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In structural engineering, a tensile structure is a construction of elements carrying only tension and no 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 structures.
Most tensile structures are supported by some form of compression or bending elements, such as masts (as in The O2, formerly the Millennium Dome), compression rings or beams.
A 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.
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 tents, where the guy ropes 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 structures 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 German architect and engineer Frei Otto, whose first use of the idea was in the construction of the 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, tensile structures have been promoted by designers and engineers such as Ove Arup, Buro Happold, Frei Otto, Mahmoud Bodo Rasch, Eero Saarinen, Horst Berger, Matthew Nowicki, 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.
Common materials for doubly curved fabric structures are PTFE-coated fiberglass and PVC-coated polyester. These are woven materials with different strengths in different directions. The 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. [1]
Cables can be of mild steel, high strength steel (drawn carbon steel), stainless steel, polyester or 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 kN/mm2 (or 150±10 GPa) and come in sizes from 3 to 90 mm diameter.[ citation needed ] 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 kN/mm2 and comes in sizes from 20 mm to 160 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:
Cable material | E (GPa) | UTS (MPa) | 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% |
Air-supported structures 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.
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:
where:
Lines of principal curvature have no twist and intersect other lines of principal curvature at right angles.
A geodesic or 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 .
In a soap film surface tensions are uniform in both directions, so R1 = −R2.
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:
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.
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 anticlastic saddles, various radial, conical tent forms and any combination of them.
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. [2] [3]
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.
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).
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).
By equilibrium:
The horizontal and vertical reactions :
By geometry:
The length of the cable:
The tension in the cable:
By substitution:
The tension is also equal to:
The extension of the cable upon being loaded is (from Hooke's Law, where the axial stiffness, k, is equal to ):
where E is the Young's modulus of the cable and A is its cross-sectional area.
If an initial pretension, is added to the cable, the extension becomes:
Combining the above equations gives:
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 .
A similar solution to that above can be derived where:
By equilibrium:
By geometry:
This gives the following relationship:
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, and load, W.
The fundamental natural frequency, f1 of tensioned cables is given by:
where T = tension in newtons, m = mass in kilograms and L = span length.
The Construction Specifications Institute (CSI) and Construction Specifications Canada (CSC), MasterFormat 2018 Edition, Division 05 and 13:
CSI/CSC MasterFormat 1995 Edition:
In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.
Tensegrity, tensional integrity or floating compression is a structural principle based on a system of isolated components under compression inside a network of continuous tension, and arranged in such a way that the compressed members do not touch each other while the prestressed tensioned members delineate the system spatially.
A helix is a shape like a cylindrical coil spring or the thread of a machine screw. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word helix comes from the Greek word ἕλιξ, "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called a helicoid.
Fracture is the appearance of a crack or complete separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If a displacement develops perpendicular to the surface, it is called a normal tensile crack or simply a crack; if a displacement develops tangentially, it is called a shear crack, slip band, or dislocation.
Structural analysis is a branch of solid mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on the physical structures and their components. In contrast to theory of elasticity, the models used in structure analysis are often differential equations in one spatial variable. Structures subject to this type of analysis include all that must withstand loads, such as buildings, bridges, aircraft and ships. Structural analysis uses ideas from applied mechanics, materials science and applied mathematics to compute a structure's deformations, internal forces, stresses, support reactions, velocity, accelerations, and stability. The results of the analysis are used to verify a structure's fitness for use, often precluding physical tests. Structural analysis is thus a key part of the engineering design of structures.
In structural engineering, buckling is the sudden change in shape (deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have buckled. Euler's critical load and Johnson's parabolic formula are used to determine the buckling stress of a column.
Prestressed concrete is a form of concrete used in construction. It is substantially "prestressed" (compressed) during production, in a manner that strengthens it against tensile forces which will exist when in service.
In continuum mechanics, the maximum distortion energy criterion states that yielding of a ductile material begins when the second invariant of deviatoric stress reaches a critical value. It is a part of plasticity theory that mostly applies to ductile materials, such as some metals. Prior to yield, material response can be assumed to be of a linear elastic, nonlinear elastic, or viscoelastic behavior.
Euler–Bernoulli beam theory is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral loads only. By ignoring the effects of shear deformation and rotatory inertia, it is thus a special case of Timoshenko–Ehrenfest beam theory. It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution.
In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed, some fraction of the deformation will be permanent and non-reversible and is known as plastic deformation.
Macaulay's method (the double integration method) is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams. Use of Macaulay's technique is very convenient for cases of discontinuous and/or discrete loading. Typically partial uniformly distributed loads (u.d.l.) and uniformly varying loads (u.v.l.) over the span and a number of concentrated loads are conveniently handled using this technique.
In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof.
A T-beam, used in construction, is a load-bearing structure of reinforced concrete, wood or metal, with a T-shaped cross section. The top of the T-shaped cross section serves as a flange or compression member in resisting compressive stresses. The web of the beam below the compression flange serves to resist shear stress. When used for highway bridges the beam incorporates reinforcing bars in the bottom of the beam to resist the tensile stresses which occur during bending.
Dynamic relaxation is a numerical method, which, among other things, can be used to do "form-finding" for cable and fabric structures. The aim is to find a geometry where all forces are in equilibrium. In the past this was done by direct modelling, using hanging chains and weights, or by using soap films, which have the property of adjusting to find a "minimal surface".
The capstan equation or belt friction equation, also known as Euler–Eytelwein formula, relates the hold-force to the load-force if a flexible line is wound around a cylinder.
Concrete has relatively high compressive strength, but significantly lower tensile strength. The compressive strength is typically controlled with the ratio of water to cement when forming the concrete, and tensile strength is increased by additives, typically steel, to create reinforced concrete. In other words we can say concrete is made up of sand, ballast, cement and water.
The stretched grid method (SGM) is a numerical technique for finding approximate solutions of various mathematical and engineering problems that can be related to an elastic grid behavior. In particular, meteorologists use the stretched grid method for weather prediction and engineers use the stretched grid method to design tents and other tensile structures.
A shell is a three-dimensional solid structural element whose thickness is very small compared to its other dimensions. It is characterized in structural terms by mid-plane stress which is both coplanar and normal to the surface. A shell can be derived from a plate in two steps: by initially forming the middle surface as a singly or doubly curved surface, then by applying loads which are coplanar to the plate's plane thus generating significant stresses. Materials range from concrete to fabric.
In Einstein's theory of general relativity, the interior Schwarzschild metric is an exact solution for the gravitational field in the interior of a non-rotating spherical body which consists of an incompressible fluid and has zero pressure at the surface. This is a static solution, meaning that it does not change over time. It was discovered by Karl Schwarzschild in 1916, who earlier had found the exterior Schwarzschild metric.
This page is a glossary of Prestressed concrete terms.