Editing Tensile structure (section)

== 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).