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 fle-

xible structures may attract more attention than the ear-

thquake resistant design. Therefore, a detailed investiga-

tion on the pressure distribution over the roof surface is

essential and important. For the past few decades, re-

searches regarding curvatures of the roofs or levels of

Reynolds numbers have been made in the viewpoint of

investigating the stagnation point, the occurrence of sep-

aration, 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 he-

mispherical 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

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 height-

span 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. Let-

chford and Sarkar [5] investigated the effect of surface

roughness on the pressure distributions and the conse-

quent 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 in-

vestigate the Reynolds number effects on the aerodyna-

mic 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 se-

veral installations of splitter plates of circular cylinder

under smooth and turbulent flows and compared the dis-

tributions of pressure coefficients under different Rey-

Journal of Applied Science and Engineering, Vol. 19, No. 2, pp. 125�134 (2016) DOI: 10.6180/jase.2016.19.2.03

*Corresponding author. E-mail: yllo@mail.tku.edu.tw

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 pa-

rameters 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 pres-

sures 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 oc-

curs due to the curvature effect or Reynolds number ef-

fect, non-Gaussian wind load effect results in extremely

large local loading. Through the convenient applications

of moment-based Hermite models, peak factors of non-

Gaussian wind effect can be estimated in comparison with

the empirical and the Gaussian ones. The extreme pres-

sures on the domed roofs are then calculated with the

properly assumed mean and RMS pressure coefficients.

2. Wind Pressures on Domed Roofs

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 pro-

perly equipped spires and roughness blocks as shown in

Photo 1. The index of the power law, �, is about 0.27 and

the turbulence intensities at model heights varies from

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 nomencla-

ture 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 di-

rection for the wind pressures generally remain isobaric

and perpendicular to the wind direction [1�4,6,7]. In-

stantaneous 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 calcu-

lation of the blockage ratio is defined by the ratio of the

projected area of the model to the cross sectional area of

126 Yuan-Lung Lo

Photo 1. Experimental setup.Figure 2. Geometric symbols of roof and circular base mo-

dels.

Figure 1. Simulated flow profiles (� represents the boundaryheight).

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%. Rey-

nolds 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 mo-

ments of wind pressures, normalized aerodynamic pres-

sure coefficient is calculated in advanced as Equation

(1):

(1)

in which pj represents instantaneous pressure at moment

j; pref represents the reference static pressure; � repre-

sents 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. Cp and �Cp are the mean and

RMS pressure coefficients. N is the number of the re-

cord length. It is especially noticed that in this research,

each sample record is conducted 1-sec moving averaged

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 kur-

tosis coefficients. From Figure 3, most skewness coeffi-

cients vary between 0.5 and -0.5. Apparent large nega-

tive 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 ten-

dencies may contribute to the subsequent estimation of

positive or negative extreme pressures. Figure 4 shows

the variations of excess kurtosis coefficients. Similar ten-

Investigation on Non-Gaussian Peak Factors for Wind Pressures on Domed Roof Structures 127

Table 1. Blockage ratios of testing models

f/D = 0.0 ~ 0.4 f/D = 0.5

h/D = 0.0 < 1.78% 1.78%

h/D = 0.1 < 2.01% 2.01%

h/D = 0.2 < 2.24% 2.24%

h/D = 0.3 < 2.47% 2.47%

h/D = 0.4 < 2.69% 2.69%

h/D = 0.5 < 2.92% 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 themeridian lines.

dencies can be indicated at the locations where the sepa-

ration occurs. It is interesting that, excess kurtosis coef-

ficients 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

through a memoryless monotonic translation form as

(Grigoriu [11])

(3)

where y and u are standardized non-Gaussian and Gaus-

sian random variables; g() is the translation form; FY ()and () 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 (�4 > 0):

(4a)

(4b)

where Hen(u) represents the Hermite polynomial func-

tion of u and hn represents the Hermite polynomial co-

efficients. The parameters h3 and h4 in Equation (4) are

proposed [12] as

(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.

128 Yuan-Lung Lo

Figure 4. Distributions of excess kurtosis coefficients alongthe meridian lines.

Kwon and Kareem [13] presented an analytical ex-

pression for non-Gaussian peak factor based on the con-

cept of translation process (Grigoriu [11]), the moment-

based Hermite model (Winterstein and Kashef [12]),

and the frame work of Gaussian peak factor (Davenport

[14]).

(6)

where � = 2 0ln( )v T ; v0 = m m2 0/ ; mi = n S n dni

y ( )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 narrow-

band processes [15]; n is frequency in Hz; T is duration

of a record in second; � is Euler’s constant (� 0.5772);

�3, �4, �, 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 (�4

< 0) as

(7a)

(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

limits of softening and hardening processes and the prac-

tical 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

(1.25�3)2 � �4 (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 prac-

tical limit for any pair of skewness and excess kurtosis

coefficients is as Equation (9)

(9)

which can be derived from Equation (2).

Figure 5 shows the distribution of the paired empiri-

cal higher moments in comparison with Equation (8) and

(9), the non-monotonic limit (�4 = 0) and the Gaussian

line (�3 = 0). It is demonstrated first that, the pairs of

higher moments (�3 and �4) almost exhibit non-zero values

indicating the non-Gaussian characteristics. Most pairs

locate inside or follow the monotonic limit of the soften-

ing 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-

Investigation on Non-Gaussian Peak Factors for Wind Pressures on Domed Roof Structures 129

Figure 5. Distributions of pairs of empirical higher moments.

ing process. However in this research, the softening pro-

cess and the hardening process are both adopted accord-

ing 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 transla-

tion 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 Da-

venport [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 �3 > 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 con-

trary, those with �3 < 0 demands a larger negative peak

130 Yuan-Lung Lo

Figure 6. Comparison results of empirical and estimated peak factors due to f/D effect.

factor. Compared to Gaussian peak factors, �3 < 0 gives

a slight overestimation in negative peak factors and a sig-

nificant underestimation in positive peak factors; mean-

while �3 > 0 gives the estimation in the other way around.

And as supposed, it is difficult from Figure 6 to find a

clear tendency when �4 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 per-

formance of translation process by moment-based Her-

mite model.

4.2 Estimation of Extreme Wind Pressures

By substituting the mean and RMS values of pressure

coefficients, empirical and estimated extremes are calcu-

lated 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 underestima-

tion 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

Investigation on Non-Gaussian Peak Factors for Wind Pressures on Domed Roof Structures 131

Figure 7. Comparison results of empirical and estimated peak factors due to h/D effect.

empirical ones and generally in a very small deviation er-

ror. For those Gaussian extremes, apparent overestima-

tion in the positive extremes in the middle area and un-

derestimation in the negative extremes in the upstream

are indicated. In the practical design, the over- or under-

estimation 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 estima-

tions of the negative extremes in the middle and down-

stream 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 pres-

sures in the practical design.

5. Conclusions

In this paper, 120 runs of wind pressure measure-

ments over the 35 roof surfaces were conducted and

analyzed through the calculation of aerodynamic coef-

ficients and statistical higher moments. The distribution

of skewness and excess kurtosis coefficients was dis-

cussed and the f/D effect was pointed out to be more ap-

parent than the h/D effect. Non-Gaussian translation pro-

132 Yuan-Lung Lo

Figure 8. Comparison results of empirical and estimated extremes due to f/D effect.

cess based on moment-based Hermite polynomial mo-

dels were introduced to estimate the non-Gaussian peak

factors based on the empirical skewness and kurtosis

coefficients. The estimation results show that, the esti-

mated non-Gaussian peak factors were found in a very

good agreement with the empirical ones and the signif-

icant differences were indicated caused in the non-Gaus-

sian area which coincides the locations where the separa-

tion usually occurs. Gaussian and non-Gaussian extreme

pressures were calculated for comparison and demon-

strated the accuracy of the adopted moment-based Her-

mite polynomial models for mildly non-Gaussian wind

pressures. For strongly non-Gaussian wind pressures,

less accurate peak factors may be obtained due to the in-

sufficiency in describing the distribution tail by skew-

ness 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 charac-

teristics in detail.

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Manuscript Received: Jul. 1, 2015

Accepted: Nov. 6, 2015

134 Yuan-Lung Lo