CABLE 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 , 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.

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.

These antigravity 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.

The basic types of cable-supported structures are as follows:

Single-layer, cable-suspended structures: single-curvature and dish-shaped (synclastic) hanging roofs

Prestressed tensile membranes and cablenets (see Ch. 12.5) edge-supported saddle roofs

mast-supported conical saddle roofs

arch-supported saddle roofs

air-supported structures; 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: combinations of the preceding systems

Some historically important

cable structures

Suspended Theater Roof, 1824, Friedrich Schnirch

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

Tower Bridge, London, 1894

different cables

for different

load cases

19th century examples

Experiments with structure,

Russian Constructivism

Pavilion, Chicago, 1933, Bennett & Associates

Dymaxion House, 1923, Buckminster Fuller

Shabolovka tower, Vladimir Shukhov, 1922, Moscow

proposal Palazzo del Congress, Venice, 1969, Louis Kahn

Golden Gate Bridge (one 2224 ft), San

Francisco, 1936, C.H. Purcell

Kenneth Snelson, Needle Tower, 1968,

Hirshorn Museum, Washington; this 60-ft

high (18 m) tower explores the spatial

interaction of tension and compression.

A network of continuous cables is

prestressed into shape by discontinuous

compression struts which never touch

each other. Buckminster Fuller explained

tensegrity as tensile integrity, as

islands of compression in a sea of

tension

Examples of cable structures

The 22-story, 100-m high, BMW Building in Munich,

Germany (1972, Karl Schwanzer) consists of four suspended

cylinders. Here, four central prestressed suspended huge

concrete hangers are supported by a post - tensioned bracket

cross at the top that cantilevers from the concrete core.

Secondary perimeter columns are carried in tension or

compression by story-high radial cantilevers at the

mechanical floor level. Cast aluminum cladding is used as

skin.

Westcoast Transmission Tower, Vancouver, Canada, 1969

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

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

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

Lufthansa Hangar (153 m), Munich, 1992, Buechl + Angerer

TU Munich

Incheon International Airport, Seoul

proposal Shanghai-Pudong Museum, von Gerkan, Marg and Partner

Temporary American Center, Paris,

1991, Nasrin Seraji

Newark air terminal C, USA

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

Examples of cable-supported columns

World Trade Center, Amsterdam, 2003 (?), Kohn,

Pedersen & Fox

project by Eric Owen Moss Architects (EOMA)

Luxembourg, 2007

Lehrter Bahnhof, Berlin, 2006, von Gerkan, Marg and Partners

Olympic Stadium, Munich, Germany, 1972, Frei Otto, Leonhardt-Andrae

CABLE STRUCTURES

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 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.

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.

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)

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.

Cables refer to flexible tension members consisting of rods or one, or

more groups of wires,

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

strands

ropes

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 = P = 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

SINGLE-LAYER, SIMPLY

SUSPENDED CABLE ROOFS SINGLE-CURVATURE and DISH-SHAPED,

SYNCLASIC, 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.

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

Paper factory Burgo, Mantua, 1962, Pier Luigi Nervi

Maison de la Culture, Firminy, 1965,

Le Corbusier

Braga Stadium, Braga, Portugal, 2004, Eduardo

Souto de Moura , AFA Associados with Arup

Trade Fair Hannover, Hall 9, von Gerkan

Marg and Partners, 1997, Schlaich

Trade Fair Center, Stuttgart, 2007,

Wulf & Partners

Suspended roof, Hohenems,

Vorarlberg, Austria

Portuguese

Pavilion, Expo 98,

Lisbon, Alvaro Siza

Lufthansa-

maintanance hangar

V, Frankfurt, Germany,

1972, ABB Architects,

Dyckerhoff and

Widmann

The David L. Lawrence Convention Center, Pittsburgh, PA, 2003, R. Vinoly

L = 140

30'

14'14'

f = 9.33'

H

H

V

Tmax

o

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 -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 o (roll down angle); this angle also represents the slope of the curvature at the reactions.

sin o= (L/2)/R =70/267.26 = 0.262, o = 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 = d2/4 = 3.80 or d 2.20 in

Try 2 -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 = (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 HALL 26, HANOVER,

1996, HERZOG ARCH, SCHLAICH

45'

53'

30'

15'

30'198'

213'

04

R = 207 ft

Suspended Roof Structure

45

'5

3.3

5'

30'

15'

30'193.70'

208.70'

150'

63'

Dulles Airport, Washington, 1962, Eero Saarinen/ Fred Severud, 161-ft (49 m)

suspended tensile vault

AWD-Dome

(Stadthalle), Bremen,

Germany, 1964,

Klumpp, Dyckerhoff &

Widmann AG

PRESTRESSED TENSILE MEMBRANES

and CABLE NETS

edge-supported saddle roofs

arch-supported saddle roofs

MAST-SUPPORTED CONICAL SADDLE ROOFS anticlastic surface structures tensioned by cables and masts

HYBRID SURFACE STRUCTURES

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

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

Olympic Stadium, Munich, Germany, 1972, Gnther Behnisch architect +

Frei Otto, Leonhardt-Andrae

Olympic Stadium, Munich, Germany, 1972, Frei Otto, Leonhardt-Andrae

Haj Terminal, King Abdul Aziz International Airport, Jeddah, Saudi Arabia, 1982, SOM

CABLE-SUPPORTED STRUCTURES

Cable-Supported Beams and Arched Beams

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.

Golden Gate Bridge (one 2224 ft), San

Francisco, 1936, C.H. Purcell

Akashi-Kaikyo-Bridge,

Japan, 1998, 1990 m span

curved suspension bridge, Bochum,

Germany, 2003, von Gerkan Marg

Burgo Paper Mill,

Mantua, Italy, 1963,

Pier Luigi Nervi

Airport hangar Biala Podlaska, Poland

Milleneum Bridge,

London, 2000, Foster,

Arup

Old Federal Reserve Bank Building, Minneapolis, 1973, Gunnar Birkerts, 273-ft (83

m) span truss at top

Cable-supported

structures

German Museum of

Technology Berlin,

2001, Helge Pitz and

Ulrich Wolff Architects

Auditorium of the Technical University, Munich, Germany

Wilkhahn Factory, Bad Muender, Germany,

Herzog Arch., 1992

Integrated urban buildings, Linkstr. Potsdamer Platz,Berlin, 1998, Richard Rogers

Mercedes-Benz Center am Salzufer, Berlin, 2000,

Lamm, Weber, Donath und Partner

Cable supported bridge, Berlin

Shopping Center, Stuttgart

Shopping street in Wolfsburg, Germany

Shopping street in Bauzen, Germany

Surrey Central City Galleria roof,,Surrey, British Columbia, 2002, Bing Thom

Architects, StructureCraft

Concord Sales Pavilion, Vancouver,2000, Busby + Associates Architects,

StructureCraft

Debis Theater, Marlene Dietrich Platz, Berlin, 1998, Germany, Renzo Piano

World Trade Center, Amsterdam, 2003 (?), Kohn,

Pedersen & Fox

Bus shelter, Schweinfurt, Germany

StructureCraft, Vancouver, Canada

Cable-Supported Beams

a

b

c

d

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

The parabolic spatial arch structure with its 42-m cantilevers is supported on only two monumental

conical concrete-filled steel pipe columns spaced at 124 m. The main span of the roof structure (which

is about 12 m deep at midspan) consists of a pair of 1.2 m tubular inclined steel arches that span

124 m between the columns and curve up in half-arches in the cantilever portion. A series of 16

tension rods inversely curved to the compression arches complete the beam action. The layout of the

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.

Kempinski Hotel, Munich,

Germany, 1997, H. Jahn/Schlaich:

the elegance and lightness of the the

40-m (135-ft) span glass and steel

lattice roof is articulated through the

transparency of roof skin and the

almost non-existence of the diagonal

arches which are cable- supported

by a single post at their

intersection at center span. This

new technology features construction

with its own aesthetics reflecting a

play between artistic, architectural

mathematical, and engineering

worlds. The depth of the box arches

is reduced by the central

compression strut (flying column)

carried by the suspended tension

rods. The arches, in turn, are

supported by tubular trusses on each

side, which separate the roof from

the buildings.

Museum Courtyard Roof (1989), Hamburg, glass-covered grid shell over L-shaped

courtyard, Architect von Gerkan Marg und Partner

ba

4'

4'

c

4'

4'

40'

4'

4'

Cable-Supported Arched Beams

the Living Bridge, Limerick University ,

Ireland, 2007, Wilkinson Eyre Architects

Kansai International Airport,

Renzo Piano, 1994

Munich Airport Center, Munich, Germany, 1997, Helmut Jahn Arch.: the open public atrium as

transition, building blocks form walled boundaries to a square which is covered by a transparent roof

hanging from stayed cables, with a minimum of structure that gives a strong identity to space - the

new technology features construction with its own aesthetics reflecting a play between artistic,

architectural mathematical, and engineering worlds.

Lehrter Bahnhof, Berlin, 2006, von Gerkan, Marg and Partners

Lehrter Bahnhof, Berlin, 2002, Gerkan, Marg & Partner, Mero

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

5.86'

27.32'10'

4.29'

10.1

0 k

7.70 k

Mmax

Mmin

Waterloo Terminal, London, 1993, Nicholas Grimshaw

+ Anthony Hunt

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

b e

c f

L = 40'

10'

6'

12'

10'

AD E

B C

Cable-stayed bridge systems

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.

Marcaibo Bridge, Venezuela ,

1962, Riccardo Morandi

Oberkassel Rhine Bridge,

Germany, 1973

Friedrich-Ebert-

Bridge, Bonn,

Germany, 1967

3rd Orinoco Brcke, Venezuela, 2010, Harrer Ingenieure GmbH

New Mississippi River Bridge

Record-breaking cable stayed road bridge Currently under detailed design

Advanced 3D nonlinear, dynamic and staged construction analyses with LUSAS Bridge

When built, the New Mississippi River Bridge will be a record-breaking, cable-stayed structure linking the States of Illinois and

Missouri in the USA, helping to relieve traffic on other bridges across the river. The designer, Modjeski and Masters, was

chosen by the Illinois and Missouri Departments of Transportation to perform both the bridge-type study and to provide

preliminary and final design for the proposed bridge.

Facts and figures

At 222 feet (68m) in width, the Mississippi River Bridge will be the worlds widest cable-stayed structure. It will carry eight traffic lanes with shoulders that provide for four additional lanes in the future.

The total length of the bridge is approximately 3,150 feet (961m).

The main span of 2,000 feet (610m) will be the longest clear span across the Mississippi River, the longest cable-stayed span

in the Western hemisphere and the fifth-longest cable-stayed span in the world.

Two 510 foot (155m) high single pylon towers will soar 435 feet (133m) above the roadway.

It will be the first major cable-stayed bridge to use three planes of cables in the main span.

Speyer Rhine Bridge,

Germany, 1975

Alamillo Bridge,

Sevilla, Spain,1992,

Santiago Calatrava

Three bridges over the Hoofdvaart Haarlemmermeer, the

Netherland, 2004, Santiago Calatrava

Erasmus Bridge, Rotterdam, 1996, architect Ben Van Berkel

Bangkok

Zakim Bunker Hill Bridge,

Boston, 2003

Bridge, Hoofddorp, Netherlands,

2004, Santiago Calatrava

Willemsbridge, Rotterdam, 1981, is a double suspension bridge, C.Veeling

Pedestrian Bridge Bad Homburg, 2002,

Architect Schlaich

Miho Museum Bridge, Shiga,

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

Robertrson

Ruck-a-chucky Bridge, Myron Goldsmith/ SOM, T.Y. Lin

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

Fiumicino Airport (Alitalia Hangar) , Rome International Airport, Rome, 1970,

Riccardo Morandi

Ice Hockey Rink, Squaw

Valley, CA, 1960, Corlett &

Spackman

Lufthansa Hangar (153 m), Munich, 1992, Buechl + Angerer

INMOS microprocessor factory, Newport, Gwent , 1987, Richard Rogers

& Partners, Hunt

Convention Center Trade Fair Hanover, 1989, H. Storch & W. Ehlers

Fleetguard Factory, Quimper, France, 1981, Richard Rogers

Waking Pool, 1989, F. Browns

The University of Chicago Gerald Ratner

Athletic Center, Cesar Pelli, 2002

HYDRAULIC JACKING SYSTEM USED TO TENSION CABLES

Cable-Stayed Roof

Structures

b

c

d

W14 x 30

W14 x 22

P6

P10

P8

W14 x 43

P8

P5

W14 x 26

P8

P5

a

Patcenter, Princeton, 1984, Richard Rogers

Peter Rice of Ove Arup & Partners

The principle of a central support with large free spaces on either side was

established after early discussions with the architect. The structural frame has

four separate elements. These are the tension support element, which consists of

a compression A-frame with hangers supporting a horizontal roof beam on either

side. This horizontal roof beam spans 25 m and the suspension system is placed

at 9 m centres. The A-frame sits on the second element, which is a portal frame

designed to resist the horizontal load and the vertical asymmetric load transmitted

to it by the A-frame. The third component in the system is the tie-down columns

which support the two suspended beams and resist uplift. These beams will

themselves resist uplift under wind load through the tension support members

and the beam acting together as an uplift beam. The fourth component is the

suspended platforms for the services capsules and the longitundinal bracing. The

structural solution attempted to achieve four aims:

1. All the horizontal forces associated with the vertical support system are

resolved at roof level. This means that only the external horizontal loads (wind

loads) are transferred to ground level, and this is done through the central portal,

so there is the minimum interruption to flow of space across the centre of the

plan. This is all that is required even when vertical loads on one side {such as

drifting snow loads) give large asymmetric horizontal forces in upper triangular

frame.

2. The uplift loads of wind are separately catered for to ensure that the roof

would be truly lightweight. The uplift beam and the suspension system work

together.

3. The stability of the support frame normal to its plane avoids direct stabilising

members to the ground. The stability is provided by ensuring that the way the

compression members of the A-frame rotate out of plane produces restoring

forces on the frame.

4. The bulk of the steel weight is in standard steel construction, with only some

special visible external elements being designed in non- conventional rods and

pin-ended columns. This is important in the environment of the American

construction industry which penalises non- conventional construction heavily.

The early sketches did not have the suspended plant room capsules. This meant

that although the A-frame was basically stable geometrically, it felt unsafe

visually. By using the hangers of suspended platforms to stabilise top of the A-

frame, the frame was stiffened, and the assembly felt visually more stable.

Also in the early sketches of the tension support system, a symmetrical

arrangement of supports was used. It was found in the detailed analysis that

these did not remain in tension in all load cases. To solve this the geometry was

changed to that which is now to be built.

Bangkok

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

Schlaich Bergermann

City Manchester Soccer Stadium,

Manchester, UK, 2003, ARUP

Architects and Engineers

The most visible features of the stadium are the 12

support masts (shown in blue, above). Tensile forces

are maintained in the cable net under

all loading conditions.

Millenium Dome (365 m), London, 1999, Rogers + Happold

TENSEGRITY TRIPOD

TENSEGRITY

tensile integrity

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

Open tensegrity structures

planar open 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.

Twist unit: X Piece

Kenneth Snelson, Needle Tower, 1968,

Hirshorn Museum, Washington; this 60-ft

high (18 m) tower explores the spatial

interaction of tension and compression.

A network of continuous cables is

prestressed into shape by discontinuous

compression struts which never touch

each other. Buckminster Fuller explained

tensegrity as tensile integrity, as

islands of compression in a sea of

tension

Tensegrity sculptures by K. Snelson

SPHRERICAL ASSEMBLY OF TENSEGRITY TRIPODS

DOUBLE - LAYER TENSEGRITY DOME

The Skylon tower (172.8 m)

at the Festival of Britain,

London, 1951, Hidalgo Moya,

Philip Powell Arch

Warnowturm Rostock,

Rostock, Germany, 2003,

Gerkan, Marg & Partner

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 Fullers aspension (ascending suspension) tensegrity domes.

Cable Beams

a

b

c

b

a

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Cable-Supported Columns (spatial units)

Planar open tensegrity structures

Cable frames

Fullers tensegrity dome

FULLERS TENSEGRITY DOME GEIGERS CABLE DOME

Spatial open tensegrity structures

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 Fullers 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.

Geigers 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 Fullers 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.

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.

Olympic Fencing and Gymnastics Arenas,

Seoul, 1989, Geiger

Olympic Fencing and Gymnastics Arenas, Seoul, 1989, Geiger

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,

Georgia Dome, Atlanta, 1995,

Weidlinger, Structures such as the

Hypar-Tensegrity Dome, 234 m x 186 m

Georgia Dome, Atlanta, 1992, HYPAR TENSEGRITY DOME

The worlds largest cable dome is currently Atlantas 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 Geigers 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.

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 roofs 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.

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.

Shanghai-Pudong International Airport, 2001, Paul Andreu principal architect,

Coyne et Bellier structural engineers

Petersbogen shopping center,

Leipzig, 2001, HPP Hentrich-

Petschnigg

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.

World Trade Center,

Amsterdam, 2003, Kohn,

Pedersen & Fox

Underground shopping Xidan Beidajie, Xichangan Jie, Beijing

Utica Memorial Auditorium, Utica, New York, 1965, Lev Zetlin

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

Sony Center, Potzdamer Platz,

Berlin, 2000, Helmut Jahn

Arch., Ove Arup USA Struct.

Eng

Cable Beams

a

b

c

Newark air terminal C, USA

MUDAM, Museum of Modern Art, Luxembourg, 2006, I.M. Pei

Chongqing shopping center

Shopping Center Dalian, China

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

Cable-Supported Columns

a. b. c. d. e.

P3

P2 P

2

P1

.5

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n.

rod

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 cables 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.

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