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, (1) The Design of Building Structures (Vol.1, Vol. 2), rev. ed., PDF eBook by Wolfgang Schueller, 2016, published originally by Prentice Hall, 1996, (2) Building Support Structures, Analysis and Design with SAP2000 Software, 2nd ed., eBook by Wolfgang Schueller, 2015. The SAP2000V15 Examples and Problems SDB files are available on the Computers & Structures, Inc. (CSI) website: http://www.csiamerica.com/go/schueller
Introduction Most tensile structures are very flexible in comparison to conventional structures. This is particularly true for the current, fashionable, minimal structures, where all the members want to be under axial forces. Here, repetitive members with pinned joints are tied together and stabilized by cables or rods. Not only the low stiffness of cables, but also the nature of hinged frame construction, make them vulnerable to lateral and vertical movements. To acquire the necessary stiffness, special construction techniques have been developed, such as spatial networks, as well as the prestressing of tension members so that they remain in tension under any loading conditions. Because of the lightweight and flexible nature of cable-stayed roof structures they may be especially vulnerable with respect to vertical stiffness, wind uplift, lateral stability, and dynamic effects; redundancy must also be considered in case of tie failure. Temperature effects are critical when the structure is exposed to environmental changes. The movement of the exposed structure must be compatible with the enclosure. In the partially exposed structure, differential movement within the structure must be considered; slotted connections may be used to relieve thermal movement.
In traditional gravity-type structures the inherent massiveness of material transmits a feeling of stability and protection.
In contrast, tensile structures seem to be weightless and to float in the air; their stability is dependent on induced tension and on an intricate, curved three-dimensional geometry in which the skin is pre-stretched.
Antigravity roof structures require a new aesthetics; now the curve rather than the straight line, is the generator of space. The aesthetics is closely related to biological structures and natural forms – there is no real historical precedent for the complex forms of membrane structures. Fabric structures are forms in tension – as nearly weightless structures they are pure, essential, and minimal. Spatial, curved geometry, together with induced tension is necessary for structural integrity.
CABLES in STRUCTURES Lateral bracing Suspended highrise structures (tensile columns) Single-layer, simply suspended cable roofs Single-curvature and dish-shaped (synclastic) hanging roofs
Prestressed tensile membranes and cable nets (see Surface Structures) Edge-supported saddle roofs Mast-supported conical saddle roofs Arch-supported saddle roofs Air supported structures and air-inflated structures (air members)
Cable-supported structures cable-supported beams and arched beams cable-stayed bridges cable-stayed roof structures
Tensegrity structures Planar open and closed tensegrity systems: cable beams, cable trusses, cable frames Spatial open tensegrity systems: cable domes Spatial closed tensegrity systems: polyhedral twist units
Hybrid structures Combination of the above systems
In typical cable-suspended structures the cables form the roof surface structure, whereas in cable-supported structures cables give support to other members. Tensile structures such as tensile membranes and tensegrity structures are pretensioned structures so they can resist compression forces, however, guyed structures may also be prestressed structures.
Cables form tensile beams and membranes, or assist beams, columns, surface structures or other member types as inclined stays or suspended members. Today, the principle is applied to cranes, ships, television towers, bridges, roof structures, the composite tensile cladding systems of glass and stainless steel, and to entire buildings.
In cable structures, tensile members, such as ropes, strands, rods, W-shapes , prestressed concrete members, chains, or other member types, are main load-bearing elements; they can be an integral part of a structural system and can give primary support to linear members, surfaces, and volumes from above or below, as well as brace buildings against lateral forces; cables have low bending and torsional stiffness compared to their axial tensile stiffness.
Cables refer to flexible tension members consisting of,
rods, plates, W-sections, tubes, etc. strands, ropes, tensile reinforced concrete columns wood members Wires are laid helically around a center wire to produce a strand, while ropes are formed by strands laid helically around a core (e.g. wire rope or steel strand).
STRAND An assembly of wires Around a central core
Z-lock CABLE
WIRE ROPE Assembly of strands
Steel strand and wire rope are inherently redundant members since they consist of individual wires. The minimum ultimate tensile strength Fu of strands and ropes is in the range of 200 to 220 ksi (1379 to 1517 MPa) depending on the coating class (and 270 ksi =1862 MPa for prestressing strand). The strand has more metallic area than the rope of the same diameter and hence is stronger and stiffer. The minimum modulus of elasticity of wire rope is 20,000 ksi (138,000 MPa) and 24,000 ksi = 165,000 MPa for strands of nominal diameters up to 2 9/16 in. (65 mm) and 23,000 ksi (159,000 MPa) for the larger diameters. The cable capacity can be obtained from the manufacturer's catalogues, but for rough preliminary design purposes of cable sizes assume a metallic cable area As of roughly 60 percent of its nominal gross area An for ropes and 75 percent for strands. The ultimate tensile force is, Pu = γP = 2.2P. Hence the required nominal cross-sectional cable area as based on 67 percent increase of the required gross area An for ropes and 33 percent for strand, is
Some historically significant cable structures
19th century examples
Suspended Theater Roof, 1824, Friedrich Schnirch
The first suspended roof: prototype, Banska Bystrica, Slovacia, 1826, Bedrich Schnirch Arch
Bollman Iron Truss Bridge, Savage, MD, 1869, Wendel Bollman
Tower Bridge, London, 1894, Horace Jones Arch, John Wolfe Barry Struct. Eng
different cables for different load cases
Transat Chair, 1927, Eileen Gray Designer
Iakov Chernikhov’ s experiments with architectural structures, 19251932, Russian Constructivism
Pavilion, Chicago, 1933, Bennett & Associates
Dymaxion House, 1923, Buckminster Fuller
Shabolovka tower, Moscow, 1922, Vladimir Shukhov
Golden Gate Bridge (longest span 4200 FT), San Francisco, 1937, Joseph Strauss, Irving Morrow and Charles Ellis Designers
Lateral tensile bracing Highrise suspension buildings (tensile columns)
Reliance Controls factory, Swindon, 1967, Team 4, Anthony Hunt Struct. Eng
Stansted Airport, London, 1991, Norman Foster Arch, Ove Arup Struct. Eng.
Sainsbury Centre for the Arts, Norwich, England, 1977, Norman Foster Arch
Newark air terminal C, USA
Peek & Cloppenburg, Cologne, Germany, 2005, Renzo Piano Arch, Knippers Helbig Struct. Eng (façade)
Pavilion of the Future, Seville, Spain, 1992, Peter Rice/Arup Struct. Eng
Highrise suspension structures
Tivoli Stadion, Aachen, Germany, 2009, Paul Niederberghaus + Hellmich Arch
Sainsburys Store, Camden Town, London, 1988, Nicholas Grimshaw Arch, Kenchington Little Struct. Eng
Centre Georges Pompidou, Paris, France, 1977, Piano & Rogers Arch, Peter Rice/Ove Arup and Edmund Happold Struct.Eng
Office building of the European Investment Bank, 2009, Luxembourg, Ingenhoven Architects, Werner Sobek Struct. Eng
Fondation Avicienne (Maison de l'Iran), Cité Internationale Universitaire, Paris, 1969, Claude Parent + Moshen Foroughi et Heydar Ghiai Arch
Media TIC Building, Barcelona, Spain, 2010, Enric Ruiz-Geli Arch, Agusti Obiol – BOMA Struct. Eng
Ludwig Erhard Haus, Berlin, Germany, 1999, Nick Grimshaw Arch
Exchange House, London, 1990, SOM Arch + Strct. Eng
Poly Corporation Headquarters, Beijing, China, 2007, SOM Arch + Struct. Eng
Old Federal Reserve Bank Building, Minneapolis, 1973, Gunnar Birkerts, 273-ft (83 m) span truss at top
Laboratory building, Heidelberg, Germany, Rossmann & Partner Arch
German Museum of Technology Berlin, 2001, Helge Pitz and Ulrich Wolff Architects
House (World War 2 bunker), Aachen, Germany
Auditorium of the Technical University, Munich, Germany
TU Munich
Shanghai-Pudong Museum, Shanghai-Pudong, China, 2005, von Gerkan, Marg & Partner Arch, Schlaich Bergermann und Partner Struct. Eng
German Museum of Technology, Berlin, 2001, Helge Pitz and Ulrich Wolff Architects
Standard Bank Centre, Johannesburg, South Africa, 1970, Hentrich-Petschnigg Arch
Westcoast Transmission Company Tower, Vancouver, Canada, 1969, Rhone & Iredale Arch, Bogue Babicki Struct.
BMW Towers, Munich, Germany, 1972, Karl Schwanzer Arch, Helmut Bomhard Struct. Eng
Hospital tower of the University of Cologne, Germany, Leonard Struct. Eng.
Visual study of Olivetti Building, Florence, Italy, 1973, Alberto Galardi
Olivetti Building, Florence, Italy, 1973, Alberto Garlardi Arch
Kleefelder Hängehaus (Norcon-Haus), Hannover, Germaqny, 1984, Schuwirth & Erman Arch
Torhaus am Aegi, Hanover, Germany, 2006, Storch Ehlers Arch, Eilers & Vogel Struct. Eng
Turning Torso, Malmö, Sweden, 2005, Santiago Calatrava Arch + Struct. Eng
Collserola Tower, Barcelona, Spain, 1992, Norman Foster Arch, Chris Wise/Arup Struct. Eng
Lookout Tower Killesberg (40 m), Stuttgart, 2001, Jörg Schlaich designer
The Single Cable • • • • • •
Funicular cables Cable action under transverse loads Parabolic cable Cubic parabolic cable Cable action under radial loads Prestretched cable
The deformation of a cable under its loads takes the shape of a funicular curve that is produced by only axial forces since a cable has negligible bending strength: polygonal and curved shapes (e.g. catenary shapes, parabolic shapes, circular shapes)
Funicular tension lines
The simple, flexible, suspended cable takes different shapes under different loading conditions; in other words, the cable shape and length are a function of loading and state of stress: • Polygonal shape (kinked shape) is a function of concentrated loads. • Curved shape is a function of uniform loads, a situation that is most typical in suspended roof structures. • Second degree parabolic shape is a function of constant uniform load, w, on the horizontal projection of the roof. This situation applies for live loads on shallow suspended roof structures (where the cables are arranged in a parallel fashion), in accordance with code requirements, and occurs in suspension bridges where the suspended cables carry the roadway. • Catenary shape or hyperbolic cosine (cosh) curve is a function of uniform load along the cable length (e.g., self weight). For small sag-to-span ratios of n ≤ 1:10, the geometry of a catenary and a parabola are practically the same so that the simpler parabola can be used. • Cubic parabolic shape is a function of uniformly distributed, tapered, transverse loads along the cable's horizontal base, such as a triangular, or trapezoidal-shaped load. These situations usually occur where cables are arranged in a radial fashion, such as in a typical circular suspension roof. • Circular shape is a function of constant uniform radial pressure, p. The radial forces cause cable forces of constant magnitude that are proportional to the radius of curvature. When these radial forces, however, are not constant and increase uniformly from a minimum at the center to a maximum at the edge, the cable takes an elliptical shape.
Polygonal cable
Prestretching cable
Cable vibration
The geometry of the loaded cable depends on the type of loading. Because typical computer programs only consider linear behavior that is small deflection theory, the cable geometry should not change too much under loading; it is important to define the cable geometry to be close to what is expected after the structure is loaded. For that reason it may be necessary to correct the cable geometry after one or more preliminary runs that determine the shape of the cable under the P-Delta load combination (e.g. dead and live loads for the typical gravity load case). However, keep in mind that for designing the cables, for example, in cable beams, gravity cannot act by itself since then the members have to be designed as compression members! Consider load combinations of gravity, wind loads, pre-stress, and temperature decrease of the cables, which produces shortening and causes significant axial forces. If the stretching of the cable is large it may not be possible to obtain meaningful results with a P-Delta load combination. The P-Delta effect can be a very important contributor to the stiffness of cable structures.
WHY IS IT NONLINEAR? Linear Elastic Theory approximates the length change of a bar by the dot product of the direction vector and the displacement. But in this situation, you can see from the figure above, that they are perpendicular to each other therefore dot product = 0. This would mean that the bar did not change length, which from observation is untrue. It is therefore necessary to use nonlinear analysis.
The Effects of Prestress The geometry of the structure itself is unstable as opposed to a structure shown at the right. The effects of prestress on the structure make it stronger. It is now able to counter the external forces.
The sum of the forces : 2T*(2d/L) = P P = (4T/L)d
Modeling of Cables Cable structures are flexible structures where the effect of large deflections on the magnitude of the member forces must be considered. Cable elements are tension-only members, where the axial forces are applied to the deflected shape. You can not just apply, for instance transverse loads, to a suspended cable with small moments of inertia using a linear analysis, all you get is a large deflection with no increase in axial forces because the change in geometry occurs after all the loads have been applied. To take the effect of large deflections into account, a P-Delta analysis that is a non-linear analysis has to be performed. Here the geometry change due to the deflections, , and the effect of the applied loads, P, along the deformed geometry is called the P- effect. The P-Delta effect only affects transverse stiffness, not axial stiffness. Therefore, frame elements representing a cable can carry compression as well as tension; this type of behavior is generally unrealistic. You should review the analysis results to make sure that this does not occur.
In SAP use cable elements for modeling. First define the material properties then model cable behavior by providing for each frame element section properties with small but realistic bending and torsional stiffness (e.g. use 1-in. dia. steel rods or a small value such as 1.0, for the moment of inertia). Do not use moment end-releases because otherwise the structure may be unstable; disregard moments and shear. Apply concentrated loads only at the end nodes of the elements, where the cable kinks occur. For uniform loads sufficient frame elements are needed to form a polygon composed of frame elements. SAP provides for the modeling of curved cables, Keep as Single Object or Break in Multiple Equal Length Objects.
Tensile structures (e.g. cable beams, tensile membranes) may have to be prestressed by applying external prestress forces, or temperature forces.
To perform the P-DELTA ANALYSIS in SAP, unlock the model after you have performed the linear analysis. Click Define > Analysis Cases > Modify/Show Case > in the Analysis Type area select the Nonlinear option. In the Other Parameters area, check the Modify/Show button for Results Saved and select Multiple States, then check the Modify/Show button for the Nonlinear Parameters edit box > in that form select the P-Delta with Large Displacements option in the Geometric Nonlinearity Parameters area then click the OK buttons and proceed with analysis as before. In other words, click Analyze > Set Analysis Options > select XZ Plane > click OK > click Run Analysis > click Run Now (i.e. click Run Analysis button). Notice, the educational version of SAP will run only the small displacement case with P-Delta.
Single-layer, cable-suspended structures: single-curvature and dish-shaped (synclastic) hanging roofs
Simply suspended or hanging roofs include cable roofs of single curvature and synclastic shape, that is cylindrical roofs with parallel cable arrangement, and polygonal dishes with radial cable pattern or cable nets. The simply suspended cables may be of the singleplane, double-flange, or double-layer type. The concept of simply suspended roofs has surely been influenced by suspension bridge construction. Most buildings using the suspended roof concept are either rectangular or round; in other words, the cable arrangement is either parallel or radial. However, in free-form buildings, the roof geometry is not a simple inverted cylinder or dish and the cable layout is irregular.
Simply suspended structures
proposal Palazzo del Congress, Venice, 1969, Louis Kahn
Portuguese Pavilion, Expo 98, Lisbon, Alvaro Siza Arch, Cecil Balmond (Arup) Struct. Eng.
Braga Stadium, Braga, Portugal, 2004, Eduardo Souto de Moura , AFA Associados with Arup
Lufthansa-maintanance hangar V, Frankfurt, Germany, 1972, ABB Architects, Dyckerhoff and Widmann
Trade Fair Hannover, Hall 9, von Gerkan Marg and Partners, 1997, Schlaich
Grand Hall, Stuttgart Trade Fair Centre, Stuttgart, Germany, 2007, Wulf Arch, Mayr Ludescher Struct. Eng.
Essingen stressed-ribbon footbridge over Main-Danube Canal, 1986, Richard Johann Dietrich Arch, Heinz Brüninghoff Struct. Eng
In the typical suspended roof the cables (or other member types such as W-sections, metal sheets, prestressed concrete strips) are integrated with the roof structure. Here, one distinguishes whether single- or double-layer cable systems are used. Simple, single-layer, suspended cable roofs must be stabilized by heavyweight or rigid members. Sometimes, prestressed suspended concrete shells are used where during erection they act as simple suspended cable systems, while in the final state they behave like inverted prestressed concrete shells. In simple, double-layer cable structures, such as the typical bicycle wheel roof, stability is achieved by secondary cables prestressing the main suspended cables.
The suspended cable adjusts its shape under load action so it can respond in tension. It is helpful to visualize the deflected shape of the cable (i.e. cable profile) as the shape of the moment diagram of an equivalent, simply supported beam carrying the same loads as the cable. The moment analogy method is useful since the magnitude of the moment, Mmax, can be readily obtained from handbooks. Hence, the horizontal thrust force, H, at the reaction for a simple suspended cable with supports at the same level and cable sag, f, is H = Mmax /f
Parabolic cable
Tmax V
H
θo
f = 9.33' H 30'
14'
L = 140 ‘
Suspended Roof Structure
14'
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 where, AD = πd2/4 = 3.80
or
(4.8)
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.
R = 207 ft
40
53'
45'
15' 30'
± 213' ± 198'
30'
Asymmetrical Suspended Roof Structure
53.35'
150'
45'
63'
15' 30'
208.70' 193.70'
30'
Maison de la Culture, Firminy, 1965, Le Corbusier
Dulles Airport, Washington, 1962, Eero Saarinen/ Fred Severud, 161-ft (49 m) suspended tensile vault
AWD-Dome (Stadthalle), Bremen, Germany, 1964, Klumpp Arch, Dyckerhoff & Widmann AG
Suspended roof, Hohenems, Vorarlberg, Austria, Reinhard Drexel Arch, Merz Kaufmann Struct. Eng
The David L. Lawrence Convention Center, Pittsburgh, PA, 2003, R. Vinoly Arch, Dewhurst MacFarlane Struct. Eng
Cable action under radial loads
Cubic parabolic cable
Suspended dished roof, axial force diagram
Prestressed tensile membranes and cable nets: edge-supported saddle roofs mast-supported conical saddle roofs arch-supported saddle roofs air-supported structures; air-inflated structures (air members) Hybrid surface structures
Tensile membrane structures
Kagawa Prefectural Gymnasium, Kagawa, Japan, 1964, Kenzo Tange Arch
Yoyogi National Gymnasium, Tokyo, 1964, Kenzo Tange Arch, Yoshikatsu Tsuboi Struct. Eng
Small Olympic Stadium, 1964, Tokyo, Kenzo Tange/ Y. Tsuboi
David S. Ingalls Skating Rink, New Haven, USA, 1958, Eero Saarinen Arch, Fred N. Severud Struct Eng
Jaber Al Ahmad Stadium Kuwait, Kuwait, 2005, Weidleplan Arch, Schlaich Bergemann Struct. Eng.
Khan Shatyr Entertainement Center, Astana, Kazakhstan, 2010, Norman Foster Arch, Bureau Happold Struct. Eng
The Great Flight Cage, The National Zoo, Washington DC, 1965, Richard Dimon (DMJM)
Cable-supported structures cable-supported beams and arches suspended cable-supported roof structures cable-stayed bridges cable-stayed roof structures
Cable-supported beams and roofs In contrast to cable-stayed roof structures, where cables give support to the roof framing from above, here the many possibilities of supporting framework from below are briefly investigated.
The conventional king-post and queen- post trusses, which represent single-strut and double-strut cable-supported beams, are familiar. These systems form composite truss-like structures with steel or wood compression members as top chords, steel tension rods as bottom chords, and compression struts as web members. Single-strut, cable-supported beams can also be overlapped in plane or spatially .
Subtensioned structures range from simple parallel to two-way and complex spatial systems, which however, are beyond the scope of this context.
Cable-supported structures
Single-strut and multistrut 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-BrauchleRing, 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
a
b
c
d
Cable-Supported Beams
a
b
c
d
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
4' 4'
4'
a
4'
4'
4'
b
c 40'
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
AWM Carport, Munich, 2012, Ackermann Arch + Struct Eng
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!
Pressure lines in elevation
Plan view
Asymmetrical arch
10
.10
k
Mmax
Mmin
7.70 k 5.86'
4.29'
10'
27.32'
EXAMPLE: 9.2: Asymmetrical composite arches
Waterloo Terminal, London, 1993, Nicholas Grimshaw Arch, Anthony Hunt Struct. Eng
2.68'
C. 10'
30 deg 17.32'
60 deg
Bh Bv 10' 30 deg
a.
Ah
20'
Av
17.32'
2.68'
7.32' 5.86' 17.32'
4.29'
b.
10'
27.32'
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 prestressing 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.
10'
a
d 6'
12'
b
e c
f
10'
L = 40'
g
Problem 9.1: Braced arches
10'
a
d 6'
12'
b
e c 10'
L = 40'
f
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 prestressing 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
B
C
Braced arches D
E
20'
10'
a
500 0
0
50
50
50 0
b
c
0
50
d
e
50 0
Introducing to the semicircular arch a horizontal tie rod (Problem 9.3) at midheight, 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 archsupported 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
a
c
b
e
d
f
Cable-Stayed Bridges
CABLE – STAYED ROOF STRUCTURES • Cable-stayed, double-cantilever roofs for central spinal buildings
• Cable-stayed, single-cantilever roofs as used for hangars and grandstands • Cable-stayed beam structures supported by masts from the outside • Spatially guyed, multidirectional composite roof structures
Cable-supported structures
Alitalia Hangar, Rom, Italy, 1960, Riccardo Morandi Arch,
Ice Hockey Rink, Squaw Valley, CA, 1960, Corlett & Spackman
Airport Munich Hangar 1 (153 m), Munich, 1992, Günter Büschl Arch, Fred Angerer Struct. Eng
Frankenstadion (Grundig), Nürnberg, Germany, 2005, Hentrich, Petschnigg Arch, K+S Struct Eng
INMOS microprocessor factory, Newport, Gwent , 1987, Richard Rogers & Partners, Anthony Hunt Struct. Eng
Fleetguard Factory, Quimper, France, 1981, Richard Rogers Arch, Peter Rice/Arup Struct. Eng
Renault Distribution Center, Swindon, England, 1982, Norman Foster Arch, Ove Arup Struct. Eng
Railway Station, Tilburg, Holland, 1965, Koen van der Gaast Arch
PATCenter, Princeton, USA, 1984, Richard Rogers Arch, Ove Arup Struct. Eng
Igus Headquarters & Factory, Cologne, Germany, 2000, Nicholas Grimshaw Arch, Whitby Bird Struct. Eng
Sainsbury’s supermarket, Canterbury, UK, 1984, Ahrends Burton Koralek Arch, Ernest Green Struct. Eng
Italian Industry Pavilion at Expo '70, Osaka, Japan, 1970, Renzo Piano Arch
The Sydney Convention And Exhibition Centre, 1986, Cox, Richardson, Taylor and Partners
The University of Chicago Gerald Ratner Athletic Center, Cesar Pelli, 2002
Temporary American Center, Paris, 1991, Nasrin Seraji Arch
Bangkok
Retractable roof, Castle Kufstein, 2006, Kugel + Rein, Architects und Engineers
New Waldstadion, Frankfurt/Main, 2005, von Gerkan-Marg Arch, SchlaichBergermann Struct Eng
Hannover 96 Arena, Hannover, Germany, 2004, Schulitz Arch, RFR Struct Eng
Stadium BC Place, Vancouver, Canada, 2011, Stantec/Cannon Arch, Geiger Berger+Schlaich Bergemann Struct Eng
National Stadium Warsaw, Poland, 2012, Gerkan Marg Arch, Schlaich Bergemann Struct Eng
Imtech Arena (Volksparkstadion), Hamburg, 1998, Mos Arch, Sclaich Bergermann Struct Eng (roof)
Rothenbaum Tennis Stadium, Hamburg, 1999, ASP Schweger Arch, Werner Sobek Struct Eng
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 Phenomenart, Fukuoka, 1989, Shoei Yoh + Architects
Westfalenstadion Dortmund, 2003, Schröder Schulte-Ladbeck Arch, Engels Struct Eng
RheinEnergie Football Stadium, Koeln, 2003, Van Gerkan-Marg Arch, Schlaich-Bergemann Struct Eng
Jeonju World Cup Stadium, Jeonju, South Korea, 2001, Pos A.C Arch, CS Struct. Eng
Inland Revenue Centre Amenity Building, Nottingham, UK, 1994, Michael Hopkins Arch, Ove Arup 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
W14 x 26 P5 P8
20'
a
W14 x 43 20'
P5 P8
b 20'
20'
20'
80'
W14 x 30 P6
P8
20'
5'
Typical Cable-supported Roof (beam) Structures
30'
c
10'
5'
10'
20'
W14 x 22
d P10 50'
80'
50'
Force flow in cable-supported roofs
Tensile Membrane Structures (typically cable nets with coated fabrics)
The basic prestressed tensile membranes are as follows: Pneumatic structures of domical and cylindrical shape (i.e., synclastic shapes) • Air-supported structures • Air-inflated structures (i.e., air members) • Hybrid air structures Anticlastic prestressed membrane structures • Edge-supported saddle roofs • Mast-supported conical saddle roofs • Arch-supported saddle roofs • Corrugate tensile roofs (radial, linear) Membrane surfaces as cladding Hybrid tensile surface structures (possibly including tensegrity)
Classification of tensile membranes
Pneumatic Structures Pneumatic structures may be organized as follows:
• Air-supported structures high-profile, ground-mounted air structures, and berm- or wall-mounted, low-profile roof membranes • Air-inflated structures (i.e., air members) Tubular systems (line elements) Dual-wall systems or airmats (surface elements) • Hybrid air structures
Classification of pneumatic structures
Pneumatic structures
Low-profile , long-span pneumatic roof structures
Effect of internal air pressure on geometry
Soap bubbles
In air-supported structures the tensile membrane floats like a curtain on top of the enclosed air, whose pressure exceeds that of the atmosphere; only a small pressure differential is needed. The typical normal operating pressure for air-supported membranes is in the range of 4.5 to 10 psf (0.2 kN/m2 to 0.5 kN/m2 = 0.5 kPa) or 2 mbar to 5 mbar, or roughly 1.0 to 2.0 inches of water as read from a water-pressure gage.
See also packing of soap bubbles
Kiss the Frog: the Art of Transformation, inflatable pavilion for Norway’s National Galery, Oslo, 2001, Magne Magler Wiggen Architect,
Traveling exhibition
Effect of wind loading on spherical membrane shapes
Air-inflated members and Example 9.14
Air-supported cylindrical membrane structure
p
T = pR
T = pR
Lense-shaped pneumatic bubble structure
Lense-shaped pneumatic bubble structure
Air Cushion Roof, F22 Diagram (COMB1)
Roman Arena Inflated Roof, Nimes, France, 1988, Architect Finn Geipel, Nicolas Michelin, Paris; Schlaich Bergermann und Partne; internal pressure 0.4…0.55 kN/m2
Expo 02 , Neuchatel, Switzerland, Multipack Arch, air cussion, ca 100 m dia.
US Pavilion, EXPO 70, Osaka, Davis-Brody Arch, Geiger – Berger Struct. Eng.
US Pavilion, EXPO 70, Osaka, DavisBrody
Pontiac Metropolitan Stadium , Detroit, 1975, O'Dell/Hewlett & Luckenbach Arch, Geiger Berger Struct. Eng.
Metrodome, Minneapolis, 1982, SOM Arch, Geiger-Berger Struct. Eng
Typical membrane roof details
Tensile foundation principles
Tension foundations
Anticlastic Prestressed Membrane Structures Membrane structures may be organized either according to their surface form or their support condition: • Saddle-shaped and stretched between their boundaries, representing orthogonal anticlastic surfaces with parallel fabric patterns. • Conical-shaped and center supported at high or low points, representing radial anticlastic surfaces with radial fabric patterns. • The combination of these basic surface forms yields an infinite number of new forms.
The following organization is often used based on support conditions: • • • •
Edge-supported saddle surface structures Arch-supported saddle surface structures Mast-supported conical (including point-hung) membrane structures (tents) Hybrid structures, including tensegrity nets
Tent architecture
Methods for stabilizing cable structures
Anchorage of tensile forces
Point-supported tents
Edge supports for cable nets
Examples 9.9 and 9.10
Arched, prestress membrane force
wp
f T1
T1 w
T2 Suspended, load-carrying membrane force
Anticlastic Tensile Membrane Forces
f
T2
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 nontouching 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.
DOUBLE - LAYER TENSEGRITY DOME
TENSEGRITY TRIPOD
Twist unit: X Piece
Tensegrity sculptures by K. Snelson
SPHRERICAL ASSEMBLY OF TENSEGRITY TRIPODS
The Skylon tower (172.8 m) at the Festival of Britain, London, 1951, Hidalgo Moya, Philip Powell Arch, Felix Samuely Struct. Eng
Warnow tower, Rostock, Rostock, Germany, 2003, Gerkan, Marg Arch
In contrast, open tensegrity structures are stabilized at the supports. Therefore, no diagonal compression members are required and shorter struts can be used. Open tensegrity structures can form planar or spatial structures. • Examples of planar systems include: cable beams, cable trusses, cable frames as shown in Fig.s 11.18a, 11.19 and 11.22. These structures can also form spatial units as shown in Fig.s 11.18c and Fig.11.21. • Examples of spatial systems include: flat or bent roof structures. Examples of the spatial open tensegrity systems are the tensegrity domes. David Geiger invented a new generation of low-profile domes, which he called cable domes. He derived the concept from Buckminster Fuller’s aspension (ascending suspension) tensegrity domes.
David Geiger invented a new generation of low-profile domes after his air domes, which he called cable domes. He derived the concept from Buckminster Fuller’s aspension (ascending suspension) tensegrity domes, which are triangle based and consist of discontinuous radial trusses tied together by ascending concentric tension rings; but the roof was not conceived as made of fabric. Geiger’s prestressed domes, in contrast, appear in plan like simple, radial Schwedler domes with concentric tension hoops. His domes consist of radioconcentric spatial cable network and vertical compression struts. In other words, radial cable trusses interact with concentric floating tension rings (attached to the bottom of the posts) that step upward toward the crown in accordance with Fuller’s aspension effect. The trusses get progressively thinner toward the center, similar to a pair of cantilever trusses not touching each other; the heaviest member occur at the perimeter of the span. In section, the radial trusses appear as planar and the missing bottom chords give the feeling of instability, which however, is not the case since they are replaced by the hoop cables that the the cables together.
Fuller’s tensegrity dome Spatial open tensegrity structures
The cable dome concept can also be perceived as ridge cables radiating from the central tension ring to the perimeter compression ring. They are held up by the short compression struts, which in turn, are supported by the concentric hoop (or ring) cables and are braced by the intermediate tension diagonals, as well as by the radial cables. A typical diagonal cable is attached to the top of a post and to the bottom of the next post. The pie-shaped fabric panels span from ridge cable to ridge cable and then are tensioned by the valley cables, thus being shaped into anticlastic surfaces; they contribute to the overall stiffness of the dome. The maximum radial cable spacing is limited by the strength of the fabric and detail considerations. The number of tension hoop is a function of the dome span. The sequence of erection of the roof network, which is done without scaffolding, is critical, that is, the stressing sequence of the posttensioned roof cables to pull the dome up into place.
The first tensegrity domes built were the gymnastics and fencing stadiums for the 1988 Summer Olympics in Seoul, South Korea. The 393-ft span dome for the gymnastics stadium required three tension hoops and has a structural weight of merely 2 psf. The 688-ft span Florida Suncoast Dome in St. Petersburg (1989) is one of the largest cable domes in the world. The dome is a four-hoop structure with 24 cable trusses and has a structural weight of only 5 psf. The dome weight is 8 psf, which includes the steel cables, posts, center tension ring, the catwalks supported by the hoop cables, lighting, and fabric panels. The translucent fabric consists of the outer Teflon-coated fiberglass membrane, the inner vinyl-coated polyester fabric, and an 8-in. thick layer of fiberglass insulation sandwiched between them. The dome has a 6o tilt and rests on all-precast, prestressed concrete stadium structure,
Olympic Fencing and Gymnastics Stadiums, Seoul, 1989, David Geiger Struct. Eng
The world’s largest cable dome is currently Atlanta’s Georgia Dome (1992), designed by engineer Mattys Levy of Weidlinger Associates. Levy developed for this enormous 770- x 610-ft oval roof the hypar tensegrity dome, which required three concentric tension hoops. He used the name because the triangular-shaped roof panels form diamonds that are saddle shaped. In contrast to Geiger’s radial configuration primarily for round cable domes, Levy used triangular geometry, which works well for noncircular structures and offers more redundancy, but also results in a more complex design and erection process. An elliptical roof differs from a circular one in that the tension along the hoops is not constant under uniform gravity load action. Furthermore, while in radial cable domes, the unbalanced loads are resisted first by the radial trusses and then distributed through deflection of the network, in triangulated tensegrity domes, loads are distributed more evenly.
Georgia Dome, Atlanta, 1992, Scott W. Braley Arch, Matthys P. Levy/ Weidlinger Struct. Eng.
The oval plan configuration of the roof consists of two radial circular segments at the ends, with a planar, 184-ft long tension cable truss at the long axis that pulls the roof’s two foci together. The reinforced-concrete compression ring beam is a hollow box girder 26 ft wide and 5 ft deep that rests on Teflon bearing pads on top of the concrete columns to accommodate movements. The Teflon-coated fiberglass membrane, consisting of the fused diamondshaped 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 prestress 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.
12'
P
4'
4'
Planar open tensegrity structures
a
a ±4' ±4'
b
12'
c
b
¾-in. rod
c 8'
40'
8'
P3
P2
P2
P1.5
12'
4' 2' 4'
Cable Beams
Cable frames
a.
b.
c.
d.
e.
Cable-Supported Columns (spatial units)
Shopping Center, Stuttgart
Cologne/Bonn Airport, Germany, 2000, Helmut Jahn Arch., Ove Arup USA Struct. Eng.
Suspended glass skins form a composite system of glass and stainless steel. Here, glass panels are glued together with silicone and supported by lightweight cable beams. Typically, the lateral wind pressure is carried by the glass panels in bending to the suspended vertical cable support structures that act as beams. The tensile beams are laterally stabilized by the glass or braced by stainless steel rods.
The dead loads are usually transferred from the glass panels to vertical tension rods, or each panel is hung directly from the next panel above; in other words, the upper panels carry the deadweight of the lower panels in tension. The structural and thermal movements in the glass wall are taken up by the resiliency of the glass-to-glass silicone joints and, for example, by balljointed 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
Sony Center, Berlin, 2000, H. Jahn Arch, Ove Arup Struct Eng
World Trade Center, Amsterdam, 2003, Kohn, Pedersen & Fox
World Trade Center, Amsterdam, 2002, Kohn Pedersen Fox Arch
World Trade Center, Amsterdam, 2003 (?), Kohn, Pedersen & Fox
Underground shopping Xidan Beidajie, Xichang’an Jie, Beijing
Clouds of the Great Arch of La Défense, Paris, France, 1989, Johan Otto von Spreckelsen Designer, Peter Rice/Arup Struct. Eng
Cable beams
a
b
c
Cable Beams
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
b.
c.
¾-in. rod P3
P2
P2
P1.5
a.
d.
Cable-Supported Columns
e.
Petersbogen shopping center, Leipzig, 2001, HPP Hentrich-Petschnigg
Kansai International Airport, 1994, Renzo Piano Arch, Ove Arup Struct Eng
Cité des Sciences et de l'Industrie, Paris, 1986, Peter Rice/Arup
OZ Building, Tel Aviv, Israel, 1995, Avram Yaski Arch, Octatube
Greenhouse Pavillons Parc Citroen, Paris, France, 1992, Patric Berger Arch, Peter Rice/Arup Struct. Eng
Unileverhaus Hamburg, Germany , 2009, Behnisch Architekten, Weber Poll Struct. Eng
Ringseildächer mit CFK-Zugelementen, Bautechnik 91(10) · September 2014, Mike Schlaich, Yue Liu*, Bernd Zwingmann
Cable beams
Utica Memorial Auditorium, Utica, New York, 1960, Gehron & Seltzer and Frank Delle Cese Arch, Lev Zetlin Struct. Eng
Maracanã Stadium Roof Structure, Maracanã, Rio de Janeiro, 2013, Schlaich Bergermann Arch and Struct. Eng
Mercedes Benz Arena, Stuttgart, Germany, 1993, Asp Arch, Schlaich Bergermann Struct Eng
Waldstadion, Frankfurt, Germany, 2005, von Gerkan, Marg 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.
4' 4' 12'
P
12'
±4' ±4'
a
12'
4' 2' 4'
b
Planar tensegrity frames
c 8'
40'
8'
Cases: Gravity, Prestress, Gravity + Prestress Planar tensegrity frames
de Young Memorial Museum, San Francisco, 2005, Jacques Herzog+Pierre de Meuron Arch, Rutherford & Chekene Struct Eng
