for the type of roof (F — flat roof), the second stands for the type of region (G — Gaussian, NG — non-Gaussian) and the number (1 or 2) stands for the type of spectra in that zone. Parameters of the spectral shapes corresponding to Fig. 6 are provided in Table 2. Note that the position constant (a ) 2 of spectra from Gaussian regions is lower than that of spectra from non-Gaussian regions which indicates that the spectra from non-Gaussian regions are located above spectra from Gaussian regions in lower frequencies (see Fig. 6). The shape constants (c and c ) of spectra from Gaussian regions are also lower than those of spectra from 1 2 non-Gaussian regions which shows that the reduction rate of spectral amplitudes from Gaussian regions is generally lower than that of spectral amplitudes from non-Gaussian regions. The effect of such variations of parameters on spectral shapes is clear from Fig. 6. Similar results have been obtained in case of other roof geometries.
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
671
Fig. 5. Measured and fitted wind pressure spectra (flat roof, non-Gaussian zone).
Table 1 Comparison of spectral properties (Flat roof, non-Gaussian zone) Sample
N 0
N 1
e
b
Fit 1 2 3 4 5
80.2 78.5 70.8 75.7 64.9 79.5
166.7 155.4 160.2 163.7 154.9 155.4
0.48 0.50 0.44 0.46 0.42 0.51
0.88 0.86 0.90 0.89 0.91 0.86
It is interesting to note that the parameters obtained in the above fitting are all positive and therefore, the derivative of Eq. (1) at any point is a negative quantity which reveals that the fitted curve has a downward slope. As a result, this function represents only mild growth in spectra and needs modifications to fit the predominant spectral growth observed in some cases such as sample 3 shown in Fig. 2. In such cases (e.g. farwind as well as leeward non-Gaussian regions of monoslope roof) spectra show a clear hump and the proposed Eq. (1) requires an additional term a e~c3f to fit 3 the data efficiently; the parameters a and c can be established using the fitting 3 3 procedure as previously discussed.
672
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
Fig. 6. Standard spectral shapes for flat roof.
Table 2 Proposed spectral parameters for flat roof Parameters (]10~2)
S FG1 S FNG1 S FNG2
a 1
a 2
c 1
c 2
0.9756 1.207 0.8162
1.956 11.61 12.05
0.9725 1.977 1.084
16.52 30.15 38.23
6. Application The application of the developed standard spectral shapes is briefly described here. It is shown that for practical purposes, from the knowledge of variance of pressure fluctuations at a specific roof location, spectra of pressure fluctuations at the same location can be synthetically generated using standard spectral shapes provided in this study. Thereafter, the simulation scheme proposed by Suresh Kumar and Stathopoulos [2] can be utilized to generate Gaussian as well as non-Gaussian pressure fluctuations on roofs. The measured spectrum and its empirical fit shown in Fig. 3 are used to demonstrate the efficiency of the synthetic spectra in time series simulation. The simulations have been carried out based on the approach discussed in Ref. [2]. The first four moments (mean, variance, skewness and kurtosis) of the
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
673
Table 3 Statistics of simulated signals
Simulation-I Simulation-II
Mean
Variance
Skewness
Kurtosis
!0.8 !0.8
0.28 0.28
!1.92 !1.96
9.87 10.22
Table 4 Number of peaks and down-crossings of simulated signals Peaks
Simulation-I Simulation-II
1854 1939
Level crossing below mean in terms of p 0
!1
!2
!3
!4
!5
!6
617 629
362 374
189 193
100 100
43 45
22 25
12 12
simulated signal using measured spectra (Simulation-I) has been compared with those of the simulated signal using synthetic spectra (Simulation-II) in Table 3. The number of peaks and crossings of the same simulated signals are compared in Table 4. The corresponding values of both signals are close and similar results prevail in other cases. 7. Concluding remarks An extensive investigation of power spectra of wind-induced pressures acting on low building roofs has registered the variation of spectra with respect to location, wind direction, roof geometry and terrain conditions. However, similarities among normalized spectra observed in many cases suggest an appropriate empirical form for the synthetic generation of normalized spectra. Thereafter, normalized spectra are categorized and the standard spectral shapes associated with various zones of each roof and their parameters are established, and provided for the case of flat square roofs. The application of the proposed spectral shapes is also briefly discussed. Finally, it must be emphasized that the provided spectral shapes can be adopted in codal provisions for the generation of pressure fluctuations on roofs; these are required for the design of roofs against extreme wind pressures and fatigue. References [1] ASCE, Wind tunnel studies of buildings and structures, J. Aerospace Eng. ASCE 9 (1) (1996) 19—36. [2] K. Suresh Kumar, T. Stathopoulos, Computer simulation of fluctuating wind pressures on low building roofs, Proc. 3rd Int. Col. on Bluff Body Aerodyn. and Appl., Blacksburg, Virginia, July 28—August 1 1996, J. Wind Eng. Ind. Aerodyn. 69—71 (1997) 485—495.
674
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
[3] T. Stathopoulos, D. Surry, A.G. Davenport, Effective wind loads on flat roofs, J. Struct. Eng. ASCE 107 (2) (1981) 281—298. [4] M. Kasperski, H. Koss, Beatrice joint project wind action on low-rise buildings. Part 2: Analysis in the frequency domain, Proc. 3rd Int. Col. on Bluff Body Aerodyn. and Appl., Blacksburg, Virginia, July 28—August 1 1996. [5] MATLAB, Optimization ToolBox User’s Guide, The Mathworks Inc., Mass., 1994. [6] E. Vanmarcke, Random Fields: Analysis and Synthesis, The MIT Press, Cambridge, 1983. [7] NBCC, User’s Guide — NBC 1995, Structural Commentaries (Part 4), National Research Council of Canada, Ottawa, Canada, 1995.
Power spectra of wind pressures on low building roofs K. Suresh Kumar1, T. Stathopoulos* School for Building, Concordia University, Montreal, Quebec, Canada H3G 1M8
Abstract Wind pressure spectra were measured at various tap locations on the roofs of several low building models placed in two types of terrain in order to determine their characteristic shape and derive a suitable analytical model for their representation. The results show that though spectra vary under different conditions, appropriately normalized spectra are found to be identical in many situations. On this basis, a suitable empirical form has been derived for the synthetic generation of normalized spectra. An attempt has also been made to classify complex spectral patterns, and suggest standard spectral shapes associated with various zones of each roof and their parameters. The synthetically generated pressure spectra can be utilized for the simulation of wind pressure fluctuations on roofs not only for the evaluation of extreme pressures in a risk-consistent way but also for fatigue design purposes. ( 1998 Published by Elsevier Science Ltd. All rights reserved. Keywords: Low building roofs; Power spectra; Wind pressures
1. Introduction Past experience reveals that low building roofs are severely affected by repeated high-pressure fluctuations during wind storms. More recently, the ASCE manual of practice for wind tunnel studies [1] refers to the growing concern in low-rise building industry regarding the local failure of fasteners and cladding due to the stress reversals caused by the fluctuating pressures acting on low building roofs. The simplification of pressure fluctuations to traditional pressure loading cycles is not appropriate in many cases. On the other hand, collection of long time histories using wind tunnels is time-consuming and expensive. Within this context, a new simulation scheme which can generate both Gaussian and non-Gaussian pressure fluctuations, based on the
* Corresponding author. 1 Present address: Faculty of Building, Architecture and Planning, FAGO, Eindhoven University of Technology, Postbus 513, 5600 MB Eindhoven, The Netherlands. 0167-6105/98/$19.00 ( 1998 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 6 0 - 9
666
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
characteristics of local pressure data obtained from low building roofs, has been suggested recently [2]. Spectra of wind pressures required for this simulation is the subject of this paper. In contrast with many studies that have been conducted with respect to the wind velocity spectrum (identification and analytical description), wind pressure spectra have received only limited attention perhaps due to their complex nature. Wind pressure spectra representing energy content in pressure fluctuations receive contributions not only from mechanical but also from building-generated turbulence, which varies depending upon the building configuration, exposure condition and architectural details. As a result, most of the previous studies focused only on isolated measurements for specific cases; however, the classification of spectra of spaceaveraged pressures over flat roof panels [3] and the more recent attempt to describe the basic shapes of spectra [4] are notable exceptions. In order to acquire systematic information on spectra of wind pressures acting on low building roofs, an extensive investigation has been carried out in this area. This paper presents the results of a study with the main objective to establish the overall characteristics of wind pressure spectra acting on low building roofs and provide an empirical model for their description. Thereafter, following normalization and classification of complex spectral patterns, standard spectral shapes in terms of their parameters are established for various roof geometries.
2. Measurement details In the present study, experimental measurements were carried out in the boundary layer wind tunnel of the Centre for Building Studies (CBS) of Concordia University. Appropriately scaled (1 : 400) plexiglass models of flat-roof, monoslope roof and gable-roof buildings 12—15 m high were tested for several wind angles in open and suburban terrain conditions. The wind speed at gradient height was approximately 11 m/s. Roof pressure fluctuations were measured by using SETRA 237 pressure transducers (0.1 psd range) placed in a scanivalve. A conventional restricted tube system with a flat frequency response up to 100 Hz was used as the transfer medium between the tap and the Setra transducer. Frequencies of pressure fluctuations at most tappings are expected to be well below this value. Pressure data were acquired in blocks of 8192 samples each at a sampling rate of 500 Hz using a waveform analyzer (DATA 6000) after each signal passed through a low-pass filter with a cut-off at Nyquist frequency of 250 Hz. Simultaneously, the pressure spectrum was also evaluated with the help of DATA 6000 analyzer. Smoothening of the spectra has been carried out by ensemble averaging for 16 records.
3. Characteristics of pressure spectra Typical samples of measured pressure spectra are shown in Figs. 1 and 2. The form of the spectrum adopted in this study is normalized (S( f )/p2) which is shown as
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
667
Fig. 1. Sample wind pressure spectra on flat roof.
Fig. 2. Evolution of spectral shapes on monoslope roof.
a function of the reduced frequency, F"f h/», where, f is the frequency, h is a typical building dimension (mean height of the building is chosen in this study), » is the mean velocity at the height of the building and p is the standard deviation of fluctuating pressure. As expected, wind pressure spectra measured on low building roofs vary with wind direction, roof geometry and tap location. At higher frequencies, the amplitudes of spectra seem to be similar; however, they are more different at lower frequencies in which most of the energy lies. Measurement results show that, in general, the amplitude of pressure spectrum decreases as the frequency increases.
668
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
However, a small growth in spectral amplitudes at dimensionless frequencies (F) between 0.1 and 0.2 ( f+20—40 Hz) has been observed at taps located in farwind (i.e., the windward edge region away from the windward corner in case of oblique wind angle) as well as leeward regions of the roof. These observed humps on spectra are possibly associated with shedding of dominant eddies in those regions. Typical examples are shown in Figs. 1 and 2 for flat (sample S2) and monoslope roofs (sample 3), respectively. Further, Fig. 2 presents the evolution of spectral shapes and the transition from one to the other for a monoslope roof building. It is also noted from experimental results that the pressure spectra normalized with respect to the variance (p2) corresponding to the same tap at open and suburban terrain conditions seem to be similar. Further, it can be seen from roof pressure spectra that significant energy is distributed over a wide frequency range indicating the broadband nature of the corresponding pressure fluctuations. In summary, the results show that although the spectra change for different roof geometries, terrain conditions, wind attack angle and tap location, they can be classified into two categories: (1) spectra whose amplitude decreases as the frequency increases and (2) spectra whose amplitude decreases up to a certain frequency and then grow to have another hump. Since the variance of pressure fluctuations also varies for different configurations, normalization of power spectra by the variance contributes to their similarity and assists in the generalization process.
4. Empirical model for pressure spectra Based on the similarities found in normalized spectra for various conditions, it may be appropriate to suggest a suitable analytical model for pressure spectra. Within this context, several well-known curve-fitting techniques have been employed to extract a suitable empirical equation for spectra. However, a different approach by using trial and error proved to be successful. The code is written in MATLAB environment with the help of available built-in functions. The independent variable ( f ), the dependent variable (S( f )/p2), initial values of parameters and the function to be fitted are inputs to the program. Traditional optimization methods have been used in the program to estimate the optimum parametric values iteratively by minimizing the sum of the squared residuals (i.e., the sum of the squares of the differences between the fitted spectral ordinates and the measured spectral ordinates). In this study, the Levenberg—Marquardt algorithm known for its robustness along with cubic polynomial interpolation line search method has been used to carry out least-squares optimization [5]. Several functions representing approximately the same shape as the roof pressure spectra have been tried and the following exponential function was found to be more appropriate: S( f )/p2"a e~c1f#a e~c2f, (1) 1 2 where S( f ) is the spectral ordinate, p2 is the variance, f is the frequency, a and a are 1 2 the position constants and c and c are the shape constants. The position constants 1 2
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
669
Fig. 3. Measured and fitted wind pressure spectra.
control the location of the spectra, whilst the shape constants control their shape. An example of the measured and fitted wind pressure spectra is shown in Fig. 3. The evolution of spectral curve is shown by plotting a e~c1f and a e~c2f separately. It is 1 2 clear that the term a e~c1f controls the position and shape of spectra at higher1 frequency region, while the term a e~c2f controls the position and shape of spectra at 2 lower-frequency region. The spectral fit appears satisfactory; for instance, the spectral fit has zero up-crossing or down-crossing rate (N )"57, positive or negative peak 0 rate (N )"150, irregularity factor (e)"0.38 and bandwidth parameter (b)"0.92 1 which are close to the corresponding target values of 55, 150, 0.37 and 0.93, respectively (N , N , e and b are estimated from corresponding spectra using the expressions 0 1 provided in Ref. [6]). Adequate performance of the proposed empirical expression has been established for the various cases examined. Though the fit is based on minimizing the squared residuals, the level of accuracy of the fit is quantified based on four spectral statistics (N , N , e and b); the fit which represents each of the target spectral 0 1 statistics within 10% is considered reasonable for practical applications.
5. Towards generalization It may be possible to categorize normalized spectra on a roof based on their similarities which is a gross simplification of the actual complex spectral patterns. It was decided, first, to classify the zones of Gaussian as well as non-Gaussian pressure fluctuations due to the difference in simulation methodologies used for each case. The statistics of pressure fluctuations was carefully checked and the corresponding zones have been identified for various ranges of wind direction. A particular zone is
670
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
Fig. 4. Gaussian and non-Gaussian zones for flat roofs.
considered non-Gaussian if the absolute values of skewness and kurtosis of pressure fluctuations at various taps are greater than 0.5 and 3.5, respectively. Fig. 4 shows the approximate Gaussian and non-Gaussian zones for a flat roof. Variable z is assumed to be 10% of least horizontal dimension or 40% of lower eave height whichever is less [7]. The non-Gaussian region is divided into two zones since, in general, two types of spectra have been found there. The first type observed on windward locations has generally continuous decreasing amplitudes while the second, observed on farwind or leeward locations, has another hump — see Fig. 1 (samples S1 and S2). A group of normalized spectra (measured) corresponding to farwind non-Gaussian region of a flat roof building under open and suburban exposures is shown in Fig. 5. Since spectra appear similar, they have been averaged and fitted by Eq. (1). The statistics of the fitted and observed spectra presented in Table 1 indicates that the fitted spectrum is a good representative of the observed spectra. In the case of Gaussian region, the normalized spectra measured at several locations on a flat roof at different conditions appear also similar. Therefore, they are averaged and fitted to obtain the most suitable spectra for that zone. A similar exercise has been carried out for various roof geometries with successful results. The final outcome of this exercise is a set of standard spectral shapes and their parameters associated with different regions of each roof. The standard spectral shapes of a flat roof building are presented in Fig. 6; each spectrum is assigned a subscript where the first letter stands for the type of roof (F — flat roof), the second stands for the type of region (G — Gaussian, NG — non-Gaussian) and the number (1 or 2) stands for the type of spectra in that zone. Parameters of the spectral shapes corresponding to Fig. 6 are provided in Table 2. Note that the position constant (a ) 2 of spectra from Gaussian regions is lower than that of spectra from non-Gaussian regions which indicates that the spectra from non-Gaussian regions are located above spectra from Gaussian regions in lower frequencies (see Fig. 6). The shape constants (c and c ) of spectra from Gaussian regions are also lower than those of spectra from 1 2 non-Gaussian regions which shows that the reduction rate of spectral amplitudes from Gaussian regions is generally lower than that of spectral amplitudes from non-Gaussian regions. The effect of such variations of parameters on spectral shapes is clear from Fig. 6. Similar results have been obtained in case of other roof geometries.
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
671
Fig. 5. Measured and fitted wind pressure spectra (flat roof, non-Gaussian zone).
Table 1 Comparison of spectral properties (Flat roof, non-Gaussian zone) Sample
N 0
N 1
e
b
Fit 1 2 3 4 5
80.2 78.5 70.8 75.7 64.9 79.5
166.7 155.4 160.2 163.7 154.9 155.4
0.48 0.50 0.44 0.46 0.42 0.51
0.88 0.86 0.90 0.89 0.91 0.86
It is interesting to note that the parameters obtained in the above fitting are all positive and therefore, the derivative of Eq. (1) at any point is a negative quantity which reveals that the fitted curve has a downward slope. As a result, this function represents only mild growth in spectra and needs modifications to fit the predominant spectral growth observed in some cases such as sample 3 shown in Fig. 2. In such cases (e.g. farwind as well as leeward non-Gaussian regions of monoslope roof) spectra show a clear hump and the proposed Eq. (1) requires an additional term a e~c3f to fit 3 the data efficiently; the parameters a and c can be established using the fitting 3 3 procedure as previously discussed.
672
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
Fig. 6. Standard spectral shapes for flat roof.
Table 2 Proposed spectral parameters for flat roof Parameters (]10~2)
S FG1 S FNG1 S FNG2
a 1
a 2
c 1
c 2
0.9756 1.207 0.8162
1.956 11.61 12.05
0.9725 1.977 1.084
16.52 30.15 38.23
6. Application The application of the developed standard spectral shapes is briefly described here. It is shown that for practical purposes, from the knowledge of variance of pressure fluctuations at a specific roof location, spectra of pressure fluctuations at the same location can be synthetically generated using standard spectral shapes provided in this study. Thereafter, the simulation scheme proposed by Suresh Kumar and Stathopoulos [2] can be utilized to generate Gaussian as well as non-Gaussian pressure fluctuations on roofs. The measured spectrum and its empirical fit shown in Fig. 3 are used to demonstrate the efficiency of the synthetic spectra in time series simulation. The simulations have been carried out based on the approach discussed in Ref. [2]. The first four moments (mean, variance, skewness and kurtosis) of the
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
673
Table 3 Statistics of simulated signals
Simulation-I Simulation-II
Mean
Variance
Skewness
Kurtosis
!0.8 !0.8
0.28 0.28
!1.92 !1.96
9.87 10.22
Table 4 Number of peaks and down-crossings of simulated signals Peaks
Simulation-I Simulation-II
1854 1939
Level crossing below mean in terms of p 0
!1
!2
!3
!4
!5
!6
617 629
362 374
189 193
100 100
43 45
22 25
12 12
simulated signal using measured spectra (Simulation-I) has been compared with those of the simulated signal using synthetic spectra (Simulation-II) in Table 3. The number of peaks and crossings of the same simulated signals are compared in Table 4. The corresponding values of both signals are close and similar results prevail in other cases. 7. Concluding remarks An extensive investigation of power spectra of wind-induced pressures acting on low building roofs has registered the variation of spectra with respect to location, wind direction, roof geometry and terrain conditions. However, similarities among normalized spectra observed in many cases suggest an appropriate empirical form for the synthetic generation of normalized spectra. Thereafter, normalized spectra are categorized and the standard spectral shapes associated with various zones of each roof and their parameters are established, and provided for the case of flat square roofs. The application of the proposed spectral shapes is also briefly discussed. Finally, it must be emphasized that the provided spectral shapes can be adopted in codal provisions for the generation of pressure fluctuations on roofs; these are required for the design of roofs against extreme wind pressures and fatigue. References [1] ASCE, Wind tunnel studies of buildings and structures, J. Aerospace Eng. ASCE 9 (1) (1996) 19—36. [2] K. Suresh Kumar, T. Stathopoulos, Computer simulation of fluctuating wind pressures on low building roofs, Proc. 3rd Int. Col. on Bluff Body Aerodyn. and Appl., Blacksburg, Virginia, July 28—August 1 1996, J. Wind Eng. Ind. Aerodyn. 69—71 (1997) 485—495.
674
K.S. Kumar, T. Stathopoulos/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 665–674
[3] T. Stathopoulos, D. Surry, A.G. Davenport, Effective wind loads on flat roofs, J. Struct. Eng. ASCE 107 (2) (1981) 281—298. [4] M. Kasperski, H. Koss, Beatrice joint project wind action on low-rise buildings. Part 2: Analysis in the frequency domain, Proc. 3rd Int. Col. on Bluff Body Aerodyn. and Appl., Blacksburg, Virginia, July 28—August 1 1996. [5] MATLAB, Optimization ToolBox User’s Guide, The Mathworks Inc., Mass., 1994. [6] E. Vanmarcke, Random Fields: Analysis and Synthesis, The MIT Press, Cambridge, 1983. [7] NBCC, User’s Guide — NBC 1995, Structural Commentaries (Part 4), National Research Council of Canada, Ottawa, Canada, 1995.