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

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St a

bilit

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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




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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N e t 4

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http://slidepdf.com/reader/full/07-membranes 6/79 Anticlastic membranes Copyright Prof Schierle 2012 6

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A r c h s h

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P o i n

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C a b l e s t a y e d M

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B a d s u p p o r t

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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



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 #



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



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



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 #



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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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



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



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



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



R e

l i a n t

S t a d

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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



Olympic Stadium London 2012 Architect: Populous, Engineer: Bureau Happold

Wave shape PVC coated Polyester fabric



A r c h s h a p

e

S d d l



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)


Size 60’x80’ - capacity 600

Longest span fabric roof 1968

fabric tensile strength 720 pli

Skating Rink Munich




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




Bangkok Airport Terminal


Engineer: Ove Arup




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

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From last slide: Fa = 150pli




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




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




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




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




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 #




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






Saddle shape Wave shape Arch shape Pont shape

Anticlastic Stability




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




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




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




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




Fabric Properties

* Self-cleaning

St t




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




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




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




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




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

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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




Edge Conditions

Edge Cable (tension)

Edge Arch (compression)

Edge Beam (bending)



Raleigh Arena North Carolina (1953)n




g ( )

Architect: Novicki and DeitrickEngineer: Severud Elstad Krueger

Edge arch / cable roof



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




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




Saddle shapes Wave shapes

S u

r f a c e c o n d i t i o n s




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




Saddle shapes

Expo ‘64 Lausanne




p Architect: Saugey / Schierle

Engineer: Froidevaux et Weber

26 restaurant pavilions:

Featured Swiss regional cuisines

Symbolizing sailing and mountains

Design example




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




Wave shapes

Computer model

San Diego




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




Denver Airport

Architect: FentressPhoto: David Benbennick




Denver AirportPhoto: David Benbennick

Sony Center Berlin




y


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




Arch shapes

Study model






Size: 30’x40’

First twin fabric with thermal insulation

Theater pavilion Armonk (1968)


Size 60’x80’ - capacity 600

Longest span fabric roof 1968

fabric tensile strength 720 pli

Skating Rink MunichArchitect: Ackermann




Architect: AckermannEngineer: Schlaich / Bergermann

Prismatic arch truss supportstranslucent PVC fabric on woodslats and cable net

Arch truss (L=328’)

detail

Point Shapes




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




Architect: G G SchierleEngineer: ASI, Advanced Structures Inc



Erection




Erection

Color lighting

Layout

Erection

German Pavilion

E 67 M t l




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.




Balance Forces

Unbalanced

Balanced



Design Process




Computer Aided

• Form-finding

• Analysis

• Pattern design

Computer modelComputer model

Load shapedotted lines

CAD patterns by triangulation

Optimization




Edge & surface curvature(Schierle, 1971)

Usual optimum L/f = 10L = spanf = sag

L

f

Watts Towers

C lt l C t




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




Architect: Kulla, Herr und PartnerEngineer: Schlaich Bergermann

from cantilever cable trusses




Grid Shell Mannheim, 1975 Architect: Mutschler / Otto

Form-finding model




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/




anticlasticfabric




C

u r v e d w a l l t o r e s i s t w

i n d

Speaker Speaker



Prof G G Schierle, PhD, FAIA

USC - School of ArchitectureLos Angeles, CA 90089-0291

T 213-740-4590

F 213-740-8888

[email protected]

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

[email protected]

http://www.usc.edu/structures

htt // ll /titl / hi l /

thank youthank you

07 Membranes

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