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Geometrically Nonlinear Structures: or why it's good to be a lightweight

Editorial Type: Case Study     Date: 07-2015    Views: 2718      










Modern lightweight structures, whether in fabric, cable, timber, concrete or stone, are nonlinear, long-spanning, flexible, highly efficient, and environmentally friendly. They can be used to create structures that soar. Oasys GSA structural analysis software brings the ideas they inspire to life, and here Oasys application specialist Peter Debney shares his enthusiasm...

With traditional linear structures the loads are resisted by the stiffness in the beams, columns, and walls; with lightweight tension-only and compression-only nonlinear structures the overall form of the structure becomes critical. Get the form right and the structure can span huge distances with minimal material. Get the form wrong and you are in trouble.

The real truth is that all structures are nonlinear; it's just that the simpler linear analysis usually gives answers that are close enough for the majority of engineering design challenges. But new lightweight structures call for more thorough analysis.

So what makes nonlinear different to linear analysis? One of the most important things to remember is that while with linear analysis you establish equilibrium of the forces on the original geometry, with a nonlinear analysis you get equilibrium of the forces on the deformed geometry. The problem is, you don't know what the deformation is until you have resolved the forces and you cannot resolve the forces until you know the deformed shape. All nonlinear analyses thus requires the iteration which is afforded by the power and speed of Oasys GSA.

CABLES
While linear structures resist lateral loads with bending stiffness, lightweight nonlinear tensile structures work by deflecting until the forces are in balance.

A single loaded cable, describing a catenary (the curve that an idealised hanging chain or cable assumes under its own weight when supported only at its ends, familiar to us from suspension bridges) is stabilised out-of-plane by gravity and possibly additional factors such as a bridge deck. However, such structures are still vulnerable to sway, whether induced by wind or pedestrians. A solution to this problem (though not normally for bridges!) is to have cables going in multiple directions, so sway in one direction is resisted by cables at other angles, giving what is called a cable net. These are actually common structures in nature, as spun by spiders.

If such a cable net is horizontal and loaded it will deflect down with an essentially catenary shape, giving resistance to gravity loads, but there is still a problem: what about uplift forces on such a cable net when it is clad? Suction would be resisted purely by the self-weight of the structure, but that is potentially minimal with lightweight structures. The solution is to have the cables in one direction curved down to resist gravity loads, and those in the other direction curved upward to resist suction loads. This double-curved hyperbolic surface is characteristic of many cable nets and the shape naturally gives stiffness in all directions.

So what does such a double curved surface look like? An excellent example is Expedition's award winning Velodrome for the London 2012 Olympics, famous for its "Pringle" shaped roof.

FABRICS
The modern science and engineering of fabric structures was pioneered by Frei Otto, with his roof to the Munich Olympic Games. Rejecting the heavy wartime architecture of Nazi Germany, Otto aspired to make modern architecture as light as possible, in both senses of the word: the Munich roof achieved this by using both a minimum of material and maximum glazing.

Fabrics are woven, and this gives rise to warp-weft interaction. So fabrics are thus sensitive to the balance of pre-stresses in the two principle directions; and the fabric will wrinkle as a whole if the pre-stress is much higher in one direction than the other. A correctly tensioned fabric will be smooth; unbalanced tensions will wrinkle the surface as you can see here in Arup's Marsyas sculpture during and after erection at the Tate Modern in London.

Fabrics require an edge support, which can either be solid, such as a beam, or flexible, such as a cable. With flexible edges, the cable's curvature is dependent on the balance in the pre-stress between the cable and fabric and calculating this, as Oasys GSA does, has quite an impact on the aesthetics and geometry of the fabric structure.



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