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Transcript of 07 Membranes
8/10/2019 07 Membranes
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Anticlastic membranes Copyright Prof Schierle 2012 2
1 Two stressed strings stabilize a point in space
2 Two sets of strings form a stable membrane
3 Without prestress, convex string gets slack,
causing instability
4 Flat strings deform greatly under load,
causing instabilityA
nti
cl
a sti c
St a
bilit
y
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PrestressThe effect of prestress on cable structures is
shown on a wire with and without prestress,
subject to a load P applied at its center.
1 Wire without prestress
resists load P in upper link only
Wire force F = P
2 Wire with prestress PS
resists load P in upper and lower link.
Upper link increases: F = PS + P/2
Lower link decreases: F = PS –P/2
Prestress reduces deflection to half
3 Stress/strain diagram
A Stress/strain without prestress
B Stress/strain with prestress
C Prestress reduced to zero (PS = 0)
D Prestressed wire after PS = 0
Note:
Prestress should be half the stress under load + a reserve for thermal variation
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Anticlastic membranes Copyright Prof Schierle 2012 4
Minimal surface vs. Hyperbolic Paraboloid
1 Minimal surfaceof squareplan
2 Minimal surfaceof rhomboidplan
(membranecenter is belowmid-height)
3 Hyperbolic Paraboloidof squareplan
4 Hyperbolic Paraboloid of rhomboid plan
(membrane center is at mid-height)
Minimal surface equations (Schierle, 1977 *)Y= f1(X/S1)(f1+f2)/f1+ X tan
Y= f2 (Z/S2)(f1+f2)/f2
* First published 1977 in
Journal of Optimization Theory and Applicationhttp://www.springerlink.com/content/j7310q6651450w86/ M
i n i m
a l
S u
r f a c e
The minimal surface is defined as follows:
• Minimum surface area between any boundary
• Equal and opposite curvature at any point• Uniform stress throughout the surface
• f1/f2 = A/B (Schierle, 1977 *)
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E d
g e c o n d
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E d g e
b e a m
E d g e a r c h
E d g e c a b l e
S u
r f a c e c o n d i t i o n
s
P o i n t s h a p e
A r c h s h a p e
W a v e s h a p e
S a d d l e s h a p e
N e
t c abl e
o rfib e
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o n
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N e t 3
: s m a l l d e f l e c t i o
n S q u a r e / r h o
m b o i d m e s h
V e r t i c a l p l a n e o r i e n t a t i o n
N e t 4
: l a r g e d e f l e c t i o
n
S t r a i g h t g e
n e r a t i n g l i n e s
P r i n c i p l e c
u r v a t u r e
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W
a v e
s h a p e
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A r c h s h
a p e
P o i n
t s h
a p e
C a b l e s t a y e d M
u l t i - m a s t
D i s
h
R i n g
B a d s u p p o r t
P r o p p e d
M
u l t i - m a s t
E y
e
L o o p
R a d
i a l c a b l e s
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Olympic stadium Munich
Architect: Guenter Behnisch
Engineer: Leonhardt und Andrae
The architect’s metaphor was a spiderweb floating over the landscape.
The roof consists of seven saddle-shape
cable nets.
Anticlastic curvature provides stability:• Concave cables support gravity
• Convex cables resist wind uplift
Cable nets are supported by masts at rear
and a giant ring cable in front which issuspended from the masts by guy cables
The cable net of 75 cm (2.5’) meshes was
manufactured as square meshes that form
rhomboids to assume anticlastic curvature.
S a
d d l e s h
a p e
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Assume
Allowable cable stress (210/3) Fa= 70 ksi
DL = 5 psf 5 psf
LL = 20 psf Wind uplift 21 psf = 25 psf Net uplift 16 psf
Uniform load (cables spaced 75 cm = 2.5’)
Gravity
w = 25 psf x 2.5’ w = 62.5 plf Wind uplift
p = 16 psf x 2.5’ p = 40 plf
GRAVITY LOAD
Global momentM = w L2/8= 62.5 x 1972/8 M = 303,195 #’
Horizontal reaction
H = M/f = 303,197 #” / 39’ H = 7,774 #
Vertical reaction
R = w L/2 = 62.5 x 197’/2 R = 6,156 #
Gravity tension (10% residual prestress)
T = 1.1(H2+R2)1/2
T = 1.1(7,7742+6,1562)1/2 T = 10,908 #
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Cable stressf= T/Am=10,908/(0.28x1000) f = 39 ksi
39<70, OK
Note: Twin cables provide concentric joints toassume square meshes to form rhomboids toassume anticlastic curvatureRoof of 10’ acrylic panels with rubber joints
Gravity T (from previous page) T = 10,908 #
Wind tension(normal pressure + 10% residual prestress)
T=1.1pr = 1.1x40x226’ Wind T = 9,944 #10,908 > 9,944 Gravity governs
Metallic area (twin ½” net cables, 70% metallic) Am=2x0.7r 2=2x0.7(0.5/2)2 Am=0.28 in2
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Fabric structure alternate Assume:
Allowable fabric stress
(tensile strength /4)Fa = 800/4 Fa = 200 lb/in
(lb/in - convention for fabric)
DL = 1 psf 1 psf
LL = 12 psf wind uplift 21 psf
= 13 psf net uplift 20 psf
Cable details from
Alan Holgate (in Arch library) :
The Art of Structural Engineering, page 72
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Fabric stress
f= T / 12 = 4,972 / 12 f = 414 pli
414 > 200 NOT OK
Note: Structure would have to be designed withsmaller panels or more curvature to reduce stress
Wind load (normal to surface, T= p r)
T= 1.1p r= 1.1x20x226’ Wind T = 4,972 #
4,972 > 2,063 Wind governs
Uniform load (analyze 1’ wide strip)
Gravity w = 13 psf x 1’ w= 13 plf
Wind uplift p = 20 psf x 1’ p = 20 plf
GRAVITY LOADGlobal moment
M= w L2/8= 13 x 1972/8 M = 63,065 #’
Horizontal reaction
H= M/f = 63,065 #” / 39’ H = 1,617 #
Vertical reaction
R= w L/2 = 13 x 197’/2 R = 1,281 #
Gravity tension (10% residual prestress)T= 1.1(H2+R2)1/2= 1.1(16172+12812)1/2 T = 2,063 #
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Arena AmazoniaManaus, Brazil Architect: GMPEngineer: Schlaich Bergermann
FIFA World CUP 2014 facilityPTFE membrane supported by
cantilevering steel framedesigned also for rain drainage
W H ll H b 1963
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W
a v e
s h a
p e
Wave Hall Hamburg 1963
Architect: Frei Otto
The exhibit hall for the Garden Show Hamburg 1963,
features a wave shape with saddle shapes (left) access.
Assume:Span C-D L = 60’
Ridge cable sag f = 7.5
Valley cable span L = 60’
Valley cable sag f = 15’Bay size 40’ x 67’
Mast height difference h = 10’
Gravity load 20 psf (including DL)
Wind uplift 30 psf (including DL deduction)
Gravity load:
w = (20 psf)(40’)/1000 w = 0.8 klf
H = wL2/(8f) = 0.8x602/(8x7.5) H = 48 k
Left mast Rc= H (2f+h/2)/L/2
Rc = (48)(15+5)/30 Rc = 32 kRidge cable tension T = (482+322)1/2 T = 58 k
Wind load:
p = (30 psf) (40’)/1000 p = 1.2 klf
H = 1.2x60
2
/(8x15) H = 36 kR = p L/2 = 1.2x60/2 R = 36 k
Valley cable tension T = (362+362)1/2 T = 51 k
San Diego Convention Center
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San Diego Convention Center
Architect: Arthur Erickson
Engineer / fabric roof design: Horst Berger
Allowable cable stress Fa = 70 ksi
Bay width e = 60’Span L = 300’Sag f = 30’
Wind uplift p = 10 psf
Design valley cableCable load w=10x60’/1000 w = 0.6 klf
Horizontal reaction
H = 0.6x3002/(8x30) H = 225 k
Vertical reaction
R = 0.6x300/2 R = 90 kCable tension
T = (2252+902)1/2 T = 242 k
Cable size (70% metallic)
A = 242/70/0.7 A = 4.94 in2
=2(A/)1/2 =2(4.94/)1/2 = 2.5“
f=30’
L=300’
e=60’
S C t B li
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Sony Center Berlin
Architect: Helmut Jahn
Engineer: Ove Arup
Canopy features:
• Outer truss compression ring
• Top tension rings
• Flying buttress supports top ring
• Radial stays support flying buttress
• Radial cables support skylightand shading fabric
Analysis steps:
• Flying buttress gravity load
• Vertical reaction per stay cable• Tension in stay cables (vectors)
• Horizontal component H per stay
• Compression ring force: C = R H/e
(e = stay spacing at ring)
• Tension in tension ring: T = R H/e
• Global moment in radial cables
• H and R in radial cables: H = M/f
R = H(h/2+2f)/(L/2)
• Tension in radial cablesT=(H2+R2)1/2
Reliant Stadium Houston
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Reliant Stadium Houston
Architect: HOK SportsEngineer: Walter More and AssociatesFabric roof: Birdair
The Reliant Stadium features:• Removable roof, 2 panels: 240’x385’• Teflon-coated fiberglass 25% translucent• Fabric stabilized by 2” valley cables• Convex prismatic trusses span 240’
Wimbledon Center Court retractable roof
Architect: Sir Nicholas GrimshawEngineer: Moog
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R e
l i a n t
S t a d
i u m
H o u s
t o n
R e
t r a c t a b l e
r o o f
f e a t u
r e s
Ol i St di L d 2012
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Olympic Stadium London 2012 Architect: Populous, Engineer: Bureau Happold
Wave shape PVC coated Polyester fabric
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A r c h s h a p
e
S d d l
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Study model
EFL portable classroom (1968)
Architect: G G SchierleEngineer: Nick Forell
Size: 30’x40’
First twin fabric with thermal insulation
Theater pavilion Armonk (1968)
Architect: G G SchierleEngineer: Nick Forell
Size 60’x80’ - capacity 600
Longest span fabric roof 1968
fabric tensile strength 720 pli
Skating Rink Munich
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Skating Rink Munich Architect: AckermannEngineer: Schlaich / Bergermann
Prismatic arch truss supportstranslucent PVC fabric on woodslats and cable net
Arch truss (L=328’)
detail
Bangkok Airport Terminal
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Bangkok Airport Terminal
Architect: Helmut Jahn
Engineer: Ove Arup
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Sea-World tent, Vallejo Architect: SchierleEngineer: ASI
The membrane, open on all sides, supportedby 7 steel tripods and a central steel mast.
Edge cables transfer membrane stressto the steel anchors.
A loop cable and radial guy cables transfersmembrane stress to mast top.
The loop diameter need to be large enough to
keep membrane stress at allowable limits.
Design started with a stretch fabric modelfollowed by computer analysis
Assume Allowable fabric stress Fa = 150pli(Fa = tensile strength 600/ 4)Floor area A ~ 6000 sq. ft.DL = 1 psf
LL = 12 psf w = 13psf
P
o i n t
s h a
p e
From last slide: Fa = 150pli
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From last slide: Fa = 150pliFloor area A ~ 6,000ft2
Roof load w = 13psf
Mast reaction R (use 10% residual prestress)
R = A w = 1.1 x 6000 x 13 / 1000 R = 86 k
Fabric tension (from vector triangle) T = 104 k
Fabric strength per footFa’ = Fa x 12”/ft = 150 pli x 12” Fa’ = 1800 plf
Required ring length LL = T / Fa’ = 104,000 # / 1800 L = 58’
Ring radius
= L / = 58’ / 3.14 = 18.5’
Use double fabric at 2 = 20’Use tension ring 1 = 2 / 2 1 = 10’
Edge cable tension Te (radius R = 60’)Te = T R = 0.15 klf x 60’ Te = 9 k
Wire rope (~ 60% metallic)
Ag = Te /(Fa x 0.6) = 9/(70x0.6) Ag = 0.21 in2
Rope size = 2(Ag/)1/2 = 2(0.21/)1/2 = 0.52 Use 5/8”
Fabric stress at base (400’ circumference)T = 60k/ 400’ T = 0.15 klf
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Erection
Colored lighting
C o r n e r d e t a i l
F a b r i c p a t t e r n s e e m s
T u r n b u c k l e s t o p r e
s t r e s s
D o u b
l e f a b r i c a t s t r e s s f o c u s
W e b b i n g t o h o l d f a
b r i c
Oklahoma Mall
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Oklahoma MallFabric roof Architect: Adams + Associates
Contractor: Structureflex
PVC fabric Ferrari 1002 T2supported by 60’ mastsprotects from sun and rain
open for natural ventilationFabric Architecture images
Incheon Stadium South Korea
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Incheon Stadium, South Korea
Architect: AdomeEngineer: Schlaich / Bergermann
Built as one of the stadiums for the 2002 FIFAworld soccer cup, the Incheon Stadium features:
Capacity 57,179
Point shape units supported by 24 masts
Inner tension ring
Exterior compression truss and edge cables
Teflon-coated fiber glass fabric, 15% translucent
Fabric prestress considers thermal expansion
Test model
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Test model
Assume
Fabric stress is in lbs/in (thickness ignored)
Geometric scale Sg = 1:50
Strain scale Ss = 1:1
Model E-modulus Em = 2 pli
Original E-modulus Eo = 6000 pli
Since stress is measured in pli (f = P/L rather than f = P/A)
Am/Ao = Sg (geometric scale), hence
Force scale
Sf = Am Em /(Ao Eo)
Sf = SgEm/Eo= (1/50)(2/6000) = 0.0000067 Sf = 1:150,000
DL = 1 psf
LL = 12 psf
w = 13 psf
Original floor area A ~ 6000 ft2
Original load
Po = A w = 6000 x 13 Po = 78,000 #
Model load
Pm = Po Sf = 78,000/150,0000 Pm = 0.52 #
Use 10 caps Pc = 0.52 /10 Pc = 0.052 #
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Prof. G G Schierle, PhD, FAIA
Design of Fabric Structures
Session T33, Thursday, 04/30, 2 – 3:30 PM
Prof. G G Schierle, PhD, FAIA
Design of Fabric Structures
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Design of Fabric Structures
Saddle shape Wave shape Arch shape Pont shape
Anticlastic Stability
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Anticlastic Stability
• Two stressed strings stabilize a point in space
• Two sets of strings form a stable surface
• Without prestress, convex fiber gets slack,causing instability
• Flat fiber deform greatly under load,causing instability
• Triangular panels are flat & unstable ( AVOID)//
P t
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Prestress
Prestress (PS) effect on a string
F = force, P = load, = deflection1 Without prestress top link resists all
Assume: = 1
2 With prestress = 1/2Top link increase: F=PS+P/2
Lower link decrease: F=PS–P/2
3 Stress / strain diagram f/
A without prestress
B with prestress
C Prestress reduced to PS = 0
D Prestressed string after PS = 0
Cable nets need about 50% prestress
Fabric structures need about 30% prestresshttp://www-classes.usc.edu/architecture/structures/papers/GGS-Yin.pdf
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Minimal SurfaceCriteria:
• Minimum surface area• Equal stress throughout
• Equal +- curvature at any point
Governing Equations (Schierle 1977*)
*First published 1977 inJournal of Optimization Theory
and Applications
F1/F2 = A/B
Y = F1(X/S1)K/F1+ X tan
Y = F2(Z/S2)K/F2
K= F1+F2
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3 : s m a l l d e f l e c t i o n
P r i n c i p a l c u r v a t u r e
4 : l a
r g e d e f l e c t i o n
S t r a
i g h t g e n e r a t i n g l i n e
Fiber orientationGood Flawed
F b i P ti
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Fabric Properties
* Self-cleaning
St t
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Maximum spans Assuming:Live load LL = 20 psf Safety factor Sf = 4
Span/sag ratio L/f = 10Fabric breaking strength Max. span600 pli (lb/in) ~ 60 ft800 pli (lb/in) ~ 80 ft
Design stress (tensile strength / 4)Tensile strength Design stress
400 pli 100 pli
600 pli 150 pli
800 pli 200 pli
Costs
Type Cost / sq. ft
Prefab PVC $15 to $20
Custom
PVC $30 to $60
PTFE Teflon-coated fiberglass $60 to $180
Note:costs exclude foundations
Structure
Design / Analysis
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RL
RR
H
H
TR
TL
W= w L
w
h
RL
RR
L/2L
ff
H
Symmetric suspensionHorizontal reaction H = w L2/(8f)
Vertical reaction R = w L/2Max fabric tension T = 1.35 w L
Asymmetric suspension
Vector methodTotal load W = w LFabric tensions TR TL
Horizontal reaction H
Vertical reactions RL RR
w
Design / AnalysisRadial loadEdge cable tension T = R p
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Lateral Load
Seismic (not critical)V = Cs W (base shear)V = seismic base shear Cs = Seismic coefficient
W = mass (dead load)Example (V / ft2, Cs = 0.2, w = 1 psf)V = 0.2 x1 V = 0.2 psf
LDG: Lateral Design GraphSample: 100’ x 50’ x 20’
Wind (critical)Velocity
• 90 mph (most USA)• 150 mph (Golf coast)
Gust factors (G= 0.85 for rigid structures)G ~ 1.5 for fabric structuresExample (V per ft2, wind pressure p = 20 psf)
V = p G = 20 x 1.5 V = 30 psf
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Acoustics
• Thin fabrics provide little sound insulation
• Micro-perforated foils absorb sound(suspended under structural fabric)
• Fabric form defines acoustics:
• Anticlastic forms disperse sound• Synclastic forms focus sound
Lighting
Daylight sunny days ~75000 lux
Daylight overcast ~25000 lux10% translucent fabric ~2500 - 7500 lux
Typical office lighting ~1000 lux
Thermal
While fabric has low R-values
Thermal reflection is very good
e
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S u
r f a c e c o n d i t i o n s
P o i n
t s h a p e A r c h s h a p e
W a v e s h
a p e S a d d l e s h a p e
E d g e c o n
d i t i o n s
E d g e
b e a m
E d g e a
r c h
E d g e c a b l e
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Edge Conditions
Edge Cable (tension)
Edge Arch (compression)
Edge Beam (bending)
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Raleigh Arena North Carolina (1953)n
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g ( )
Architect: Novicki and DeitrickEngineer: Severud Elstad Krueger
Edge arch / cable roof
EFL portable classroom (1968)
Architect: G G SchierleEngineer: Nick Forell
Edge arch / anticlastic Fabric
Sony Center Berlin (2000) Architect: Helmut JahnEngineer: Ove Arup
Edge ring / radial cables and fabric
E d
g e A r c
h / R i n
g – c o
m p r e s
s i o n
Horticultural Center
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o t cu tu a Ce teGallaway Gardens, Georgia
By ODC
Dining Pavil ion
Saddlebrook Florida
By Helios Industries
Note:
Edge beams facilitate groupings
E d
g e B e
a m – b
e n d i n g
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Saddle shapes Wave shapes
S u
r f a c e c o n d i t i o n s
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Arch shapes
S t a y
e d
M a s t s
D i s h
R i n g
P u n c t u
r e
P r o p p e d M a s t s
E y e
L o o p
R a d i a l
Point shapes
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Saddle shapes
Expo ‘64 Lausanne
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p Architect: Saugey / Schierle
Engineer: Froidevaux et Weber
26 restaurant pavilions:
Featured Swiss regional cuisines
Symbolizing sailing and mountains
Design example
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L=120’
f=12’
A A
B
BSection B-B
Design example
Assume:
Wind pressure p = 30 psf
Allowable fabric stress Fa= 200 pli
Available canvass stress Fa= 50 pli
Wind load (normal to fabric)T = p R = (30)(100) T = 3000 #
Fabric stress per inch
f = 3000/12 f = 250 pli
Fabric NOT OK 250 > 200 > 50
Cable net was required
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Wave shapes
Computer model
San Diego
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g
Convention Center
Architect: Arthur EricksonEngineer: Horst Berger
Fabric design: Horst Berger
Concrete pylons at 60’ supportridge, valley, and guy cables that
span 300’ between pylons
Translucent Teflon coated fiberglass fabric provides daylight
Ridge cables support gravity load
Valley cables support wind upliftGuy cables support
Flying buttresses
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Denver Airport
Architect: FentressPhoto: David Benbennick
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Denver AirportPhoto: David Benbennick
Sony Center Berlin
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y
Architect: Helmut Jahn
Engineer: Ove Arup
• Truss compression ring ø 335’
• Flying buttress mast supports
top tension ring
• Radial guy cables support mast
• Radial roof cables hold fabric
• Translucent fabric
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Arch shapes
Study model
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EFL portable classroom (1968)
Architect: G G SchierleEngineer: Nick Forell
Size: 30’x40’
First twin fabric with thermal insulation
Theater pavilion Armonk (1968)
Architect: G G SchierleEngineer: Nick Forell
Size 60’x80’ - capacity 600
Longest span fabric roof 1968
fabric tensile strength 720 pli
Skating Rink MunichArchitect: Ackermann
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Architect: AckermannEngineer: Schlaich / Bergermann
Prismatic arch truss supportstranslucent PVC fabric on woodslats and cable net
Arch truss (L=328’)
detail
Point Shapes
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p
S t a
y e d
M a s
t s
D i s h
R i n g
P u n c t u
r e
P r o
p p e d
M a s t s
E y e
L o o p
R a d i a
l
Olympic Stadium London 2012
Oklahoma City Mall PCV cover
Sea-World Pavil ion VallejoArchitect: G G Schierle
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Architect: G G SchierleEngineer: ASI, Advanced Structures Inc
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Erection
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Erection
Color lighting
Layout
Erection
German Pavilion
E 67 M t l
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Expo 67 Montreal Architect: Gutbrod & Otto
Engineer: Leonhardt & Andrae
Translucent fabric for natural
lighting suspended from cable
net on 3-D adjustable hangers.
Prefab panels assembled on
site with lacing.
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Balance Forces
Unbalanced
Balanced
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Design Process
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Computer Aided
• Form-finding
• Analysis
• Pattern design
Computer modelComputer model
Load shapedotted lines
CAD patterns by triangulation
Optimization
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Edge & surface curvature(Schierle, 1971)
Usual optimum L/f = 10L = spanf = sag
L
f
Watts Towers
C lt l C t
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Cultural Center (2002) Architect: Ado / Schierle
Engineer: ASI
Removable fabric and cable truss
Stadium Oldenburg GermanyArchitect: Kulla Herr und Partner
Anticlastic fabric panels suspendedfrom cantilever cable trusses
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Architect: Kulla, Herr und PartnerEngineer: Schlaich Bergermann
from cantilever cable trusses
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Grid Shell Mannheim, 1975 Architect: Mutschler / Otto
Form-finding model
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Engineer: Ove ArupGrid shell of 50 cm square, 50 mmtwin slats form rhomboids in space;covered with translucent fabric.http://en.wikipedia.org/wiki/Gridshellhttp://www.smdarq.net/case-study-mannheim-multihalle/
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anticlasticfabric
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C
u r v e d w a l l t o r e s i s t w
i n d
Speaker Speaker
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Prof G G Schierle, PhD, FAIA
USC - School of ArchitectureLos Angeles, CA 90089-0291
T 213-740-4590
F 213-740-8888
http://www.usc.edu/structures
Prof G G Schierle, PhD, FAIA
USC - School of ArchitectureLos Angeles, CA 90089-0291
T 213-740-4590
F 213-740-8888
http://www.usc.edu/structures
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thank youthank you