Editing Tensile structure (section)

=== 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.