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Flow over spherical inflated buildings
Journal of Wind Engineering and Industrial Aerodynamics, 17 (1984) 305--327
305
Elsevier Science Publishers B.V., Amsterdam - - P r i n t e d in The Netherlands
FLOW OVER SPHERICAL INFLATED BUILDINGS
B.G. NEWMAN and U. GANGULI*
Department of Mechanical Engineering, McGill University, Montreal, Quebec (Canada) S.C. S HR IVAS TAV A
Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, Quebec (Canada) (Received September 21, 1983; accepted in revised form May 23, 1984)
Summary The pressure distributions on three domes of different heights (h/c = 0.5, 0.37 and 0.25) have been measured in a boundary-layer wind tunnel in which the wind flow over a sparsely wooded area was simulated. The tests were made in the wall-law region of the boundary layer and the results are therefore first normalized using the undisturbed skin friction r w . The tensions in inflated membraries of the same geometry were determined using the SAP IV finite-element program assuming that the effects of membrane weight and air friction are unimportant. The principal tensions then indicate where buckling would first take place and what would be the minimum internal pressure to just prevent it. The minimum internal gauge pressure to prevent buckling, which is shown to occur on the upwind side of the building, is 180 (h/c) r w ' This conclusion is compatible with the recommendations of the National Standards of Canada (1981) [1]. The location, orientation and magnitude of the largest principal tension indicate that rupture of the membrane would first occur near the top of the building.
Notation a
A , B , B1 c
length parameter in shear-flow representation of boundary layer constants in linearized hot-wire-anemometer calibration base diameter of dome (P external -- p® ) / (1/ 2 )p
d
D(f) e(~) E(o~) f h
principal membrane tension (p U~c/2) zero-offset displacement in law of the wall power-spectral density fluctuating component of linearized anemometer signal time-averaged linearized anemometer signal frequency height of dome
*Present address: Division o f Building Research, National Research Council o f Canada, M-20, Montreal Rd., Ottawa, K I A 0R6 Canada.
0167-6105/84/$03.00
© 1984 Elsevier Science Publishers B.V.
306 h L
1/n Pi P~ q R Re S t T To
%
Train Tm~ U U~ Ue Uh U~ Ueff
,u-r
Y Yo 8 0 OG K
p 0
rW
constant in longitudinal cooling law longitudinal integral scale of turbulence exponent in power-law representation of wind velocity profile internal pressure within dome freestream static pressure dynamic pressure at y = h radius of dome skin-friction Reynolds number (Urc/V) scale factor thickness of membrane local membrane tension per unit length (N m - ' ) T in polar direction T in azimuthal direction least of minimum principal tensions greatest of maximum principal tensions mean velocity freestream velocity velocity at edge of boundary layer velocity at y = h skin-friction velocity ((rw/P)l/,) effective cooling velocity normal to h o t wire temporal means of turbulence velocity fluctuations distance from ground roughness-length parameter in law of the wall angle between normal to hot wire and mean flow direction boundary-layer height or reference height colatitude measured from top of dome angle subtended at ground by dome kinematic viscosity of air yon Karman constant (0.41) density of air mass of membrane/unit area skin friction at ground longitude measured from plane of symmetry
Superscripts *
full-scale values
1. Introduction This paper is concerned with wind flow over inflated spherical domes. Such air-supported structures are usually made from relatively thin, flexible, impervious material which is sometimes reinforced with cables. The air pressure within the structure must be sufficient to prevent buckling and collapse in high winds, b u t not so large that the membrane is torn by the large tensions which may develop.
307 Air-supported structures are used typically as temporary shelters, greenhouses, warehouses, and also to cover athletics facilities [2]. The first proposal for such a structure was made b y Lanchester [3] in 1917 when he patented detailed plans for an inflated field-hospital. In 1950 Bird [4] designed and produced inflated spherical radomes to cover early-warning radar antennae which were m o u n t e d on cylindrical towers in the DEW line, northern Canada. Notable examples of large inflated buildings are the following: the Pontiac Silverdome stadium in Detroit [5] (base area 4 X 104 m 2, height 60 m) and the similar stadium at British Columbia Place (220 X 168 m elliptical base, height 30 m); the Pan American exhibition building at the Brussels World Fair, 1962 [ 3 ] (spherical dome, height/base diameter = 0.75); the U.S. Atomic Energy Commission travelling exhibition " A t o m s for Peace" (spherical dome, height/base diameter = 0.40); the stadium at Dalhousie University [6] (86 X 66 m elliptical base, height 11 m). Two of the above buildings have spherical shapes. Typical spherical buildings have height-to-base ratios between 1/3 and 1/2, and the base often exceeds 100 m in diameter [5]. Models of these structures have been tested in a wind tunnel to determine the minimum internal pressure to prevent buckling. In tests on a model radome consisting of a three-quarters sphere on a cylindrical tower, the full wind dynamic pressure q was necessary [4, 7]. For a hemisphere m o u n t e d on the ground, 0.7q was required. These tests were made in uniform flow; the boundary layer on the ground was relatively small and therefore not representative of the boundary layer in the natural wind. Aeroelastic tests on other more complicated shapes have been made by Niemann [8] and also by the group at CSTB Nantes [ 9 - - 1 2 ] , who have studied the dynamics of stretchable semicylindrical models with one-quarter spherical ends. In an even more idealized situation, Newman and Tse [13] investigated flow past a lenticular inflated aerofoil at zero incidence, which may be interpreted as a reflection-plane model for the flow across a very long inflated building in the absence of boundary-layer effects on the ground. For buildings of small height it was found that very little internal gauge pressure was required to avoid collapse, and indeed the internal pressure might be allowed to drop to negative values before the membrane became unstable. Newman and Goland [14] followed up this investigation by testing two
308 ined. The domes are spherical and the membranes forming them are considered to be fully flexible, so that bending m o m e n t s in the m em brane are u n i m p o r t a n t . The m e m b r a n e is also taken to be inextensible, so that, in contrast to the planar two~limensional configuration studied by Newman and Goland, the geometry of the dom e is unaffect ed by the air flow as long as buckling is avoided. Three rigid plexiglas models with height-to-diameter ratios of 0.25, 0.37 and 0.5 were tested in a b o u n d a r y layer which was approximately an order o f magnitude thicker than the height of the models. The pressure distribution was measured and a finite-element analysis for thin shells was used to determine tensions and shearing forces in an equivalent inflated m e m b r a n e conforming to the assumption t ha t the m e m b r a n e is impervious, inextensible and flexible. With this i nf or m at i on the principal tensions were determined everywhere. In particular, t he place where the m i ni m um principal tension first becomes zero was determined: this is where buckling first occurs. The orientation of the tension also establishes the direction of t h e crease. The predictions were c om par e d with additional tests on flexible inflated domes as the internal pressure was gradually reduced. Thus the minimum internal pressure to avoid buckling was established and the associated m a x i m u m principal tensions in ot her parts of the dom e were determined. The flow r o u n d each model was also studied visually by using tufts on the main rigid models and by using smoke on smaller rigid models m o u n t e d in a smoke tunnel. 2. Experiments 2.1 Wind t u n n e l and boundary-layer simulation
The 2 m × 1.5 m blower wind tunnel in the Aerodynamics L a b o r a t o r y at McGill University was used. The test b o u n d a r y layer was developed as "roughness 2 " by Newman and Goland [14] over the 10 m working section to simulate wind flow over a sparsely w o o d e d area (power-law e x p o n e n t 1In = 0.24 [15, 1 6 ] , or, in a logarithmic representation, Y0 -~ 0.08 mm). Spires and roughness were used upstream following the design procedure o f Standen [17] and Campbell and Standen [ 1 8 ] . Experiments were c o n d u c t e d at one tunnel speed only. This speed was near the m a x i m u m for the wind tunnel and had a magnitude of 7.5 m s -1 near the roof. The Reynolds n u m b e r for t he experiments was t herefore effectively fixed. Its value based on the skin-friction velocity (0.365 m s -~) and the base diameter of t he structures was R e = Urc/v = 11 000. 2.2 W i n d - t u n n e l m o d e l s
Three model sizes were used, as shown in Table 1. Inflated spherical airsupported structures suffer no change in shape when made o f fabric which does n o t stretch. This p r o p e r t y facilitates measurements o f the external
309
pressure distribution using solid models. Three plexiglas domes of the dimensions given in Table 1 and ranging in thickness from 6 to 13 m m , with static pressure taps drilled radially outward, were used to obtain the external pressure distribution. The domes were moulded in plexiglas and then machined to the required spherical shape. The wind-tunnel arrangement, the coordinate system and a typical model are shown in Figs. 1--3, respectively. Each model contained four rows of pressure taps at rightangles to each other. The symmetrical nature of the structures allowed the use of one main row of taps drilled at equal increments of colatitude 0, since each structure was mounted on a circular base which could be rotated to various values of longitude ¢. The main pressure taps were spaced 3.5 ° apart for the shallowest d o m e and 5 ° apart for the other two models. The other rows of taps (11--15 ° apart) were used to check the symmetry of the models and the flow. TABLE
1
Wind-tunnel model sizes
Model
1 2 3
Base diameter Height
Height-to-base ratio
(m)
(m)
(h/c)
0.463 0.455 0.457
0.116 0.168 0.229
0.25 0.37 0.50
;table
,•i
screen
ires
mode
roughness (sol d m o d e l s ) nclined m a n o m e t e r
pressure tap~ (tunnelreference)
Iblowina side of vacuum c l e a n e r ( i n f l a t e d models )
Fig. 1. Schematic diagram of wind-tunnel arrangement.
310 The inflated models were made of impervious, effectively unstretchable light cloth (Stablekote II rip-stop nylon, 38 g m -2, which is normally used for spinnaker sails). The three flexible models were each made as close as possible to the dimensions of t he solid models.
•
_.________~ X
wi/~d~ ~ /
!
Fig. 2. Coordinate system for spherical dome.
Fig. 3. Model dome.
j.z
Since a spherical surface c a n n o t be " d e v e l o p e d " , Gothic-arch elements were used to appr oxi m a t e the curved surface. The widths at various distances along th e surfaces of the elements were calculated and sixteen pieces were cu t and sewn together for the cons truct i on of each flexible model. The angle subtended at the vertex of each "triangular" element was therefore 22.5 °. Glue was spread bet w e e n the adjacent overlapping flaps in order to prevent leakage. The flexible models were m o u n t e d on circular w o o d e n bases. The lower edge o f each model had extra material which was t ucked u n d er the base and sealed of f with tape. The models could be rotated, so t h a t various sections could be checked for buckling and the dependence o f the results on geometric imperfections could be determined. An adjustable supply o f air from t he blowing side of a 650 W domestic vacuum cleaner was used to inflate the models. The air was i nt roduced u n i f o r m l y to the inside o f the models via a plastic t ube c o n n e c t e d to the center of the base. The inflated models were marked at equal increments o f 0. Thus the location o f buckling could be identified.
2.3 Smoke-tunnel models In order to obtain an understanding of the flow around the spherical inflated domes the flow patterns a bout similar smaller solid models in a smoke tunnel were studied, T he relevant dimensions are given in Table 2.
311
2.4 Instrumentation Standard Disa 55D hot-wire e q u i p m e n t was used t oget her with a linearizer. Normal h o t wires were used t o measure the mean velocity in the b o u n d a r y layer and t he longitudinal turbulence. The spectrum o f t he latter was d e t e r m i n e d using a Hewlett-Packard analyzer (5420A). Slanting wires were used to measure the turbulence shearing stress in the b o u n d a r y layer in t h e absence o f t he m ode l [ 1 9 , 2 0 ] . A c o m b o f p i t o t tubes was also used to measure the mean velocity in t he tunnel b o u n d a r y layer. TABLE 2 Smoke-tunnel model sizes Model
Base diameter Height (ram) (mm)
Height-to-base ratio
1 2 3
101.6 101.6 101.6
0.25 0.37 0.50
25.4 37.6 50.8
2.5 Solid-model measurements The axial s y m m e t r y of the flow over t he solid models was checked by rotating th e mo de l to bring each meridional row of holes into the centerline position. The m a x i m u m variation of local gauge pressure due to asymm e t r y o f th e spherical surfaces and variations in the pressure taps was less t h a n -+ 3%. Pressure readings f r o m the primary rows of pressure taps were taken at values o f ~ = 0 °, 10 °, 20 °, ...., 180 °. At each position the pressure taps were c o n n e c t e d in tu r n to a single-tube reservoir m a n o m e t e r inclined at 1:25; the use o f a mu ltit ube m a n o m e t e r at this inclination would have given less accurate results.
2.6 Wind-tunnel interference effects The largest ratio of t he frontal area o f the model to the cross-sectional area o f th e wind tunnel was 0.03, and thus blockage effects were expect ed t o be small. The solid blockage correction (in a uni form stream) was estim a t e d to be 0.5% [ 2 1 ] , and the wake blockage was 0.4% [ 2 2 ] . Thus interference effects were small and corrections were n o t applied to t he measured results. 3. Numerical m e t h o d A finite-element program was used to evaluate t he stresses in the membrane from th e measured pressure distribution. These stresses were then used to predict the stabilizing internal pressure and the position and orienta-
312 tion of first buckling when the internal pressure is reduced slightly. The spherical domes were discretized into small elements by lines of colatitude (0 = constant) and longitude (¢ = constant), the intersections of which correspond to the positions a t which the external pressure distribution was measured. The finite-element program used in the analysis was the structural analysis program SAP IV developed by Bathe et al. [25]. The element type employed was a thin-shell quadrilateral, requiring four nodes for its definition. The Cartesian coordinate system is shown in Fig. 4. Since the y--z plane is one of symmetry of loading, only one-half of the dome needed to be analyzed. Y
~
F--constant O
O
~
// wind
~
:
-
-
-
~
n ° d e =x
z
Fig. 4. Quadrilateral elements used in numerical analysis. With the exception of the boundary nodes each node had three translational plus three rotational degrees of freedom. The latter are defined as the three components of angular displacement of the normal to the middle surface at a node. The nodes at the base of the structure were fixed in translation but rotations were permitted. The nodes in the y--z plane were allowed translational m o t i o n in the y- and z
313
more than sufficient accuracy was achieved using a discretization in which the ~ curves were 10 ° apart and the 0 curves were 3.5 ° apart for the smallest dome and 5 ° apart for the two larger domes. These values were also selected for convenience, since they coincided with the positions of the pressure taps. An a t t e m p t was made to predict the pressure distribution over a dome by assuming that the dome could be represented by a distribution of sources and sinks, as in the m e t h o d of Hess and Smith [ 2 7 ] , and representing the boundary layer by a flow with uniform vorticity and slip at the upstream boundary wall. This is the m e t h o d which was used successfully by Newman and Goland [14] to predict the flow over planar two-dimensional inflated buildings. The best matching m e t h o d equated the velocity and slope of the velocity profile at the top of the dome. This is the same as m e t h o d 2 used by Newman and Goland [14]. In the present application the theory, which assumes that the vorticity does not change direction when flowing over the dome, can be expected to work only when the shear flow is weak and the dome is shallow. When compared with experiment the predictions were not obviously best for the lowest dome ( h / c = 0.25). However, for no case were they sufficiently good to be used for the purpose of predicting the membrane tension. In general, the predicted pressure in the central plane (~ = 0 °) was too low over most of the building, particularly near the top, for all domes, and too high near the upwind face of the tallest domes. 4. Results and discussion 4.1 Boundary
layer on the tunnel floor
The p i t o t ¢ o m b survey of the boundary layer on the floor of the e m p t y wind tunnel was made at a nominal tunnel speed U e -- 7.5 m s -1. The boundary layer was found to have a thickness 8 = 1 m. The velocity profile is conveniently represented by a power law:
V/Ve =
(y/~)l/n
The results were plotted logarithmically and showed fair agreement with 1 / n = 0.24 on the tunnel centerline. Measurements 0.2 m on either side of the centerline gave 1 / n = 0.23 and 0.25. The skin-friction velocity U 7 = ( r w / P ) ~ was measured in two ways: (i) a slanting h o t wire was used to measure --h~- at various heights up to 0.30 m, giving substantially constant values: the average value was U r = 0.365 m s -1, which compares very favorably with the value of 0.360 m s -1 obtained by Newman and Goland [14] for essentially the same experimental situation; (ii) a normal h o t wire was used to obtain accurate measurements of the time-average velocity at various heights y, and these values were then fitted to a logarithmic law of the form U / U r = ( l / K ) ln[(y - - d ) / y o ]
314 where K is the von Karman constant, 0.41: d was taken as zero because the measurements were made on the s m oot h surface 0.5 building diameters downstream o f the roughness carpet, while Y0 depends on the surface roughness and increases with it. The results gave Ur = 0.375 m s -1 and Y0 -~ 0.08 mm. Measurements of the typical longitudinal scale L of the turbulence eddies were obtained from the power-spectral density D(f) of the longitudinal turbulence fluctuations at 24 values of y up to y = 0.65 m on the centerline and at the offset station, when 1In = 0.25 (f is the f r e q u e n c y in Hz).
o
06 _
~ ~1/A
//
0.0
L 0.5
I
y - 3 0 cm s U 6.6m/s
v\
fl = 19s'1 s=s,o
i___±
i _ _
50 f
Fig. 5. Typical power-spectral-density frequency distribution fD(f) as a function of log f.
T h e signals were processed on the Hewlett--Packard system t o give [ D(f) as a f u n c t i o n o f logf. An example is shown in Fig. 5. The distribution was m a t c h e d to the standard spectral distribution a d o p t e d by ESDU for the Earth b o u n d a r y layer [26, 14] . The c o m p u t e d values of the eddy scalef a cto r S ranged f r om 440 to 550, with an average value of ~ 500.
4.2 Model experiments The measured pressure distributions, referred to ambient pressure (i.e., t h e pressure at the m odel position when the tunnel is e m p t y ) , are presented for th e three solid models in Fig. 6(a)--(c). These results are presented nondimensionally as Cp~ = (p --p~)/(1/2)p U~, where Ur was taken as 0.365 m s -1. Th ey are shown as a f u n c t i o n o f colatitude 0 for various planes of longitude ¢ (Fig. 2). Values for ¢ < 90 ° (i.e., on the upwind faces) are p l o t t e d on the left-hand side of t he figures, and for ¢ > 90 ° on the righthand side. Contours of c ons t ant pressure are shown in a side view of the t h ree models in Fig. 7(a)--(c). Regions of positive, negative and const ant negative Cp~ on the leeward side are also indicated. It is seen t h a t as the
315
height of the dome increases the highest value of CpT at the front of the model increases and moves upwards. The minimum pressure on the top of the dome decreases and the constant pressure in the separated regions on the leeward side decreases. It may be noted that the isobars do not lie on circles at right-angles to the flow, which was the simplifying assumption made by Bird when calculating the tension in an equivalent inflated membrane. However, this assumption does appear to be best for the highest dome, which has a geometry closest to the radome which Bird was investigating [ 4 ]. A qualitative view of the flow is revealed in the smoke-tunnel pictures shown in Fig. 8, which show the flow over a solid dome of height h / c --0.37. For these tests the Reynolds number was low (6000) and the upstream boundary layer was thin. Nevertheless, the broad features will be similar to the main experiments. The pictures are for a plane of streamlines at heights y / h = 0.25, 0.50 and 0.75. The flow appears to be greatly affected by a horseshoe vortex which wraps around the dome. This vortex, by convecting freestream flow towards the surface, helps to reduce the size of the separated region downstream. The m o v e m e n t of the windward stagnation point upwards accounts for the relatively high upstream pressures when compared with the theory in Section 3. The downstream separation is the other dominant feature which is not accurately represented in the theory.
(a)
h/c --0.25
•
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0 o G
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O •-e--
~-
180 170
1 0
• -- 1 ~ 0
-40
I-v.. t'-
140 130 120 O-- 1 1 0 o- 100
~'-SO
• -60 0-70
0-80
-90
Fig. 6. Pressure distributions o n solid models. (a)
•
|,:
-so-
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"51o
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h/c = 0.25.
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317 T h e f i n i t e - e l e m e n t p r o g r a m SAP IV was used t o d e t e r m i n e t h e distribut i o n o f t e n s i o n in inextensible, i m p e r v i o u s inflated m e m b r a n e s o f t h e same spherical g e o m e t r y as t h e t h r e e solid models. This r e q u i r e d t h a t t h e c h o s e n Y o u n g ' s m o d u l u s s h o u l d be high: a value o f 1040 N m -2 was chosen. Poisson's ratio was t a k e n as zero. T h e actual thickness tic o f t h e m o d e l m e m -
ind
lllllliiiNiiilliiiJi[i-iiiii; .-. (a)
h/c
(b)
0.25
h/c
-= 0 . 3 7
in__d
,|ll
(c)
II IIl'JI
h/C = 0 - 5 0
Fig. 7. Contours of constant pressure on solid models: (a) h/c (c) h/c = 0.50.
= 0.25; (b)
h/c
ffi
0.37;
318
Y -- 0.25 -E
Y - - 0.50
--Y -- 0.75 C Fig. 8. Smoke-tunnel pictures of flow over a m o d e l d o m e , to left).
h/c
=
0.37 (flow from right
319 branes was 0.22 X 10 -3. Trials with t/c = 0.22 × 10 -I not surprisingly gave results which violated the m e m b r a n e equation T e + T~ = h p R
For a very thin membrane ( t / c = 0.22 X 10 -1°) the displacements became large and the geometry changed significantly. The value tic = 0.22 X 10 -3 seemed to be satisfactory and was used in all the subsequent calculations. Despite the large value of Young's modulus, the change of tension across the membrane was less than 1%. The effect of a change in Poisson's ratio from 0 to 0.3 produced similar insignificant changes of tension. These results confirmed that in an inextensible membrane shell the stresses are independent of material constants. As a check on the program the overall balance of base tensions and pressures for each complete dome was confirmed to an accuracy of 2%. The membrane equation was well satisfied for all elements with the exception of those nearest to the base, presumably because the imposed boundary conditions were not the appropriate membrane conditions there [28]. Since the change of membrane geometry is negligible, it is permissible to superimpose two solutions: (i) a numerical solution using the measured external pressure distribution and an internal pressure which is atmospheric; and (ii) the simple solution T = (1/2)(p i - - p ~ ) R where Pi is the inside pressure and p~ is atmospheric pressure applied uniformly to the outside o f the dome. Solution (i) was therefore first obtained using the measured external distribution of gauge pressures and the principal tensions were determined at the center of each element. The least of these m i n i m u m tensions Train was identified. Since it was negative, the minimum internal gauge pressure t o just prevent buckling and to stabilize the envelope is
(Pi --P~ )rain
= 2(--Tmin)/R
The validity of this superposition was also confirmed numerically by calculating a case with a positive internal gauge pressure. In addition, it may be remarked t h a t results were also obtained by smoothing the experimental data before putting the values into the finiteelement program. However, no difference could be discerned between these results and those obtained by using the raw experimental data. The raw data for pressure were used subsequently. The results for solution (i) are shown in Fig. 9(a)--(c), where the minim u m principal tension has been appropriately normalized using (1/2)p U~c. The solid curves are plotted against ~b for various values of 0 and the single dashed curve against 0 for ¢ = 0 °. For all three domes buckling occurs for ¢ = 0°; in other words, it occurs on the windward side of the model
320
in the plane o f s y m m e t r y . The orientation o f this m i m i m u m tension is circumferential in all cases. Thus buckling is predicted to occur first with vertical creases. The results for the three domes are shown in Table 3, where the stabilizing pressures are given normalized in terms of Uh, the wind velocity corresponding to the t op of the dome. 0 G is the value of 0 at the ground,
h/c F~(~ // -:
..........
l~]c .37~l
39 48 50
///
i
~-
.25
1o ~
10CTz 0
~ / -
7 2.5
C~
-10 o
-20
/
-30
-3O "40
(o)
ckling
,/
[% J
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10
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20
30
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40
/
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e
h/c
I _
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20
J
I
40 e
I
I
60
1
.50 !
CTI
(C) l
[ 20
i
k 40 e
~
i 60
i~_ 80
Fig. 9. M i n i m u m p r i n c i p a l t e n s i o n w i t h a t m o s p h e r i c i n t e r n a l pressure: (a) h/c = 0 . 2 5 ; (b) h/c = 0 . 3 7 ; (c) h/c = 0.50.
321
TABLE 3 Predicted m i n i m u m internal pressure Pi t~ prevent buckling and associated m a x i m u m tension Tmax near top o f model
Model (pi--poo)min/ (1/2)p U~
Uh/Ur
0.25 0.37 0.50
14.38 16.49 16.66
h/c
0.44 0.49 0.65
(Pi--P~)min ] ebuckl~g (1/2)p U¢
Obuckling/OG Tvaax/ Tmax/
91 133 180
0.90 0.83 0.80
48 ° 62.50 72.5 °
(1/2)p
(1/2)p
0.34 0.33 0.42
71 89 116
v~c
v¢c
w h i c h is t h e base o f t h e d o m e . N o t e t h a t at t h e p o i n t o f b u c k l i n g 0/0 G decreases w i t h increasing h/c. A useful, a l t h o u g h e m p i r i c a l result, is t h a t f o r i n c i p i e n t buckling: [(Pi --p®)/(1/2)p U~] (c/h) = 3 6 0 (-+ 1%) T h e m a x i m u m t e n s i o n s w h e n t h e i n t e r n a l p r e s s u r e is j u s t s u f f i c i e n t to p r e v e n t b u c k l i n g are also o f practical interest. T h e values are p l o t t e d against 0 f o r various values o f ¢ in Fig. 1 0 ( a ) - - ( c ) . T h e m a x i m u m values are o r i e n t e d in t h e l o n g i t u d i n a l d i r e c t i o n a n d are n e a r ¢ = 90 ° a n d 0 = 1 0 - - 2 0 °, i.e., crossw i n d a n d n e a r t h e t o p o f t h e d o m e . With t h e stabilizing p r e s s u r e a p p l i e d inside t h e d o m e t h e values o f Tma x are also given in T a b l e 3. T h e i n f l a t e d m e m b r a n e m o d e l s w e r e e x a m i n e d closely in t h e w i n d - t u n n e l f l o w as t h e internal p r e s s u r e was altered, w i t h r e s p e c t t o t h e c a l c u l a t e d pressures in T a b l e 3. D u e t o difficulties o f m a n u f a c t u r e , t h e m o d e l s w e r e n o t e x a c t l y spherical a n d a x i s y m m e t r i c . E a c h m o d e l was t h e r e f o r e r o t a t e d t o bring t h e best possible s h a p e i n t o t h e u p w i n d region, w h e r e b u c k l i n g was p r e d i c t e d to occur. T h e d i r e c t i o n o f b u c k l i n g was d i f f i c u l t t o determ i n e . T h e b u c k l e did n o t m a n i f e s t itself as a m e r i d i o n a l crease b u t instead was a small u n s t e a d y circular d e n t o n t h e surface o f t h e m o d e l w h i c h grew in size as t h e internal p r e s s u r e was l o w e r e d . All such d e n t s w e r e first observed n e a r ¢ = 0 °. T h e a p p r o x i m a t e p o s i t i o n s a n d t h e c o r r e s p o n d i n g pressures are s h o w n in T a b l e 4. T h e stabilizing pressures and t h e p o s i t i o n s o f TABLE 4 Buckling positions and internal pressures Pi observed on inflated models Model
Buckling position 0 (~ = 0 °)
h/c
Theory
Experiment
0.25 0.37 0.50
48 ° 62.5 ° 72.5 °
35--40 ° 60 ° 70--75 °
(Pi--P~ )vain(exp')/(Pi --Poo )vain (theor.) 1.0--1.3 1.1 1.5
322 h/c
/
0.25 1
10(
I
~ , L
I t
9ol-
8(
/ / / / ]~ -
.37
F---ID
85,95 45
CT¢
10~h/c:
/ ,//
= 95 85
175
45 175
/ : Ji 70
6(3
4G
(a)l 10
0
I 20
I
I 30
40
50
0
9
h/c
.50
,
120 ?
10
20 30 40 50 60
70
e
0
f. . . . . . lit I / / /~
85 95 472 0 = 135
CO0 C~ (c) I
0
I
L
_
[
i
10 20 30 40 50 50
J
I
70 80 90
0
Fig. 10. M a x i m u m
p r i n c i p a l t e n s i o n w h e n b u c k l i n g is just a v o i d e d : (a) h/c = 0 . 2 5 ; (b)
/tic = 0 . 3 7 ; (c) h/c = 0.50.
buckling are seen to be predicted reasonably well for the two lower models. The highest model was not a true spherical dome and due to expansion of the seams was noticeably flat-topped. This is considered to be the main cause of the relatively high experimental stabilizing pressure for this model. It is therefore concluded that the m e t h o d of prediction which has been
323
developed in this paper gives a fairly good estimate of the internal pressure required to prevent buckling. The behavior of the models was also observed at other internal pressures. The results with the internal pressure expressed as a proportion of the theoretical stabilizing pressure (Pi - - P - ) r a i n may be summarized as follows: Pi -- P -
Behavior
(Pi --Po. )min 6 4
membrane generally t a u t and steady on all models very slight fluttering or panting o f the membrane on the windward side of the centerline for the two highest models (h/c = 0.5 and 0.37) very slight fluttering observed at a similar position on the lowest model (h/c = 0.25) fluttering extended to ¢ = +30 ° on the windward side and amplitude increased
3 2--1.5
The m o d e of collapse as the pressure was reduced below the theoretical stabilizing pressure is indicated in Fig. 11. Photographs showing the m o d e onset of buckling, observed in close proximity to the point of maximum pressure
1.
2. shape of dome at an internal pressure of 0.8 x t i m e s the stabilizing
I
pressure
x\
\
X
.
,
\.~
""-'-:':4
(h/c--0.37)
3. shape at an internal pressure of 0.4 x s t a b i l i z i n g
4.
shape
pressure
at zero gauge
internal
pressure Fig. 11. M o d e of collapse as pressure is reduced below that which just prevents buckling.
324
(p~-p~) =1,0 ( P i " Poo)min. theory
( Pi - Pco) : 0.7 ( Pi- PoO) rain.theory
(p~-p~) (Pi - Poo) rain. theory = 0 . 6
Fig. 12. Photographs showing mode of collapse at subcritical pressures,
h/c
=
0.5.
325 of buckling for such subcritical pressures are shown in Fig. 12(a)--(c) for the tallest inflatable model (h/c = 0.5). 4.3 R e l a t i o n o f m o d e l e x p e r i m e n t s to full-scale buildings
The applicability of the present model tests (c = 0.45 m) to full-scale buildings is a m o o t question. Since the longitudinal scale of the eddies was ~ 1/500 that of typical wind eddies, the size of the full-scale building is strictly 320 m, although variation from this value by factors of perhaps 2 or 3 m a y be permissible if only static effects are of interest. However, for rounded buildings such as domes even the average positions of flow separation m a y be sensitive to turbulence scale as well as to turbulence intensity, so that large departures from a scaling factor of 500 are considered to be undesirable. Aerodynamic scale effects, i.e., sensitivity to changes of Reynolds number, are dealt with partly by the fact the models were tested within the wall-law region, for which the upstream flow is determined b y Tw, p and the roughness height for a fully rough boundary layer. Making model size proportional to roughness height then establishes similitude. However, the flow over the d o m e itself is likely to depend on viscosity, since the dome is smooth and has no sharp corners to fix the positions of separation. Buckling is associated with stagnation on the upwind face of the dome, which in turn is determined by separation of the upwind fully rough boundary layer. Thus the predictions for buckling should be applicable to full-scale buildings, and the agreement with the National Code seems to support this contention. The associated maximum tensions near the top of the building, however, may be t o o low in the model tests. The nondimensional mass of the membrane, a/pc, where a is the mass per unit area, was ~ 25--50 times bigger for the models than it would be at full scale. Thus the oscillations of the model are unrealistically large. Experiments on flags [29] and bluff membranes [30] exhibit the same trends. Moreover, the frequencies for a given wavenumber also depend on the value of a / p c [ 3 1 ] . Thus the present experiments on inflated models do n o t address dynamic effects, and no a t t e m p t has been made to simulate this aspect of full-scale behavior.
5. Conclusions Winds over a sparsely w o o d e d area for which the longitudinal scale factor was 500 have been simulated in a boundary-layer wind tunnel. Pressure measurements on model domes have indicated that buckling of an equivalent inflated dome would occur on the plane of s y m m e t r y and upwind when the internal inflation gauge pressures were respectively 0.65, 0.49 and 0.44 of the reference dynamic pressure q at the t o p of the dome for h/c = 0.5, 0.37 and 0.25. Niemann [8] measured a similar value of 0.7 for h/c = 0.5. The corresponding values specified by the National Standard of Canada
326
[1] are 0.7 for h / c = 0.5 and 0.6 for h / c = 0.35. These are slightly higher, and they may be desirable in practice to avoid subbuckling oscillation of the membrane, which was observed on the present inflated models. A convenient empirical formula for the minimum gauge pressure (Pi - - P ~ ) to prevent buckling is
(Pi --P~)/rW
-- 1 8 0
h/c
The corresponding m a x i m u m principal tensions are longitudinal and occur near the top of the dome in planes nearly at right-angles to the flow. They vary slightly from 0.42 q c for the highest dome to 0.33 q c for the others. As the wind speed increases and with it r w the internal pressure must be increased to prevent buckling, and the m a x i m u m tension at the top of the building increases in proportion. The membrane in this area may therefore require reinforcement. The base of the building, where the imposed boundary conditions are somewhat uncertain, is also a sensitive area. The present model tests are considered to predict adequately the upwind initial buckling of a full-scale inflated building when the base diameter of the building is ~ 100 m. The prediction of the associated m a x i m u m tension at the top of the building is less certain, because of aerodynamic scale effects. Dynamic effects associated with oscillation of the membrane at subbuckling pressures have not been investigated. References 1 National Standard of Canada, Air Supported Structures, CAN3-5367-M81, 1981. 2 W. McQuade, A new air age in construction, Fortune Mag., (October 1977) 228-231. 3 Cedric Price Architects, F. Newby, R.H. Suan, J.S. Felix and Partners, Air Structures: A Survey, Department of the Environment, London, HMSO, 1971. 4 W.W. Bird, Design manual for spherical radomes (revised), Cornell Aeronaut. Lab., Rep. UB-909-D2 (1965). 5 American Society of Civil Engineers, Air Supported Structures, State of the Art Report, New York, Am. Soc. Cir. Eng., 1979. 6 J. Springfield and D. Sinoski, The air supported steel membrane roof at Dalhousie University, Halifax, Nova Scotia, Can. Struct. Eng. Conf., Canadian Steel Construction Council, Willowdale, Ont., 1980, pp. 1--29. 7 G. Beger and E. Macher, Results of wind tunnel tests on some pneumatic structures, Proc. 1st Int. Colloq. o n Pneumatic Structures, Stuttgart, 1967, pp. 142--146. 8 H.-J. Niemann, Wind tunnel experiments on aeroelastic models of air supported structures: results and conclusions, Proc. 2nd Int. Colloq. on Pneumatic Structures, Delft, The Netherlands, 1972, p. HJN-I--12. 9 J. Bietry, Etude des effets du vent sur les structures gonflables: conditions de similitude, CSTB, Nantes, Rapp. EN. ADYM-75-8-R (1975). 10 G. Grillaud and J. Gandemer, Etude des effets du vent sur les structures gonflables: ~tude experimentale en soufflerie sur maquette aeroelastique, CSTB, Nantes, Rapp. EN. ADYM-76.7.L (1976). 11 H. Maille, Effet du vent sur les structures gonflables: equipement experimental, CSTB, Nantes, Rapp. EN. ADYM-77.9.L (1977).
327 12 13 14 15 16 17
18
19 20
21 22 23 24 25 26 27 28 29 30 31
G. Grillaud and J. Gandemer, Etude de la r~ponse dynarnique d'une structure gonflable: ~tude en soufflerie,CSTB, Nantes, Rapp. EN. A D Y M - 7 7 (1977). B.G. N e w m a n and M.-C. Tse, Flow past a thin, inflated lenticular aerofoil, J. Fluid Mech., 100 (1980) 673--689. B.G. Newman and D. Goland, Two-dimensional inflated buildings in a crosswind, J. Fluid Mech., 117 (1981) 507--530. A . G . Davenport, The relationship of wind structure to wind loading, Proc. 16th Syrup. on Buildings and Structures, National Physical Laboratory, Teddington, 1963, p. 5 4 - 6 5 . E.J. Plate, Atmospheric boundary layers, USAEC, Rep. TID-25465 (1971). N.M. Standen, A spire array for generating thick turbulent shear layers for natural wind simulation in wind tunnels, Natl. Res. Counc. (Canada), Rep. LTR-LA-94 (1972). S. Campbell and N.M. Standen, Simulation of Earth's boundary layer by artificially thickened wind tunnel boundary layers, Natl. Res. Counc. (Canada), Rep. LTRLA-37 (1969). J.O. Hinze, Turbulence, McGraw-Hill, New York, 1959, Chap. 2. H.P.A.H. Irwin, The longitudinal cooling correction for wires inclined to the prongs and some turbulence measurements in fully developed pipe flow, McGill University, M.E.R.L. TN-72-1 (1971). R.C. Pankurst and D.W. Holder, Wind Tunnel Technique, Pitman, London, 1952. E.C. Maskell, A theory of the blockage effects on bluff bodies and stalled wings in a closed wind tunnel, Aerodyn. Res. Counc., Rep. & Memo. 3400 (1963). D. Goland, Two-dimensional inflated buildings in a cross wind, M. Eng. Thesis, Department o f Mechanical Engineering, McGill University (1980). R.P. Patel, Reynolds stresses in fully developed turbulent flow down a circular pipe, McGill University, M.E.R.L. Rep. 68-7 (1968). K. Bathe, E. Wilson and F. Peterson, A structural analysis program for static and dynamic response of linear systems, University of California, Berkeley, CA, Rep. EERC 73-11 (1973). N.J. Cook, Determination of the model scale factor in wind tunnel simulations of the adiabatic atmospheric boundary layer, J. Ind. Aerodyn., 2 (1978) 311--321. J.L. Hess and A.M.O. Smith, Calculation of potential flow about arbitrary bodies, Prog. Aeronaut. Sci., 8 (1969) 1--138. V.V. Novozhilov, The Theory of Thin Shells, Noordhoff, Groningen, 1964. R.A. Fairthorne, Drag of flags, Aerodyn. Res. Counc., Rep. & Memo. 1345 (1930). B.G. Newman and H.T. Low, Two-dimensional flow at right angles to a flexible membrane, Aeronaut. Q., 32 (1981) 2 4 3 - 2 6 9 . B.G. Newman, The aerodynamics of flexible membranes, Proc. 8th C.A.N.C.A.M., Moncton, 1981, p. 63 (published in enlarged form in Proc. Indian Acad. Sci., 5
(1982) 107--129).
305
Elsevier Science Publishers B.V., Amsterdam - - P r i n t e d in The Netherlands
FLOW OVER SPHERICAL INFLATED BUILDINGS
B.G. NEWMAN and U. GANGULI*
Department of Mechanical Engineering, McGill University, Montreal, Quebec (Canada) S.C. S HR IVAS TAV A
Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, Quebec (Canada) (Received September 21, 1983; accepted in revised form May 23, 1984)
Summary The pressure distributions on three domes of different heights (h/c = 0.5, 0.37 and 0.25) have been measured in a boundary-layer wind tunnel in which the wind flow over a sparsely wooded area was simulated. The tests were made in the wall-law region of the boundary layer and the results are therefore first normalized using the undisturbed skin friction r w . The tensions in inflated membraries of the same geometry were determined using the SAP IV finite-element program assuming that the effects of membrane weight and air friction are unimportant. The principal tensions then indicate where buckling would first take place and what would be the minimum internal pressure to just prevent it. The minimum internal gauge pressure to prevent buckling, which is shown to occur on the upwind side of the building, is 180 (h/c) r w ' This conclusion is compatible with the recommendations of the National Standards of Canada (1981) [1]. The location, orientation and magnitude of the largest principal tension indicate that rupture of the membrane would first occur near the top of the building.
Notation a
A , B , B1 c
length parameter in shear-flow representation of boundary layer constants in linearized hot-wire-anemometer calibration base diameter of dome (P external -- p® ) / (1/ 2 )p
d
D(f) e(~) E(o~) f h
principal membrane tension (p U~c/2) zero-offset displacement in law of the wall power-spectral density fluctuating component of linearized anemometer signal time-averaged linearized anemometer signal frequency height of dome
*Present address: Division o f Building Research, National Research Council o f Canada, M-20, Montreal Rd., Ottawa, K I A 0R6 Canada.
0167-6105/84/$03.00
© 1984 Elsevier Science Publishers B.V.
306 h L
1/n Pi P~ q R Re S t T To
%
Train Tm~ U U~ Ue Uh U~ Ueff
,u-r
Y Yo 8 0 OG K
p 0
rW
constant in longitudinal cooling law longitudinal integral scale of turbulence exponent in power-law representation of wind velocity profile internal pressure within dome freestream static pressure dynamic pressure at y = h radius of dome skin-friction Reynolds number (Urc/V) scale factor thickness of membrane local membrane tension per unit length (N m - ' ) T in polar direction T in azimuthal direction least of minimum principal tensions greatest of maximum principal tensions mean velocity freestream velocity velocity at edge of boundary layer velocity at y = h skin-friction velocity ((rw/P)l/,) effective cooling velocity normal to h o t wire temporal means of turbulence velocity fluctuations distance from ground roughness-length parameter in law of the wall angle between normal to hot wire and mean flow direction boundary-layer height or reference height colatitude measured from top of dome angle subtended at ground by dome kinematic viscosity of air yon Karman constant (0.41) density of air mass of membrane/unit area skin friction at ground longitude measured from plane of symmetry
Superscripts *
full-scale values
1. Introduction This paper is concerned with wind flow over inflated spherical domes. Such air-supported structures are usually made from relatively thin, flexible, impervious material which is sometimes reinforced with cables. The air pressure within the structure must be sufficient to prevent buckling and collapse in high winds, b u t not so large that the membrane is torn by the large tensions which may develop.
307 Air-supported structures are used typically as temporary shelters, greenhouses, warehouses, and also to cover athletics facilities [2]. The first proposal for such a structure was made b y Lanchester [3] in 1917 when he patented detailed plans for an inflated field-hospital. In 1950 Bird [4] designed and produced inflated spherical radomes to cover early-warning radar antennae which were m o u n t e d on cylindrical towers in the DEW line, northern Canada. Notable examples of large inflated buildings are the following: the Pontiac Silverdome stadium in Detroit [5] (base area 4 X 104 m 2, height 60 m) and the similar stadium at British Columbia Place (220 X 168 m elliptical base, height 30 m); the Pan American exhibition building at the Brussels World Fair, 1962 [ 3 ] (spherical dome, height/base diameter = 0.75); the U.S. Atomic Energy Commission travelling exhibition " A t o m s for Peace" (spherical dome, height/base diameter = 0.40); the stadium at Dalhousie University [6] (86 X 66 m elliptical base, height 11 m). Two of the above buildings have spherical shapes. Typical spherical buildings have height-to-base ratios between 1/3 and 1/2, and the base often exceeds 100 m in diameter [5]. Models of these structures have been tested in a wind tunnel to determine the minimum internal pressure to prevent buckling. In tests on a model radome consisting of a three-quarters sphere on a cylindrical tower, the full wind dynamic pressure q was necessary [4, 7]. For a hemisphere m o u n t e d on the ground, 0.7q was required. These tests were made in uniform flow; the boundary layer on the ground was relatively small and therefore not representative of the boundary layer in the natural wind. Aeroelastic tests on other more complicated shapes have been made by Niemann [8] and also by the group at CSTB Nantes [ 9 - - 1 2 ] , who have studied the dynamics of stretchable semicylindrical models with one-quarter spherical ends. In an even more idealized situation, Newman and Tse [13] investigated flow past a lenticular inflated aerofoil at zero incidence, which may be interpreted as a reflection-plane model for the flow across a very long inflated building in the absence of boundary-layer effects on the ground. For buildings of small height it was found that very little internal gauge pressure was required to avoid collapse, and indeed the internal pressure might be allowed to drop to negative values before the membrane became unstable. Newman and Goland [14] followed up this investigation by testing two
308 ined. The domes are spherical and the membranes forming them are considered to be fully flexible, so that bending m o m e n t s in the m em brane are u n i m p o r t a n t . The m e m b r a n e is also taken to be inextensible, so that, in contrast to the planar two~limensional configuration studied by Newman and Goland, the geometry of the dom e is unaffect ed by the air flow as long as buckling is avoided. Three rigid plexiglas models with height-to-diameter ratios of 0.25, 0.37 and 0.5 were tested in a b o u n d a r y layer which was approximately an order o f magnitude thicker than the height of the models. The pressure distribution was measured and a finite-element analysis for thin shells was used to determine tensions and shearing forces in an equivalent inflated m e m b r a n e conforming to the assumption t ha t the m e m b r a n e is impervious, inextensible and flexible. With this i nf or m at i on the principal tensions were determined everywhere. In particular, t he place where the m i ni m um principal tension first becomes zero was determined: this is where buckling first occurs. The orientation of the tension also establishes the direction of t h e crease. The predictions were c om par e d with additional tests on flexible inflated domes as the internal pressure was gradually reduced. Thus the minimum internal pressure to avoid buckling was established and the associated m a x i m u m principal tensions in ot her parts of the dom e were determined. The flow r o u n d each model was also studied visually by using tufts on the main rigid models and by using smoke on smaller rigid models m o u n t e d in a smoke tunnel. 2. Experiments 2.1 Wind t u n n e l and boundary-layer simulation
The 2 m × 1.5 m blower wind tunnel in the Aerodynamics L a b o r a t o r y at McGill University was used. The test b o u n d a r y layer was developed as "roughness 2 " by Newman and Goland [14] over the 10 m working section to simulate wind flow over a sparsely w o o d e d area (power-law e x p o n e n t 1In = 0.24 [15, 1 6 ] , or, in a logarithmic representation, Y0 -~ 0.08 mm). Spires and roughness were used upstream following the design procedure o f Standen [17] and Campbell and Standen [ 1 8 ] . Experiments were c o n d u c t e d at one tunnel speed only. This speed was near the m a x i m u m for the wind tunnel and had a magnitude of 7.5 m s -1 near the roof. The Reynolds n u m b e r for t he experiments was t herefore effectively fixed. Its value based on the skin-friction velocity (0.365 m s -~) and the base diameter of t he structures was R e = Urc/v = 11 000. 2.2 W i n d - t u n n e l m o d e l s
Three model sizes were used, as shown in Table 1. Inflated spherical airsupported structures suffer no change in shape when made o f fabric which does n o t stretch. This p r o p e r t y facilitates measurements o f the external
309
pressure distribution using solid models. Three plexiglas domes of the dimensions given in Table 1 and ranging in thickness from 6 to 13 m m , with static pressure taps drilled radially outward, were used to obtain the external pressure distribution. The domes were moulded in plexiglas and then machined to the required spherical shape. The wind-tunnel arrangement, the coordinate system and a typical model are shown in Figs. 1--3, respectively. Each model contained four rows of pressure taps at rightangles to each other. The symmetrical nature of the structures allowed the use of one main row of taps drilled at equal increments of colatitude 0, since each structure was mounted on a circular base which could be rotated to various values of longitude ¢. The main pressure taps were spaced 3.5 ° apart for the shallowest d o m e and 5 ° apart for the other two models. The other rows of taps (11--15 ° apart) were used to check the symmetry of the models and the flow. TABLE
1
Wind-tunnel model sizes
Model
1 2 3
Base diameter Height
Height-to-base ratio
(m)
(m)
(h/c)
0.463 0.455 0.457
0.116 0.168 0.229
0.25 0.37 0.50
;table
,•i
screen
ires
mode
roughness (sol d m o d e l s ) nclined m a n o m e t e r
pressure tap~ (tunnelreference)
Iblowina side of vacuum c l e a n e r ( i n f l a t e d models )
Fig. 1. Schematic diagram of wind-tunnel arrangement.
310 The inflated models were made of impervious, effectively unstretchable light cloth (Stablekote II rip-stop nylon, 38 g m -2, which is normally used for spinnaker sails). The three flexible models were each made as close as possible to the dimensions of t he solid models.
•
_.________~ X
wi/~d~ ~ /
!
Fig. 2. Coordinate system for spherical dome.
Fig. 3. Model dome.
j.z
Since a spherical surface c a n n o t be " d e v e l o p e d " , Gothic-arch elements were used to appr oxi m a t e the curved surface. The widths at various distances along th e surfaces of the elements were calculated and sixteen pieces were cu t and sewn together for the cons truct i on of each flexible model. The angle subtended at the vertex of each "triangular" element was therefore 22.5 °. Glue was spread bet w e e n the adjacent overlapping flaps in order to prevent leakage. The flexible models were m o u n t e d on circular w o o d e n bases. The lower edge o f each model had extra material which was t ucked u n d er the base and sealed of f with tape. The models could be rotated, so t h a t various sections could be checked for buckling and the dependence o f the results on geometric imperfections could be determined. An adjustable supply o f air from t he blowing side of a 650 W domestic vacuum cleaner was used to inflate the models. The air was i nt roduced u n i f o r m l y to the inside o f the models via a plastic t ube c o n n e c t e d to the center of the base. The inflated models were marked at equal increments o f 0. Thus the location o f buckling could be identified.
2.3 Smoke-tunnel models In order to obtain an understanding of the flow around the spherical inflated domes the flow patterns a bout similar smaller solid models in a smoke tunnel were studied, T he relevant dimensions are given in Table 2.
311
2.4 Instrumentation Standard Disa 55D hot-wire e q u i p m e n t was used t oget her with a linearizer. Normal h o t wires were used t o measure the mean velocity in the b o u n d a r y layer and t he longitudinal turbulence. The spectrum o f t he latter was d e t e r m i n e d using a Hewlett-Packard analyzer (5420A). Slanting wires were used to measure the turbulence shearing stress in the b o u n d a r y layer in t h e absence o f t he m ode l [ 1 9 , 2 0 ] . A c o m b o f p i t o t tubes was also used to measure the mean velocity in t he tunnel b o u n d a r y layer. TABLE 2 Smoke-tunnel model sizes Model
Base diameter Height (ram) (mm)
Height-to-base ratio
1 2 3
101.6 101.6 101.6
0.25 0.37 0.50
25.4 37.6 50.8
2.5 Solid-model measurements The axial s y m m e t r y of the flow over t he solid models was checked by rotating th e mo de l to bring each meridional row of holes into the centerline position. The m a x i m u m variation of local gauge pressure due to asymm e t r y o f th e spherical surfaces and variations in the pressure taps was less t h a n -+ 3%. Pressure readings f r o m the primary rows of pressure taps were taken at values o f ~ = 0 °, 10 °, 20 °, ...., 180 °. At each position the pressure taps were c o n n e c t e d in tu r n to a single-tube reservoir m a n o m e t e r inclined at 1:25; the use o f a mu ltit ube m a n o m e t e r at this inclination would have given less accurate results.
2.6 Wind-tunnel interference effects The largest ratio of t he frontal area o f the model to the cross-sectional area o f th e wind tunnel was 0.03, and thus blockage effects were expect ed t o be small. The solid blockage correction (in a uni form stream) was estim a t e d to be 0.5% [ 2 1 ] , and the wake blockage was 0.4% [ 2 2 ] . Thus interference effects were small and corrections were n o t applied to t he measured results. 3. Numerical m e t h o d A finite-element program was used to evaluate t he stresses in the membrane from th e measured pressure distribution. These stresses were then used to predict the stabilizing internal pressure and the position and orienta-
312 tion of first buckling when the internal pressure is reduced slightly. The spherical domes were discretized into small elements by lines of colatitude (0 = constant) and longitude (¢ = constant), the intersections of which correspond to the positions a t which the external pressure distribution was measured. The finite-element program used in the analysis was the structural analysis program SAP IV developed by Bathe et al. [25]. The element type employed was a thin-shell quadrilateral, requiring four nodes for its definition. The Cartesian coordinate system is shown in Fig. 4. Since the y--z plane is one of symmetry of loading, only one-half of the dome needed to be analyzed. Y
~
F--constant O
O
~
// wind
~
:
-
-
-
~
n ° d e =x
z
Fig. 4. Quadrilateral elements used in numerical analysis. With the exception of the boundary nodes each node had three translational plus three rotational degrees of freedom. The latter are defined as the three components of angular displacement of the normal to the middle surface at a node. The nodes at the base of the structure were fixed in translation but rotations were permitted. The nodes in the y--z plane were allowed translational m o t i o n in the y- and z
313
more than sufficient accuracy was achieved using a discretization in which the ~ curves were 10 ° apart and the 0 curves were 3.5 ° apart for the smallest dome and 5 ° apart for the two larger domes. These values were also selected for convenience, since they coincided with the positions of the pressure taps. An a t t e m p t was made to predict the pressure distribution over a dome by assuming that the dome could be represented by a distribution of sources and sinks, as in the m e t h o d of Hess and Smith [ 2 7 ] , and representing the boundary layer by a flow with uniform vorticity and slip at the upstream boundary wall. This is the m e t h o d which was used successfully by Newman and Goland [14] to predict the flow over planar two-dimensional inflated buildings. The best matching m e t h o d equated the velocity and slope of the velocity profile at the top of the dome. This is the same as m e t h o d 2 used by Newman and Goland [14]. In the present application the theory, which assumes that the vorticity does not change direction when flowing over the dome, can be expected to work only when the shear flow is weak and the dome is shallow. When compared with experiment the predictions were not obviously best for the lowest dome ( h / c = 0.25). However, for no case were they sufficiently good to be used for the purpose of predicting the membrane tension. In general, the predicted pressure in the central plane (~ = 0 °) was too low over most of the building, particularly near the top, for all domes, and too high near the upwind face of the tallest domes. 4. Results and discussion 4.1 Boundary
layer on the tunnel floor
The p i t o t ¢ o m b survey of the boundary layer on the floor of the e m p t y wind tunnel was made at a nominal tunnel speed U e -- 7.5 m s -1. The boundary layer was found to have a thickness 8 = 1 m. The velocity profile is conveniently represented by a power law:
V/Ve =
(y/~)l/n
The results were plotted logarithmically and showed fair agreement with 1 / n = 0.24 on the tunnel centerline. Measurements 0.2 m on either side of the centerline gave 1 / n = 0.23 and 0.25. The skin-friction velocity U 7 = ( r w / P ) ~ was measured in two ways: (i) a slanting h o t wire was used to measure --h~- at various heights up to 0.30 m, giving substantially constant values: the average value was U r = 0.365 m s -1, which compares very favorably with the value of 0.360 m s -1 obtained by Newman and Goland [14] for essentially the same experimental situation; (ii) a normal h o t wire was used to obtain accurate measurements of the time-average velocity at various heights y, and these values were then fitted to a logarithmic law of the form U / U r = ( l / K ) ln[(y - - d ) / y o ]
314 where K is the von Karman constant, 0.41: d was taken as zero because the measurements were made on the s m oot h surface 0.5 building diameters downstream o f the roughness carpet, while Y0 depends on the surface roughness and increases with it. The results gave Ur = 0.375 m s -1 and Y0 -~ 0.08 mm. Measurements of the typical longitudinal scale L of the turbulence eddies were obtained from the power-spectral density D(f) of the longitudinal turbulence fluctuations at 24 values of y up to y = 0.65 m on the centerline and at the offset station, when 1In = 0.25 (f is the f r e q u e n c y in Hz).
o
06 _
~ ~1/A
//
0.0
L 0.5
I
y - 3 0 cm s U 6.6m/s
v\
fl = 19s'1 s=s,o
i___±
i _ _
50 f
Fig. 5. Typical power-spectral-density frequency distribution fD(f) as a function of log f.
T h e signals were processed on the Hewlett--Packard system t o give [ D(f) as a f u n c t i o n o f logf. An example is shown in Fig. 5. The distribution was m a t c h e d to the standard spectral distribution a d o p t e d by ESDU for the Earth b o u n d a r y layer [26, 14] . The c o m p u t e d values of the eddy scalef a cto r S ranged f r om 440 to 550, with an average value of ~ 500.
4.2 Model experiments The measured pressure distributions, referred to ambient pressure (i.e., t h e pressure at the m odel position when the tunnel is e m p t y ) , are presented for th e three solid models in Fig. 6(a)--(c). These results are presented nondimensionally as Cp~ = (p --p~)/(1/2)p U~, where Ur was taken as 0.365 m s -1. Th ey are shown as a f u n c t i o n o f colatitude 0 for various planes of longitude ¢ (Fig. 2). Values for ¢ < 90 ° (i.e., on the upwind faces) are p l o t t e d on the left-hand side of t he figures, and for ¢ > 90 ° on the righthand side. Contours of c ons t ant pressure are shown in a side view of the t h ree models in Fig. 7(a)--(c). Regions of positive, negative and const ant negative Cp~ on the leeward side are also indicated. It is seen t h a t as the
315
height of the dome increases the highest value of CpT at the front of the model increases and moves upwards. The minimum pressure on the top of the dome decreases and the constant pressure in the separated regions on the leeward side decreases. It may be noted that the isobars do not lie on circles at right-angles to the flow, which was the simplifying assumption made by Bird when calculating the tension in an equivalent inflated membrane. However, this assumption does appear to be best for the highest dome, which has a geometry closest to the radome which Bird was investigating [ 4 ]. A qualitative view of the flow is revealed in the smoke-tunnel pictures shown in Fig. 8, which show the flow over a solid dome of height h / c --0.37. For these tests the Reynolds number was low (6000) and the upstream boundary layer was thin. Nevertheless, the broad features will be similar to the main experiments. The pictures are for a plane of streamlines at heights y / h = 0.25, 0.50 and 0.75. The flow appears to be greatly affected by a horseshoe vortex which wraps around the dome. This vortex, by convecting freestream flow towards the surface, helps to reduce the size of the separated region downstream. The m o v e m e n t of the windward stagnation point upwards accounts for the relatively high upstream pressures when compared with the theory in Section 3. The downstream separation is the other dominant feature which is not accurately represented in the theory.
(a)
h/c --0.25
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180 170
1 0
• -- 1 ~ 0
-40
I-v.. t'-
140 130 120 O-- 1 1 0 o- 100
~'-SO
• -60 0-70
0-80
-90
Fig. 6. Pressure distributions o n solid models. (a)
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317 T h e f i n i t e - e l e m e n t p r o g r a m SAP IV was used t o d e t e r m i n e t h e distribut i o n o f t e n s i o n in inextensible, i m p e r v i o u s inflated m e m b r a n e s o f t h e same spherical g e o m e t r y as t h e t h r e e solid models. This r e q u i r e d t h a t t h e c h o s e n Y o u n g ' s m o d u l u s s h o u l d be high: a value o f 1040 N m -2 was chosen. Poisson's ratio was t a k e n as zero. T h e actual thickness tic o f t h e m o d e l m e m -
ind
lllllliiiNiiilliiiJi[i-iiiii; .-. (a)
h/c
(b)
0.25
h/c
-= 0 . 3 7
in__d
,|ll
(c)
II IIl'JI
h/C = 0 - 5 0
Fig. 7. Contours of constant pressure on solid models: (a) h/c (c) h/c = 0.50.
= 0.25; (b)
h/c
ffi
0.37;
318
Y -- 0.25 -E
Y - - 0.50
--Y -- 0.75 C Fig. 8. Smoke-tunnel pictures of flow over a m o d e l d o m e , to left).
h/c
=
0.37 (flow from right
319 branes was 0.22 X 10 -3. Trials with t/c = 0.22 × 10 -I not surprisingly gave results which violated the m e m b r a n e equation T e + T~ = h p R
For a very thin membrane ( t / c = 0.22 X 10 -1°) the displacements became large and the geometry changed significantly. The value tic = 0.22 X 10 -3 seemed to be satisfactory and was used in all the subsequent calculations. Despite the large value of Young's modulus, the change of tension across the membrane was less than 1%. The effect of a change in Poisson's ratio from 0 to 0.3 produced similar insignificant changes of tension. These results confirmed that in an inextensible membrane shell the stresses are independent of material constants. As a check on the program the overall balance of base tensions and pressures for each complete dome was confirmed to an accuracy of 2%. The membrane equation was well satisfied for all elements with the exception of those nearest to the base, presumably because the imposed boundary conditions were not the appropriate membrane conditions there [28]. Since the change of membrane geometry is negligible, it is permissible to superimpose two solutions: (i) a numerical solution using the measured external pressure distribution and an internal pressure which is atmospheric; and (ii) the simple solution T = (1/2)(p i - - p ~ ) R where Pi is the inside pressure and p~ is atmospheric pressure applied uniformly to the outside o f the dome. Solution (i) was therefore first obtained using the measured external distribution of gauge pressures and the principal tensions were determined at the center of each element. The least of these m i n i m u m tensions Train was identified. Since it was negative, the minimum internal gauge pressure t o just prevent buckling and to stabilize the envelope is
(Pi --P~ )rain
= 2(--Tmin)/R
The validity of this superposition was also confirmed numerically by calculating a case with a positive internal gauge pressure. In addition, it may be remarked t h a t results were also obtained by smoothing the experimental data before putting the values into the finiteelement program. However, no difference could be discerned between these results and those obtained by using the raw experimental data. The raw data for pressure were used subsequently. The results for solution (i) are shown in Fig. 9(a)--(c), where the minim u m principal tension has been appropriately normalized using (1/2)p U~c. The solid curves are plotted against ~b for various values of 0 and the single dashed curve against 0 for ¢ = 0 °. For all three domes buckling occurs for ¢ = 0°; in other words, it occurs on the windward side of the model
320
in the plane o f s y m m e t r y . The orientation o f this m i m i m u m tension is circumferential in all cases. Thus buckling is predicted to occur first with vertical creases. The results for the three domes are shown in Table 3, where the stabilizing pressures are given normalized in terms of Uh, the wind velocity corresponding to the t op of the dome. 0 G is the value of 0 at the ground,
h/c F~(~ // -:
..........
l~]c .37~l
39 48 50
///
i
~-
.25
1o ~
10CTz 0
~ / -
7 2.5
C~
-10 o
-20
/
-30
-3O "40
(o)
ckling
,/
[% J
I
10
J
20
30
I
40
/
50
e
h/c
I _
I
20
J
I
40 e
I
I
60
1
.50 !
CTI
(C) l
[ 20
i
k 40 e
~
i 60
i~_ 80
Fig. 9. M i n i m u m p r i n c i p a l t e n s i o n w i t h a t m o s p h e r i c i n t e r n a l pressure: (a) h/c = 0 . 2 5 ; (b) h/c = 0 . 3 7 ; (c) h/c = 0.50.
321
TABLE 3 Predicted m i n i m u m internal pressure Pi t~ prevent buckling and associated m a x i m u m tension Tmax near top o f model
Model (pi--poo)min/ (1/2)p U~
Uh/Ur
0.25 0.37 0.50
14.38 16.49 16.66
h/c
0.44 0.49 0.65
(Pi--P~)min ] ebuckl~g (1/2)p U¢
Obuckling/OG Tvaax/ Tmax/
91 133 180
0.90 0.83 0.80
48 ° 62.50 72.5 °
(1/2)p
(1/2)p
0.34 0.33 0.42
71 89 116
v~c
v¢c
w h i c h is t h e base o f t h e d o m e . N o t e t h a t at t h e p o i n t o f b u c k l i n g 0/0 G decreases w i t h increasing h/c. A useful, a l t h o u g h e m p i r i c a l result, is t h a t f o r i n c i p i e n t buckling: [(Pi --p®)/(1/2)p U~] (c/h) = 3 6 0 (-+ 1%) T h e m a x i m u m t e n s i o n s w h e n t h e i n t e r n a l p r e s s u r e is j u s t s u f f i c i e n t to p r e v e n t b u c k l i n g are also o f practical interest. T h e values are p l o t t e d against 0 f o r various values o f ¢ in Fig. 1 0 ( a ) - - ( c ) . T h e m a x i m u m values are o r i e n t e d in t h e l o n g i t u d i n a l d i r e c t i o n a n d are n e a r ¢ = 90 ° a n d 0 = 1 0 - - 2 0 °, i.e., crossw i n d a n d n e a r t h e t o p o f t h e d o m e . With t h e stabilizing p r e s s u r e a p p l i e d inside t h e d o m e t h e values o f Tma x are also given in T a b l e 3. T h e i n f l a t e d m e m b r a n e m o d e l s w e r e e x a m i n e d closely in t h e w i n d - t u n n e l f l o w as t h e internal p r e s s u r e was altered, w i t h r e s p e c t t o t h e c a l c u l a t e d pressures in T a b l e 3. D u e t o difficulties o f m a n u f a c t u r e , t h e m o d e l s w e r e n o t e x a c t l y spherical a n d a x i s y m m e t r i c . E a c h m o d e l was t h e r e f o r e r o t a t e d t o bring t h e best possible s h a p e i n t o t h e u p w i n d region, w h e r e b u c k l i n g was p r e d i c t e d to occur. T h e d i r e c t i o n o f b u c k l i n g was d i f f i c u l t t o determ i n e . T h e b u c k l e did n o t m a n i f e s t itself as a m e r i d i o n a l crease b u t instead was a small u n s t e a d y circular d e n t o n t h e surface o f t h e m o d e l w h i c h grew in size as t h e internal p r e s s u r e was l o w e r e d . All such d e n t s w e r e first observed n e a r ¢ = 0 °. T h e a p p r o x i m a t e p o s i t i o n s a n d t h e c o r r e s p o n d i n g pressures are s h o w n in T a b l e 4. T h e stabilizing pressures and t h e p o s i t i o n s o f TABLE 4 Buckling positions and internal pressures Pi observed on inflated models Model
Buckling position 0 (~ = 0 °)
h/c
Theory
Experiment
0.25 0.37 0.50
48 ° 62.5 ° 72.5 °
35--40 ° 60 ° 70--75 °
(Pi--P~ )vain(exp')/(Pi --Poo )vain (theor.) 1.0--1.3 1.1 1.5
322 h/c
/
0.25 1
10(
I
~ , L
I t
9ol-
8(
/ / / / ]~ -
.37
F---ID
85,95 45
CT¢
10~h/c:
/ ,//
= 95 85
175
45 175
/ : Ji 70
6(3
4G
(a)l 10
0
I 20
I
I 30
40
50
0
9
h/c
.50
,
120 ?
10
20 30 40 50 60
70
e
0
f. . . . . . lit I / / /~
85 95 472 0 = 135
CO0 C~ (c) I
0
I
L
_
[
i
10 20 30 40 50 50
J
I
70 80 90
0
Fig. 10. M a x i m u m
p r i n c i p a l t e n s i o n w h e n b u c k l i n g is just a v o i d e d : (a) h/c = 0 . 2 5 ; (b)
/tic = 0 . 3 7 ; (c) h/c = 0.50.
buckling are seen to be predicted reasonably well for the two lower models. The highest model was not a true spherical dome and due to expansion of the seams was noticeably flat-topped. This is considered to be the main cause of the relatively high experimental stabilizing pressure for this model. It is therefore concluded that the m e t h o d of prediction which has been
323
developed in this paper gives a fairly good estimate of the internal pressure required to prevent buckling. The behavior of the models was also observed at other internal pressures. The results with the internal pressure expressed as a proportion of the theoretical stabilizing pressure (Pi - - P - ) r a i n may be summarized as follows: Pi -- P -
Behavior
(Pi --Po. )min 6 4
membrane generally t a u t and steady on all models very slight fluttering or panting o f the membrane on the windward side of the centerline for the two highest models (h/c = 0.5 and 0.37) very slight fluttering observed at a similar position on the lowest model (h/c = 0.25) fluttering extended to ¢ = +30 ° on the windward side and amplitude increased
3 2--1.5
The m o d e of collapse as the pressure was reduced below the theoretical stabilizing pressure is indicated in Fig. 11. Photographs showing the m o d e onset of buckling, observed in close proximity to the point of maximum pressure
1.
2. shape of dome at an internal pressure of 0.8 x t i m e s the stabilizing
I
pressure
x\
\
X
.
,
\.~
""-'-:':4
(h/c--0.37)
3. shape at an internal pressure of 0.4 x s t a b i l i z i n g
4.
shape
pressure
at zero gauge
internal
pressure Fig. 11. M o d e of collapse as pressure is reduced below that which just prevents buckling.
324
(p~-p~) =1,0 ( P i " Poo)min. theory
( Pi - Pco) : 0.7 ( Pi- PoO) rain.theory
(p~-p~) (Pi - Poo) rain. theory = 0 . 6
Fig. 12. Photographs showing mode of collapse at subcritical pressures,
h/c
=
0.5.
325 of buckling for such subcritical pressures are shown in Fig. 12(a)--(c) for the tallest inflatable model (h/c = 0.5). 4.3 R e l a t i o n o f m o d e l e x p e r i m e n t s to full-scale buildings
The applicability of the present model tests (c = 0.45 m) to full-scale buildings is a m o o t question. Since the longitudinal scale of the eddies was ~ 1/500 that of typical wind eddies, the size of the full-scale building is strictly 320 m, although variation from this value by factors of perhaps 2 or 3 m a y be permissible if only static effects are of interest. However, for rounded buildings such as domes even the average positions of flow separation m a y be sensitive to turbulence scale as well as to turbulence intensity, so that large departures from a scaling factor of 500 are considered to be undesirable. Aerodynamic scale effects, i.e., sensitivity to changes of Reynolds number, are dealt with partly by the fact the models were tested within the wall-law region, for which the upstream flow is determined b y Tw, p and the roughness height for a fully rough boundary layer. Making model size proportional to roughness height then establishes similitude. However, the flow over the d o m e itself is likely to depend on viscosity, since the dome is smooth and has no sharp corners to fix the positions of separation. Buckling is associated with stagnation on the upwind face of the dome, which in turn is determined by separation of the upwind fully rough boundary layer. Thus the predictions for buckling should be applicable to full-scale buildings, and the agreement with the National Code seems to support this contention. The associated maximum tensions near the top of the building, however, may be t o o low in the model tests. The nondimensional mass of the membrane, a/pc, where a is the mass per unit area, was ~ 25--50 times bigger for the models than it would be at full scale. Thus the oscillations of the model are unrealistically large. Experiments on flags [29] and bluff membranes [30] exhibit the same trends. Moreover, the frequencies for a given wavenumber also depend on the value of a / p c [ 3 1 ] . Thus the present experiments on inflated models do n o t address dynamic effects, and no a t t e m p t has been made to simulate this aspect of full-scale behavior.
5. Conclusions Winds over a sparsely w o o d e d area for which the longitudinal scale factor was 500 have been simulated in a boundary-layer wind tunnel. Pressure measurements on model domes have indicated that buckling of an equivalent inflated dome would occur on the plane of s y m m e t r y and upwind when the internal inflation gauge pressures were respectively 0.65, 0.49 and 0.44 of the reference dynamic pressure q at the t o p of the dome for h/c = 0.5, 0.37 and 0.25. Niemann [8] measured a similar value of 0.7 for h/c = 0.5. The corresponding values specified by the National Standard of Canada
326
[1] are 0.7 for h / c = 0.5 and 0.6 for h / c = 0.35. These are slightly higher, and they may be desirable in practice to avoid subbuckling oscillation of the membrane, which was observed on the present inflated models. A convenient empirical formula for the minimum gauge pressure (Pi - - P ~ ) to prevent buckling is
(Pi --P~)/rW
-- 1 8 0
h/c
The corresponding m a x i m u m principal tensions are longitudinal and occur near the top of the dome in planes nearly at right-angles to the flow. They vary slightly from 0.42 q c for the highest dome to 0.33 q c for the others. As the wind speed increases and with it r w the internal pressure must be increased to prevent buckling, and the m a x i m u m tension at the top of the building increases in proportion. The membrane in this area may therefore require reinforcement. The base of the building, where the imposed boundary conditions are somewhat uncertain, is also a sensitive area. The present model tests are considered to predict adequately the upwind initial buckling of a full-scale inflated building when the base diameter of the building is ~ 100 m. The prediction of the associated m a x i m u m tension at the top of the building is less certain, because of aerodynamic scale effects. Dynamic effects associated with oscillation of the membrane at subbuckling pressures have not been investigated. References 1 National Standard of Canada, Air Supported Structures, CAN3-5367-M81, 1981. 2 W. McQuade, A new air age in construction, Fortune Mag., (October 1977) 228-231. 3 Cedric Price Architects, F. Newby, R.H. Suan, J.S. Felix and Partners, Air Structures: A Survey, Department of the Environment, London, HMSO, 1971. 4 W.W. Bird, Design manual for spherical radomes (revised), Cornell Aeronaut. Lab., Rep. UB-909-D2 (1965). 5 American Society of Civil Engineers, Air Supported Structures, State of the Art Report, New York, Am. Soc. Cir. Eng., 1979. 6 J. Springfield and D. Sinoski, The air supported steel membrane roof at Dalhousie University, Halifax, Nova Scotia, Can. Struct. Eng. Conf., Canadian Steel Construction Council, Willowdale, Ont., 1980, pp. 1--29. 7 G. Beger and E. Macher, Results of wind tunnel tests on some pneumatic structures, Proc. 1st Int. Colloq. o n Pneumatic Structures, Stuttgart, 1967, pp. 142--146. 8 H.-J. Niemann, Wind tunnel experiments on aeroelastic models of air supported structures: results and conclusions, Proc. 2nd Int. Colloq. on Pneumatic Structures, Delft, The Netherlands, 1972, p. HJN-I--12. 9 J. Bietry, Etude des effets du vent sur les structures gonflables: conditions de similitude, CSTB, Nantes, Rapp. EN. ADYM-75-8-R (1975). 10 G. Grillaud and J. Gandemer, Etude des effets du vent sur les structures gonflables: ~tude experimentale en soufflerie sur maquette aeroelastique, CSTB, Nantes, Rapp. EN. ADYM-76.7.L (1976). 11 H. Maille, Effet du vent sur les structures gonflables: equipement experimental, CSTB, Nantes, Rapp. EN. ADYM-77.9.L (1977).
327 12 13 14 15 16 17
18
19 20
21 22 23 24 25 26 27 28 29 30 31
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