design wind force coefficients for hyperbolic paraboloid free roofs

8/12/2019 Design Wind Force Coefficients for Hyperbolic Paraboloid Free Roofs

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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

D


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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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Design Wind Force Coefficients for Hyperbolic Paraboloid Free Roofs 3

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



Mem: Membrane


11/19


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


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


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


0.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6

Stressratio

Apex

Ca1 Ca2 Ca3Ca4 Ca5


12/19


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


13/19


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


14/19


(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


15/19


16/19


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


17/19


(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


18/19


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


19/19


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)

design wind force coefficients for hyperbolic paraboloid free roofs

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