Journal of Applied Science and Engineering, Vol. 19, No. 2, pp. 125-134 (2016)
DOI: 10.6180/jase.2016.19.2.03
Investigation on Non-Gaussian Peak Factors for Wind
Pressures on Domed Roof Structures
Yuan-Lung Lo
Department of Civil Engineering, Tamkang University,
Tamsui, Taiwan 251, R.O.C.
Abstract
Roof curvature and Reynolds number effect significantly define the characteristics of wind
pressures on domed roof structures. Once the flow separates from the roof surface, apparent changes in
aerodynamic coefficients or non-Gaussian wind-induced pressure spectra are expected. This paper
investigated the non-Gaussian peak factors of wind pressure coefficients on domed roofs under a
simulated suburban boundary layer flow. By estimating the higher statistic moments, skewness and
kurtosis coefficients, and adopting the moment-based Hermite polynomial translation techniques,
including the softening and hardening processes, comparisons were made based on the empirical and
simulated results. The fairly good estimation results by translation process were demonstrated and the
influence on the estimation of wind pressure extremes was discussed in a practical viewpoint.
Key Words: Non-Gaussian, Hermite, Extremes
1. Introduction
Domed roof structures have been commonly used
for their good performance in many aspects. Because of
their less-weighted material and large span geometrical
characteristics, wind resistant design of this kind of flexible structures may attract more attention than the earthquake resistant design. Therefore, a detailed investigation on the pressure distribution over the roof surface is
essential and important. For the past few decades, researches regarding curvatures of the roofs or levels of
Reynolds numbers have been made in the viewpoint of
investigating the stagnation point, the occurrence of separation, reattachment of flow, or wake behaviors by means
of wind tunnel tests of pressure measurement or direct
visual observation technologies.
Maher [1] conducted the mean wind pressures on hemispherical domes with various height-span ratios. The
effect of surface roughness of hemispherical domes was
investigated in the range of Re = 6 ´ 105~1.8 ´ 106. Toy
*Corresponding author. E-mail: [email protected]
et al. [2] conducted surface pressures on hemispherical
domes under two turbulent flows. Taylor [3] measured
the mean and root-mean-square (RMS) values of wind
pressures on the hemispherical domes with various heightspan ratios under oncoming turbulence of 10%~20% and
the Reynolds number of 1.1 ´ 105~3.1 ´ 105. Ogawa et
al. [4] investigated the mean and RMS wind pressures
and spectrum characteristics of three height-span ratios
of domed roofs under laminar and turbulent flows. Letchford and Sarkar [5] investigated the effect of surface
roughness on the pressure distributions and the consequent overall drag and lift forces. Uematsu and Tsuruishi
[6] proposed a computer-assisted wind load evaluation
system for the claddings of hemispherical domes. Cheng
and Fu [7] preceded a series of wind tunnel tests to investigate the Reynolds number effects on the aerodynamic characteristics of hemispherical dome in smooth and
turbulent boundary layer flows by adjusting the dome
scale and the flow velocities. Qiu et al. [8] simulated several installations of splitter plates of circular cylinder
under smooth and turbulent flows and compared the distributions of pressure coefficients under different Rey-
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Yuan-Lung Lo
nolds numbers. Qiu et al. [9] applied the empirical model
of mean wind pressure coefficient proposed by Yeung
[10] to regress the relationship between the model parameters and the Reynolds numbers. So far, investigations
on pressures on the domed roofs are still attractive issues
and have not been fully understood.
This paper focuses on the estimation of extreme pressures on the domed roofs which significantly dominates
the wind resistant design of claddings or sub-structures.
As it is generally known, once the separation of flow occurs due to the curvature effect or Reynolds number effect, non-Gaussian wind load effect results in extremely
large local loading. Through the convenient applications
of moment-based Hermite models, peak factors of nonGaussian wind effect can be estimated in comparison with
the empirical and the Gaussian ones. The extreme pressures on the domed roofs are then calculated with the
properly assumed mean and RMS pressure coefficients.
2. Wind Pressures on Domed Roofs
18% to 23% as indicated in Figure 1. Acrylic test models
are manufactured and composed of the roof model and
the circular base model, each of which varies with a ratio
of its characteristic length to the roof span. For the roof
model, f/D varies from 0.0 to 0.5; while for the base
model, h/D varies from 0.0 to 0.5. Geometric nomenclature is given in Figure 2 except for the non-existing case,
f/D = 0 with h/D = 0.
Pressure taps are equally installed along the central
meridian line of the roof surface parallel to the wind direction for the wind pressures generally remain isobaric
and perpendicular to the wind direction [1-4,6,7]. Instantaneous pressures are recorded simultaneously by
the micro-pressure scanning system at a sampling rate of
1000 Hz and corrected by the numerical way to reduce
the tubing effect. 120 runs of recording, each of which
represents a 10-minute sample length in field scale, are
carried out for reliable statistical estimation. The calculation of the blockage ratio is defined by the ratio of the
projected area of the model to the cross sectional area of
2.1 Wind Tunnel Tests
Systematic tests are conducted in a wind tunnel with
a cross section of 12.5 m in length, 1.8 m in height and
1.8 m in width. The turbulent flow is simulated by properly equipped spires and roughness blocks as shown in
Photo 1. The index of the power law, a, is about 0.27 and
the turbulence intensities at model heights varies from
Figure 1. Simulated flow profiles (d represents the boundary
height).
Photo 1. Experimental setup.
Figure 2. Geometric symbols of roof and circular base models.
Investigation on Non-Gaussian Peak Factors for Wind Pressures on Domed Roof Structures
the wind tunnel. Table 1 lists the blockage ratios of the
testing models in this research. The blockage effect is
ignored for that the largest blockage ratio caused by the
case of f/D = 0.5 with h/D = 0.5 is about 2.92%. Reynolds numbers among all tests are calculated in the range
of 1.12~1.56 ´ 105, which are near the critical range;
however, the Reynolds number effect do not affect the
aerodynamic behavior due to the oncoming turbulence
from the approaching wind [7].
2.2 Characteristics of Estimated Higher Moments
To investigate the geometric effect on higher moments of wind pressures, normalized aerodynamic pressure coefficient is calculated in advanced as Equation
(1):
(1)
127
in order to estimate equivalently 1-second peak factors
as well as extremes.
Figures 3 and 4 are plotted to show the variations of
higher moments along the central meridian lines. Here
the kurtosis coefficients are reduced by 3 as excess kurtosis coefficients. From Figure 3, most skewness coefficients vary between 0.5 and -0.5. Apparent large negative values are indicated at the front area in the cases of
f/D = 0.0 and 0.1, as well as indicated around 110° ~130°
in the cases of f/D = 0.2~0.5, where the separation of
flow is considered occurred. For the cases of f/D = 0.2~
0.5, the locations with positive mean pressures show
positive skewness coefficients while the negative mean
pressures show negative skewness coefficients. Such tendencies may contribute to the subsequent estimation of
positive or negative extreme pressures. Figure 4 shows
the variations of excess kurtosis coefficients. Similar ten-
in which pj represents instantaneous pressure at moment
j; pref represents the reference static pressure; r represents the air density; UH represents the mean wind speed
at model height. Higher moments, skewness and kurtosis
coefficients are then estimated from sample records of
pressure coefficients as Equation (2):
(2)
where i = 3 and i = 4 represent skewness and kurtosis
coefficients respectively. C p and C p¢ are the mean and
RMS pressure coefficients. N is the number of the record length. It is especially noticed that in this research,
each sample record is conducted 1-sec moving averaged
Table 1. Blockage ratios of testing models
h/D = 0.0
h/D = 0.1
h/D = 0.2
h/D = 0.3
h/D = 0.4
h/D = 0.5
f/D = 0.0 ~ 0.4
f/D = 0.5
< 1.78%
< 2.01%
< 2.24%
< 2.47%
< 2.69%
< 2.92%
1.78%
2.01%
2.24%
2.47%
2.69%
2.92%
Note: The cross-sectional area of the wind tunnel is 2.2 ´
1.8 = 3.96 m2.
Figure 3. Distributions of skewness coefficients along the
meridian lines.
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Yuan-Lung Lo
through a memoryless monotonic translation form as
(Grigoriu [11])
(3)
where y and u are standardized non-Gaussian and Gaussian random variables; g(×) is the translation form; FY (×)
and F(×) are cumulative probability density functions
(CDFs) of Y and U respectively.
3.2 Non-Gaussian Peak Factor for a Softening
Process
To describe the nonlinear form of g(×), Winterstein
and Kashef [12] presented a third order Hermite model
for the softening translation process (g4 > 0):
(4a)
(4b)
where Hen(u) represents the Hermite polynomial function of u and h n represents the Hermite polynomial coefficients. The parameters h3 and h4 in Equation (4) are
proposed [12] as
Figure 4. Distributions of excess kurtosis coefficients along
the meridian lines.
dencies can be indicated at the locations where the separation occurs. It is interesting that, excess kurtosis coefficients along the central meridian line sometimes show
negative values, which means that both the softening
process and the hardening process may be necessary to
be adopted if adopting moment-based Hermite model.
From both figures, it is obvious that the h/D value does
not make clear differences on the variations of higher
moments and it shall be more emphasized on skewness
coefficients.
3. Simulation of Non-Gaussian Peak Factors
3.1 Translation Process Concept
The standardized non-Gaussian process Y can be
related to an underlying standard Gaussian process U
(5a)
(5b)
(5c)
The model proposed by Winterstein and Kashef [12]
shows good practical applicability since it eliminates
solving the nonlinear and coupled equations while in
translation process. However, the skewness and kurtosis
coefficients are still limited in some application ranges
in order to remain the monotonic increasing feature. The
empirical results shown in Figures 3 and 4 are tested later
to show their being located within the limits of Equations
(4) and (5) for a softening process.
Investigation on Non-Gaussian Peak Factors for Wind Pressures on Domed Roof Structures
Kwon and Kareem [13] presented an analytical expression for non-Gaussian peak factor based on the concept of translation process (Grigoriu [11]), the momentbased Hermite model (Winterstein and Kashef [12]),
and the frame work of Gaussian peak factor (Davenport
[14]).
limits of softening and hardening processes and the practical limits for pairs of estimated higher moments. With
the improvement of approximation of Hermite models,
Winterstein and Kashef [12] has presented the narrower
monotonic limit for softening process as
(1.25g3)2 £ g4
(6)
129
(8)
A symmetric limit can be derived from Equation (8)
based on the similar concept indicated by Choi and
Sweetman [17] for the hardening process. And the practical limit for any pair of skewness and excess kurtosis
coefficients is as Equation (9)
¥
where b = 2 ln(v 0T ); v0 = m2 / m0 ; mi = ò n i S y ( n) dn;
0
Sy(n) is one-sided spectral density of the standardized
non-Gaussian variable y, used to define the crossing rate
v0 according to the theory of the statistics of narrowband processes [15]; n is frequency in Hz; T is duration
of a record in second; g is Euler’s constant (» 0.5772);
g3, g4, k, h3, and h4 are defined as Equations (2), (4), and
(5) respectively.
3.3 Non-Gaussian Peak Factor for a Hardening
Process
Huang et al. [16] presented a similar analytical form
of non-Gaussian peak factor for a hardening process (g4
< 0) as
(7a)
(9)
which can be derived from Equation (2).
Figure 5 shows the distribution of the paired empirical higher moments in comparison with Equation (8) and
(9), the non-monotonic limit (g4 = 0) and the Gaussian
line (g3 = 0). It is demonstrated first that, the pairs of
higher moments (g3 and g4) almost exhibit non-zero values
indicating the non-Gaussian characteristics. Most pairs
locate inside or follow the monotonic limit of the softening process. Even some cases fall outside the softening
limit, the estimation based on Equation (6) is expected
to provide a fairly good approximation. On the contrary,
although some pairs locate in the hardening process,
whose peak factors can be estimated by Equation (7),
the deviation from the non-monotonic limit is less than
0.5 and may be simply considered applicable to a soften-
(7b)
where a1 = (1 - 3h4)/h4 and a2 = h3/h4. Equation (7) was
derived based on the inversion form of the hardening
process proposed by Choi and Sweetman [17].
3.4 Limits of Softening and Hardening Processes
Choi and Sweetman [17] showed the monotonic
Figure 5. Distributions of pairs of empirical higher moments.
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Yuan-Lung Lo
ing process. However in this research, the softening process and the hardening process are both adopted according to the values of excess kurtosis coefficients.
4. Discussions
4.1 Simulation Results of Peak Factors
Six representative cases out of 35 models are selected
and plotted in Figure 6 to show the comparison results of
empirical and estimated peak factors based on translation process. Each pressure tap is drawn by the mean of
120 estimates plus one standard deviation as shown. For
reference, Gaussian peak factors are plotted as well. Since
Figures 3 and 4 show the less significant effect from h/D
values, it is supposed that the estimation of peak factors
shall be emphasized on the f/D effect. From Figure 6, it
is indicated that the Gaussian peak factors based on Davenport [14] can only meet the empirical ones at certain
locations where correspond to small values of empirical
higher moments. The mean of 120 estimates along the
central meridian line generally coincides with the mean
of 120 empirical ones based on Equations (6) and (7).
For those with g3 > 0, the probability density function
contains a longer tail in the positive extreme and hence a
larger positive peak factor shall be expected; on the contrary, those with g3 < 0 demands a larger negative peak
Figure 6. Comparison results of empirical and estimated peak factors due to f/D effect.
Investigation on Non-Gaussian Peak Factors for Wind Pressures on Domed Roof Structures
factor. Compared to Gaussian peak factors, g3 < 0 gives
a slight overestimation in negative peak factors and a significant underestimation in positive peak factors; meanwhile g3 > 0 gives the estimation in the other way around.
And as supposed, it is difficult from Figure 6 to find a
clear tendency when g4 changes.
Cases of f/D = 0.5 are also plotted in Figure 7 to
show the highly similar distributions in different h/D
values. From Figure 7, extremely large peak factors are
indicated at locations where the separation is expected
to occur. Fairly small estimation errors can be indicated
in both Figures 6 and 7, which represents the good performance of translation process by moment-based Her-
131
mite model.
4.2 Estimation of Extreme Wind Pressures
By substituting the mean and RMS values of pressure
coefficients, empirical and estimated extremes are calculated in Figure 8. For the flat roof (f/D = 0.0) with h/D =
0.1, estimated extremes meet quite well to the empirical
extremes, even in the strong non-Gaussian area. Those
calculated extremes based on Gaussian peak factors are
overestimated in the positive extremes and underestimation in the negative, which may result in improper design
wind loads in the cladding in this area. For the cases with
f/D = 0.1~0.5, the estimated extremes fit very well to the
Figure 7. Comparison results of empirical and estimated peak factors due to h/D effect.
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Yuan-Lung Lo
empirical ones and generally in a very small deviation error. For those Gaussian extremes, apparent overestimation in the positive extremes in the middle area and underestimation in the negative extremes in the upstream
are indicated. In the practical design, the over- or underestimation in extremes caused by Gaussian assumption
may be acceptable for the cladding design of wind loads
on domed roofs since it is more focused on the estimations of the negative extremes in the middle and downstream area and the positive extremes in the upstream
area. It is also worth mentioning that, even the significant
differences have been introduced in the distributions of
non-Gaussian peak factors in Figures 6 and 7, the mean
and RMS values of pressure coefficients may reduce the
non-Gaussian effect on the estimation of extreme pressures in the practical design.
5. Conclusions
In this paper, 120 runs of wind pressure measurements over the 35 roof surfaces were conducted and
analyzed through the calculation of aerodynamic coefficients and statistical higher moments. The distribution
of skewness and excess kurtosis coefficients was discussed and the f/D effect was pointed out to be more apparent than the h/D effect. Non-Gaussian translation pro-
Figure 8. Comparison results of empirical and estimated extremes due to f/D effect.
Investigation on Non-Gaussian Peak Factors for Wind Pressures on Domed Roof Structures
cess based on moment-based Hermite polynomial models were introduced to estimate the non-Gaussian peak
factors based on the empirical skewness and kurtosis
coefficients. The estimation results show that, the estimated non-Gaussian peak factors were found in a very
good agreement with the empirical ones and the significant differences were indicated caused in the non-Gaussian area which coincides the locations where the separation usually occurs. Gaussian and non-Gaussian extreme
pressures were calculated for comparison and demonstrated the accuracy of the adopted moment-based Hermite polynomial models for mildly non-Gaussian wind
pressures. For strongly non-Gaussian wind pressures,
less accurate peak factors may be obtained due to the insufficiency in describing the distribution tail by skewness and kurtosis coefficients. However, such significant
biases may be somehow reduced when the peak factor
is incorporated with the mean and RMS wind pressures
for extreme pressures. To provide a practical design of
local wind loads on claddings or sub-structures, it is
strongly suggested to estimate the wind pressure characteristics in detail.
References
[1] Maher, F. J., “Wind Loads on Basic Dome Shapes,” J.
Struct. Div. ASCE ST3, pp. 219-228 (1965).
[2] Toy, N., Moss, W. D. and Savory, E., “Wind Tunnel
Studies on a Dome in Turbulent Boundary Layers,” J.
Wind Eng. Ind. Aerodyn, Vol. 11, pp. 201-212 (1983).
doi: 10.1016/0167-6105(83)90100-9
[3] Taylor, T. J., “1991 Wind Pressures on a Hemispherical
Dome,” J. Wind Eng. Ind. Aerodyn, Vol. 40, pp. 199213 (1992). doi: 10.1016/0167-6105(92)90365-H
[4] Ogawa, T., Nakayama, M., Murayama, S. and Sasaki,
Y., “Characteristics of Wind Pressures on Basic Structures with Curved Surfaces and their Response in Turbulent Flow,” J. Wind Eng. Ind. Aerodyn, Vol. 38, pp.
427-438 (1991). doi: 10.1016/0167-6105(91)90060-A
[5] Letchford, C. W. and Sarkar, P. P., “Mean and Fluctuating Wind Loads on Rough and Smooth Parabolic
Domes,” J. Wind Eng. Ind. Aerodyn., Vol. 88, pp. 101117 (2000). doi: 10.1016/S0167-6105(00)00030-1
[6] Uematsu, Y. and Tsuruishi, R., “Wind Load Evaluation
System for the Design of Roof Cladding of Spherical
133
Domes,” J. Wind Eng. Ind. Aerodyn., Vol. 96, pp.
2054-2066 (2008). doi: 10.1016/j.jweia.2008.02.051
[7] Cheng, C. M. and Fu, C. L., “Characteristics of Wind
Loads on a Hemispherical Dome in Smooth Flow and
Turbulent Boundary Layer Flow,” J. Wind Eng. Ind.
Aerodyn., Vol. 98, pp. 328-344 (2010). doi: 10.1016/j.
jweia.2009.12.002
[8] Qiu, Y., Sun, Y., Wu, Y. and Tamura, Y., “Effects of
Splitter Plates and Reynolds Number on the Aerodynamic Loads Acting on a Circular Cylinder,” J. Wind
Eng. Ind. Aerodyn., Vol. 127, pp. 40-50 (2014). doi:
10.1016/j.jweia.2014.02.003
[9] Qiu, Y., Sun, Y., Wu, Y. and Tamura, Y., “Modeling the
Mean Wind Loads on Cylindrical Roofs with Consideration of the Reynolds Number Effect in Uniform
Flow with Low Turbulence,” J. Wind Eng. Ind. Aerodyn.,
Vol. 129, pp. 11-21 (2014). doi: 10.1016/j.jweia.2014.
02.011
[10] Yeung, W. W. H., “Similarity Study on Mean Pressure
Distributions of Cylindrical and Spherical Bodies,” J.
Wind Eng. Ind. Aerodyn., Vol. 95, No. 4, pp. 253-266
(2007). doi: 10.1016/j.jweia.2006.06.014
[11] Grigoriu, M., “Crossings of Non-Gaussian Translation
Processes,” J. Eng. Mech., Vol. 110, No. 4, pp. 610620 (1984). doi: 10.1061/(ASCE)0733-9399(1984)110:
4(610)
[12] Winterstein, S. R. and Kashef, T., “Moment-based
Load and Response Models with Wind Engineering
Applications,” J. Sol. Energy Eng., Trans., ASME, Vol.
122, No. 3, pp. 122-128 (2000). doi: 10.1115/1.12880
28
[13] Kwon, D. K. and Kareem, A., “Peak Factors for NonGaussian Load Effects Revisited,” J. Structural Eng.,
Vol. 137, No. 12, pp. 1611-1619 (2011). doi: 10.1061/
(ASCE)ST.1943-541X.0000412
[14] Davenport, A. G., “Note on the Distribution of the
Largest Value of a Random Function with Application
to Gust Loading,” ICE Proc., Vol. 28, No. 2, pp. 187196 (1964). doi: 10.1680/iicep.1964.10112
[15] Newland, D. E., An Introduction to Random Vibrations and Spectral & Wavelet Analysis (3rd Ed.), Dover Publications, Inc, Mineola, New York (1993).
[16] Huang, M. F., Lou, W., Pan, X., Chan, C. M. and Li, Q.
S., “Hermite Extreme Value Estimation of Non-Gaussian Wind Load Process on a Long-span Roof Struc-
134
Yuan-Lung Lo
ture,” J. Struct. Eng., Vol. 140, pp. 04014061-104014061-12 (2014). doi: 10.1061/(ASCE)ST.1943541X.0000962
[17] Choi, M. and Sweetman, B., “The Hermite Moment
Model for Highly Skewed Response with Application
to Tension Leg Platforms,” J. Offshore Mech. Arctic
Eng., Vol. 132 (2010). doi: 10.1115/1.4000398
Manuscript Received: Jul. 1, 2015
Accepted: Nov. 6, 2015