Editing Tensile structure

{{short description|Structure whose members are only in tension}}
{{more footnotes needed|date=September 2011}}
[[Image:Tensile Steel Lattice Shell of Oval Pavilion by Vladimir Shukhov 1895.jpg|thumb|200px|The world's first tensile [[steel]] [[Thin-shell structure|shell]] by [[Vladimir Shukhov]] (during construction), [[Nizhny Novgorod]], 1895]]
[[Image:Sidney Myer Music Bowl Aerial View.jpg|thumb|200px|The [[Sidney Myer Music Bowl]] in [[Kings Domain, Melbourne]]]]

In [[structural engineering]], a '''tensile structure''' is a [[construction]] of elements carrying only [[tension (physics)|tension]] and no [[compression (physical)|compression]] or [[bending]]. The term ''tensile'' should not be confused with [[tensegrity]], which is a structural form with both tension and compression elements. Tensile structures are the most common type of [[thin-shell structure]]s.

Most tensile structures are supported by some form of compression or bending elements, such as masts (as in [[The O2 (London)|The O<sub>2</sub>]], formerly the [[Millennium Dome]]), compression rings or beams.

A [[Tension fabric building|tensile membrane structure]] is most often used as a [[roof]], as they can economically and attractively span large distances. Tensile membrane structures may also be used as complete buildings, with a few common applications being sports facilities, warehousing and storage buildings, and exhibition venues.

== History ==

This form of construction has only become more rigorously analyzed and widespread in large structures in the latter part of the twentieth century. Tensile structures have long been used in [[tent]]s, where the [[guy rope]]s and tent poles provide pre-tension to the fabric and allow it to withstand loads.

Russian engineer [[Vladimir Shukhov]] was one of the first to develop practical calculations of stresses and deformations of tensile structures, shells and membranes. Shukhov designed eight tensile structures and [[thin-shell structure]]s exhibition pavilions for the [[Nizhny Novgorod Fair of 1896]], covering the area of 27,000 square meters. A more recent large-scale use of a membrane-covered tensile structure is the [[Sidney Myer Music Bowl]], constructed in 1958.

[[Antonio Gaudi]] used the concept in reverse to create a compression-only structure for the [[Colonia Guell Church]]. He created a hanging tensile model of the church to calculate the compression forces and to experimentally determine the column and vault geometries.

The concept was later championed by [[Germany|German]] architect and engineer [[Frei Otto]], whose first use of the idea was in the construction of the [[Expo 67 pavilions#National pavilions|West German pavilion at Expo 67]] in Montreal. Otto next used the idea for the roof of the Olympic Stadium for the [[1972 Summer Olympics]] in [[Munich]].

Since the 1960s, [[tension (mechanics)|tensile]] structures have been promoted by [[design]]ers and [[engineer]]s such as [[Ove Arup]], [[Buro Happold]], [[Frei Otto]], [[Mahmoud Bodo Rasch]], [[Eero Saarinen]], [[Horst Berger]], [[Matthew Nowicki]], [[Jorg Schlaich|Jörg Schlaich]], and [[David Geiger]].

Steady technological progress has increased the popularity of fabric-roofed structures. The low weight of the materials makes construction easier and cheaper than standard designs, especially when vast open spaces have to be covered.

== Types of structure with significant tension members ==

=== Linear structures ===

*[[Suspension bridges]]
*[[Stressed ribbon bridge]]
*Draped cables
*[[Cable-stayed]] [[beam (structure)|beams]] or [[truss]]es
*Cable trusses
*Straight tensioned cables

=== Three-dimensional structures ===

*[[Bicycle wheel]] (can be used as a roof in a horizontal orientation)
*3D cable trusses
*[[Tensegrity]] structures

=== Surface-stressed structures ===

*Prestressed membranes
*Pneumatically stressed membranes
* [[Gridshell]]
* [[Fabric structure]]

== Cable and membrane structures ==

[[Image:Membrane Roof and Tensile Lattice Shell of Shukhov Rotunda 1895.jpg|thumb|left|The world's first steel membrane roof and lattice steel shell in the [[Shukhov Rotunda]], [[Russia]], 1895]]

=== Membrane materials ===

Common materials for doubly curved fabric structures are [[PTFE]]-coated [[fiberglass]] and [[Polyvinyl chloride|PVC]]-coated [[polyester]]. These are woven materials with different strengths in different directions. The [[Warp (weaving)|warp]] fibers (those fibers which are originally straight—equivalent to the starting fibers on a loom) can carry greater load than the [[weft]] or fill fibers, which are woven between the warp fibers.

Other structures make use of [[ETFE]] film, either as single layer or in cushion form (which can be inflated, to provide good insulation properties or for aesthetic effect—as on the [[Allianz Arena]] in [[Munich]]). ETFE cushions can also be etched with patterns in order to let different levels of light through when inflated to different levels.

In daylight, fabric membrane translucency offers soft diffused naturally lit spaces, while at night, artificial lighting can be used to create an ambient exterior luminescence. They are most often supported by a structural frame as they cannot derive their strength from double curvature.<ref>{{cite web |title=Sprung |url=https://www.army-technology.com/contractors/field/sprung-structures2/ |website=Army Technology}}</ref>

[[Image:IRB-7-MUDDY2.jpg|thumb|left|Simple suspended bridge working entirely in tension]]

=== Cables ===

Cables can be of [[mild steel]], [[High-strength low-alloy steel|high strength steel]] (drawn carbon steel), [[stainless steel]], [[polyester]] or [[Aramid|aramid fibres]]. Structural cables are made of a series of small strands twisted or bound together to form a much larger cable. Steel cables are either spiral strand, where circular rods are twisted together and "glued" using a polymer, or locked coil strand, where individual interlocking steel strands form the cable (often with a spiral strand core).

Spiral strand is slightly weaker than locked coil strand. Steel spiral strand cables have a [[Young's modulus]], ''E'' of 150±10&nbsp;kN/mm<sup>2</sup> (or 150±10 [[gigapascal|GPa]]) and come in sizes from 3 to 90&nbsp;mm diameter.{{Citation needed|reason=The information is probably country/manufacturer specific and needs a source.|date=April 2016}} Spiral strand suffers from construction stretch, where the strands compact when the cable is loaded. This is normally removed by pre-stretching the cable and cycling the load up and down to 45% of the ultimate tensile load.

Locked coil strand typically has a Young's Modulus of 160±10&nbsp;kN/mm<sup>2</sup> and comes in sizes from 20&nbsp;mm to 160&nbsp;mm diameter.

The properties of the individual strands of different materials are shown in the table below, where UTS is [[ultimate tensile strength]], or the breaking load:

{| class="wikitable"
|-
! Cable material
! ''E'' (GPa)
! UTS (MPa)
! [[Strain (materials science)|Strain]] at 50% of UTS
|-
| Solid steel bar
| 210
| 400–800
| 0.24%
|-
| Steel strand
| 170
| 1550–1770
| 1%
|-
| Wire rope
| 112
| 1550–1770
| 1.5%
|-
| Polyester fibre
| 7.5
| 910
| 6%
|-
| Aramid fibre
| 112
| 2800
| 2.5%
|}

=== Structural forms ===

[[Air-supported structure]]s are a form of tensile structures where the fabric envelope is supported by pressurised air only.

The majority of fabric structures derive their strength from their doubly curved shape. By forcing the fabric to take on double-curvature the fabric gains sufficient [[stiffness]] to withstand the loads it is subjected to (for example [[wind]] and [[snow]] loads). In order to induce an adequately doubly curved form it is most often necessary to [[pretension]] or prestress the fabric or its supporting structure.

=== Form-finding ===

The behaviour of structures which depend upon prestress to attain their strength is non-linear, so anything other than a very simple cable has, until the 1990s, been very difficult to design. The most common way to design doubly curved fabric structures was to construct scale models of the final buildings in order to understand their behaviour and to conduct form-finding exercises. Such scale models often employed stocking material or tights, or soap film, as they behave in a very similar way to structural fabrics (they cannot carry shear).

Soap films have uniform stress in every direction and require a closed boundary to form. They naturally form a minimal surface—the form with minimal area and embodying minimal energy. They are however very difficult to measure. For a large film, its weight can seriously affect its form.

For a membrane with curvature in two directions, the basic equation of equilibrium is:

: <math>w = \frac{t_1}{R_1} + \frac{t_2}{R_2}</math>

where:

*''R''<sub>1</sub> and ''R''<sub>2</sub> are the principal radii of curvature for soap films or the directions of the warp and weft for fabrics
*''t''<sub>1</sub> and ''t''<sub>2</sub> are the tensions in the relevant directions
*''w'' is the load per square metre

Lines of [[principal curvature]] have no twist and intersect other lines of principal curvature at right angles.

A [[geodesic]] or [[geodesy|geodetic]] line is usually the shortest line between two points on the surface. These lines are typically used when defining the cutting pattern seam-lines. This is due to their relative straightness after the planar cloths have been generated, resulting in lower cloth wastage and closer alignment with the fabric weave.

In a pre-stressed but unloaded surface ''w'' = 0, so <math>\frac{t_1}{R_1} = -\frac{t_2}{R_2}</math>.

In a soap film surface tensions are uniform in both directions, so ''R''<sub>1</sub> = −''R''<sub>2</sub>.

It is now possible to use powerful [[non-linear]] [[numerical analysis]] programs (or [[finite element analysis]]) to formfind and design fabric and cable structures. The programs must allow for large deflections.

The final shape, or form, of a fabric structure depends upon:

*shape, or pattern, of the fabric
*the geometry of the supporting structure (such as masts, cables, ringbeams etc.)
*the pretension applied to the fabric or its supporting structure

[[Image:HyperbolicParaboloid.svg|thumb|right|[[Hyperbolic paraboloid]]]]

It is important that the final form will not allow [[ponding]] of water, as this can deform the membrane and lead to local failure or progressive failure of the entire structure.

Snow loading can be a serious problem for membrane structure, as the snow often will not flow off the structure as water will. For example, this has in the past caused the (temporary) collapse of the [[Hubert H. Humphrey Metrodome]], an air-inflated structure in [[Minneapolis, Minnesota]]. Some structures prone to [[ponding]] use heating to melt snow which settles on them.

[[Image:Saddle point.svg|thumb|left|Saddle Shape]]

There are many different doubly curved forms, many of which have special mathematical properties. The most basic doubly curved from is the saddle shape, which can be a [[hyperbolic paraboloid]] (not all saddle shapes are hyperbolic paraboloids). This is a double [[ruled surface]] and is often used in both in lightweight shell structures (see [[hyperboloid structures]]). True ruled surfaces are rarely found in tensile structures. Other forms are [http://en.wiktionary.org/wiki/anticlastic anticlastic] saddles, various radial, conical tent forms and any combination of them.

=== Pretension ===

'''Pretension''' is tension artificially induced in the structural elements in addition to any self-weight or imposed loads they may carry. It is used to ensure that the normally very flexible structural elements remain stiff under all possible loads.<ref>{{Cite journal|last1=Quagliaroli|first1=M.|last2=Malerba|first2=P. G.|last3=Albertin|first3=A.|last4=Pollini|first4=N.|date=2015-12-01|title=The role of prestress and its optimization in cable domes design|url=http://www.sciencedirect.com/science/article/pii/S0045794915002503|journal=Computers & Structures|language=en|volume=161|pages=17–30|doi=10.1016/j.compstruc.2015.08.017|issn=0045-7949|url-access=subscription}}</ref><ref>{{Citation|last1=Albertin|first1=A|title=Prestress optimization of hybrid tensile structures|date=2012-06-21|url=https://doi.org/10.1201/b12352-256|work=Bridge Maintenance, Safety, Management, Resilience and Sustainability|pages=1750–1757|publisher=CRC Press|doi=10.1201/b12352-256|isbn=978-0-415-62124-3|access-date=2020-06-30|last2=Malerba|first2=P|last3=Pollini|first3=N|last4=Quagliaroli|first4=M|doi-broken-date=2024-11-12|url-access=subscription}}</ref>

A day to day example of pretension is a shelving unit supported by wires running from floor to ceiling. The wires hold the shelves in place because they are tensioned – if the wires were slack the system would not work.

Pretension can be applied to a membrane by stretching it from its edges or by pretensioning cables which support it and hence changing its shape. The level of pretension applied determines the shape of a membrane structure.

=== Alternative form-finding approach ===

The alternative approximated approach to the form-finding problem solution is based on the total energy balance of a grid-nodal system. Due to its physical meaning this approach is called the [[stretched grid method]] (SGM).

== Simple mathematics of cables ==

=== Transversely and uniformly loaded cable ===

A uniformly loaded cable spanning between two supports forms a curve intermediate between a [[catenary]] curve and a [[parabola]]. The simplifying assumption can be made that it approximates a circular arc (of radius ''R'').

[[Image:Catenary cable diagram.svg|700px|left]]
{{Clear}}

'''By [[mechanical equilibrium|equilibrium]]:'''

The horizontal and vertical reactions :

:<math>H = \frac{wS^2}{8d}  </math>
:<math>V = \frac{wS}{2}</math>

'''By [[geometry]]:'''

The length of the cable:

:<math>L = 2R\arcsin\frac{S}{2R}</math>

The tension in the cable:

:<math>T = \sqrt{H^2+V^2}</math>

By substitution:

:<math>T = \sqrt{\left(\frac{wS^2}{8d}\right)^2 + \left(\frac{wS}{2}\right)^2}</math>

The tension is also equal to:

:<math>T = wR</math>

The extension of the cable upon being loaded is (from [[Hooke's law]], where the axial stiffness, ''k,'' is equal to <math>k = \frac{{EA}}{{L}}</math>):

:<math>e = \frac{TL}{EA}</math>

where ''E'' is the [[Young's modulus]] of the cable and ''A'' is its cross-sectional [[area]].

If an initial pretension, <math>T_0</math> is added to the cable, the extension becomes:

:<math>e = L - L_0 = \frac{L_0(T-T_0)}{EA}</math>

Combining the above equations gives:

:<math>\frac{L_0(T-T_0)}{EA} + L_0 = \frac{2T\arcsin\left(\frac{wS}{2T}\right)}{w}</math>

By plotting the left hand side of this equation against ''T,'' and plotting the right hand side on the same axes, also against ''T,'' the intersection will give the actual equilibrium tension in the cable for a given loading ''w'' and a given pretension <math>T_0</math>.

=== Cable with central point load ===

[[Image:point-loaded cable.svg|800px|left]]
{{Clear}}
A similar solution to that above can be derived where:

'''By equilibrium:'''

:<math>W = \frac{4Td}{L}</math>

:<math>d = \frac{WL}{4T}</math>

'''By geometry:'''

:<math>L = \sqrt{S^2 + 4d^2} = \sqrt{S^2 + 4\left(\frac{WL}{4T}\right)^2}</math>

This gives the following relationship:

:<math>L_0 + \frac{L_0(T-T_0)}{EA} = \sqrt{S^2 + 4\left(\frac{W(L_0+\frac{L_0(T-T_0)}{EA})}{4T}\right)^2}</math>

As before, plotting the left hand side and right hand side of the equation against the tension, ''T,'' will give the equilibrium tension for a given pretension, <math>T_0</math> and load, ''W''.

== Tensioned cable oscillations ==

The fundamental [[natural frequency]], ''f''<sub>1</sub> of tensioned cables is given by:

:<math>f_1=\frac{\sqrt{\left(\frac{T}{m}\right)}}{2L}</math>

where ''T'' = tension in [[newton (unit)|newton]]s, ''m'' = [[mass]] in kilograms and ''L'' = span length.

== Notable structures ==

{{stack|
[[File:Rotunda by Vladimir Shukhov Nizhny Novgorod 1896.jpg|thumb|Rotunda by Vladimir Shukhov Nizhny Novgorod 1896]][[File:Rotunda and rectangular pavilion by Vladimir Shukhov in Nizhny Novgorod 1896.jpg|thumb|Rotunda and rectangular pavilion by Vladimir Shukhov in Nizhny Novgorod 1896]]
}}
*[[Shukhov Rotunda]], [[Russia]], 1896
*[[Canada Place]], [[Vancouver]], [[British Columbia]] for [[Expo '86]]
*[[Yoyogi National Gymnasium]] by [[Kenzo Tange]], [[Yoyogi Park]], [[Tokyo]], [[Japan]]
*[[Ingalls Rink]], [[Yale University]] by [[Eero Saarinen]]
*[[Khan Shatyr Entertainment Center]], [[Astana, Kazakhstan]]
*[[Tropicana Field]], [[St. Petersburg, Florida|St. Petersburg]], [[Florida]]
*[[Olympiapark, Munich|Olympiapark]], [[Munich]] by [[Frei Otto]]
*[[Sidney Myer Music Bowl]], [[Melbourne]]
*[[The O2 (London)|The O<sub>2</sub>]] (formerly the [[Millennium Dome]]), [[London]] by [[Buro Happold]] and [[Richard Rogers|Richard Rogers Partnership]]
*[[Denver International Airport]], [[Denver]]
*[[Dorton Arena]], [[Raleigh, North Carolina|Raleigh]]
*[[Georgia Dome]], [[Atlanta]], [[Georgia (U.S. state)|Georgia]] by [[Heery International|Heery]] and [[Weidlinger Associates]] (demolished in 2017)
*[[Grantley Adams International Airport]], [[Christ Church, Barbados|Christ Church]], [[Barbados]]
*[[Pengrowth Saddledome]], [[Calgary, Alberta|Calgary]] by [[Graham McCourt Architects]] and [[Jan Bobrowski and Partners]]
*[[Scandinavium]], [[Gothenburg]], [[Sweden]]
*[[Hong Kong Museum of Coastal Defence]]
*Modernization of the [[Central Railway Station, Sofia|Central Railway Station]], [[Sofia]], [[Bulgaria]]
*[[Redbird Arena]], [[Illinois State University]], [[Normal, Illinois]]
*Retractable Umbrellas, [[Al-Masjid an-Nabawi]], Medina, [[Saudi Arabia]]
*[[Killesberg Tower]], [[Stuttgart]]

== Gallery of well-known tensile structures ==

<gallery>
 File:Olympic park 12.jpg|The roof tensile structures by [[Frei Otto]] of the [[Olympiapark, Munich|Olympiapark]], [[Munich]]
 File:Canary.wharf.and.dome.london.arp.jpg|The [[Millennium Dome]] (now The O<sub>2</sub>), [[London]], by [[Buro Happold]] and [[Richard Rogers]]
 File:Denver International Airport terminal.jpg|[[Denver International Airport]] terminal
 File:Thtr300 kuehlturm.jpg|The [[THTR-300]] cable-net dry [[cooling tower]], [[hyperboloid structure]] by [[Schlaich Bergermann & Partner]]
 File:Killesberg Tower.jpg|Killesberg Tower, Stuttgart, by [[Schlaich Bergermann Partner]]
 File:GeorgiaDome md.jpg|[[Georgia Dome]] in [[Atlanta]]
 File:Stamps of Kazakhstan, 2010-08.jpg|Daytime computer [[Architectural rendering|render]] of [[Khan Shatyr Entertainment Center]], the highest tensile structure in the world
</gallery>

== Classification numbers ==

The [[Construction Specifications Institute]] (CSI) and Construction Specifications Canada (CSC), [[MasterFormat]] 2018 Edition, Division 05 and 13:

* 05 16 00 – Structural Cabling
* 05 19 00 - Tension Rod and Cable Truss Assemblies
* 13 31 00 – Fabric Structures
* 13 31 23 – Tensioned Fabric Structures
* 13 31 33 – Framed Fabric Structures

CSI/CSC [[MasterFormat]] 1995 Edition:

* 13120 – Cable-Supported Structures
* 13120 – Fabric Structures

== See also ==

* [[Buckminster Fuller]]
* [[Gaussian curvature]]
* [[Geodesic dome]]
* [[Geodesics]]
* [[Hyperboloid structure]]
* [[Kārlis Johansons]]
* [[Kenneth Snelson]]
* [[Suspended structure]]
* [[Suspension bridge]]
* [[Tensairity]]
* [[Tensegrity]]
* [[Wire rope]]

== References ==

{{reflist}}

== Further reading ==

{{commons category|Tensile structures}}
* "The Nijni-Novgorod exhibition: Water tower, room under construction, springing of 91 feet span", [[The Engineer (UK magazine)|"The Engineer"]], № 19.3.1897, P.292-294, London, 1897.
*[[Horst Berger]], ''Light structures, structures of light: The art and engineering of tensile architecture'' (Birkhäuser Verlag, 1996) {{ISBN|3-7643-5352-X}}
*Alan Holgate, ''The Art of Structural Engineering: The Work of Jorg Schlaich and his Team'' (Books Britain, 1996) {{ISBN|3-930698-67-6}}
* [http://spec.lib.vt.edu/IAWA/inventories/English.html Elizabeth Cooper English]:  [http://repository.upenn.edu/dissertations/AAI9989589/ "Arkhitektura i mnimosti": The origins of Soviet avant-garde rationalist architecture in the Russian mystical-philosophical and mathematical intellectual tradition"], a dissertation in architecture, 264 p., University of Pennsylvania, 2000.
* "Vladimir G. Suchov 1853–1939. Die Kunst der sparsamen Konstruktion.", Rainer Graefe, Jos Tomlow und andere, 192 S., Deutsche Verlags-Anstalt, Stuttgart, 1990, {{ISBN|3-421-02984-9}}.
* [[Conrad Roland]]: ''[[Frei Otto]] – Spannweiten. Ideen und Versuche zum Leichtbau''. Ein Werkstattbericht von Conrad Roland. Ullstein, Berlin, Frankfurt/Main und Wien 1965.
* Frei Otto, Bodo Rasch: Finding Form - Towards an Architecture of the Minimal, Edition Axel Menges, 1996, {{ISBN|3930698668}}
* Nerdinger, Winfried: Frei Otto. Das Gesamtwerk: Leicht Bauen Natürlich Gestalten, 2005, {{ISBN|3-7643-7233-8}}

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[[Category:Roofs]]
[[Category:Russian inventions]]
[[Category:Structural system]]
[[Category:Tensile architecture]]
[[Category:Tensile membrane structures]]