A tensile structure is a
construction
Construction is a general term meaning the art and science to form objects, systems, or organizations,"Construction" def. 1.a. 1.b. and 1.c. ''Oxford English Dictionary'' Second Edition on CD-ROM (v. 4.0) Oxford University Press 2009 and com ...
of elements carrying only
tension
Tension may refer to:
Science
* Psychological stress
* Tension (physics), a force related to the stretching of an object (the opposite of compression)
* Tension (geology), a stress which stretches rocks in two opposite directions
* Voltage or el ...
and no
compression
Compression may refer to:
Physical science
*Compression (physics), size reduction due to forces
*Compression member, a structural element such as a column
*Compressibility, susceptibility to compression
* Gas compression
*Compression ratio, of a ...
or
bending
In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.
The structural element is assumed to ...
. The term ''tensile'' should not be confused with
tensegrity
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 (usually ...
, 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
The Millennium Dome was the original name of the large dome-shaped building on the Greenwich Peninsula in South East (London sub region), South East London, England, which housed a major exhibition celebrating the beginning of the third millenn ...
), compression rings or beams.
A
tensile membrane structure
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 st ...
is most often used as a
roof
A roof ( : roofs or rooves) is the top covering of a building, including all materials and constructions necessary to support it on the walls of the building or on uprights, providing protection against rain, snow, sunlight, extremes of temper ...
, 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.
History
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
tent
A tent () is a shelter consisting of sheets of fabric or other material draped over, attached to a frame of poles or a supporting rope. While smaller tents may be free-standing or attached to the ground, large tents are usually anchored using gu ...
s, where the
guy rope
A guy-wire, guy-line, guy-rope, or stay, also called simply a guy, is a tensioned cable designed to add stability to a free-standing structure. They are used commonly for ship masts, radio masts, wind turbines, utility poles, and tents. A thi ...
s and tent poles provide pre-tension to the fabric and allow it to withstand loads.
Russian engineer
Vladimir Shukhov
Vladimir Grigoryevich Shukhov (russian: link=no, Влади́мир Григо́рьевич Шу́хов; – 2 February 1939) was a Russian Empire and Soviet engineer-polymath, scientist and architect renowned for his pioneering works on new ...
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
The Sidney Myer Music Bowl is an outdoor bandshell performance venue in Melbourne, Victoria, Australia. It is located in the lawns and gardens of Kings Domain on Linlithgow Avenue close to the Arts Centre and the Southbank entertainment precinct ...
, constructed in 1958.
Antonio Gaudi
Antonio is a masculine given name of Etruscan origin deriving from the root name Antonius. It is a common name among Romance language-speaking populations as well as the Balkans and Lusophone Africa. It has been among the top 400 most popular male ...
used the concept in reverse to create a compression-only structure for the
Colonia Guell Church
Colonia may refer to:
Arts and entertainment
*Colonia (music group), a Croatian dance music group
* ''Colonia'' (Autopsia album), 2002
* ''Colonia'' (A Camp album), 2009
* ''Colonia'' (film), a 2015 historical romantic thriller
Places
* Coloni ...
. 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
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
architect and engineer
Frei Otto
Frei Paul Otto (; 31 May 1925 – 9 March 2015) was a German architect and structural engineer noted for his use of lightweight structures, in particular tensile and membrane structures, including the roof of the Olympic Stadium in Munich for ...
, 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
The 1972 Summer Olympics (), officially known as the Games of the XX Olympiad () and commonly known as Munich 1972 (german: München 1972), was an international multi-sport event held in Munich, West Germany, from 26 August to 11 September 1972. ...
in
Munich
Munich ( ; german: München ; bar, Minga ) is the capital and most populous city of the States of Germany, German state of Bavaria. With a population of 1,558,395 inhabitants as of 31 July 2020, it is the List of cities in Germany by popu ...
.
Since the 1960s,
tensile
In physics, tension is described as the pulling force transmitted axially by the means of a string, a rope, chain, or similar object, or by each end of a rod, truss member, or similar three-dimensional object; tension might also be described as t ...
structures have been promoted by
design
A design is a plan or specification for the construction of an object or system or for the implementation of an activity or process or the result of that plan or specification in the form of a prototype, product, or process. The verb ''to design'' ...
ers and
engineer
Engineers, as practitioners of engineering, are professionals who invent, design, analyze, build and test machines, complex systems, structures, gadgets and materials to fulfill functional objectives and requirements while considering the l ...
s such as
Ove Arup
Sir Ove Nyquist Arup, CBE, MICE, MIStructE, FCIOB (16 April 1895 – 5 February 1988) was an English engineer who founded Arup Group Limited, a multinational corporation that offers engineering, design, planning, project management, and ...
,
Buro Happold
Buro Happold (previously ''BuroHappold Engineering'') is a British professional services firm that provides engineering consultancy, design, planning, project management, and consulting services for buildings, infrastructure, and the environment ...
, Walter Bird of
Birdair, Inc.,
Frei Otto
Frei Paul Otto (; 31 May 1925 – 9 March 2015) was a German architect and structural engineer noted for his use of lightweight structures, in particular tensile and membrane structures, including the roof of the Olympic Stadium in Munich for ...
,
Mahmoud Bodo Rasch
Mahmoud Bodo Rasch (born 12 May 1943) is a German architect who specializes in the construction of large convertible umbrellas and lightweight structures. He is founder and owner of SL Rasch GmbH Special and Lightweight Structures with branches i ...
,
Eero Saarinen
Eero Saarinen (, ; August 20, 1910 – September 1, 1961) was a Finnish-American architect and industrial designer noted for his wide-ranging array of designs for buildings and monuments. Saarinen is best known for designing the General Motors ...
,
,
Matthew Nowicki
Matthew Nowicki (in Poland known as Maciej Nowicki) (26 June 1910 – 1 September 1950) was a Polish architect. He was chief architect of the new Indian city of Chandigarh.
Career
Nowicki was born in Chita in Siberia. After the Second Worl ...
,
Jörg Schlaich
Jörg Schlaich (17 October 1934 – 4 September 2021) was a German structural engineer and is known internationally for his ground-breaking work in the creative design of bridges, long-span roofs, and other complex structures. He was a co-founde ...
, the duo of
Nicholas Goldsmith &
Todd Dalland at
FTL Design & Engineering Studio and
David Geiger
David H. Geiger (1935 – October 3, 1989) was an American engineer who invented the air-supported fabric roof system that at the time of his death was in use at almost half the domed stadiums in the world.
Geiger was born in Philadelphia, Pennsy ...
.
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.
Types of structure with significant tension members
Linear structures
*
Suspension bridges
A suspension bridge is a type of bridge in which the deck is hung below suspension cables on vertical suspenders. The first modern examples of this type of bridge were built in the early 1800s. Simple suspension bridges, which lack vertical ...
*
Stressed ribbon bridge
A stressed ribbon bridge (also stress-ribbon bridge or catenary bridge) is a tension structure (similar in many ways to a simple suspension bridge). The suspension cables are embedded in the deck, which follows a catenary arc between supports. ...
*Draped cables
*
Cable-stayed
A cable-stayed bridge has one or more ''towers'' (or ''pylons''), from which cables support the bridge deck. A distinctive feature are the cables or stays, which run directly from the tower to the deck, normally forming a fan-like pattern ...
beams or
truss
A truss is an assembly of ''members'' such as beams, connected by ''nodes'', that creates a rigid structure.
In engineering, a truss is a structure that "consists of two-force members only, where the members are organized so that the assembl ...
es
*Cable trusses
*Straight tensioned cables
Three-dimensional structures
*
Bicycle wheel
A bicycle wheel is a wheel, most commonly a wire wheel, designed for a bicycle. A pair is often called a wheelset, especially in the context of ready built "off the shelf" performance-oriented wheels.
Bicycle wheels are typically designed ...
(can be used as a roof in a horizontal orientation)
*3D cable trusses
*
Tensegrity
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 (usually ...
structures
Surface-stressed structures
*Prestressed membranes
*Pneumatically stressed membranes
*
Gridshell
A gridshell is a structure which derives its strength from its double curvature (in a similar way that a fabric structure derives strength from double curvature), but is constructed of a grid or lattice.
The grid can be made of any material, ...
*
Fabric structure
A fabric structure is a structure made of fabric, with or without a structural frame. The technology provides end users a variety of aesthetic free-form building designs. Custom-made structures are engineered and fabricated to meet worldwide struc ...
Cable and membrane structures
Membrane materials
Common materials for doubly curved fabric structures are
PTFE
Polytetrafluoroethylene (PTFE) is a synthetic fluoropolymer of tetrafluoroethylene that has numerous applications. It is one of the best-known and widely applied PFAS. The commonly known brand name of PTFE-based composition is Teflon by Chemour ...
-coated
fiberglass
Fiberglass (American English) or fibreglass (Commonwealth English) is a common type of fiber-reinforced plastic using glass fiber. The fibers may be randomly arranged, flattened into a sheet called a chopped strand mat, or woven into glass cloth ...
and
PVC-coated
polyester
Polyester is a category of polymers that contain the ester functional group in every repeat unit of their main chain. As a specific material, it most commonly refers to a type called polyethylene terephthalate (PET). Polyesters include natural ...
. These are woven materials with different strengths in different directions. The
warp
Warp, warped or warping may refer to:
Arts and entertainment Books and comics
* WaRP Graphics, an alternative comics publisher
* ''Warp'' (First Comics), comic book series published by First Comics based on the play ''Warp!''
* Warp (comics), a ...
fibers (those fibers which are originally straight—equivalent to the starting fibers on a loom) can carry greater load than the
weft
Warp and weft are the two basic components used in weaving to turn thread or yarn into fabric. The lengthwise or longitudinal warp yarns are held stationary in tension on a frame or loom while the transverse weft (sometimes woof) is draw ...
or fill fibers, which are woven between the warp fibers.
Other structures make use of
ETFE
Ethylene tetrafluoroethylene (ETFE) is a fluorine-based plastic. It was designed to have high corrosion resistance and strength over a wide temperature range. ETFE is a polymer and its source-based name is poly(ethene-co-tetrafluoroethene). It i ...
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
Allianz Arena (; known as Fußball Arena München for UEFA competitions) is a football stadium in Munich, Bavaria, Germany with a 70,000 seating capacity for international matches and 75,000 for domestic matches. Widely known for its exterior ...
in
Munich
Munich ( ; german: München ; bar, Minga ) is the capital and most populous city of the States of Germany, German state of Bavaria. With a population of 1,558,395 inhabitants as of 31 July 2020, it is the List of cities in Germany by popu ...
). 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.
Cables
Cables can be of
mild steel
Carbon steel is a steel with carbon content from about 0.05 up to 2.1 percent by weight. The definition of carbon steel from the American Iron and Steel Institute (AISI) states:
* no minimum content is specified or required for chromium, cobalt ...
,
high strength steel
High-strength low-alloy steel (HSLA) is a type of alloy steel that provides better mechanical properties or greater resistance to corrosion than carbon steel. HSLA steels vary from other steels in that they are not made to meet a specific chemica ...
(drawn carbon steel),
stainless steel
Stainless steel is an alloy of iron that is resistant to rusting and corrosion. It contains at least 11% chromium and may contain elements such as carbon, other nonmetals and metals to obtain other desired properties. Stainless steel's corros ...
,
polyester
Polyester is a category of polymers that contain the ester functional group in every repeat unit of their main chain. As a specific material, it most commonly refers to a type called polyethylene terephthalate (PET). Polyesters include natural ...
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
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
, ''E'' of 150±10 kN/mm² (or 150±10
GPa
Grading in education is the process of applying standardized measurements for varying levels of achievements in a course. Grades can be assigned as letters (usually A through F), as a range (for example, 1 to 6), as a percentage, or as a numbe ...
) and come in sizes from 3 to 90 mm diameter. 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
Ultimate tensile strength (UTS), often shortened to tensile strength (TS), ultimate strength, or F_\text within equations, is the maximum stress that a material can withstand while being stretched or pulled before breaking. In brittle materials t ...
, or the breaking load:
Structural forms
Air-supported structure
An air-supported (or air-inflated) structure is any building that derives its structural integrity from the use of internal pressurized air to inflate a pliable material (i.e. structural fabric) envelope, so that air is the main support of the str ...
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
Stiffness is the extent to which an object resists deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness, k, of a b ...
to withstand the loads it is subjected to (for example
wind
Wind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few hou ...
and
snow
Snow comprises individual ice crystals that grow while suspended in the atmosphere—usually within clouds—and then fall, accumulating on the ground where they undergo further changes.
It consists of frozen crystalline water throughout ...
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:
:
where:
*''R''
1 and ''R''
2 are the principal radii of curvature for soap films or the directions of the warp and weft for fabrics
*''t''
1 and ''t''
2 are the tensions in the relevant directions
*''w'' is the load per square metre
Lines of
principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by ...
have no twist and intersect other lines of principal curvature at right angles.
A
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
or
geodetic
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...
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 ''R''
1 = −''R''
2.
It is now possible to use powerful
non-linear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
programs (or
finite element analysis
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
) 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
It is important that the final form will not allow
ponding Ponding is the (typically) unwanted pooling of water, typically on a flat roof or roadway. Ponding water accelerates the deterioration of many materials, including seam adhesives in single-ply roof systems, steel equipment supports, and particularly ...
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
The Hubert H. Humphrey Metrodome (commonly called the Metrodome) was a domed sports stadium located in downtown Minneapolis, Minnesota. It opened in 1982 as a replacement for Metropolitan Stadium, the former home of the National Football League ...
, an air-inflated structure in
Minneapolis, Minnesota
Minneapolis () is the largest city in Minnesota, United States, and the county seat of Hennepin County. The city is abundant in water, with thirteen lakes, wetlands, the Mississippi River, creeks and waterfalls. Minneapolis has its origins ...
. Some structures prone to
ponding Ponding is the (typically) unwanted pooling of water, typically on a flat roof or roadway. Ponding water accelerates the deterioration of many materials, including seam adhesives in single-ply roof systems, steel equipment supports, and particularly ...
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
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.
Every plane ...
(not all saddle shapes are hyperbolic paraboloids). This is a double
ruled surface
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the ...
and is often used in both in lightweight shell structures (see
hyperboloid structures
Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet. Often these are tall structures, such as towers, where the hyperboloid geometry's structural strength is used to support an object high above the grou ...
). True ruled surfaces are rarely found in tensile structures. Other forms ar
anticlasticsaddles, 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.
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
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 meth ...
(SGM).
Simple mathematics of cables
Transversely and uniformly loaded cable
A uniformly loaded cable spanning between two supports forms a curve intermediate between a
catenary
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.
The catenary curve has a U-like shape, superficia ...
curve and a
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
. The simplifying assumption can be made that it approximates a circular arc (of radius ''R'').
By
equilibrium:
The horizontal and vertical reactions :
:
:
By
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
:
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
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
, where the axial stiffness, ''k,'' is equal to
):
:
where ''E'' is the
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
of the cable and ''A'' is its cross-sectional
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
.
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
.
Cable with central point load
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''.
Tensioned cable oscillations
The fundamental
natural frequency
Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force.
The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all par ...
, ''f''
1 of tensioned cables is given by:
:
where ''T'' = tension in
newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* ''Newton'' (film), a 2017 Indian film
* Newton ( ...
s, ''m'' =
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
in kilograms and ''L'' = span length.
Notable structures
*
Shukhov Rotunda
Shukhov Rotunda was a round exhibition pavilion built for All-Russia Exhibition 1896 in Nizhny Novgorod, Russia. It was built in 1896 with a diagrid hanging cover (tensile gridshell – diagrid roof, Russian Empire patent No. 1894, dated Marc ...
,
Russia
Russia (, , ), or the Russian Federation, is a List of transcontinental countries, transcontinental country spanning Eastern Europe and North Asia, Northern Asia. It is the List of countries and dependencies by area, largest country in the ...
, 1896
*
Canada Place
Canada Place is a building situated on the Burrard Inlet in Vancouver, British Columbia, Canada. It is home to the Vancouver Convention Centre, the Pan Pacific Vancouver Hotel, the Vancouver World Trade Centre, and the virtual flight ride FlyOve ...
,
Vancouver
Vancouver ( ) is a major city in western Canada, located in the Lower Mainland region of British Columbia. As the List of cities in British Columbia, most populous city in the province, the 2021 Canadian census recorded 662,248 people in the ...
,
British Columbia
British Columbia (commonly abbreviated as BC) is the westernmost province of Canada, situated between the Pacific Ocean and the Rocky Mountains. It has a diverse geography, with rugged landscapes that include rocky coastlines, sandy beaches, ...
for
Expo '86
The 1986 World Exposition on Transportation and Communication, or simply Expo 86, was a World's Fair held in Vancouver, British Columbia, Canada from May 2 until October 13, 1986. The fair, the theme of which was "Transportation and Communicatio ...
*
Yoyogi National Gymnasium
Yoyogi National Gymnasium, officially is an indoor arena located at Yoyogi Park in Shibuya, Tokyo, Japan, which is famous for its suspension roof design.
It was designed by Kenzo Tange and built between 1961 and 1964 to house swimming and d ...
by
Kenzo Tange
is a common masculine Japanese given name.
Possible writings
Kenzō can be written using different kanji characters and can mean:
*賢三, "wise, three"
*健三, "healthy, three"
*謙三, "humble, three"
*健想, "healthy, concept"
*建造, "bu ...
,
Yoyogi Park
is a park in Shibuya, Tokyo, Japan. It is located adjacent to Harajuku Station and Meiji Shrine in Yoyogikamizonochō. The park is a popular Tokyo destination, especially on Sundays when it is used as a gathering place for Japanese rock music ...
,
Tokyo
Tokyo (; ja, 東京, , ), officially the Tokyo Metropolis ( ja, 東京都, label=none, ), is the capital and largest city of Japan. Formerly known as Edo, its metropolitan area () is the most populous in the world, with an estimated 37.468 ...
,
Japan
Japan ( ja, 日本, or , and formally , ''Nihonkoku'') is an island country in East Asia. It is situated in the northwest Pacific Ocean, and is bordered on the west by the Sea of Japan, while extending from the Sea of Okhotsk in the north ...
*
Ingalls Rink
David S. Ingalls Rink is a hockey rink in New Haven, Connecticut, designed by architect Eero Saarinen and built between 1953 and 1958 for Yale University. It is commonly referred to as The Whale, due to its shape. The building was constructed fo ...
,
Yale University
Yale University is a private research university in New Haven, Connecticut. Established in 1701 as the Collegiate School, it is the third-oldest institution of higher education in the United States and among the most prestigious in the wo ...
by
Eero Saarinen
Eero Saarinen (, ; August 20, 1910 – September 1, 1961) was a Finnish-American architect and industrial designer noted for his wide-ranging array of designs for buildings and monuments. Saarinen is best known for designing the General Motors ...
*
Khan Shatyry Entertainment Center,
Nur-Sultan, Kazakhstan
Astana, previously known as Akmolinsk, Tselinograd, Akmola, and most recently Nur-Sultan, is the capital city of Kazakhstan.
The city lies on the banks of the Ishim (river), Ishim River in the north-central part of Kazakhstan, within the Akmo ...
*
Tropicana Field
Tropicana Field (commonly known as the Trop) is a Multi-purpose stadium, multi-purpose domed stadium located in St. Petersburg, Florida, United States. The stadium has been the home of the Tampa Bay Rays of Major League Baseball (MLB) since the ...
,
St. Petersburg
Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...
,
Florida
Florida is a state located in the Southeastern region of the United States. Florida is bordered to the west by the Gulf of Mexico, to the northwest by Alabama, to the north by Georgia, to the east by the Bahamas and Atlantic Ocean, and to ...
*
Olympiapark
The Olympiapark (English: Olympic Park) in Munich, Germany, is an Olympic Park which was constructed for the 1972 Summer Olympics. Located in the Oberwiesenfeld neighborhood of Munich, the Park continues to serve as a venue for cultural, social, ...
,
Munich
Munich ( ; german: München ; bar, Minga ) is the capital and most populous city of the States of Germany, German state of Bavaria. With a population of 1,558,395 inhabitants as of 31 July 2020, it is the List of cities in Germany by popu ...
by
Frei Otto
Frei Paul Otto (; 31 May 1925 – 9 March 2015) was a German architect and structural engineer noted for his use of lightweight structures, in particular tensile and membrane structures, including the roof of the Olympic Stadium in Munich for ...
*
Sidney Myer Music Bowl
The Sidney Myer Music Bowl is an outdoor bandshell performance venue in Melbourne, Victoria, Australia. It is located in the lawns and gardens of Kings Domain on Linlithgow Avenue close to the Arts Centre and the Southbank entertainment precinct ...
,
Melbourne
Melbourne ( ; Boonwurrung/Woiwurrung: ''Narrm'' or ''Naarm'') is the capital and most populous city of the Australian state of Victoria, and the second-most populous city in both Australia and Oceania. Its name generally refers to a met ...
*
The O2 (formerly the
Millennium Dome
The Millennium Dome was the original name of the large dome-shaped building on the Greenwich Peninsula in South East (London sub region), South East London, England, which housed a major exhibition celebrating the beginning of the third millenn ...
),
London
London is the capital and largest city of England and the United Kingdom, with a population of just under 9 million. It stands on the River Thames in south-east England at the head of a estuary down to the North Sea, and has been a majo ...
by
Buro Happold
Buro Happold (previously ''BuroHappold Engineering'') is a British professional services firm that provides engineering consultancy, design, planning, project management, and consulting services for buildings, infrastructure, and the environment ...
and
Richard Rogers Partnership
RSHP is a British architectural firm, founded in 1977 and previously known as the Richard Rogers Partnership which became Rogers Stirk Harbour + Partners. in 2007. The firm rebranded from Rogers Stirk Harbour + Partners to simply RSHP on 30 Jun ...
*
Denver International Airport
Denver International Airport , locally known as DIA, is an international airport in the Western United States, primarily serving metropolitan Denver, Colorado, as well as the greater Front Range Urban Corridor. At , it is the largest airport in ...
,
Denver
Denver () is a consolidated city and county, the capital, and most populous city of the U.S. state of Colorado. Its population was 715,522 at the 2020 census, a 19.22% increase since 2010. It is the 19th-most populous city in the Unit ...
*
Dorton Arena
J. S. Dorton Arena is a 7,610-seat multi-purpose arena located in Raleigh, North Carolina, on the grounds of the North Carolina State Fair. It opened in 1952.
Architect Maciej Nowicki of the North Carolina State University Department of Architec ...
,
Raleigh
Raleigh (; ) is the capital city of the state of North Carolina and the seat of Wake County in the United States. It is the second-most populous city in North Carolina, after Charlotte. Raleigh is the tenth-most populous city in the Southeas ...
*
Georgia Dome
The Georgia Dome was a domed stadium in the Southeastern United States. Located in Atlanta between downtown to the east and Vine City to the west, it was owned and operated by the State of Georgia as part of the Georgia World Congress Center ...
,
Atlanta
Atlanta ( ) is the capital and most populous city of the U.S. state of Georgia. It is the seat of Fulton County, the most populous county in Georgia, but its territory falls in both Fulton and DeKalb counties. With a population of 498,715 ...
,
Georgia
Georgia most commonly refers to:
* Georgia (country), a country in the Caucasus region of Eurasia
* Georgia (U.S. state), a state in the Southeast United States
Georgia may also refer to:
Places
Historical states and entities
* Related to the ...
by
Heery and
Weidlinger Associates
Weidlinger Associates, Inc., was a U.S.-based structural engineering firm that designs and rehabilitates buildings, bridges, and infrastructure and provides special services in applied science, forensics, and physical security. Weidlinger's clien ...
(demolished in 2017)
*
Grantley Adams International Airport
Grantley Adams International Airport (GAIA) is the international airport of Barbados, located in Seawell, Christ Church. It is the only designated port of entry for persons arriving and departing by air in Barbados and operates as one of t ...
,
Christ Church,
Barbados
Barbados is an island country in the Lesser Antilles of the West Indies, in the Caribbean region of the Americas, and the most easterly of the Caribbean Islands. It occupies an area of and has a population of about 287,000 (2019 estimate). ...
*
Pengrowth Saddledome
Scotiabank Saddledome is a multi-use indoor arena in Calgary, Alberta, Canada. Located in Stampede Park in the southeast end of downtown Calgary, the Saddledome was built in 1983 to replace the Stampede Corral as the home of the Calgary Flames ...
,
Calgary
Calgary ( ) is the largest city in the western Canadian province of Alberta and the largest metro area of the three Prairie Provinces. As of 2021, the city proper had a population of 1,306,784 and a metropolitan population of 1,481,806, makin ...
by
Graham McCourt Architects and
Jan Bobrowski and Partners
Jan, JaN or JAN may refer to:
Acronyms
* Jackson, Mississippi (Amtrak station), US, Amtrak station code JAN
* Jackson-Evers International Airport, Mississippi, US, IATA code
* Jabhat al-Nusra (JaN), a Syrian militant group
* Japanese Article Numb ...
*
Scandinavium
Scandinavium () is an indoor arena located in Gothenburg, Sweden. Construction on Scandinavium began in 1969 after decades of setbacks, and was inaugurated on 18 May 1971.
Scandinavium has been selected as a championship arena at least fifty t ...
,
Gothenburg
Gothenburg (; abbreviated Gbg; sv, Göteborg ) is the second-largest city in Sweden, fifth-largest in the Nordic countries, and capital of the Västra Götaland County. It is situated by the Kattegat, on the west coast of Sweden, and has ...
,
Sweden
Sweden, formally the Kingdom of Sweden,The United Nations Group of Experts on Geographical Names states that the country's formal name is the Kingdom of SwedenUNGEGN World Geographical Names, Sweden./ref> is a Nordic country located on ...
*
Hong Kong Museum of Coastal Defence
The Hong Kong Museum of Coastal Defence is a museum in Hong Kong, located in a former coastal defence fort overlooking the Lei Yue Mun channel, near Shau Kei Wan on Hong Kong Island. The fort was built by the British in 1887, intended to def ...
*Modernization of the
Central Railway Station,
Sofia
Sofia ( ; bg, София, Sofiya, ) is the capital and largest city of Bulgaria. It is situated in the Sofia Valley at the foot of the Vitosha mountain in the western parts of the country. The city is built west of the Iskar river, and ha ...
,
Bulgaria
Bulgaria (; bg, България, Bǎlgariya), officially the Republic of Bulgaria,, ) is a country in Southeast Europe. It is situated on the eastern flank of the Balkans, and is bordered by Romania to the north, Serbia and North Macedon ...
*
Redbird Arena
CEFCU Arena, formerly known as Redbird Arena, is a 10,200-seat multi-purpose arena located in Normal, Illinois, on the campus of Illinois State University. Built in 1989, the building is notable for its use of a Teflon-coated roof that gives off ...
,
Illinois State University
Illinois State University (ISU) is a public university in Normal, Illinois. Founded in 1857 as Illinois State Normal University, it is the oldest public university in Illinois. The university emphasizes teaching and is recognized as one of th ...
,
Normal, Illinois
Normal is a town in McLean County, Illinois, United States. As of the 2020 census, the town's population was 52,736. Normal is the smaller of two principal municipalities of the Bloomington–Normal metropolitan area, and Illinois' seventh most ...
*Retractable Umbrellas,
Al-Masjid an-Nabawi
Al-Masjid an-Nabawi (), known in English as the Prophet's Mosque, is a mosque built by the Islamic prophet Muhammad in the city of Medina in the Al Madinah Province of Saudi Arabia. It was the second mosque built by Muhammad in Medina, after Qub ...
, Medina,
Saudi Arabia
Saudi Arabia, officially the Kingdom of Saudi Arabia (KSA), is a country in Western Asia. It covers the bulk of the Arabian Peninsula, and has a land area of about , making it the fifth-largest country in Asia, the second-largest in the A ...
*
Killesberg Tower
The Killesberg Tower (called 'Killesbergturm' in German language, German) is a 40-meter high observation tower located in the Killesberg Park in Stuttgart, Stuttgart, Germany. Originally planned for the 1993 World Horticultural Exposition, an inte ...
,
Stuttgart
Stuttgart (; Swabian: ; ) is the capital and largest city of the German state of Baden-Württemberg. It is located on the Neckar river in a fertile valley known as the ''Stuttgarter Kessel'' (Stuttgart Cauldron) and lies an hour from the ...
Gallery of well-known tensile structures
File:Olympic park 12.jpg, The roof tensile structures by Frei Otto
Frei Paul Otto (; 31 May 1925 – 9 March 2015) was a German architect and structural engineer noted for his use of lightweight structures, in particular tensile and membrane structures, including the roof of the Olympic Stadium in Munich for ...
of the Olympiapark
The Olympiapark (English: Olympic Park) in Munich, Germany, is an Olympic Park which was constructed for the 1972 Summer Olympics. Located in the Oberwiesenfeld neighborhood of Munich, the Park continues to serve as a venue for cultural, social, ...
, Munich
Munich ( ; german: München ; bar, Minga ) is the capital and most populous city of the States of Germany, German state of Bavaria. With a population of 1,558,395 inhabitants as of 31 July 2020, it is the List of cities in Germany by popu ...
File:Canary.wharf.and.dome.london.arp.jpg, The Millennium Dome
The Millennium Dome was the original name of the large dome-shaped building on the Greenwich Peninsula in South East (London sub region), South East London, England, which housed a major exhibition celebrating the beginning of the third millenn ...
(now The O2), London
London is the capital and largest city of England and the United Kingdom, with a population of just under 9 million. It stands on the River Thames in south-east England at the head of a estuary down to the North Sea, and has been a majo ...
, by Buro Happold
Buro Happold (previously ''BuroHappold Engineering'') is a British professional services firm that provides engineering consultancy, design, planning, project management, and consulting services for buildings, infrastructure, and the environment ...
and Richard Rogers
Richard George Rogers, Baron Rogers of Riverside (23 July 1933 – 18 December 2021) was a British architect noted for his modernist and Functionalism (architecture), functionalist designs in high-tech architecture. He was a senior partner a ...
File:Denver International Airport terminal.jpg, Denver International Airport
Denver International Airport , locally known as DIA, is an international airport in the Western United States, primarily serving metropolitan Denver, Colorado, as well as the greater Front Range Urban Corridor. At , it is the largest airport in ...
terminal
File:Thtr300 kuehlturm.jpg, The THTR-300
The THTR-300 was a thorium cycle high-temperature nuclear reactor rated at 300 MW electric (THTR-300) in Hamm-Uentrop, Germany. It started operating in 1983, synchronized with the grid in 1985, operated at full power in February 1987 and was shu ...
cable-net dry cooling tower
A cooling tower is a device that rejects waste heat to the atmosphere through the cooling of a coolant stream, usually a water stream to a lower temperature. Cooling towers may either use the evaporation of water to remove process heat and ...
, hyperboloid structure
Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet. Often these are tall structures, such as towers, where the hyperboloid geometry's structural strength is used to support an object high above the gro ...
by Schlaich Bergermann & Partner
Schlaich bergermann partner is a nationally and internationally active structural engineering and consulting firm with headquarters in Stuttgart, Germany and branch offices in Berlin, New York City, São Paulo, Shanghai and Paris.
History
The ...
File:Killesberg Tower.jpg, Killesberg Tower, Stuttgart, by Schlaich Bergermann Partner
Schlaich bergermann partner is a nationally and internationally active structural engineering and consulting firm with headquarters in Stuttgart, Germany and branch offices in Berlin, New York City, São Paulo, Shanghai and Paris.
History
The f ...
File:GeorgiaDome md.jpg, Georgia Dome
The Georgia Dome was a domed stadium in the Southeastern United States. Located in Atlanta between downtown to the east and Vine City to the west, it was owned and operated by the State of Georgia as part of the Georgia World Congress Center ...
in Atlanta
Atlanta ( ) is the capital and most populous city of the U.S. state of Georgia. It is the seat of Fulton County, the most populous county in Georgia, but its territory falls in both Fulton and DeKalb counties. With a population of 498,715 ...
File:Stamps of Kazakhstan, 2010-08.jpg, Daytime computer render of Khan Shatyr Entertainment Center
Khan Shatyr ( kk, Хан Шатыр, Han Şatyr, Royal Marquee) is a transparent tent located in Astana, the capital city of Kazakhstan. Built in a neo-futurism, neo-futurist style, the architectural project was unveiled by the first President of ...
, the highest tensile structure in the world
Classification numbers
The
Construction Specifications Institute
The Construction Specifications Institute (CSI) is a United States national association of more than 8,000 construction industry professionals who are experts in building construction and the materials used therein. The institute is dedicated to ...
(CSI) and Construction Specifications Canada (CSC),
MasterFormat MasterFormat is a standard for organizing specifications and other written information for commercial and institutional building projects in the U.S. and Canada. Sometimes referred to as the "Dewey Decimal System" of building construction, MasterFo ...
2018 Edition, Division 05 and 13:
* 05 16 00 – Structural Cabling
* 05 19 00 - Tension Rod and Cable Truss Assemblies
* 13 31 00 – Fabric Structures
* 13 31 23 – Tensioned Fabric Structures
* 13 31 33 – Framed Fabric Structures
CSI/CSC
MasterFormat MasterFormat is a standard for organizing specifications and other written information for commercial and institutional building projects in the U.S. and Canada. Sometimes referred to as the "Dewey Decimal System" of building construction, MasterFo ...
1995 Edition:
* 13120 – Cable-Supported Structures
* 13120 – Fabric Structures
See also
*
Buckminster Fuller
Richard Buckminster Fuller (; July 12, 1895 – July 1, 1983) was an American architect, systems theorist, writer, designer, inventor, philosopher, and futurist. He styled his name as R. Buckminster Fuller in his writings, publishing more t ...
*
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
*
Geodesic dome
A geodesic dome is a hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron. The triangular elements of the dome are structurally rigid and distribute the structural stress throughout the structure, making geodesic dom ...
*
Geodesics
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
*
Hyperboloid structure
Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet. Often these are tall structures, such as towers, where the hyperboloid geometry's structural strength is used to support an object high above the gro ...
*
Kārlis Johansons
Karl Johansson (January 16, 1890 in Cēsis, Latvia, as Kārlis Johansons in his native Latvian; Russian: Карл Вольдемарович Иогансон, ''Karl Voldemarovich Ioganson;'' German: ''Karl Ioganson'' – October 18, 1929 in Mos ...
*
Kenneth Snelson
Kenneth Duane Snelson (June 29, 1927 – December 22, 2016) was an American contemporary sculptor and photographer. His sculptural works are composed of flexible and rigid components arranged according to the idea of 'tensegrity'. Snelson prefer ...
*
Suspended structure
A suspended structure is a structure which is supported by cables coming from beams or trusses which sit atop a concrete center column or core. The design allows the walls, roof and cantilevered floors to be supported entirely by cables and a cen ...
*
Suspension bridge
A suspension bridge is a type of bridge in which the deck (bridge), deck is hung below suspension wire rope, cables on vertical suspenders. The first modern examples of this type of bridge were built in the early 1800s. Simple suspension bridg ...
*
Tensairity
Tensairity is a trademarked term for a light weight structural concept that uses low pressure air to stabilize compression elements against buckling.[Tensegrity
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 (usually ...]
*
Wire rope
Steel wire rope (right hand lang lay)
Wire rope is several strands of metal wire twisted into a helix forming a composite
''rope'', in a pattern known as ''laid rope''. Larger diameter wire rope consists of multiple strands of such laid rope in a ...
References
Further reading
* "The Nijni-Novgorod exhibition: Water tower, room under construction, springing of 91 feet span",
"The Engineer", № 19.3.1897, P.292-294, London, 1897.
*
, ''Light structures, structures of light: The art and engineering of tensile architecture'' (Birkhäuser Verlag, 1996)
*Alan Holgate, ''The Art of Structural Engineering: The Work of Jorg Schlaich and his Team'' (Books Britain, 1996)
Elizabeth Cooper English"Arkhitektura i mnimosti": The origins of Soviet avant-garde rationalist architecture in the Russian mystical-philosophical and mathematical intellectual tradition" a dissertation in architecture, 264 p., University of Pennsylvania, 2000.
* "Vladimir G. Suchov 1853–1939. Die Kunst der sparsamen Konstruktion.", Rainer Graefe, Jos Tomlow und andere, 192 S., Deutsche Verlags-Anstalt, Stuttgart, 1990, .
*
Conrad Roland
Conrad Roland (''Konrad Roland Lehmann''; 1934 – 25 September 2020) was a German architect and pioneer of the construction of space nets (tensile structures), which primarily are to be found as rope climbing frames on playgrounds. In 1978 he ...
: ''
Frei Otto
Frei Paul Otto (; 31 May 1925 – 9 March 2015) was a German architect and structural engineer noted for his use of lightweight structures, in particular tensile and membrane structures, including the roof of the Olympic Stadium in Munich for ...
– Spannweiten. Ideen und Versuche zum Leichtbau''. Ein Werkstattbericht von Conrad Roland. Ullstein, Berlin, Frankfurt/Main und Wien 1965.
* Frei Otto, Bodo Rasch: Finding Form - Towards an Architecture of the Minimal, Edition Axel Menges, 1996,
* Nerdinger, Winfried: Frei Otto. Das Gesamtwerk: Leicht Bauen Natürlich Gestalten, 2005, {{ISBN, 3-7643-7233-8
Structural system
Roofs
Russian inventions