design wind force coefficients for hyperbolic paraboloid free roofs
TRANSCRIPT
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Journal of Physical Science and Application 4 (1) (2014) 1-19
Design Wind Force Coefficients for Hyperbolic
Paraboloid Free Roofs
Fumiyoshi Takeda1, Tatsuya Yoshino
1and Yasushi Uematsu
2
1. Technical Research Center, R&D Division, Taiyo Kogyo Corporation 3-20, Syodai-Tajika, Hirakatashi, Osaka, Japan
2. Department of Architecture and Building Science, Tohoku University, Sendai, Japan
Received: August 11, 2013 / Accepted: September 04, 2013 / Published: January 15, 2014.
Abstract:This study discusses the wind force coefficients used to design hyperbolic paraboloid free roofs, which are obtained from
wind tunnel experiments, computational fluid dynamics, and structural analyses. Design wind force coefficients are proposed on the
basis of the wind tunnel experiment results with rigid models. The proposed wind force coefficients are compared with the
specifications of the Australia/New Zealand Standard from the viewpoint of load effect. In addition, the application of the proposed
wind force coefficients to membrane structures is investigated. Moreover, the effect of the deformation of membrane structures on
wind force is verified. As a result, it is clarified that the proposed wind force coefficients need to be improved when designing
membrane roofs, and show future work.
Key words:Wind force coefficients, hyperbolic paraboloid roof, membrane structure, wind tunnel experiment, computational fluid
dynamics, structural analysis.
Nomenclature
CD, CL: Drag and lift coefficients, respectively
CMx, CMy: Moment coefficients about the x and y axes,respectively
CNW, CNL: Wind force coefficients on the windward andleeward halves of the roof, respectively
CNW0, CNL0: Basic values of CNWand CNL, respectively
CNW, CNL: Design wind force coefficients on the windward
and leeward halves of the roof, respectivelyD: Drag
F1, F2: Frame models 1 and 2 for structural analysis,respectively
Gf: Gust effect factor
H: Mean roof height
Iu: Turbulence intensity
L: Lift
Lx: Longitudinal length scale of turbulence
Mx,My: Aerodynamic moments about the x and y axes,
respectivelyN: Axial force in a column
N*: Non-dimensional axial force in a column(=N/(qHa
2/4))
Nmean: Mean value ofN*values
Corresponding author: Fumiyoshi Takeda, researchengineer, research fields: building science, civil engineering.
E-mail: [email protected].
NW,NL: Normal wind force on the windward and leewardhalves, respectively
N1,N2,N3: Concentrated loads on the leaf springs of the
force balanceS: Projection area of the roof
S1: Suspension model 1 for structural analysis
UH: Mean wind speed at the mean roof heightH
WD1: The wind direction range of = 0 45
WD2: The wind direction range of = 90 45
a: Horizontal (projection) width of the roof
gf: Peak factor
h: Difference in height of the roof
qH: Reference velocity pressure at the mean roof
height
Greek Letters: Power law exponent for the mean velocity profile
: Correction factor for considering oblique winds
n:Maximum displacement at the n-thstep ofstructural analysis
ij: Absolute value of the deference between themaximum displacements at the iandjsteps ofstructural analysis
: Wind direction
: Correction factor of wind force coefficients for
membrane structures
DAVID PUBLISHING
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Design Wind Force Coefficients for Hyperbolic Paraboloid Free Roofs2
1. Introduction
Hyperbolic paraboloid (HP) free roofs are widely
used for structures providing shade and weather
protection in public spaces such as parks, playgrounds,and shopping areas. Fig. 1 shows an example of such
a roof. Membrane structures are often used in the
design of these roofs [1], because they are generally
very lightweight structures. However, they are also
vulnerable to wind loading. In practice, such roofs
often experience damage during windstorms.
Therefore, their resistance to wind is one of the
greatest concerns for structural engineers when
designing these roofs. Moreover, the wind force
coefficients are important parameters in the design.
The Australia/New Zealand (AS/NZ) Standard [2]
has specified the design wind force coefficients on
HP-shaped free roofs. However, the range of roof
shapes for which the wind force coefficients are
specified is rather limited. Regarding the
wind-induced response of HP-shaped free roofs, Pun
and Letchford reported the analytical results of an
HP-shaped tension membrane roof subjected to
fluctuating wind loads [3]. However, to the best of ourknowledge few studies of wind loads on HP-shaped
free roofs have been conducted.
The purpose of this study is to propose the wind
force coefficients on HP-shaped free roofs for the
design of structural elements such as columns, posts,
beams, cables, and membranes. The paper consists of
six chapters. Following the Introduction, Chapter 2
presents the roof shapes and definitions of wind force
coefficients on the HP-shaped free roofs. Chapter 3
explains the wind tunnel experiments using rigid
models. The arrangement, procedure, and results of the
wind tunnel experiments are presented in Section 3.1.
On the basis of these results, we propose the design
wind force coefficients in Section 3.2, assuming that
the roof is rigid and supported by four corner columns
(Fig. 2). The wind force coefficients are represented as
equivalent static loads, in which the dynamic load
effect is considered in the evaluation of the gust effect
factor. In Section 3.3, a comparison is made between
the proposed design wind force coefficients and
specified values in the AS/NZ Standard. Furthermore,
the application of the proposed design wind force
coefficients to membrane structures is described in
Section 3.4.
In practical membrane structures, there are several
roof supporting systems such as a post and guy cable
system shown in Fig. 1. Moreover, because the
membrane roof is not rigid but rather flexible, the roof
deformation becomes larger than that of conventional
roofs such as metal roofs. Therefore, when the
proposed wind force coefficients are used for the
design of membrane structures, the following questionsmay arise. (1) Can we apply them to membrane
structures with other supporting systems? The load
path may change depending on the roof supporting
system. (2) Can we apply them to flexible roofs that
Fig. 1 HP-shaped membrane free roofs.
Fig. 2 HP-shaped roof supported by four corner columns.
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deform under wind loading? The roof deformation
may change the wind forces significantly. (3) Can we
apply them to suspension structures for which the
boundary may also deform? Therefore, in Chapter 4,
the above subjects are investigated by computational
fluid dynamics (CFD) and structural analyses for three
roof models with different structural systems.
Finally, in Chapter 5, we discuss the wind force
coefficients to be applied to membrane roofs on the
basis of the results obtained in Chapter 4. Note that
the present paper is an extended and revised version of
our previous papers [4-6].
2. Wind Force and Moment Coefficients on
HP-shaped Free Roofs
2.1 Roof Shape
Three models (Models A-C) with different rise/span
(or sag/span) ratios are investigated in the present
study (Fig. 3a). The layout of the roof is a square of
15 m 15 m, and the mean roof height (H) is 8 m
(Fig. 3b).
2.2 Definition of Wind Force Coefficients
Fig. 4 shows the definition of the aerodynamic
forces and moments acting on the roof, whereDandL
represent drag and lift and Mx and My represent the
moments about the x and yaxes, respectively. These
values are normalized as follows:
(1)
(2)
(3)
(4)
where qH represents the reference velocity pressure at
the mean roof height H,h represents the difference in
height of the roof (Fig. 3a),a represents the horizontal
(projection) width of the roof, and Srepresents the wind
force coefficients of the projection area of the roof. For
(a)
(b)
Fig. 3 Roof shapes: (a) rise/span ratio (b) dimension of
Model A
Fig. 4 Definition ofD,L,MxandMy.
7.5 m
5.0 m
2.5 m
h= a/2
h= a/3
h= a/6
a: horizontal (projection) width of the roof
h: difference in height of the roof
21.2 m
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simplicity, the design roofs are specified by the two
uniformly distributed values (CNW and CNL) over the
windward and leeward halves (Fig. 5), respectively,
which are defined as follows:
(5)
/ (6)
where, NW and NL represent the normal wind forces
(positive downward) on the windward and leeward
halves, respectively. The coefficients CNWand CNLfor
the wind directions = 0 and 90 can be provided by
CLand CMyas follows [5].
= 0:
32 (7)
32 (8)= 90: 32 (9) 32 (10)
3. Design Wind Force Coefficients
3. 1 Wind Tunnel Experiments
3.1.1 Experimental Arrangement
The experiments were conducted in a boundary layer
wind tunnel with a working section 1.4 m wide, 1.0 m
high, and 6.5 m long at the Department of Architecture
and Building Science, Tohoku University, Japan. A
turbulent boundary layer with a power law exponent
of= 0.18 for the mean velocity profile was generated
on the wind tunnel floor. The turbulence intensity Iu
and longitudinal length scaleLxof the flow at a height
ofz= 100 mm were 0.17 and 0.16 m, respectively. The
wind tunnel model was made of nylon resin with ageometric scale of 1/100. The thickness of the nylon
resin was 1 mm. Fig. 6 shows a model mounted on a
Y-shaped force balance designed, built, and gauged for
this experiment.
The force balance was made of 1.2-mm-thick
phosphor bronze to measure the liftL and aerodynamic
momentsMx and My (Fig. 7). The aluminum column
base was pin-jointed to the end of a leaf spring. The
Fig. 5 Definition of CNLand CNW.
Fig. 6 Model mounted on a force balance (Model A).
Fig. 7 Y-shaped force balance.
bending stress at the base of each leaf spring was
measured by strain gauges, from which the
concentrated load at the end of each arm was computed.
The liftL and aerodynamic momentsMxandMyabout
thex andy axes were computed from the concentrated
loadsN1toN3as follows:
(11) (12) (13)
Low
Low
HighHighy
CNL CNW
CNW CNL NW
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The definitions ofx1,x2, andx3are shown in Fig. 7.
Note that the computed Mx and My values include
additional moments induced by the drag. The origin of
thexandyaxes is not located at the center of the roof
surface. The distance between the origin and roof
center causes moments about the x and y axes.
Therefore, we estimated the effects of additional
moments on the axial forces induced in the corner
columns supporting the roof. Our estimation indicates
that the eccentricity of the origin overestimates the
maximum load effect by up to approximately 11% for
Model A, 8% for Model B and 3 % for Model C [4].
3.1.2 Experimental Procedure
The measurements were performed at a wind speedof UH6 m/s and at a mean roof height ofH = 80 mm.
The design wind speed is assumed to be 31.5 m/s,
which is a typical value of strong wind events;
therefore, the velocity scale is approximately 1/5.25.
The geometric scale of the models (1/100) and this
velocity scale yield a time scale of approximately 1/19.
The wind direction was changed from 0 to 90 with
increments of 15. The outputs of the strain meters
were sampled simultaneously at a rate of 200 Hz for a
period of 32 s, which approximately corresponds to 10
min in full scale. The measurements were repeated six
times under the same condition. The statistics of
aerodynamic coefficients were evaluated by applying
the ensemble average to the results of six consequent
runs.
3.1.3 Experimental Results
Fig. 8 shows the statistical values of the lift and
moment coefficients as a function of for Model A
(h/a = 1/2); the mean and the maximum and minimumpeak values are plotted in each figure. The lift
coefficient is maximum (upward) when 0 and
minimum (downward) when 90. This feature is
related to increased wind velocity along the convex
surface, i.e., the top surface for 0 and the bottom
surface for 90. The magnitude of the negative
peak value of CMxis maximum when 90, whereas
that of CMyis maximum when 0. The values of CMx
(a)
(b)
(c)
Fig. 8 Statistics of lift and moment coefficients (Model A).
(a) CL (b) CMxand (c) CMy.
for 0 and those of CMyfor 90 are relatively
small in magnitude. The variation in CMx with is
opposite to that of CMy.
3.1.4 Load Effects
The axial forceN induced in each column (Fig. 9) is
computed from the time histories of CL, CMx,and CMy,
assuming that the roof is rigid and supported by four
corner columns [4]. In the present study, we focus on
-1.00-0.75-0.50-0.25
0.000.250.500.751.00
0 15 30 45 60 75 90
CL
(deg.)
CLmean
CLmax
CLmin
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0 15 30 45 60 75 90
CMx
(deg.)
CMxmean CMxmax CMxmin
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0 15 30 45 60 75 90
CMy
(deg.)
CMymean CMymax CMyminCMxmaxCMxminCMxmax
CLmean
CLmaxCLmin
CMxmaxCMxmin CMxmax
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the axial forces induced in the columns by the wind
forces as the most important load effect for evaluating
the design wind force coefficients.
The maximum and minimum peak values of the
non-dimensional axial force N*(= N/(qHa
2/4)) among
the four columns are plotted in Fig. 10. The absolute
values of the maximum and minimum N* values
generally decrease with h/a. The variation in maximum
tension (positive N*) with wind direction is relatively
small, whereas that in maximum compression
(negative N*) is significant. The maximum
compression for all wind directions is induced when
90.
3.1.5 Gust Effect FactorThe gust effect factor Gfis defined as the ratio of the
maximum or minimum axial force to the mean value
induced in the column. The maximum value is used
when the mean value is positive, and the minimum
value is used when the mean value is negative. The Gf
values are computed to investigate the dynamic effect
of wind turbulence on the column axial forces. Fig. 11
shows the results for Gfplotted against the mean
reduced axial forceN*
mean. When the value of |N*mean| is
small, Gf exhibits large value with a large scatter.
However, as |N*
mean| increases, the Gfvalues collapse
into a narrow range around Gf= 2.0, which corresponds
to a peak factor ofgf2.5, on the basis of quasi-steady
assumption, i.e., Gf(1 + 2.5 0.17)2[7-9]. Agfvalue
of approximately 2.5 is somewhat smaller than that for
gable, troughed, and mono-sloped free roofs, which is
approximately 3.0 [10]. This difference may be due to
the effect of flow separation from the leading edges of
the roof on the wind loads. The turbulence induced by
the flow separation appears to be lower for HP roofs
than that for the other roofs. In the structural analyses
in Section 3.4, a Gfvalue of 2.0 is used for evaluating
the design wind forces that provide equivalent static
loads.
3.2 Proposed Design Wind Force Coefficients
The roof is divided into two areas, i.e. the windward
and leeward halves, and the design wind force
coefficients C*NW and C
*NL for these halves are
specified. The following procedure provides the design
wind force coefficients, assuming that the roof is rigid
and supported by four corner columns (Fig. 9).Step1: The basic values of wind force coefficients
CNWand CNL, denoted as CNW0and CNL0, are determined
from a combination of the lift coefficient (CL) and
moment coefficient (CMxorCMy), which produces the
maximum load effect when = 0 or 90 (Fig. 12).
Fig. 9 Four corner columns supporting the roof.
(a) (b) (c)
Fig. 10 Non-dimensional axial forces. (a) Model A (h/a= 1/2); (b) Model B (h/a= 1/3) and (c) Model C (h/a= 1/6).
-2.0
-1.0
0.0
1.0
0 30 60 90
N*
(deg.)
Max. (Tension)Min. (Compression)
-2.0
-1.0
0.0
1.0
0 30 60 90
N*
(deg.)
Max. (Tension)Min. (Compression)
-1.0
-0.5
0.0
0.5
1.0
0 30 60 90
N*
(deg.)
Max. (Tension)
Min. (Compression)
= 90
= 0
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Fig. 11 Gust effect factor based on the load effect.
Step2: Considering that the axial force induced in a
column may become maximum for oblique winds, weintroduced a correction factor , which is defined as the
ratio of the actual peak force for a wind direction range
of = 0 45 (WD1) or = 90 45 (WD2) to that
computed from the CNW0and CNL0values.
Step3: The design wind force coefficients (C*NWand
C*NL) are provided as follows:
(14) (15)
These coefficients provide equivalent static wind
loads.
Fig. 13 shows the correction factors () of Models
A-C for the two load cases I and II, which induce
maximum tension and compression in the columns,
respectively, for the wind direction ranges WD1 and
WD2. When the h/aratio is small, the value of for
WD1 is relatively large. In addition to this case, the
value of is approximately 1.0. Similar features are
observed for gable, troughed, and mono-sloped
roofs [10].
Figs. 14a and 14b show a phase-plane representation
of the CL-CMyrelation for Models B and C, respectively,
(a) = 0 (b) = 90
Fig. 12 Wind direction for basic values of C*NWand C
*NL.
Fig. 13 Correction factor .
when = 0. The circles in the figure represent the
maximum and minimum peak values of CL(CLmaxand
CLmin) during a 10 min period in full scale. Theenvelope of the trajectory appears like an ellipse with
an inclined axis, indicating a positive correlation
between CL and CMy. For Model C, the CL and CMy
values are correlated well with each other. In this case,
the CMyvalue at the instant when CLmaxor CLminoccurs
is nearly equal to the maximum or minimum value of
CMy(CMymaxor CMymin). The maximum load effect may
be given by the combination of the two peak values. On
the other hand, for Model B, the correlation between CL
and CMy is relatively low. The peak + peak
combination of CL and CMy does not necessarily
produce the maximum load effect. The maximum load
effect may be given by a certain combination of CLand
CMy. The envelope of the CL-CMytrajectory for Models
A and B is approximated by a hexagon (hereafter
Hexagon) shown in Fig. 14c. The critical condition
producing the maximum load effect may be given by
one of these six apexes. The CL-CMxrelation for = 90
exhibits a similar feature. From the combination of thelift and moment coefficients CMxor CMyobtained above,
the basic wind force coefficients CNW0 and CNL0 are
computed by Eqs. (7) and (8) or (9) and (10) for the two
wind directions =0 and 90, respectively. For each
wind direction, two sets of CNW0 and CNL0 values are
selected from the six sets corresponding to the apexes
of the Hexagon to evaluate the design wind force
coefficients, which induce maximum tension (Load
0.0
0.5
1.0
1.5
2.0
0 0.2 0.4 0.6Correctionfactor
h/a
Load case (W 1)
Load Case (WD 1)
Load Case (WD 2)
Load Case (WD 2)
= 0 = 90o
Correctionfactor
h/a
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(a)
(b)
(c)
Fig. 14 Phaseplane representation of the CL-CMyrelation
(= 0) (Hexagon): (a) model B (h/a = 1/3), (b) model C
(h/a = 1/6) and (c) model of the envelope of CL-CMy
trajectory.
case ) and maximum compression (Load case II)in the
columns.
3.3 Comparison with the Specifications of the
Australia/New Zealand Standard
The plots in Figs. 15 and 16 are the estimated values
of C*NWand C
*NLfor WD1 and WD2, respectively. In
addition, the specifications of the AS/NZ Standard
(2011) are shown by the dashed lines. The Standard
provides two values of wind force coefficients
(expressed as positive and negative) for each of the
windward and leeward halves; the h/a ratio is limited to
the range 0.1-0.3. The proposed values of C*NW for
Load cases and II are relatively close to the specified
values of the AS/NZ Standard. On the other hand,
regarding the leeward half, the proposed wind force
coefficients for the two load cases are similar to each
other and are nearly equal to one of the specified values
of the AS/NZ Standard. These features are similar to
(a) (b)Load case : Maximum tensionLoad case II: Maximum compression
Fig. 15 Wind force coefficients C*NWand C
*NL (WD1). (a)
windward half and (b) leeward half.
(a) Windward half (b) Leeward half
Load case Maximum tension
Load case IIMaximum compression
Fig. 16 Wind force coefficientsC*NWand C
*NL (WD2).
-1.0
-0.5
0.0
0.5
1.0
0 0.2 0.4 0.6
C*NW
h/a
Load case
Load case
-1.0
-0.5
0.0
0.5
1.0
0 0.2 0.4 0.6
C*NL
h/a
Load case
Load case
-1.0
-0.5
0.0
0.5
1.0
0 0.2 0.4 0.6
C*NW
h/a
Load case
Load case
-1.0
-0.5
0.0
0.5
1.0
0 0.2 0.4 0.6
C*NL
h/a
Load case
Load case
CMymax
CMymean
CMymin
CLmaxCLmeanCLmin
C y
1
CL
2
3
4 6
5
CL0.5
C y
0.00
0.05
0.10
-0.5
0.15
CMy
0.05
0.00-0.5
-0.20
1.0 CL
AS/NZ (positive)AS/NZ (positive)
AS/NZ (negative)AS/NZ (negative)
AS/NZ (positive)
AS/NZ (negative)
AS/NZ
(negative)
AS/NZ (positive)
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those observed by Uematsu et al. [10] for gable,
troughed and mono-sloped roofs.
The axial forces induced in the columns are
computed using C*NWand C
*NLand are compared with
those predicted from the AS/NZ specifications. The
results are shown in Fig. 17. As mentioned above, the
AS/NZ Standard generally provides four combinations
of the wind force coefficients on the windward and
leeward halves. The maximum and minimum axial
forces among the four values are shown in the figure.
Note that these values are consistent with the present
results for Load cases and II, respectively, despite the
existence of a difference in the wind force coefficients,
as shown in Figs. 15 and 16.
3.4 Application of the Proposed Wind Force
Coefficients to Membrane Structures
3.4.1 Structural Analysis
To investigate the application of the proposed design
wind force coefficients to membrane structures,
structural analyses were conducted using three
analytical models for the wind directions = 0 and
90. Structural analysis was performed for six pairs of
wind force coefficients (Table 1), which correspond to
the apexes of the Hexagon (Fig. 14c), including the two
design wind force coefficients proposed above. Each
structural model has the same rise/span ratio of h/a =
1/2, which is the same as that for Model A.
In practice, three structural systems are often used
for membrane structures, i.e., frame, suspension, and
air-supported types [1]. This analysis focuses on the
frame and suspension types. Fig. 18 shows the
analytical models. In Frame Model 1 (F1), the roofstructure is constructed of perimeter girders and
binding beams, which divide the roof area into 12
zones (Fig. 18a). The roof frame is covered with
pre-stressed membrane. The pre-stress is 4 kN/m in
both the warp and fill (weft) directions of the
membrane. The warp direction is shown in Fig. 18b,
which is the same for all models. The fill direction is
the membrane is assumed as 12 N/m2. Frame Model 2
(a) (b)
Fig. 17 Reduced axial force: (a) WD1 and (b) WD2.
Table 1 Wind force coefficients corresponding to the
apexes of Hexagons for Model A.
Apex= 0 (WD1) = 90 (WD2)
CNW CNL CNW CNL
1 -0.33 -0.36 -0.19 0.06
2 -0.16 -0.19 0.01 0.25
3 -0.20 -0.49 -0.45 0.32
4 0.40 -0.34 -0.28 1.16
5 0.18 -0.11 0.06 0.82
6 0.20 -0.54 -0.59 0.85
Note: Considering the correction factor due to oblique winds
(F2) consists of perimeter girders (beams) and
pre-stressed membranes (Fig. 18b). In these frame
models, the connection of the beam elements is rigid
and the roof girders are supported by four corner
columns. On the other hand, Suspension Model (S1)
consists of curved perimeter cables and a pre-stressed
membrane; the roof is supported by the posts and guy
cables at the four corners, as shown in Fig. 18c. The
column bases of the F1 and F2 models are fixed,
whereas the posts of the S1 model are pin-supported.
Therefore, not only the axial forces but also the
bending moments are induced in the columns of the F1and F2 models. On the other hand, in the S1 model, no
bending moments are induced in the columns.
The material of the columns, beams, posts and cables
is steel, whereas, that of the membranes is
polytetrafluoroethylene-coated (PTFE-coated) glass
fiber plain-weave fabric. The projection area of the S1
model is approximately 82% of that of the frame
models because of the curved perimeters. The S1 model
-1.0
-0.5
0.0
0.5
1.0
0 0.2 0.4 0.6
N*/Gf
h/a
Load case
Load case
-1.0
-0.5
0.0
0.5
1.0
0 0.2 0.4 0.6
N*/Gf
h/a
Load case
Load case
AS/NZ
(negative)
AS/NZ (positive)AS/NZ (positive)
AS/NZ (negative)
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(a)
(b)
(c)
Fig. 18 Analytical models. (a) Frame model 1 (F1); (b)
Frame model 2 (F2) and (c) Suspension model 1 (S1).
perpendicular to the warp direction. The self-weight of
is the most flexible among the three models, which
causes largest deformation to the roof under wind
loading. On the other hand, the F1 model is relatively
rigid. The roof membrane slightly deforms in the
downward direction because of the self-weight.
Therefore, the initial shapes of these three roofs are
slightly different from each other because of the
difference in the supporting system.
Structural analyses were conducted by a program
named MAGESTIC, which is a software program
developed by Taiyo Kogyo Corp., based on the finite
element method using the NewtonRaphson method,
in which geometrical nonlinearity is considered [11].
The membrane is assumed to be orthotropic and elastic.
Furthermore, it is assumed that the membrane can only
carry tension; in other words, it does not resist
compression and bending moments. As mentioned
above, the design wind speed is 31.5 m/s, which
provides a velocity pressure of 605 N/m2. Six pairs of
the wind force coefficients, which are calculated fromthe CNW0and CNL0 values for each wind direction (WD1
or WD2), are applied to the roofs to determine the
critical conditions that provide the maximum and
minimum load effects. The load effects are provided by
the combination of CL and CMy (or CMx), which
corresponds to the six apexes of the Hexagon. In the
analysis, the stresses are calculated on the basis of the
Building Standard Law of Japan [12] and Design
Standard for Steel Structures published by the
Architectural Institute of Japan [13-14].
For the membranes and cables, the tensile stresses
are calculated from the tensile forces. However, for the
beams and columns, the extreme fiber stresses were
calculated by combining the axial forces and bending
moments. For the posts of the S1 model, the axial
stresses were calculated from the axial forces. The
allowable stresses and material constants (Tables 2-4)
were also determined on the basis of the Law and
Standards [13-15]. Moreover, the ratio of thecomputed stress to allowable stress was calculated,
which is called the stress ratio in the present paper.
Figs. 19 and 20 show the results of the structural
analyses for the F1 and S1 models, respectively. In
these figures, the maximum stress ratios for the
members, i.e., the column, post, beam cable, and
membrane (Mem), are shown for the six apexes of the
Hexagon in Fig. 14c.
Mem
C1: P-406.4 6.4
C2: P-406.4 6.4
B1: P-558.8 6.4
Mem: Membrane
Warp direction Mem
C1: P-558.8 6.4
C2: P-406.4 6.4
B1: P-318.5 6.9
B2: P-216.3 5.8
B3: P-165.2 3.7
Mem: Membrane
Mem
C1: P-406.4 6.4
C2: P-216.3 4.5
Ca1: 30(7 19) StrandCa2: 42.5(1 61) Spiral
Ca3: 14(1 19) Spiral
Ca4: 18(1 19) Spiral
Ca5: 45(1 91) Spiral
Mem: Membrane
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Design Wind Force Coefficients for Hyperbolic Paraboloid Free Roofs 11
Table 2 Beam and Post.
Elastic modulus E= 2.05 108kN/m2
Poissons ratio = 0.3
Table 3 Cable.
Elastic modulusStrand rope E= 1.37 108kN/m2
Spiral rope E= 1.57 108kN/m2
Table 4 Membrane (t: Thickness).
Tensional stiffnessWarp:Ew t= 1285 kN/m
Fill:Ef t= 861 kN/m
Apparent Poissons ratio Warp: w= 0.85 Fill: f= 0. 57
Shear stiffness G t= 57 kN/m
Note: Measured by MSAJ/M-02 1995 and MSAJ/M-01 1993
in Standards of the Membrane Structures Association of Japan.
3.4.2 Application of the Proposed Design WindForce Coefficients
In Section 3.2, we proposed the design wind force
coefficients C*NW and C
*NL, focusing on the column
axial forces as the load effect, with an assumption that
the roof is rigid and supported by four corner columns.
These wind force coefficients were obtained from the
combination of CLand CMxor CMy, which provides the
maximum tension and compression in the columns.
They correspond to two apexes of the Hexagon shown
in Fig. 14(c). For example, in the case of h/a = 1/2
(Model A), Apexes 3 and 4 provide the maximum load
effects for WD1 (= 0 45), whereas Apexes 4 and
6 provides the maximum load effects for WD2 (= 90
45). However, in the case of membrane structures,
the roof is rather flexible, and not rigid. Furthermore,
the roof supporting system may be different from that
assumed when discussing the design wind force
coefficients in Section 3.2. Wind forces acting on the
roof are first transferred to the peripheral members
(beams or cables) via membrane tension, and
thereafter, they are transferred to the columns or the
post and guy cables (Fig. 18). Therefore, it is expectedthat the other load effects should be considered for
such structures. This subject is discussed below on the
basis of the structural analysis results shown in Figs.
19 and 20.
Apex 4 or 6 provides the maximum stress ratio for
all stresses of WD2, which indicates that the proposed
wind force coefficients are also applicable to the
membrane roof structures. On the other hand, Apexes 3
(a1) WD1 (a2) WD1 (b1) WD2 (b2) WD2
Fig. 19 Stress ratio for F1.
(a1) WD1 (a2) WD1 (b1) WD2 (b2) WD2
Fig. 20 Stress ratio for S1.
0.0
0.2
0.4
0.6
1 2 3 4 5 6
Stressratio
Apex
Mem(Warp) Mem(Fill)C1 C2
0.0
0.2
0.4
0.6
1 2 3 4 5 6
Stressratio
Apex
B1 B2 B3
0.0
0.2
0.4
0.6
0.8
1.0
1 2 3 4 5 6
Stressratio
Apex
Mem(Warp) Mem(Fill)C1 C2
0.0
0.2
0.4
0.6
0.8
1.0
1 2 3 4 5 6
Stressratio
Apex
B1 B2 B3
0.0
0.2
0.4
0.6
0.8
1 2 3 4 5 6
Stressratio
Apex
Mem(Warp) Mem(Fill)C1 C2
0.0
0.2
0.4
0.6
0.8
1 2 3 4 5 6
Stressratio
Apex
Ca1 Ca2 Ca3Ca4 Ca5
0.0
0.2
0.4
0.6
0.8
1.0
1 2 3 4 5 6
Stressratio
Apex
Mem(Warp) Mem(Fill)C1 C2
0.0
0.2
0.4
0.6
0.8
1.0
1 2 3 4 5 6
Stressratio
Apex
Ca1 Ca2 Ca3Ca4 Ca5
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Design Wind Force Coefficients for Hyperbolic Paraboloid Free Roofs12
or 4 do not always provide the maximum stress ratio
for WD1. For example, the maximum stress is given at
Apex 6 for the bending stress of B3 (Fig. 18a).These
results imply that the proposed design wind forcecoefficients may underestimate the responses when
applied to membrane roof structures. Table 5
summarizes the stress ratios obtained at Apexes 3, 4,
and 6 for such stresses of the members that show
maximum stress ratios at Apex 6. These members,
which include all beams (B1, B2, and B3) in the F1 and
F2 models and the peripheral cables (Ca1) in the S1
model, are connected to the membranes, and therefore,
the stresses involved in the members may maximize
because of the bending moments induced by membrane
tensions. In Table 5, the ratio of the stress ratio at Apex
6 to the larger of those at Apexes 3 and 4 is also shown.
The value of this ratio ranges from 1.0 to 1.1 for most
load effects, except for the bending stress in B3 of
Model F1, which shows a ratio of approximately 1.3.
The reason for such a large value for B3 of Model F1
may be due to the discontinuity in the distribution of
the wind force coefficient at the location of the B3
beam (center line of the roof) for WD1, i.e., thebending stress involved in this member may be affected
by the difference in the membrane tensions from the
windward and leeward halves. Such a situation was not
considered in the above discussion of design wind
force coefficients.
The above-mentioned feature, where the maximum
values of some load effects such as the bending stresses
of the beams and the membrane tensions are provided
at some apexes of the Hexagon that are different from
those for the column axial forces, implies that not only
the column axial forces but also the other load effectsshould be considered when discussing the design wind
force coefficients for membrane roof structures. To
improve the wind force coefficients, it is necessary to
identify the most important load effect for such
structures. This is the subject of our future study.
4. Effect of Roof Deformation on the Wind
Forces
4.1 Analytical Method
The present study proposed the design wind force
coefficients in Section 3.2, assuming that the roof is
rigid. In addition, we investigate the application of the
wind force coefficients to the three membrane
structures, i.e., the F1, F2, and S1 models in Section 3.4.
Because the membrane roofs are flexible enough to
deform under wind loading, roof deformation may
affect the wind forces significantly. However, the
question of whether the proposed design wind forcecoefficients can be applied to such membrane roofs
when considering the deformations still remains
unresolved. Therefore, we conducted CFD and
structural analyses using the F1, F2 and S1 models to
investigate the effect of roof deformation on the wind
loads.
First, this section shows the effectiveness of CFD
Table 5 Stress ratios of Apexes 3, 4, and 6 (WD1).
Member Stress ratio (Apex 3) Stress ratio (Apex 4) Stress ratio (Apex 6) Ratio (Apex 6 to 3 or 4)
F1
Mem (Warp) 0.34 0.25 0.34 1.02
B2 0.29 0.50 0.53 1.06
B3 0.44 0.42 0.58 1.32
F2Mem (Warp) 0.35 0.25 0.35 1.01
B1 0.42 0.46 0.47 1.03
S1
Mem (Warp) 0.30 0.26 0.31 1.03
Ca1 0.63 0.63 0.63 1.01
Ca2 0.66 0.64 0.68 1.06
Ca4 0.32 0.30 0.32 1.03
C1 0.60 0.60 0.64 1.02
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Design Wind Force Coefficients for Hyperbolic Paraboloid Free Roofs 13
analysis by comparing the mean wind force
coefficients obtained from the analysis with those
obtained from the experiments for Models A-C. CFD
analysis simulates the wind tunnel experiment.
Thereafter, CFD and structural analyses were repeated
to compare the load effects before and after roof
deformation. In practice, the membrane roofs vibrate
under dynamic wind loads. However, the present study
does not consider the influences of roof vibration. Only
the effect of deformation corresponding to the mean
(time-averaged) wind loads is considered here.
Fig. 21 shows the algorithm for estimating the effect
of roof deformation on the mean wind loads. First,CFD analysis is performed on the initial roof shape
(CFD-1), which is a three dimensional analysis with
the Reynolds-averaged Navier-Stokes (RANS) model.
Next, to obtain the roof deformation corresponding to
the mean wind loads, we conducted a structural
analysis (SA-1) using the wind force coefficients
obtained from CFD-1. This structural analysis is
similar to that described in Section 3.4.1. In the
analyses, the gust effect factor was assumed to beGf=
1.0 because the focus is on the mean wind loads. CFD
analysis (CFD-2) was repeated on the deformed roof
obtained from SA-1 to obtain the mean wind force
coefficients. Subsequently, the structural analysis
(SA-2) with the mean wind force coefficients
obtained from CFD-2 was repeated. Thus, CFD and
structural analyses were repeated until a convergence
of the response obtained. The criterion for convergence
is based on the variation in the deformed roof shape;
i.e., the following condition is used for the criterion:
1300 (16)
where n and n-1 represent the maximum
displacements at the nand n1 steps of the structural
analyses, respectively.
4.2 Effectiveness of CFD Analysis
4.2.1 Outline of CFD Analysis
In the present study, we calculate the mean wind
force and moment coefficients CL, CMxand CMy[16].
We used an open-source software program named
OpenFOAM version 1.5 [17]. The computational
domain is 1.0 m wide, 1.4 m high, and 3.0 m long, inwhich an HP-shaped roof with the same configuration
as that used in the wind tunnel experiments was placed,
as shown in Fig. 22a. Fig. 22b shows a numerical
model of Model A. Fig. 23 shows the resolution of the
model grid. The computation is based on the finite
volume method, in which the Semi-Implicit Method for
Pressure Linked Equations (SIMPLE) algorithm and
the renormalization group (RNG) k-model are used.
The boundary condition is summarized in Table 6 [18].
The turbulence intensities Iu for the analysis were
determined on the basis of the wind tunnel experiment
(Fig. 24). The wind direction changed from 0 to 90 at
increments of 15 in the same manner as in the wind
tunnel experiment.
Fig. 21 Algorithm for investigation on the effect of roof
deformation due to the wind loads.
END
Convergence?
Structural analysis
(SA-n)
CFD analysis(CFD-n)
Structural analysis(SA-1)
CFD analysis(CFD-1)
START
No
Yes
n = 2 or more
Mean wind forcecoefficients
Mean wind forcecoefficients
Deformed roof
Deformed roof
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Design Wind Force Coefficients for Hyperbolic Paraboloid Free Roofs14
(a)
(b)
Fig. 22 Numerical model (Model A). (a) Simulated wind
tunnel and (b) HP-shaped model (= 0)
(a)
(b) (c)
Fig. 23 Mesh resolution. (a) Side view of the simulated wind
tunnel model (b) Enlarged side view (c) Enlarged front view.
4.2.2 CFD Analysis Result
Figs. 25 to 27 show the results on the mean CL, CMx,
and CMyvalues for Models A, B, and C, respectively. In
these figures, the experimental results were also plotted
for comparison. In general, the CFD results agree well
with the experimental results for CLand CMy. However,
regarding the mean CMx values, the agreement is
poorer, particularly for larger values, although the CFD
Table 6 Boundary condition of simulated wind tunnel.
Surface atXmin
( Inlet )
Reference height:ZG= 0.6m
Wind velocity at the reference height:UG= 8m/sPower law index: = 0.18
Turbulence intensity: Experimental values(Fig. 24)
Surface atXmax( Outlet )
Surface pressure at outlet: 0 Pa
Surface at Ymin,
YmaxandZmaxFree-slip wall
Surface atZmin No-slip wall
HP surface No-slip wall
Fig. 24 Turbulence intensity and non-dimensional wind
velocity profile.
analysis captures the general trend of the experimental
results. Although the reason for this difference is not
yet clear, there are two possible reasons. One is related
to the experimental method. The force balance used for
measuring the wind forces may affect the wind flow
under the roof. Furthermore, the effect of the drag force,
which is unavoidable in the measurements, may affect
the results for the CMx values. The other reason is
related to the numerical model used and other factors,
such as the grid resolution around the model,
turbulence model, and the boundary condition. Further
investigations are necessary to improve the agreement
between the CFD analysis and wind tunnel experiment
results.
Furthermore, the dynamic effect should be
considered appropriately. Nevertheless, the results
show that CFD analysis is useful for the present
study.
HeightZ(mm)
Turbulence intensity
Mean wind velocity
(Experimental value)
Mean wind velocity
(Approx. expression)
0 0.2 0.4 0.6 0.8 1 1.2
0
100
200
300
400
500
600
700
IuUz/UG
UG : Mean wind velocity at a reference heightofZG = 600 mm
Iu, Uz/UG
HP-shaped roof
xy
z
xz
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Design Wind Force Coefficients for Hyperbolic Paraboloid Free Roofs16
Scale factor for displacement: 5 times
(a)
Scale factor for displacement: 5 times(b)
Scale factor for displacement: 5 times
(c)
Fig. 28 Deformation of the roof along the center line from
SA-2 for the wind direction = 0. (a) F1 model wind dir. =
0; (b) F2 model wind dir. = 0 and (c) S1model wind dir.
= 0.
respectively. The mean CDvalues from CFD-1 are also
plotted in the figures. The values for the S1 model are
corrected for the difference in the roof area. The CFD-1
results are consistent with the experimental results. Fig.
31 shows the distributions of the mean wind force
coefficients on the deformed roofs. Comparing these
results with those in Fig. 29, we observe that the region
of upward wind forces expands in the leeward direction
for the F2 and S1 models. This indicates that the roof
deformation in the upward direction causes an increase
in the wind force on the roof.
4.4 Effect of Roof Deformation on the Wind Force
Coefficients
Fig. 32 shows the comparison of the mean CD, CL,
and CMyvalues between CFD-1 and CFD-2 for = 0.
(a)
(b)
(c)
Fig. 29 Wind force coefficients obtained from CFD-1 for
the initial roof shape. (a) F1 model (Wind dir. = 0); (b) F2
model (Wind dir. = 0) and (c) S1 model (Wind dir. = 0).
1
2
3
4
5
6
7
8
9
10
0.99
0.67
0.35
0.00
+0.29
+0.61
+0.93
+1.25
+1.57
+1.89
+2.21
1
2
3
4
5
6
7
8
9
10
0.980.57
0.15
+0.27
+0.68
+1.10
+1.51
+1.93
+2.34
+2.76
+3.17
1
2
3
4
5
6
7
8
9
10
0.89
0.62
0.36
0.09
+0.18
+0.45
+0.71
+0.98
+1.25
+1.51
+1.78
1/a1/60
2/a1/53
21/a1/429
1/a1/76
2/a1/67
21/a1/600
1/a1/143
2/a1/135
21/a1/2500
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Design Wind Force Coefficients for Hyperbolic Paraboloid Free Roofs 17
(a)
(b)
Fig. 30 Comparison between the experiment and CFD-1
results for the mean aerodynamic coefficients. (a) = 0 and
(b) = 90.
In Fig. 32, the CL and CMy values obtained from
CFD-2 for the F2 and S1 models are approximately
4%18% larger than those obtained from CFD-1.
These features may have been caused by the expansion
of the upward wind force region accompanied by a
change in curvature of the membrane surface. A similar
feature was observed for = 90. Fig. 33 shows the
ratio of the maximum stress obtained from SA-2 to that
obtained from SA-1 for = 0. The values obtained
from SA-2 are generally larger than those from SA-1.
The ratios for the F1 model are up to 1.08, whereas
those for the F2 and S1 models are up to 1.13. The ratio
for column C1 of the F2 model is the largest, whereas
that for column C2 of the S1 model is the lowest. This
feature appears to be related to the structural system;
i.e., the F1 and F2 models have rigid joints and fixed
column bases, whereas the S1 model has only pin joints
and pinned supports. Therefore, the bending moments
in the columns of the F1 and F2 models appear to affect
the ratios. Regarding the ratios for Mem, the S1 model
provides the largest value. This feature is probably due
to the wind load on the leeward half of the S1 model
(Fig. 31c), which is the largest among all models.
Furthermore, this phenomenon may be related to the
(a)
(b)
(c)
Fig. 31 Wind force coefficients obtained from CFD-2 for
the initial roof shapes. (a) F1 model (= 0); (b) F2 model (
= 0) and (c) S1 model (= 0).
-0.1
0.0
0.1
0.2
0.3
0.4
1 2 3
C
oefficents
F1 F2 S1
Exp.-CL
EXP.-CMy
CFD1-CD
CFD1-CL
CFD1-CMy
-0.3
-0.2
-0.1
0.0
0.1
0.2
1 2 3
Coefficents
F1 F2 S1
Exp.-CL
EXP.-CMx
CFD1-CD
CFD1-CL
CFD1-CMx
1
2
3
4
5
6
7
8
9
10
0.99
0.67
0.35
0.00
+0.29
+0.61+0.93
+1.25
+1.57
+1.89
+2.21
1
2
3
4
5
6
7
8
9
10
0.98
0.57
0.15
+0.27
+0.68
+1.10
+1.51
+1.93
+2.34
+2.76
+3.17
0.89
0.62
0.36
0.09
+0.18
+0.45
+0.71
+0.98
+1.25
+1.51
+1.78
1
2
34
5
6
7
8
9
10
EXP. - CL
EXP. - CMy
CFD1- CD
CFD1- CL
CFD1- CMy
EXP. - CL
EXP. - CMx
CFD1- CD
CFD1- CL
CFD1- CMx
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Design Wind Force Coefficients for Hyperbolic Paraboloid Free Roofs18
Fig. 32 Comparison between CFD-1 and CDF-2 (= 0).
Fig. 33 Ratio of the maximum stress obtained from SA-2 to
that obtained from SA-1 (= 0).
member stiffness and member arrangement. In addition,
a similar feature was observed for = 90. From these
results, a correction factor for the effect of roof
deformation should be introduced in the wind force
coefficients.
5. Conclusions and Future Study
The present study discussed the wind force
coefficients for the design of HP free roofs. Three roofs
with different rise/span (or sag/span) ratios were
analyzed. First, the design wind force coefficients for
structural members were proposed on the basis of the
wind tunnel experiments with rigid models. Regarding
the wind force coefficients on HP-shaped free roofs,
the AS/NZ Standard provides the specifications.
However, the range of roof shapes, for which the wind
force coefficients are specified, is rather limited. The
proposed design wind force coefficients were
compared with those provided in the AS/NZ Standard.
In addition, structural analyses were performed for
three membrane free roofs with the same rise/span
ratio and different supporting systems, i.e., two frame
types and one suspension type. In structural analyses,
the proposed design wind force coefficients were used
to investigate their application to membrane structures.
The results suggested that the proposed design wind
force coefficients should be improved by appropriately
considering the structural system and load path.
The effect of roof deformation on the wind forces
was evaluated by an iterative analysis between CFD
and structural analyses because the membrane roof is
not rigid but rather flexible. Before starting the
iterative analysis, we showed the effectiveness of CFD
analysis using the RANS model by comparing the
mean wind force coefficients obtained from the
experiment and that obtained from CFD analysis.
Thereafter, the mean wind force coefficients for theinitial roof shape were computed by CFD analysis.
Then, we performed a structural analysis using the
mean wind force coefficients obtained from the CFD
analysis to predict the deformation of the membrane
roofs corresponding to mean wind loads. The CFD
analysis on the deformed roofs was repeated, followed
by the structural analysis using the computed mean
wind force coefficients on the deformed roof. The load
effects obtained from the second structural analyses
were compared with those obtained from the first
structural analysis. The results suggested that to
improve the wind force coefficients, it is necessary to
consider the roof deformation.
Finally, we proposed two methods for improving
the wind force coefficients. First, a correction factor
for membrane structures was introduced into the wind
force coefficients, similar to the factor introduced in
Eqs. (14) and (15) when considering the effect of wind
direction. The modified wind force coefficients are asfollows:
(17)
(18)
where, is the correction factor for membrane
structures.
Second, the basic values of CNWand CNL(CNW0and
CNL0) were themselves modified to consider the effect
-0.1
0.0
0.1
0.2
0.3
0.4
1 2 3
Meanw
indforce
coefficients
F1 F2 S1
CFD1-CD
CFD1-CL
CFD1-CMy
CFD2-CD
CFD2-CL
CFD2-CMy
1.0
1.1
1.2
1 2 3
Ratio(SA2/SA1)
F1 F2 S1
Mem(Warp)
Mem(Fill)
C1
C2
CFD1 - CDCFD1 - CL
CFD1- CMyCFD2- CD
CFD2- CLCFD2- CMy
Mem (Warp)
Mem (Fill)
C1
C2
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Design Wind Force Coefficients for Hyperbolic Paraboloid Free Roofs 19
of membrane structures. The modified coefficients are
as follows:
(19)
(20)
where, CNW0mand CNL0mrepresent the basic values of
CNW and CNL and include the effect of membrane
structures on the structural systems and roof
deformation. In both cases, we need to consider the
parameters of structural systems, roof stiffness, and
roof shape, such as aspect ratio and deformation. This
will be the subject of future study.
References[1] K. Ishii, Membrane Structures in Japan, SPS Publishing
Company, Tokyo, Japan, 1995, pp. 1-374.
[2] Standards Australia Limited/Standards New Zealand,Structural design actions Part 2: Wind actions,
Australia/New Zealand Standard, AS/NZ 1170.2 (2011).
[3] P.K.F. Pun, C.W. Letchford, Analysis of a tensionmembrane hypar roof subjected to fluctuating wind loads,
in: Third Asia-Pacific Symposium on Wind Engineering,
Hong Kong, 1993.
[4] Y. Uematsu, F. Arakatsu, S. Matsumoto, F. Takeda, Windforce coefficients for the design of a hyperbolic paraboloid
free roof, in: Proceeding of Seventh Asia-Pacific
Conference on Wind Engineering (APCWE-VII), Taipei,
Taiwan, 2009, pp. 635-638.
[5] F. Takeda, T. Yoshino, Y. Uematsu, Wind forcecoefficients for the design of a hyperbolic paraboloid free
roof, in: Proceedings of the 13th International Conference
on Wind Engineering, Amsterdam, The Netherlands,
2011.
[6] F. Takeda, T. Yoshino, Y. Uematsu, Discussion of designwind force coefficients for hyperbolic paraboloid free
roofs, in: Seventh International Colloquium on Bluff Body
Aerodynamics & Applications (BBAA7), Shanghai,
China, 2012.
[7] Y. Uematsu, E. Iizumi, T. Stathopoulos, Wind loads onfree-standing canopy roofs: Part 1. Peak wind force
coefficients for the design of cladding, Journal of Wind
Engineering 30 (105) (2005) 91-102.
[8] Y. Uematsu, Y. Iizumi, T. Stathopoulos, Wind loads onfree-standing canopy roofs: Part 2. Wind force coefficients
for the design of main force resisting systems, Journal of
Wind Engineering 31 (107) (2006) 35-49.
[9] Y. Uematsu, E. Iizumi, T. Stathopoulos, Wind loads onfree-standing canopy roofs: Part 3. Validity and
application of the proposed wind force coefficients,
Journal of Wind Engineering, 31 (109) (2006) 115-122.
[10] Y. Uematsu, E. Iizumi, T. Stathopoulos, Wind forcecoefficients for designing free-standing canopy roofs,
Journal of Wind Engineering and Industrial Aerodynamics
95 (2007) 1486-1510.
[11] K. Ishii, State-of-the-art-report on the stress deformationanalysis of membrane structures, Research Report on
Membrane Structures 4 (1990) 69-105, The Membrane
Structures Association of Japan, 1990. (in Japanese)
[12] Ministry of Land, Infrastructure and Transport, PublicNotice No.666, Japan, 2002.
[13] Architectural Institute of Japan, Design Standard for SteelStructures Based on Allowable Stress Concept, Japan,
2005. (in Japanese)
[14] Architectural Institute of Japan, Recommendations forDesign of Cable Structures, Japan, 1994. (in Japanese)
[15] Membrane Structures Association of Japan, TestingMethod for In-Plane Shear Properties of Membrane
Materials (MSAJ/M-01), Standards of the Membrane
Structures Association of Japan, 1993.
[16] F. Takeda, T. Yoshino, Y. Uematsu, Wind forcecoefficients for the design of a hyperbolic paraboloid free
roof, in: Proceeding of the International Association for
Shell and Spatial Structures (IASS) Symposium, Shanghai,
China, 2010.
[17] OpenFOAM, http://www.openfoam.com/.[18] Working group for CFD prediction of pedestrian wind
environment around building, Architectural Institute of
Japan, Guidebook for Practical Applications of CFD to
Pedestrian Wind Environment around Buildings, 2007. (in
Japanese)