THE CABLE IN STRUCTURES including SAP2000

Prof. Wolfgang Schueller

For SAP2000 problem solutions refer to “Wolfgang Schueller: Building

Support Structures – examples model files”:

https://wiki.csiamerica.com/display/sap2000/Wolfgang+Schueller%3A+Building+Su

pport+Structures+-

If you do not have the SAP2000 program get it from CSI. Students should

request technical support from their professors, who can contact CSI if necessary,

to obtain the latest limited capacity (100 nodes) student version demo for

SAP2000; CSI does not provide technical support directly to students. The reader

may also be interested in the Eval uation version of SAP2000; there is no capacity

limitation, but one cannot print or export/import from it and it cannot be read in the

commercial version. (http://www.csiamerica.com/support/downloads)

See also,

Building Support Structures, Analysis and Design with SAP2000 Software, 2nd ed.,

eBook by Wolfgang Schueller, 2015.

The SAP2000V15 Examples and Problems SDB files are available on the

Computers & Structures, Inc. (CSI) website:

http://www.csiamerica.com/go/schueller

Introduction

Most tensile structures are very flexible in comparison to conventional

structures. This is particularly true for the current, fashionable, minimal

structures, where all the members want to be under axial forces. Here,

repetitive members with pinned joints are tied together and stabilized by

cables or rods. Not only the low stiffness of cables, but also the nature of

hinged frame construction, make them vulnerable to lateral and vertical

movements. To acquire the necessary stiffness, special construction

techniques have been developed, such as spatial networks, as well as the

prestressing of tension members so that they remain in tension under any

loading conditions.

Because of the lightweight and flexible nature of cable-stayed roof structures

they may be especially vulnerable with respect to vertical stiffness, wind

uplift, lateral stability, and dynamic effects; redundancy must also be

considered in case of tie failure. Temperature effects are critical when the

structure is exposed to environmental changes. The movement of the

exposed structure must be compatible with the enclosure. In the partially

exposed structure, differential movement within the structure must be

considered; slotted connections may be used to relieve thermal

movement.

In traditional gravity-type structures the inherent massiveness of

material transmits a feeling of stability and protection.

In contrast, tensile structures seem to be weightless and to float in

the air; their stability is dependent on induced tension and on an

intricate, curved three-dimensional geometry in which the skin is

pre-stretched.

Antigravity roof structures require a new aesthetics; now the curve

rather than the straight line, is the generator of space. The

aesthetics is closely related to biological structures and natural

forms – there is no real historical precedent for the complex forms

of membrane structures.

Fabric structures are forms in tension – as nearly weightless

structures they are pure, essential, and minimal. Spatial, curved

geometry, together with induced tension is necessary for structural

integrity.

CABLES in STRUCTURES

Lateral bracing

Suspended highrise structures (tensile columns)

Single-layer, simply suspended cable roofs Single-curvature and dish-shaped (synclastic) hanging roofs

Prestressed tensile membranes and cable nets (see Surface Structures)

Edge-supported saddle roofs

Mast-supported conical saddle roofs

Arch-supported saddle roofs

Air supported structures and air-inflated structures (air members)

Cable-supported structures

cable-supported beams and arched beams

cable-stayed bridges

cable-stayed roof structures

Tensegrity structures Planar open and closed tensegrity systems: cable beams, cable trusses, cable frames

Spatial open tensegrity systems: cable domes

Spatial closed tensegrity systems: polyhedral twist units

Hybrid structures Combination of the above systems

In typical cable-suspended structures the cables form

the roof surface structure, whereas in cable-supported

structures cables give support to other members.

Tensile structures such as tensile membranes and

tensegrity structures are pretensioned structures so

they can resist compression forces, however, guyed

structures may also be prestressed structures.

Cables form tensile beams and membranes, or assist beams,

columns, surface structures or other member types as inclined

stays or suspended members. Today, the principle is applied to

cranes, ships, television towers, bridges, roof structures, the

composite tensile cladding systems of glass and stainless steel,

and to entire buildings.

In cable structures, tensile members, such as ropes, strands,

rods, W-shapes , prestressed concrete members, chains, or

other member types, are main load-bearing elements; they can

be an integral part of a structural system and can give primary

support to linear members, surfaces, and volumes from above

or below, as well as brace buildings against lateral forces;

cables have low bending and torsional stiffness compared to

their axial tensile stiffness.

Cables refer to flexible tension members consisting of,

rods, plates, W-sections, tubes, etc.

strands,

ropes,

tensile reinforced concrete columns

wood members Wires are laid helically around a center wire to produce a strand, while

ropes are formed by strands laid helically around a core (e.g. wire rope or

steel strand).

STRAND

An assembly of wires

Around a central core

Z-lock CABLE

WIRE ROPE

Assembly of strands

Steel strand and wire rope are inherently redundant members

since they consist of individual wires. The minimum ultimate

tensile strength Fu of strands and ropes is in the range of

200 to 220 ksi (1379 to 1517 MPa) depending on the coating

class (and 270 ksi =1862 MPa for prestressing strand). The

strand has more metallic area than the rope of the same

diameter and hence is stronger and stiffer. The minimum

modulus of elasticity of wire rope is 20,000 ksi (138,000 MPa)

and 24,000 ksi = 165,000 MPa for strands of nominal

diameters up to 2 9/16 in. (65 mm) and 23,000 ksi (159,000

MPa) for the larger diameters.

The cable capacity can be obtained from the manufacturer's

catalogues, but for rough preliminary design purposes of

cable sizes assume a metallic cable area As of roughly 60

percent of its nominal gross area An for ropes and 75 percent

for strands. The ultimate tensile force is, Pu = γP = 2.2P.

Hence the required nominal cross-sectional cable area as

based on 67 percent increase of the required gross area An

for ropes and 33 percent for strand, is

Some historically significant

cable structures

19th century examples

Suspended Theater Roof, 1824, Friedrich Schnirch

The first suspended roof:

prototype, Banska

Bystrica, Slovacia, 1826,

Bedrich Schnirch Arch

Bollman Iron Truss Bridge, Savage, MD, 1869, Wendel Bollman

Tower Bridge, London, 1894,

Horace Jones Arch, John Wolfe

Barry Struct. Eng

different cables

for different

load cases

Transat Chair, 1927, Eileen Gray Designer

Iakov Chernikhov’ s experiments

with architectural structures, 1925-

1932, Russian Constructivism

Pavilion, Chicago, 1933, Bennett & Associates

Dymaxion House, 1923,

Buckminster Fuller

Shabolovka tower,

Moscow, 1922,

Vladimir Shukhov

Golden Gate Bridge

(longest span 4200

FT), San Francisco,

1937, Joseph

Strauss, Irving

Morrow and Charles

Ellis Designers

Lateral tensile bracing

Highrise suspension buildings (tensile columns)

Reliance Controls factory,

Swindon, 1967, Team 4,

Anthony Hunt Struct. Eng

Stansted Airport, London, 1991, Norman Foster Arch, Ove Arup Struct. Eng.

Sainsbury Centre for the Arts,

Norwich, England, 1977, Norman

Foster Arch

Newark air terminal C, USA

Peek & Cloppenburg, Cologne,

Germany, 2005, Renzo Piano Arch,

Knippers Helbig Struct. Eng (façade)

Pavilion of the Future,

Seville, Spain, 1992,

Peter Rice/Arup

Struct. Eng

Highrise suspension structures

Tivoli Stadion, Aachen, Germany, 2009,

Paul Niederberghaus + Hellmich Arch

Sainsburys Store, Camden Town,

London, 1988, Nicholas

Grimshaw Arch, Kenchington

Little Struct. Eng

Centre Georges Pompidou, Paris, France, 1977,

Piano & Rogers Arch, Peter Rice/Ove Arup and

Edmund Happold Struct.Eng

Office building of the

European Investment Bank,

2009, Luxembourg,

Ingenhoven Architects, Werner

Sobek Struct. Eng

Fondation Avicienne (Maison de l'Iran),

Cité Internationale Universitaire, Paris,

1969, Claude Parent + Moshen Foroughi

et Heydar Ghiai Arch

Media TIC Building, Barcelona, Spain, 2010,

Enric Ruiz-Geli Arch, Agusti Obiol – BOMA

Struct. Eng

Ludwig Erhard Haus,

Berlin, Germany, 1999,

Nick Grimshaw Arch

Exchange House, London, 1990,

SOM Arch + Strct. Eng

Poly Corporation Headquarters, Beijing,

China, 2007, SOM Arch + Struct. Eng

Old Federal Reserve

Bank Building,

Minneapolis, 1973,

Gunnar Birkerts, 273-ft

(83 m) span truss at top

Laboratory building, Heidelberg,

Germany, Rossmann & Partner Arch

German Museum of

Technology Berlin,

2001, Helge Pitz and

Ulrich Wolff Architects

House (World War 2 bunker),

Aachen, Germany

Auditorium of the Technical University, Munich, Germany

TU Munich

Shanghai-Pudong Museum, Shanghai-Pudong, China, 2005, von Gerkan, Marg &

Partner Arch, Schlaich Bergermann und Partner Struct. Eng

German Museum of Technology, Berlin, 2001, Helge Pitz and Ulrich Wolff Architects

Standard Bank Centre,

Johannesburg, South Africa, 1970,

Hentrich-Petschnigg Arch

Westcoast Transmission Company Tower, Vancouver,

Canada, 1969, Rhone & Iredale Arch, Bogue Babicki Struct.

Eng

BMW Towers, Munich, Germany, 1972, Karl Schwanzer

Arch, Helmut Bomhard Struct. Eng

Hospital tower of the University of Cologne, Germany, Leonard Struct. Eng.

Visual study of Olivetti Building,

Florence, Italy, 1973, Alberto Galardi

Olivetti Building, Florence, Italy, 1973,

Alberto Garlardi Arch

Kleefelder Hängehaus (Norcon-Haus), Hannover, Germaqny, 1984,

Schuwirth & Erman Arch

Torhaus am Aegi, Hanover, Germany, 2006, Storch

Ehlers Arch, Eilers & Vogel Struct. Eng

Turning Torso, Malmö, Sweden,

2005, Santiago Calatrava Arch +

Struct. Eng

Collserola Tower, Barcelona, Spain, 1992, Norman

Foster Arch, Chris Wise/Arup Struct. Eng

Lookout Tower Killesberg (40 m), Stuttgart, 2001,

Jörg Schlaich designer

The Single Cable

• Funicular cables

• Cable action under transverse loads

• Parabolic cable

• Cubic parabolic cable

• Cable action under radial loads

• Prestretched cable

The deformation of a cable under its loads takes the shape of a funicular

curve that is produced by only axial forces since a cable has negligible

bending strength: polygonal and curved shapes (e.g. catenary shapes,

parabolic shapes, circular shapes)

Funicular tension lines

The simple, flexible, suspended cable takes different shapes under different loading

conditions; in other words, the cable shape and length are a function of loading and

state of stress:

• Polygonal shape (kinked shape) is a function of concentrated loads.

• Curved shape is a function of uniform loads, a situation that is most typical in suspended

roof structures.

• Second degree parabolic shape is a function of constant uniform load, w, on the

horizontal projection of the roof. This situation applies for live loads on shallow suspended

roof structures (where the cables are arranged in a parallel fashion), in accordance with code

requirements, and occurs in suspension bridges where the suspended cables carry the

roadway.

• Catenary shape or hyperbolic cosine (cosh) curve is a function of uniform load along the

cable length (e.g., self weight). For small sag-to-span ratios of n ≤ 1:10, the geometry of a

catenary and a parabola are practically the same so that the simpler parabola can be used.

• Cubic parabolic shape is a function of uniformly distributed, tapered, transverse loads

along the cable's horizontal base, such as a triangular, or trapezoidal-shaped load. These

situations usually occur where cables are arranged in a radial fashion, such as in a typical

circular suspension roof.

• Circular shape is a function of constant uniform radial pressure, p. The radial forces cause

cable forces of constant magnitude that are proportional to the radius of curvature. When

these radial forces, however, are not constant and increase uniformly from a minimum at the

center to a maximum at the edge, the cable takes an elliptical shape.

Polygonal cable

Prestretching cable

Cable vibration

The geometry of the loaded cable depends on the type of loading.

Because typical computer programs only consider linear behavior that is

small deflection theory, the cable geometry should not change too much

under loading; it is important to define the cable geometry to be close

to what is expected after the structure is loaded. For that reason it may

be necessary to correct the cable geometry after one or more preliminary

runs that determine the shape of the cable under the P-Delta load

combination (e.g. dead and live loads for the typical gravity load case).

However, keep in mind that for designing the cables, for example, in cable

beams, gravity cannot act by itself since then the members have to be

designed as compression members! Consider load combinations of

gravity, wind loads, pre-stress, and temperature decrease of the

cables, which produces shortening and causes significant axial forces. If

the stretching of the cable is large it may not be possible to obtain

meaningful results with a P-Delta load combination. The P-Delta effect can

be a very important contributor to the stiffness of cable structures.

WHY IS IT NONLINEAR?

Linear Elastic Theory approximates the length change of a bar by the dot product of the

direction vector and the displacement. But in this situation, you can see from the figure

above, that they are perpendicular to each other therefore dot product = 0. This would

mean that the bar did not change length, which from observation is untrue. It is therefore

necessary to use nonlinear analysis.

The Effects of Prestress

The geometry of the structure itself is unstable as opposed to a structure shown at the

right. The effects of prestress on the structure make it stronger. It is now able to counter

the external forces.

The sum of the forces : 2T*(2d/L) = P

P = (4T/L)d

Modeling of Cables

Cable structures are flexible structures where the effect of large deflections

on the magnitude of the member forces must be considered. Cable

elements are tension-only members, where the axial forces are applied to

the deflected shape. You can not just apply, for instance transverse loads,

to a suspended cable with small moments of inertia using a linear analysis,

all you get is a large deflection with no increase in axial forces because the

change in geometry occurs after all the loads have been applied.

To take the effect of large deflections into account, a P-Delta analysis that is

a non-linear analysis has to be performed. Here the geometry change due to

the deflections, , and the effect of the applied loads, P, along the deformed

geometry is called the P- effect. The P-Delta effect only affects transverse

stiffness, not axial stiffness. Therefore, frame elements representing a cable

can carry compression as well as tension; this type of behavior is generally

unrealistic. You should review the analysis results to make sure that this

does not occur.

In SAP use cable elements for modeling. First define the material

properties then model cable behavior by providing for each frame

element section properties with small but realistic bending and

torsional stiffness (e.g. use 1-in. dia. steel rods or a small value such

as 1.0, for the moment of inertia). Do not use moment end-releases

because otherwise the structure may be unstable; disregard

moments and shear. Apply concentrated loads only at the end nodes

of the elements, where the cable kinks occur. For uniform loads

sufficient frame elements are needed to form a polygon composed of

frame elements. SAP provides for the modeling of curved cables,

Keep as Single Object or Break in Multiple Equal Length Objects.

Tensile structures (e.g. cable beams, tensile membranes) may have to

be prestressed by applying external prestress forces, or temperature

forces.

To perform the P-DELTA ANALYSIS in SAP, unlock the

model after you have performed the linear analysis. Click

Define > Analysis Cases > Modify/Show Case > in the

Analysis Type area select the Nonlinear option. In the Other

Parameters area, check the Modify/Show button for Results

Saved and select Multiple States, then check the

Modify/Show button for the Nonlinear Parameters edit box >

in that form select the P-Delta with Large Displacements

option in the Geometric Nonlinearity Parameters area then

click the OK buttons and proceed with analysis as before. In

other words, click Analyze > Set Analysis Options > select

XZ Plane > click OK > click Run Analysis > click Run Now

(i.e. click Run Analysis button). Notice, the educational

version of SAP will run only the small displacement case

with P-Delta.

Single-layer, cable-suspended

structures: single-curvature and dish-shaped

(synclastic) hanging roofs

Simply suspended or hanging roofs include cable

roofs of single curvature and synclastic shape, that is

cylindrical roofs with parallel cable arrangement, and

polygonal dishes with radial cable pattern or cable nets.

The simply suspended cables may be of the single-

plane, double-flange, or double-layer type.

The concept of simply suspended roofs has surely

been influenced by suspension bridge construction.

Most buildings using the suspended roof concept are

either rectangular or round; in other words, the cable

arrangement is either parallel or radial. However, in

free-form buildings, the roof geometry is not a simple

inverted cylinder or dish and the cable layout is

irregular.

Simply suspended structures

proposal Palazzo del Congress, Venice, 1969, Louis Kahn

Portuguese Pavilion,

Expo 98, Lisbon, Alvaro

Siza Arch, Cecil

Balmond (Arup) Struct.

Eng.

Braga Stadium, Braga, Portugal,

2004, Eduardo Souto de Moura ,

AFA Associados with Arup

Lufthansa-maintanance

hangar V, Frankfurt,

Germany, 1972, ABB

Architects, Dyckerhoff

and Widmann

Trade Fair Hannover,

Hall 9, von Gerkan

Marg and Partners,

1997, Schlaich

Grand Hall, Stuttgart Trade Fair Centre, Stuttgart,

Germany, 2007, Wulf Arch, Mayr Ludescher Struct.

Eng.

Essingen stressed-ribbon

footbridge over Main-Danube

Canal, 1986, Richard Johann

Dietrich Arch, Heinz

Brüninghoff Struct. Eng

In the typical suspended roof the cables (or other member types such as

W-sections, metal sheets, prestressed concrete strips) are integrated with

the roof structure. Here, one distinguishes whether single- or double-layer

cable systems are used. Simple, single-layer, suspended cable roofs must

be stabilized by heavyweight or rigid members. Sometimes, prestressed

suspended concrete shells are used where during erection they act as

simple suspended cable systems, while in the final state they behave like

inverted prestressed concrete shells. In simple, double-layer cable

structures, such as the typical bicycle wheel roof, stability is achieved by

secondary cables prestressing the main suspended cables.

The suspended cable adjusts its shape under load action so it can respond

in tension. It is helpful to visualize the deflected shape of the cable (i.e.

cable profile) as the shape of the moment diagram of an equivalent, simply

supported beam carrying the same loads as the cable. The moment

analogy method is useful since the magnitude of the moment, Mmax, can be

readily obtained from handbooks. Hence, the horizontal thrust force, H, at

the reaction for a simple suspended cable with supports at the same level

and cable sag, f, is

H = Mmax /f

Parabolic cable

L = 140 ‘

30'

14'14'

f = 9.33'

H

H

V

Tmax

θo

Suspended Roof Structure

EXAMPLE 11.1: Suspension roof A typical cable of a single-layer suspension roof (Fig. 11.4) is investigated

for preliminary design purposes. The cables are spaced 6-ft centers and

span 140 ft and a sag-to-span ratio of 1:15 is assumed at the beginning of

the investigation. Dead and live loads are 20 and 30 psf (1.44 kPa or kN/m2)

respectively; temperature change is 500F. Run the static linear analysis first

and then run the static nonlinear analysis with P-Delta (but not using the

large displacement option in the SAP educational version) to take into

account the large cable displacements that is the change of cable geometry.

Try 2 ¼-in-diameter high-strength low-alloy steel rods A572 (Fy = 50 ksi =

345 MPa , Fu = 65 ksi = 448 MPa).

The initial cable sag is assumed as

n = f/L = 1/15 or f = 140/15 = 9.33 ft

First, the geometry input for modeling the suspended cables must be

determined. The radius, R, for the shallow arc is

R = (4h2 + L2)/8h = (4(9.33)2 + 1402)/8(9.33) = 267.26 ft

The location of the span L as related to the center of the circle is defined by

the radial angle θo (roll down angle); this angle also represents the slope of

the curvature at the reactions.

sin θo= ±(L/2)/R =70/267.26 = 0.262, θo = 15.180

The uniform load is assumed on the horizontal projection of the roof for this

preliminary manual check of the SAP results. Hence, a typical interior cable

must support

w = wD + wL = 6(0.020 + 0.030) = 0.12 + 0.18 = 0.3 k/ft

The vertical reactions are equal to each other because of symmetry and are

equal to

V = wL/2 = 0.3(140)/2 = 21 k

The minimum horizontal cable force at mid-span or the thrust force, H, at the

reaction is

H = Mmax /f = wL2 /8f = 0.3(140)2/8(9.33) = 78.78 k

The lateral thrust force according to SAP is 79.17 k as based on linear analysis

and 73.47 k as based on P-Delta analysis. The maximum cable force, Tmax, can

be determined according to Pythagoras' theorem at the critical reaction as

Tmax = 81.53 k

Or, treating the shallow cable as a circular arc, yields the following approximate

cable force of

T ≈ pR = 0.3 (267.26) = 80.18 k

Notice that there is only about 3.5% difference between the largest (Tmax) and

smallest (H) tensile force; the difference decreases as the cable profile becomes

flatter.

The SAP result of the linear analysis is 81.93 k but when performing the

nonlinear analysis that is P-Delta analysis, the maximum cable force is 76.39 k

reflecting the decrease of cable force with increase of cable sag due to large

cable displacement.

The required gross area, AD, for threaded steel rods is

AD ≥ P/0.33Fu ≈ 81.53/0.33(65) = 3.80 in2 (4.8)

where, AD = πd2/4 = 3.80 or d ≈ 2.20 in

Try 2 ¼-in-diameter steel rod.

The increase or decrease in cable length due to change in temperature is

determined as based on the span, L, rather than the cable length, l, since the

difference between the two for the shallow sag-to-span ratio is negligible,

∆l = α (∆T)l ≈ 6.5(10)-6(50)140(12) = 0.55 in

Note that the influence of temperature at this scale is relatively small as also

indicated by SAP. Keep in mind that a decrease in temperature will cause the

cable to shorten and reduce the sag, thus increasing the maximum cable

force.

Trade Fair Hannover, Hall

26, Thomas Herzog Arch,

1996, Jorg Schlaich Struct.

Eng.

45'

53'

30'

15'

30'198'

213'

04

±

±

R = 207 ft

Asymmetrical Suspended Roof Structure

45

'5

3.3

5'

30'

15'

30'193.70'

208.70'

150'

63'

Maison de la

Culture, Firminy,

1965, Le Corbusier

Dulles Airport,

Washington,

1962, Eero

Saarinen/ Fred

Severud, 161-ft

(49 m)

suspended

tensile vault

AWD-Dome (Stadthalle), Bremen,

Germany, 1964, Klumpp Arch,

Dyckerhoff & Widmann AG

Suspended roof,

Hohenems, Vorarlberg,

Austria, Reinhard Drexel

Arch, Merz Kaufmann

Struct. Eng

The David L. Lawrence Convention Center,

Pittsburgh, PA, 2003, R. Vinoly Arch, Dewhurst

MacFarlane Struct. Eng

Cable action under radial loads

Cubic parabolic cable

Suspended dished roof, axial force diagram

Prestressed tensile membranes and

cable nets:

edge-supported saddle roofs

mast-supported conical saddle roofs

arch-supported saddle roofs

air-supported structures; air-inflated structures (air

members)

Hybrid surface structures

Tensile membrane structures

Kagawa

Prefectural

Gymnasium,

Kagawa, Japan,

1964, Kenzo

Tange Arch

Yoyogi National Gymnasium, Tokyo, 1964, Kenzo Tange

Arch, Yoshikatsu Tsuboi Struct. Eng

Small Olympic Stadium, 1964, Tokyo, Kenzo Tange/ Y. Tsuboi

David S. Ingalls Skating Rink, New Haven,

USA, 1958, Eero Saarinen Arch, Fred N.

Severud Struct Eng

Jaber Al Ahmad Stadium Kuwait, Kuwait, 2005, Weidleplan Arch, Schlaich

Bergemann Struct. Eng.

Khan Shatyr Entertainement Center, Astana,

Kazakhstan, 2010, Norman Foster Arch,

Bureau Happold Struct. Eng

The Great Flight Cage, The National

Zoo, Washington DC, 1965,

Richard Dimon (DMJM)

Cable-supported structures

cable-supported beams and arches

suspended cable-supported roof structures

cable-stayed bridges

cable-stayed roof structures

Cable-supported beams and roofs

In contrast to cable-stayed roof structures, where cables give support to the roof

framing from above, here the many possibilities of supporting framework from

below are briefly investigated.

The conventional king-post and queen- post trusses, which represent single-strut

and double-strut cable-supported beams, are familiar. These systems form

composite truss-like structures with steel or wood compression members as

top chords, steel tension rods as bottom chords, and compression struts as

web members.

Single-strut, cable-supported beams can also be overlapped in plane or spatially .

Subtensioned structures range from simple parallel to two-way and complex

spatial systems, which however, are beyond the scope of this context.

Cable-supported

structures

Single-strut and multi-

strut cable-supported

beams

Integrated urban

buildings, Linkstr.

Potsdamer Platz,Berlin,

1998, Richard Rogers

Wilkhahn

Factory, Bad

Muender,

Germany,

Herzog Arch.,

1992

Cable supported bridge, Berlin

World Trade Center,

Amsterdam, 2002,

Kohn, Pedersen &

Fox Arch

World Trade Center, Amsterdam, 2002, Kohn Pedersen Fox Arch

Concord Sales Pavilion,

Vancouver,2000, Busby +

Associates Architects,

StructureCraft

U.S. Bank Stadium

(Minnesota Viking

Stadium), Minneapolis,

2016, HKS Arch, Thornton

Tomasetti Struct. Eng

Living Bridge, Limerick University , Ireland,

2007, Wilkinson Eyre Arch

Miho Museum Suspension

Bridge, Kyoto, Japan, 2009,

I.M. Pei Arch, Leslie E.

Robertson Struct. Eng

Auditorium

Paganini,

Parma, Italy,

2001,

Renzo

Piano Arch

Shopping street in Bauzen, Germany

Landeshauptstadt München,

Baureferat, Georg-Brauchle-

Ring, Munich, Germany,

Christoph Ackerman

Debis Theater, Marlene

Dietrich Platz, Berlin,

1998, Germany, Renzo

Piano Arch

Vancouver Aquarium Addition,

Vancouver, 1999, Bing Thom

Architects, PCL Engineers

Shopping street in Wolfsburg, Germany

Surrey Central City Galleria

roof,,Surrey, British Columbia,

2002, Bing Thom Architects,

StructureCraft

River Soar Bridge, Abbey Medows, Leicester, UK, Exploration

Arch., Buro Happold Struct. Eng

Milleneum Bridge, London,

2000, Foster Arch, Arup

Struct. Eng

Bus shelter,

Schweinfurt, Germany

Surrey Central City, Atrium Roof, Surrey BC, Canada, 2002, Bing Thom

Architects, StructureCraft

Cable-Supported Beams

a

b

c

d

b

c

d

a

Typical Cable-supported, Single- and Multi-Strut Beams

Lehrter Bahnhof,

Berlin, 2006, von

Gerkan, Marg and

Partners

The parabolic spatial roof arch

structure with its 42-m cantilevers is

supported on only two monumental

conical concrete-filled steel pipe

columns spaced at 124 m. The columns

taper from a maximum width of 4.5 m at

roughly 2/3 of their height to 1.3 m at

their bases and capitals, and they are

tied at the 4th and 7th floors into the

structure for reasons of lateral stability.

The glass walls are supported

laterally by 2.6-m deep free-standing

vertical cable trusses which also act

as tie-downs for the spatial roof

truss.

Tokyo International Forum,

Tokyo, Japan, 1996, Rafael

Vinoly Arch. and Kunio

Watanabe Eng

Cable-Supported Arches

When arches are braced or prestressed by tensile elements, they are

stabilized against buckling, and deformations due to various loading

conditions and the corresponding moments are minimized, which in

turn results in reduction of the arch cross-section. The stabilization of

the arch through bracing can be done in various ways.

Typical examples of braced arches with non-prestressed web members are

shown in Fig. 7.15. The most basic braced arch is the tied arch (b).

Arches may be supported by a single or multiple compression struts or

flying columns (c, d)). Slender arches may also be braced against

buckling with radial ties at center span (e) as known from the principle

of the bicycle wheel, where the thin wire spokes of the bicycle wheel are

prestressed with sufficient force so that they do not carry compression

and buckle due to external loads; the uniform radial tension produces

compression in the outer circular rim (ring) of the wheel and tension in

the inner ring. However, in the given case, the diagonal members are

not prestressed. Here, the three members at center-span are struts.

Hilton Munich Airport, Munich,

Germany, 1997, H. Jahn Arch, Jörg

Schlaich Struct. Eng

Hall 4, Hannover,

Germany, 1996, von

Gerkan Marg Arch,

Schlaich Bergermann

Struct. Eng

Ingolstadt Freight

Center, Hall Q,

Ingolstadt,

Germany

b

a

4'

4'

c

4'

4'

40'

4'

4'

Cable-Supported Arched Beams

Mercedes-Benz Center am Salzufer, Berlin,

2000, Lamm, Weber, Donath und Partner

Shanghai-Pudong International Airport, 2001, Paul Andreu Arch,

Coyne et Bellier Struct. Eng

Munich Airport Business Center, Munich,

Germany, 1997, Helmut Jahn Arch

Space Truss Arch: axial force flow

Railway Station "Lehrter Bahnhof“,

Berlin, 2003, Architect von Gerkan Marg

und Partner, Schlaich Bergerman

Structural Engineers

Berlin Central Station, Berlin, 2006, von Gerkan, Marg Arch, Schlaich

Bergerman Structural Engineers

COMPOSITE SYSTEMS AND FORM-RESISTANT STRUCTURES

An example of an asymmetrical arch system is shown in the next slide. Here the supports

are at different levels and a long-span arch and a short arch support each other, in other

words the crown hinge is located off-center.

The relatively shallow asymmetrical arch system constitutes a nearly funicular response in

compression under uniform load action since the circular geometry approaches the

parabolic one; notice that the location of the hinge is of no importance. Hence, live loading

for each arch separately must be considered in order to cause bending, while the dead load

is carried in nearly pure compression action; the long arch on the right side clearly carries

the largest moments. Superimposing the pressure lines of the two loading cases

results in a composite funicular polygon that looks like the shape of two inclined bowstring

trusses, hence suggesting a good design solution. For long-span arches the use of

triangular space trusses may be advantageous.

Under asymmetrical loading on the long arch, the long arch acts in compression and the

bottom chord in tension to resist the large positive bending moment. However, the bottom

chord of the short arch acts in compression and the top chord in tension under the negative

bending moment. But should the bottom member be straight, then it resists directly the

compression force due to the live load in funicular fashion leaving no axial force or moment

in the arch.

Under asymmetrical loading on the short arch, the bottom chord of the long truss will resist

the compression force directly, hence causing no moment or axial force in the arch if it

would be a compression member. But since it is a tension member, there must be enough

tension due to the weight of the long-span in the member to suppress the compression

force!

Plan view

Pressure lines in elevation

Asymmetrical arch

5.86'

27.32'10'

4.29'

10.1

0 k

7.70 k

Mmax

Mmin

EXAMPLE: 9.2:

Asymmetrical

composite arches

Waterloo Terminal,

London, 1993, Nicholas

Grimshaw Arch, Anthony

Hunt Struct. Eng

20' 17.32'

2.68'

10'

10'

30 deg

60 deg30 deg

17.32'

17.32'

5.86'

27.32'10'

4.29'

7.32'

a.

b.

2.68'C.

Ah

Av

Bh

Bv

PRESTRESSING TENSILE WEBS To model tensile webs of arches, the web members may have to be prestressed by applying external prestress forces, or temperature forces. With respect to external prestress forces, run the structure as if it were, say a trussed arch, and determine the compression forces in the web members, which it naturally cannot support. Then, as a new loading case, apply an external force, which causes enough tension in the compression member so that never compression can occur. With respect to temperature forces, run the structure without prestressing it, then determine the maximum compression force in the cable members which should not exist, then apply a negative thermal force (i.e. temperature decrease causes shortening) to all those members thereby pre-stressing them, so that they all will be in tension. To perform the thermal analysis in SAP, select the frame element, then click Assign, then Frame/Cable Loads, and then Temperature; in the Frame Temperature Loading dialog box select first Load Case, then Type (i.e. temperature for uniform constant temperature difference).

BRACED ARCHES

When arches are braced or prestressed by tensile elements, they are

stabilized against buckling, and deformations due to various loading

conditions and the corresponding moments are minimized, which in turn

results in reduction of the arch cross-section. The stabilization of the arch

through bracing can be done in various ways as suggested in Fig. 9.12 and

9.14.

Several typical examples of braced arches with non-prestressed web

members are shown in Fig. 9.12. The most basic braced arch is the tied

arch (b). Arches may be supported by a single or multiple compression struts

or flying columns (c, d)). Slender arches may also be braced against buckling

with radial ties at center span (e) as known from the principle of the bicycle

wheel, where the thin wire spokes of the bicycle wheel are prestressed with

sufficient force so that they do not carry compression and buckle due to

external loads; the uniform radial tension produces compression in the outer

circular rim (ring) of the wheel and tension in the inner ring. However, in the

given case, the diagonal members are not prestressed. Here, the three

members at center-span are struts.

Arches may also be supported by a dense network of overlapping diagonal

tensile members (f); notice, this case represents a pure tensile network. When

loaded on one side the diagonals under the load fold while the diagonal members

on the non-loaded side are placed under tension. SAP takes into account the

redistribution of forces by treating the cable network in case (f), for example, as

tension-only members by performing a nonlinear static analysis. In general,

however, depending on the arch proportions the tensile webbing may have to be

prestressed to act more efficiently under any loading condition and to increase the

load carrying capacity and stiffness of the arch.

The cable-braced, latticed, tied-arch in Fig. 9.12g approaches the behavior of a

truss; the cable network substantially reduces bending moments in the arch and tie

beam where the bottom loads prestress the arch. For fast approximation purposes

use the beam analogy .

The design of the unbraced arched portal frame in (a), is controlled by full

uniform gravity loading; here the lateral thrust at the frame knees is resisted

completely in bending. However, when the relatively shallow portion of the arch

is braced by a horizontal tie rod (b), the lateral displacement under full uniform

gravity loading is very much reduced, that is bending decreases substantially

although axial forces will increase. For the tied arch cases without or with flying

column supports for cases (b, c, d)), the design of the critical arch members is

controlled by gravity loading or the combination of half gravity loading together

with wind whereas the design of the web members is controlled by gravity

loading. It is apparent, as the layout of the arch webbing gets denser the arch

moments will decrease further as the structure approaches an axial system. If a

vertical load large enough is applied to the intersection of web members in case

(e) to prestress the radial rod web members, then the entire web members form

a radial tensile network. For further discussion refer to Problem 9.1.

a d

b e

c f

L = 40'

10'

6'

12'

10'

g

Problem 9.1: Braced arches

a d

b e

c f

L = 40'

10'

6'

12'

10'

Museum for Hamburg History, courtyard roof

(1989), Hamburg, Architect von Gerkan

Marg Arch, Jörg Schlaich Struct. Eng

ARCHES WITH PRESTRESSED TENSILE WEBS

The spirit of the delicate roof structure of the Lille Euro Station, Lille,

France as shown in the following conceptual drawing (1994, Jean-Marie

Duthilleul/ Peter Rice), reflects a new generation of structures aiming for

lightness and immateriality. This new technology features construction with

its own aesthetics reflecting a play between artistic, architectural,

mathematical, and engineering worlds. The two asymmetrical transverse

slender tubular steel arches (set at about 12 m or 40 ft on center) with

diameters of around one-hundredth of their span, are of different radii; the

larger arch has a span of 26 m and the smaller one 18.5 m. The arches are

braced against buckling similar to the spokes of a wheel by deceitfully

disorganized ties and rods; this graceful and light structure, in harmony with

the intimate space, was not supposed to look right but to reflect a spirit of

ambiguity. The roof does not sit directly on the arches, but on a series of

slender tubes that are resting on the arches which, in turn, carry the

longitudinal cable trusses that support the undulating metal roof. The

support structure allowed the gently curved roof almost to float or to free it

from its support, emphasizing the quality of light.

TGV Lille-Europe Station, Lille,

France, 1994, Jean-Marie

Duthilleul/ Peter Rice

PRESTRESSING TENSILE WEBS

To model tensile webs of arches, the web members may have to be

prestressed by applying external prestress forces, or temperature forces.

With respect to external prestress forces, run the structure as if it were, say

a trussed arch, and determine the compression forces in the web members,

which it naturally cannot support. Then, as a new loading case, apply an

external force, which causes enough tension in the compression member so

that never compression can occur.

With respect to temperature forces, run the structure without prestressing it,

then determine the maximum compression force in the cable members

which should not exist, then apply a negative thermal force (i.e.

temperature decrease causes shortening) to all those members thereby pre-

stressing them, so that they all will be in tension.

To perform the thermal analysis in SAP, select the frame element, then click

Assign, then Frame/Cable Loads, and then Temperature; in the Frame

Temperature Loading dialog box select first Load Case, then Type (i.e.

temperature for uniform constant temperature difference).

A

D E

B C

Braced arches

a

d e

b c

500 50 0

500

500 50 0

50 0

20

'

10

'

Introducing to the semicircular arch a horizontal tie rod (Problem 9.3) at mid-

height, reduces lateral displacement of the arches due to uniform gravity

action substantially, so that the combination of gravity load and wind load

controls now the design rather than primarily uniform gravity loading for an

arch without a tie. Also the moments due to the gravity and wind load

combination are reduced since the tie remains in tension as it transfers part of

the wind load in compression to the other side of the arch. In contrast, when

the arch is braced with a trussed network , then the arch is stiffened laterally

very much, so that the uniform gravity loading case controls the design with

the corresponding smaller moments.

Similar behavior occurs for the arch placed on the diagonal (Fig. 9.14d, e). As

a pure arch its design is controlled by bending with very small axial forces as

based on gravity loading, in other words it behaves as a flexural system.

However, when prestressed tensile webbing is introduced the moments in the

arch are substantially reduced and the axial forces increased, now the arch

approaches more the behavior of an axial-flexural structure system

requiring much smaller member sizes; also here the controlling load case is

gravity plus prestressing although the design of some members is based on

dead load and prestressing. For further discussion refer to Problem

MUDAM, Museum of Modern Art,

Luxembourg, 2006, I.M. Pei Arch

Alnwick Garden Pavilion and Visitor Centre,

Alnwick, UK, 2006, Hopkins Arch., Buro Happold

Struct. Eng.

Chiddingstone Orangery Gridshell, Kent,

UK, 2016, Peter Hulbert Arch, Buro

Happold Struct. Eng

Schlüterhof Roof, German Historical

Museum, Berlin, Germany, 2002, I.M.

Pei Arch, Schlaich Bergermann

Struct. Eng

DZ Bank including auditorium, Berlin, Germany ,2001, Frank Gehry Arch, Schlaich Bergemann Struct. Eng

Kaufmann Center for the Performing Arts,

Kansas City, MO, 2011, Moshe Safdie Arch,

Ove Arup Struct. Eng

Suspended cable- and arch-

supported bridge and roof

structures

Golden Gate Bridge

(one 2224 ft), San

Francisco, 1936,

C.H. Purcell

Akashi-Kaikyo-Bridge,

Japan, 1998, 1990 m span

Burgo Paper Mill, Mantua,

Italy, 1963, Pier Luigi Nervi

designer

Pedestrian Bridge across the Main-Danube

Canal, Kehlheim, Germany, 1986, K.

Ackermann Arch, Schlaich Bergermann

Struct. Eng

Curved suspension bridge, Bochum,

Germany, 2003, von Gerkan Marg

Dachtragwerk

Eissporthalle,

Memmingen, 1988,

Börner Pasmann Arch,

Schlaich Bergemann

Struct. Eng

Jumbo Maintenance Hangar, Deutsche Lufthansa, Hamburg Airport , van Gerkan Marg Arch,

Wupperbrücke Ohligsmühle, Wuppertal –

Elberfeld, Germany, 2002

Blennerhassett Island Bridge over the Ohio

River and Blennerhassett Island, 2008

Olympic Stadium “OAKA”,

Athens, Greece, 2004, Santiago

Calatrava

The Olympic Velodrome,

Athens, Greece, 2004,

Santiago Calatrava

Lanxess Arena, Cologne, 1998, Peter Böhm Architekten

Cable-stayed bridges

consist of the towers, cable stays, and deck structure. The stays can

give support to the deck structure only at a few points, using one,

two, three, or four cables, or the stays can be closely spaced thereby

reducing the beam moments and allowing much larger spans.

Typical multiple stays can be arranged in a fan-type fashion by

letting them start all together at the top of the tower and then spread

out. They can be arranged in a harp-type manner, where they are

arranged parallel across the height of the tower. The stay

configuration may also fall between the fan-harp types. Furthermore,

the stay configurations are not always symmetrical as indicated. In

the transverse direction, the stays may be arranged in one vertical

plane at the center or off center, in two vertical planes along the edge

of the roadway, in diagonal planes descending from a common point

to the edge deck girders, or the stays may be arranged in some other

spatial manner. In bridge design generally cables are used because

of the low live-to-dead load ratio.

Common cable-stayed bridge systems

Oberkassel Rhine Bridge,

Germany, 1976, Friedrich Tamms

Arch + Fritz Leonhardt Eng

Designers

Severins Bridge, Cologne, Germany, 1959,

Gerd Lohmer and Fritz Leonhardt designers

Friedrich-Ebert-Bridge, Bonn, Germany, 1967, Heinrich Bartmann

Arch + Hellmut Homberg Eng Designers

Maracaibo Bridge, Maracaibo, Zulia, Venezuela, 1962,

Riccardo Morandi Designer

Ganter Bridge,

Brig, Switzerland,

1980, Christian

Menn designer

Millau Viaduct, Millau, Tarn Valley, France, 2004, Michel Vilogeux

and Norman Foster Arch, Ove Arup Struct. Eng

Speyer Rhine Bridge,

Germany, 1975, Wilhelm

Tiedje Arch + Louis

Wintergerst Eng

Designers

3rd Orinoco Brücke, ,

Caicaras, Venezuela, 2010,

Harrer Ingenieure GmbH

I-70 Mississippi River

Bridge, St. Louis, MO,

2014, Modjeski and

Masters designers

Erasmus Bridge, Rotterdam, 1996, architect Ben Van Berkel

Zakim Bunker Hill Bridge,

Boston, 2003

Willemsbridge, Rotterdam,

1981, is a double suspension

bridge, C.Veeling designer

Alamillo Bridge,

Sevilla, Spain,1992,

Santiago Calatrava

Three bridges over

the Hoofdvaart

Haarlemmermeer,

the Netherland,

2004, Santiago

Calatrava

Pedestrian Bridge, Bad

Homburg, 2002, Architect

Schlaich Bergemann

Miho Museum Bridge, Shiga,

Japan,1996, I.M. Pei, Leslie e. Robertson

Ruck-a-chucky Bridge, Myron Goldsmith/

SOM, T.Y. Lin Struct. Eng, unbuilt

Cable-Stayed Bridges

a b

f

c e d

CABLE – STAYED ROOF

STRUCTURES

• Cable-stayed, double-cantilever roofs for central spinal buildings

• Cable-stayed, single-cantilever roofs as used for hangars and

grandstands

• Cable-stayed beam structures supported by masts from the outside

• Spatially guyed, multidirectional composite roof structures

Cable-supported structures

Alitalia Hangar, Rom, Italy,

1960, Riccardo Morandi Arch,

Ice Hockey Rink, Squaw

Valley, CA, 1960, Corlett &

Spackman

Airport Munich Hangar 1 (153 m), Munich, 1992, Günter Büschl

Arch, Fred Angerer Struct. Eng

INMOS microprocessor factory, Newport,

Gwent , 1987, Richard Rogers & Partners,

Anthony Hunt Struct. Eng

Fleetguard Factory, Quimper, France,

1981, Richard Rogers Arch, Peter

Rice/Arup Struct. Eng

Renault Distribution

Center, Swindon,

England, 1982, Norman

Foster Arch, Ove Arup

Struct. Eng

Railway Station, Tilburg, Holland, 1965,

Koen van der Gaast Arch

PATCenter, Princeton, USA, 1984, Richard

Rogers Arch, Ove Arup Struct. Eng

Igus Headquarters & Factory,

Cologne, Germany, 2000,

Nicholas Grimshaw Arch,

Whitby Bird Struct. Eng

Sainsbury’s supermarket, Canterbury, UK,

1984, Ahrends Burton Koralek Arch, Ernest

Green Struct. Eng

Italian Industry Pavilion at

Expo '70, Osaka, Japan,

1970, Renzo Piano Arch

The Sydney Convention And

Exhibition Centre, 1986, Cox,

Richardson, Taylor and Partners

The University of Chicago Gerald Ratner

Athletic Center, Cesar Pelli, 2002

Temporary American

Center, Paris, 1991,

Nasrin Seraji Arch

Bangkok

Horst Korber Sports Center (1990), Berlin,

Christoph Langhof Arch

Convention Center Trade Fair Hanover,

1989, H. Storch & W. Ehlers (SEP) Arch

Ontario Place, Toronto, Canada,

1971, Eberhard Zeidler Arch

Saibu Gas Museum for natural Phenomen-

art, Fukuoka, 1989, Shoei Yoh + Architects

Jeonju World Cup Stadium, Jeonju,

South Korea, 2001, Pos A.C Arch, CS

Struct. Eng

City Manchester Soccer Stadium,

Manchester, UK, 2003, ARUP Architects

and Engineers

Millenium Dome (365 m), London, 1999, Richard

Rogers Arch, Buro Happold Struct. Eng

b

c

d

W14 x 30

W14 x 22

P6

P10

P8

W14 x 43

P8

P5

a

80'50' 50'

20' 20'80'

20

'2

0'

20

'2

0'

20

'3

0'

5'

5'10' 10'

W14 x 26

P8

P5

Typical Cable-supported Roof

(beam) Structures

Force flow in cable-supported roofs

Tensile Membrane Structures (typically cable nets with coated fabrics)

The basic prestressed tensile membranes are as follows:

Pneumatic structures of domical and cylindrical shape (i.e., synclastic shapes)

• Air-supported structures

• Air-inflated structures (i.e., air members)

• Hybrid air structures

Anticlastic prestressed membrane structures

• Edge-supported saddle roofs

• Mast-supported conical saddle roofs

• Arch-supported saddle roofs

• Corrugate tensile roofs (radial, linear)

Membrane surfaces as cladding

Hybrid tensile surface structures (possibly including tensegrity)

Classification of tensile

membranes

Pneumatic Structures

Pneumatic structures may be organized as follows:

• Air-supported structures

high-profile, ground-mounted air structures, and

berm- or wall-mounted, low-profile roof membranes

• Air-inflated structures (i.e., air members)

Tubular systems (line elements)

Dual-wall systems or airmats (surface elements)

• Hybrid air structures

Classification of pneumatic structures

Pneumatic structures

Low-profile , long-span pneumatic roof structures

Effect of internal air pressure on geometry

Soap bubbles

In air-supported structures the tensile membrane floats like a curtain on top

of the enclosed air, whose pressure exceeds that of the atmosphere; only a

small pressure differential is needed. The typical normal operating pressure

for air-supported membranes is in the range of 4.5 to 10 psf (0.2 kN/m2 to 0.5

kN/m2 = 0.5 kPa) or 2 mbar to 5 mbar, or roughly 1.0 to 2.0 inches of water as

read from a water-pressure gage.

See also packing of soap bubbles

Traveling exhibition

Effect of wind loading on

spherical membrane shapes

Air-inflated

members and

Example 9.14

Air-supported cylindrical membrane

structure

p

T = pR T = pR

Lense-shaped pneumatic bubble structure

Lense-shaped

pneumatic bubble

structure

Air Cushion Roof, F22 Diagram (COMB1)

Roman Arena Inflated Roof, Nimes, France, 1988, Architect Finn Geipel, Nicolas Michelin, Paris;

Schlaich Bergermann und Partne; internal pressure 0.4…0.55 kN/m2

Expo 02 , Neuchatel, Switzerland, Multipack Arch, air cussion, ca 100 m dia.

US Pavilion, EXPO

70, Osaka, Davis-

Brody

US Pavilion, EXPO 70, Osaka,

Davis-Brody Arch, Geiger –

Berger Struct. Eng.

Pontiac Metropolitan

Stadium , Detroit, 1975,

O'Dell/Hewlett &

Luckenbach Arch, Geiger

Berger Struct. Eng.

Metrodome, Minneapolis, 1982, SOM Arch, Geiger-Berger Struct. Eng

Typical membrane roof details

Tensile foundation principles

Tension foundations

Anticlastic Prestressed Membrane

Structures

Membrane structures may be organized either according to their surface form or their

support condition:

• Saddle-shaped and stretched between their boundaries, representing orthogonal

anticlastic surfaces with parallel fabric patterns.

• Conical-shaped and center supported at high or low points, representing radial

anticlastic surfaces with radial fabric patterns.

• The combination of these basic surface forms yields an infinite number of new forms.

The following organization is often used based on support conditions:

• Edge-supported saddle surface structures

• Arch-supported saddle surface structures

• Mast-supported conical (including point-hung) membrane structures (tents)

• Hybrid structures, including tensegrity nets

Tent architecture

Methods for stabilizing

cable structures

Anchorage of tensile forces

Point-supported tents

Edge supports for cable nets

Examples 9.9 and 9.10

Suspended, load-carrying

membrane force

Arched, prestress

membrane force

f

f

wp

T2

T2

T1 T

1

w

Anticlastic Tensile Membrane Forces

Basic Saddle Shape and Deformed Shape

West Germany Pavilion at Expo 67,

Montral, 1967, Frei Otto + Rolf

Gutbrod Arch

Sidney Myer Music

Bowl, Melbourne,

1959, Australia, Barry

Patten Arch, WL Irwin

Struct. Eng

Olympic Stadium, Munich, Germany, 1972, Günther Behnisch architect + Frei Otto,

Leonhardt-Andrae Struct. Eng.

Ice Rink Roof, Munich, 1984, Architect Ackermann und Partner,

Schlaich Bergermann Struct. Eng

Saga Headquarters

Amenity Building,

Folkston, UK,

1999, Michael

Hopkins Arch, Ove

Arup Struct. Eng

Denver International Airport

Terminal, 1994, Denver, Horst

Berger/ Severud

San Diego Convention Center Roof, 1990,

Arthur Erickson Arch, Horst Berger

consultant for fabric roof

Haj Terminal, Jeddah, Saudi Arabia, 1982, SOM/ Horst Berger Arch, Fazlur Khan/SOM Struct. Eng

Schlumberger Research Center, Cambridge,

UK, 1985, Michael Hopkins Arch, Anthony

Hunt Struct. Eng

Rosa Parks Transit Center, Detroit, 2009, Parson Brinkerhoff + FTL Design and

Engineering Studio

Sony Center, Potzdamer Platz, Berlin, 2000, Helmut Jahn Arch., Ove Arup Struct. Eng

Hybrid tensile surface structures

TENSEGRITY STRUCTURES

Buckminster Fuller described tensegrity as, “small islands of compression in a sea

of tension.” Ideal tensegrity structures are self-stressed systems, where few non-

touching straight compression struts are suspended in a continuous cable network of

tension members.

Tensegrity structures may be organized as

• Closed tensegrity structures: sculptures, (e.g. polyhedral twist units)

• Open tensegrity structures

planar open and closed tensegrity structures:

cable beams, cable trusses, cable frames

spatial open tensegrity structures:

flat or bent roof structures: e.g. tensegrity domes

Tensegrity structures may form open or closed systems. In closed systems

discontinuous diagonal struts, which do not touch each other, overlap in any

projection and stabilize the structure without external help that is supports. A basic

example is the polyhedral twist unit which are generated by rotating the base

polygons; the edges are formed by tension cables and the compression struts are

contained within the body. Kenneth Snelson called his famous twist unit, X Piece

(1968), because it forms an X in elevation. This unit is often considered as the

fundamental basis of the tensegrity principle and has inspired subsequent

generations of designers.

The tensegrity sculptures by Kenneth Snelson are famous examples of the

principle as demonstrated by the, Needle Tower at the Hirshorn Museum in

Washington, DC where the compression struts do not touch. Here, the tower is

created by adding twist units with triangular basis, where the triangular module is

decreased with height in addition to changing the direction of twist. Closed

tensegrity structures have not found any practical application in building

construction till now.

TENSEGRITY TRIPOD

DOUBLE - LAYER TENSEGRITY DOME

Twist unit: X Piece

Tensegrity sculptures by K. Snelson

SPHRERICAL ASSEMBLY OF TENSEGRITY TRIPODS

The Skylon tower

(172.8 m) at the

Festival of Britain,

London, 1951,

Hidalgo Moya,

Philip Powell

Arch, Felix

Samuely Struct.

Eng

Warnow tower, Rostock,

Rostock, Germany, 2003,

Gerkan, Marg Arch

In contrast, open tensegrity structures are stabilized at the

supports. Therefore, no diagonal compression members are required and

shorter struts can be used.

Open tensegrity structures can form planar or spatial structures.

• Examples of planar systems include: cable beams, cable trusses, cable

frames as shown in Fig.s 11.18a, 11.19 and 11.22. These structures can also

form spatial units as shown in Fig.s 11.18c and Fig.11.21.

• Examples of spatial systems include: flat or bent roof structures.

Examples of the spatial open tensegrity systems are the tensegrity domes.

David Geiger invented a new generation of low-profile domes, which he called

cable domes. He derived the concept from Buckminster Fuller’s aspension

(ascending suspension) tensegrity domes.

David Geiger invented a new generation of low-profile domes after his air

domes, which he called cable domes. He derived the concept from

Buckminster Fuller’s aspension (ascending suspension) tensegrity domes,

which are triangle based and consist of discontinuous radial trusses tied

together by ascending concentric tension rings; but the roof was not

conceived as made of fabric.

Geiger’s prestressed domes, in contrast, appear in plan like simple, radial

Schwedler domes with concentric tension hoops. His domes consist of

radioconcentric spatial cable network and vertical compression struts. In other

words, radial cable trusses interact with concentric floating tension rings

(attached to the bottom of the posts) that step upward toward the crown in

accordance with Fuller’s aspension effect. The trusses get progressively

thinner toward the center, similar to a pair of cantilever trusses not touching

each other; the heaviest member occur at the perimeter of the span. In section,

the radial trusses appear as planar and the missing bottom chords give the

feeling of instability, which however, is not the case since they are replaced by

the hoop cables that the the cables together.

Fuller’s tensegrity dome

Spatial open tensegrity

structures

The cable dome concept can also be perceived as ridge cables radiating from

the central tension ring to the perimeter compression ring. They are held up

by the short compression struts, which in turn, are supported by the

concentric hoop (or ring) cables and are braced by the intermediate tension

diagonals, as well as by the radial cables. A typical diagonal cable is attached

to the top of a post and to the bottom of the next post.

The pie-shaped fabric panels span from ridge cable to ridge cable and then

are tensioned by the valley cables, thus being shaped into anticlastic

surfaces; they contribute to the overall stiffness of the dome. The maximum

radial cable spacing is limited by the strength of the fabric and detail

considerations. The number of tension hoop is a function of the dome span.

The sequence of erection of the roof network, which is done without

scaffolding, is critical, that is, the stressing sequence of the posttensioned

roof cables to pull the dome up into place.

The first tensegrity domes built were the gymnastics and fencing stadiums

for the 1988 Summer Olympics in Seoul, South Korea. The 393-ft span dome

for the gymnastics stadium required three tension hoops and has a

structural weight of merely 2 psf.

The 688-ft span Florida Suncoast Dome in St. Petersburg (1989) is one of the

largest cable domes in the world. The dome is a four-hoop structure with 24

cable trusses and has a structural weight of only 5 psf. The dome weight is 8

psf, which includes the steel cables, posts, center tension ring, the catwalks

supported by the hoop cables, lighting, and fabric panels.

The translucent fabric consists of the outer Teflon-coated fiberglass

membrane, the inner vinyl-coated polyester fabric, and an 8-in. thick layer of

fiberglass insulation sandwiched between them. The dome has a 6o tilt and

rests on all-precast, prestressed concrete stadium structure,

Olympic Fencing and Gymnastics Stadiums, Seoul, 1989, David Geiger Struct. Eng

The world’s largest cable dome is currently Atlanta’s Georgia Dome (1992),

designed by engineer Mattys Levy of Weidlinger Associates. Levy developed

for this enormous 770- x 610-ft oval roof the hypar tensegrity dome, which

required three concentric tension hoops. He used the name because the

triangular-shaped roof panels form diamonds that are saddle shaped.

In contrast to Geiger’s radial configuration primarily for round cable domes,

Levy used triangular geometry, which works well for noncircular structures

and offers more redundancy, but also results in a more complex design and

erection process. An elliptical roof differs from a circular one in that the

tension along the hoops is not constant under uniform gravity load action.

Furthermore, while in radial cable domes, the unbalanced loads are resisted

first by the radial trusses and then distributed through deflection of the

network, in triangulated tensegrity domes, loads are distributed more evenly.

Georgia Dome, Atlanta, 1992, Scott W. Braley Arch, Matthys P. Levy/

Weidlinger Struct. Eng.

The oval plan configuration of the roof consists of two radial circular

segments at the ends, with a planar, 184-ft long tension cable truss at the long

axis that pulls the roof’s two foci together. The reinforced-concrete

compression ring beam is a hollow box girder 26 ft wide and 5 ft deep that

rests on Teflon bearing pads on top of the concrete columns to accommodate

movements.

The Teflon-coated fiberglass membrane, consisting of the fused diamond-

shaped fabric panels approximately 1/16 in. thick, is supported by the cable

network but works independently of it (i.e. filler panels); it acts solely as a

roof membrane but does contribute to the dome stiffness. The total dead load

of the roof is 8 psf.

The roof erection, using simultaneous lift of the entire giant roof network from

the stadium floor to a height of 250 ft, was an impressive achievement of

Birdair, Inc.

Kurilpa Bridge (Tank Street Bridge), Brisbane, Australia, 2009, Ove Arup Struct. Eng

CABLE-BEAMS and CABLE-SUPPORTED COLUMNS

Tensile structures such as cable beams, guyed structures, tensile

membranes, tensegrity structures, etc. are pre-stressed so they can

resist compression forces which can be done by applying external pre-

stress forces and loads due temperature decrease.

Cable beams, which include cable trusses, represent the most simple case

of the family of pretensioned cable systems that includes cable nets

and tensegrity structures. Visualize a single suspended (concave)

cable, the primary cable, to be stabilized by a secondary arched

(convex) cable or prestressing cable. This secondary cable can be

placed on top of the primary cable by employing compression struts,

thus forming a lens-shaped beam (Fig. 9.4A), or it can be located below

the primary cable (either by touching or being separated at center) by

connecting the two cables with tension ties or diagonals. A combination

of the two cable configurations yields a convex-concave cable beam.

Cable beams can form simple span or multi-span structures; they also can

be cantilevers. They can be arranged in a parallel or radial fashion, or in

a rectangular or triangular grid-work for various roof forms, or they can

be used as single beams for any other application.

Cable Beams

a

b

c

b

a

c

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a. b. c. d. e.

P3

P2 P

2

P1

.5

¾-i

n.

rod

a. b. c. d. e.

P3

P2 P

2

P1

.5

¾-i

n.

rod

Cable-Supported Columns (spatial units)

Planar open tensegrity structures

Cable frames

Shopping Center, Stuttgart

Cologne/Bonn Airport, Germany, 2000, Helmut

Jahn Arch., Ove Arup USA Struct. Eng.

Suspended glass skins form a composite system of glass and stainless

steel. Here, glass panels are glued together with silicone and supported by

lightweight cable beams.

Typically, the lateral wind pressure is carried by the glass panels in bending

to the suspended vertical cable support structures that act as beams. The

tensile beams are laterally stabilized by the glass or braced by stainless

steel rods.

The dead loads are usually transferred from the glass panels to vertical

tension rods, or each panel is hung directly from the next panel above; in

other words, the upper panels carry the deadweight of the lower panels in

tension.

The structural and thermal movements in the glass wall are taken up by the

resiliency of the glass-to-glass silicone joints and, for example, by ball-

jointed metal links at the glass-to-truss connections, thereby preventing

stress concentrations and bending of the glass at the corners.

Sony Center, Potzdamer Platz,

Berlin, 2000, Helmut Jahn Arch.,

Ove Arup USA Struct. Eng

World Trade Center,

Amsterdam, 2003, Kohn,

Pedersen & Fox

World Trade Center, Amsterdam, 2002, Kohn

Pedersen Fox Arch

World Trade Center, Amsterdam, 2003 (?), Kohn, Pedersen & Fox

Underground shopping Xidan Beidajie, Xichang’an Jie, Beijing

Clouds of the Great Arch of

La Défense, Paris, France,

1989, Johan Otto von

Spreckelsen Designer,

Peter Rice/Arup Struct. Eng

Cable beams

Cable Beams

a

b

c

Shopping Center, Jiefangbei

business district, Chongqing,

China

Medical Center Library,

Vanderbilt University, Nashville,

TE, 1992, Davis Brody Arch

Commonwealth Edison

Transmission/Distribution Center, Chicago, IL,

SOM Arch – Hal Iyengar Struct. Eng

Xinghai Square shopping mall, Dalian, China

Standard Hall,

Stuttgart Trade Fair

Center, Stuttgart,

Germany, 2007, Wulf

Arch, Mayr Ludescher

Struct. Eng

Cable-Supported Columns

a. b. c. d. e.

P3

P2 P

2

P1

.5

¾-i

n.

rod

Petersbogen shopping center, Leipzig,

2001, HPP Hentrich-Petschnigg

Kansai International Airport,

1994, Renzo Piano Arch, Ove

Arup Struct Eng

Cité des Sciences et de l'Industrie, Paris, 1986, Peter Rice/Arup

OZ Building,

Tel Aviv,

Israel, 1995,

Avram Yaski

Arch,

Octatube

Greenhouse Pavillons Parc

Citroen, Paris, France, 1992,

Patric Berger Arch, Peter

Rice/Arup Struct. Eng

Unileverhaus Hamburg, Germany , 2009, Behnisch

Architekten, Weber Poll Struct. Eng

Ringseildächer mit CFK-Zugelementen,

Bautechnik 91(10) · September 2014, Mike

Schlaich, Yue Liu*, Bernd Zwingmann

Cable beams

Utica Memorial Auditorium, Utica, New York, 1960, Gehron & Seltzer and Frank

Delle Cese Arch, Lev Zetlin Struct. Eng

Maracanã Stadium Roof Structure, Maracanã,

Rio de Janeiro, 2013, Schlaich Bergermann

Arch and Struct. Eng

Mercedes Benz Arena, Stuttgart, Germany, 1993, Asp Arch, Schlaich

Bergermann Struct Eng

Tensegrity Frames

Typical planar tensegrity frames are shown in Fig. 11.21, where suspended

cables are connected to a second set of cables of reverse curvature to form

pretensioned cable trusses, which remain in tension under any loading

condition. In other words, visualize a single suspended (concave) cable, the

primary cable, to be stabilized by a secondary arched (convex) cable or

prestressing cable. This secondary cable can be placed on top of the

primary cable by employing compression struts, thus forming a lens-shaped

beam (Fig. 11.10a), or it can be located below the primary cable (either by

touching or being separated at center) by connecting the two cables with

tension ties or diagonals (c). A combination of the two cable configurations

yields a convex-concave cable beam (b).

The use of the dual-cable approach not only causes the single flexible cable to

be more stable with respect to fluttering, but also results in higher strength and

stiffness. The stiffness of the cable beam depends on the curvature of the

cables, cable size, level of pretension, and support conditions. The cable

beam is highly indeterminate from a force flow point of view; it cannot be

considered a rigid beam with a linear behavior in the elastic range. The

sharing of the loads between the cables, that is, finding the proportion of the

load carried by each cable, is an extremely difficult problem.

In the first loading stage, prestress forces are induced into the beam structure. The initial

tension (i.e. prestress force minus compression due to cable and spreader weight) in the

arched cable should always be larger than the compression forces that are induced by the

superimposed loads due to the roofing deck and live load; this is to prevent the convex cable

and web ties from becoming slack.

Let us assume that under full loading stage all the loads, w, are carried by the suspended

cables and that the forces in the arched cables are zero. Therefore, when the superimposed

loads are removed, equivalent minimum prestress loads of, w/2, are required to satisfy the

assumed condition, which in turn is based on equal cross-sectional areas of cables and equal

cable sags so that the suspended and arched cables carry the same loads.

Naturally, the equivalent prestress load cannot be zero under maximum loading conditions

since its magnitude is not just a function of strength as based on static loading and initial

cable geometry, but also of dynamic loading including damping (i.e. natural period), stiffness,

and considerations of the erection process. The determination of prestress forces requires a

complex process of analysis, which is beyond the scope of this introductory discussion. It is

assumed for rough preliminary approximation purposes that the final equivalent prestress

loads are equal to, w/2 (often designers us final prestress loads at lest equal to live loads,

wL).

It is surely overly conservative to assume all the loads to be supported by the

suspended cable, while the secondary cable’s only function is to damp the vibration of

the primary cable. Because of the small sag-to-span ratio of cable beams, it is reasonable to

treat the maximum cable force, T, as equal to the horizontal thrust force, H, for preliminary

design purposes.

b

a

c

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Planar tensegrity frames

Cases: Gravity, Prestress, Gravity + Prestress

Planar tensegrity frames