STRUCTURES
CLASSIFIED ACCORDING TO THE GENERATING FORCES
Tension
In general, the minimum amount of material is needed to transmit tensile forces and the expenditure for compressive and bending actions is higher. As Clerk Maxwell stated in his Lemma, tension is the most efficient way of using any material, as it utilises all the material at maximum efficiency rather than just the material at the extremes of the cross sectional form, as in bending. Optimal compression design brings with it the possibility of buckling, which is a phenomena that does not need to be considered in an all tensile design.
This is a somewhat idealised situation off course, as it neglects the fact that structural elements require connections and these in general will involve non-tensile stresses. In addition, tensile structures will invariably require compressive members of some form to transmit loads to "ground".
Use of suspended model to determine the load bearing behaviour of a vaulted structure:
Robert Hooke in 1670 indicated that the correct form of an arch is the inversion of the catenary.
Poleni first showed the structurally correct vault form by means of balls arranged in an inverted catenary and used this with the cross-section of the dome of St. Peter’s in Rome, to obtain the thrust line and thus was able to make his recommendation for the number of tension rings required to prevent further cracking of the dome.
Heinrich Hubsch was the first architect to use a two-dimensional suspended model for designs in 1833.
Antonio Gaudi around the turn of 20th century revived these ideas in his search for the true nature of the form. He derived arch shapes from suspended scale models so as to achieve purity of form or from an economical point of view, maximum efficiency of materials. He was the first to use a three dimensional suspended model, in the design for the Colonia Guell Church between 1898 and 1908 which. In a labour-intensive iterative process he determined the weights of the individual building units whose forms developed in the model, and then he hooked them as little bags filled with lead shot into the model which was made from linen threads.
Compression
Compared with the large number of objects whose form develops from self-forming processes under tensile stress, the number of objects formed by compressive stress is much smaller. For these structures, normally adaptation under load is not possible because the rigid material cannot shift and adapt to a different flow of forces. The structures fail because they eventually change their form under load into a form that is less favourable for the transmission of forces; which in turn means they are then subjected too much higher internal forces.
In a general sense (buckling excluded), the form of structuraly optimised compressive structures, can be determined experimentally by inverting a tensile-stressed structure which is itself optimising under a corresponding load. The thrust line of an arch is produced by inverting the suspended catenary, and the thrust surface of a shell, that is only stressed by axial pressure, by inverting a suspended form.
A rubble heap is an example of a self-forming structure under a compressive stress.
Drying sand shows typical erosion forms, which also include the forms of corbel structures, caves and arches; these are produced as self-optimising structures by taking away material subject to minor stress.
Inversion of catenary
The inverted catenary is the approximate optimum form of an arch under its own weight.
The correct arch shape is of utmost importance for materials weak in tension or where weight is an important consideration. The strong "new" materials, iron and then steel, which were developed throughout the 19th century, did not have to place too much emphasis on this aspect as they can easily resist bending. The search for a self-optimising compressive-stressed structure is difficult because the transmission of compressive forces requires a relatively rigid stressed element: the form is therefore fixed and cannot simply be adapted to the flow of forces. This is the reason why inverted forms are used. These provide the optimum form according to structural criteria by means of a structure that is self-optimising under an analogue load.
In inverted catenary arches of the same length, the greater the height, the lower horizontal force in the built in ends and in the middle of the arch. The slenderest arches are the ones with lower lateral force acting on them.
Bending
Bending by its very nature involves both tension and compression. As indicated by Clerk Maxwell, it cannot therefore be an optimum solution in the true sense of the word.
CLASSIFIED ACCORDING TO THE GENERATING FORMS
Linear elements
Catenary:
A cable carrying a uniform load along its length, such as its own weight, assumes the shape of a catenary or a hyperbolic cosine curve. Under the influence of gravity, flexible elements that are freely suspended in space develop this typical suspended form. The form is the result of the equilibrium of internal forces and is ideally suited for transmitting loads. When the loads or the boundary conditions change, an optimum form geared to the new situation is produced automatically. When a great weight much larger than the rest of the symmetric loads is applied at mid span, a funicular arch is obtained; when the loads are larger at the ends of the arch the shape of the arch approach to an elliptic arch; when the load is uniform along the length the shape is a parabola.
Example of catenary in nature: Cobweb:
The cobweb illustrates this typical suspended form in nature. It is an assembled form of catenaries. The threads running in circular direction develop into simple catenaries.
The majority of objects created by self-forming processes are determined by tensile stress during their development process. Compressive stress only determines the form in arches produced by hollowing out.
Parabola:
Under a uniform load the simple cable takes a curved shape. When this distributed transverse load is constant and acting on the horizontal projection of the cable, the cable responds with the shape of a parabola. The situation occurs in suspension bridges where the suspended cables carry the roadway by ignoring the effect of the weight of the cables. For small sag-to span ratios the geometry of a catenary and a parabola are practically the same.
The curve of the cable in a suspension bridge is a parabola. When the structure is being built and the main cables are attached to the towers, the curve is a catenary. But when the cables are attached to the deck with hangers, it is no longer a catenary. The curves of the cables then become the curve of a parabola. Both the spider web and the suspension bridge, like many other structures, are mainly empty space, because the optimum distribution of material is hardly ever as a lump, also trees spread their leaves into the light, with minimal use of material. The spider web is another structure built with extremely small quantities of material.
Example of parabolic shape in nature: Parabolic dunes:
U-shaped mounds of sand with convex noses trailed by elongated arms are parabolic dunes. Sometimes these dunes are called U-shaped, blowout, or hairpin dunes, and they are well known in coastal deserts. Unlike normal "crescent" dunes, their crests point upwind. The elongated arms of parabolic dunes follow rather than lead because they have been fixed by vegetation, while the bulk of the sand in the dune migrates forward. The longest known parabolic dune had a trailing arm 12 kilometres long.
They are characteristic of areas where strong winds are unidirectional.
Flat elements
The flat elements developing from processes subject to the laws of nature assume a diversity of forms, the geometrically exact plane being an a typical exception: it is found in crystals as a relative small prismatic surface.
· Minimal surfaces
The smallest surface within a given edge is called a minimal surface
· Nets
Nets with a different mesh form can produce anticlastically curved surfaces that are similar to minimum surfaces. Since the flow of force is bound to the directions of the fibres, differences exist. If the boundary conditions are changed, this results in the force concentrating in individual fibres that are reflected as ridges.
· Viscous liquid membranes
Many viscous liquids form membranes under the influence of tensile stress acting within the surface. The forms of the object developing from this resemble minimal surfaces.
· Pneumatic structures
The spherical envelope is the basic form of pneumatic structures; it encloses a maximum volume by a minimum surface. This form has an equal tension at any point in all directions. Pneumatic structures are relatively soft and flexible. They change their form under the influence of small changes in the forces forming or acting on them.
Spatial elements
· Formation by erosion
Large monolithic solids of rock change their form under the influence of changes in temperature. Thermal stresses result in cracks. In a damp and cold climate, the disintegration process is furthered by the effect of the penetrating and freezing water. The form of existing solids is further changed by mechanical stresses, by washing out or by the action of sand grains carried along by wind or water.
Optimisation of the form of compressively-stressed structures (including those due to bending) by reducing or removing material with litte or no stress, in a step by step iterative manner, is a recognised form of shape optimisation. In this manner, the optimum form of an arch stressed by its own weight can be formed. In this case, sufficient stability to loads attacking horizontally must be ensured. Typical arch forms develop; depending on whether a great clear height or a great span is required for the task. Their axis is the thrust line and their cross section changes from the apex to the support.
Rainbow bridge, Utah (left)
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