The cable in building structures

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

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

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

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.

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)

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.

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.

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 ,
AFAAssociados 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

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'
0
4 
±
±
R = 207 ft
Asymmetrical Suspended Roof Structure

45'53.35'
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

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

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.

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

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'
10.10k
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 deg
30 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
50
0 50 0
500
500 500
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

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

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'
20'20'20'20'20'30'
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

Low-profile , long-span pneumatic roof structures

Effect of internal air pressure on geometry

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

Effect of wind loading on
spherical membrane shapes

Air-inflated
members and
Example 9.14

Air-supported cylindrical membrane
structure

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

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

Methods for stabilizing
cable structures

Suspended, load-carrying
membrane force
Arched, prestress
membrane force
f
f
wp
T2
T2
T1 T1
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

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
12'4'4'12'4'4'12'
40'8' 8'
2'±4'±4'
P
a. b. c. d. e.
P3
P2
P2
P1.5
¾-in.rod
a. b. c. d. e.
P3
P2
P2
P1.5
¾-in.rod
Cable-Supported Columns (spatial units)
Planar open tensegrity structures
Cable frames

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

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
P2
P1.5
¾-in.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

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
12'4'4'12'4'4'12'
40'8' 8'
2'±4'±4'
P
Planar tensegrity frames

Cases: Gravity, Prestress, Gravity + Prestress
Planar tensegrity frames

The cable in building structures

The cable in building structures

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