Cantilever

# Cantilever

Discussion

Encyclopedia
A cantilever is a beam
Beam (structure)
A beam is a horizontal structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending moment.- Overview...

anchored at only one end. The beam carries the load to the support where it is resisted by moment
Moment (physics)
In physics, the term moment can refer to many different concepts:*Moment of force is the tendency of a force to twist or rotate an object; see the article torque for details. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the...

and shear stress
Shear stress
A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...

. Cantilever construction allows for overhanging structures without external bracing. Cantilevers can also be constructed with truss
Truss
In architecture and structural engineering, a truss is a structure comprising one or more triangular units constructed with straight members whose ends are connected at joints referred to as nodes. External forces and reactions to those forces are considered to act only at the nodes and result in...

es or slab
Concrete slab
A concrete slab is a common structural element of modern buildings. Horizontal slabs of steel reinforced concrete, typically between 10 and 50 centimeters thick, are most often used to construct floors and ceilings, while thinner slabs are also used for exterior paving.In many domestic and...

s.

This is in contrast to a simply supported beam such as those found in a post and lintel
Post and lintel
Post and lintel, or in contemporary usage Post and beam, is a simple construction method using a lintel, header, or architrave as the horizontal member over a building void supported at its ends by two vertical columns, pillars, or posts...

system. A simply supported beam is supported at both ends with loads applied between the supports.

## In bridges, towers, and buildings

Cantilevers are widely found in construction, notably in cantilever bridge
Cantilever bridge
A cantilever bridge is a bridge built using cantilevers, structures that project horizontally into space, supported on only one end. For small footbridges, the cantilevers may be simple beams; however, large cantilever bridges designed to handle road or rail traffic use trusses built from...

s and balconies
Balcony
Balcony , a platform projecting from the wall of a building, supported by columns or console brackets, and enclosed with a balustrade.-Types:The traditional Maltese balcony is a wooden closed balcony projecting from a...

(see corbel
Corbel
In architecture a corbel is a piece of stone jutting out of a wall to carry any superincumbent weight. A piece of timber projecting in the same way was called a "tassel" or a "bragger". The technique of corbelling, where rows of corbels deeply keyed inside a wall support a projecting wall or...

). In cantilever bridges the cantilevers are usually built as pairs, with each cantilever used to support one end of a central section. The Forth Bridge in Scotland
Scotland
Scotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the...

is an example of a cantilever truss bridge
Truss bridge
A truss bridge is a bridge composed of connected elements which may be stressed from tension, compression, or sometimes both in response to dynamic loads. Truss bridges are one of the oldest types of modern bridges...

.

Temporary cantilevers are often used in construction. The partially constructed structure creates a cantilever, but the completed structure does not act as a cantilever. This is very helpful when temporary supports, or falsework
Falsework
Falsework consists of temporary structures used in construction to support spanning or arched structures in order to hold the component in place until its construction is sufficiently advanced to support itself...

, cannot be used to support the structure while it is being built (e.g., over a busy roadway or river, or in a deep valley). So some truss arch bridge
Truss arch bridge
A truss arch bridge combines the elements of the truss bridge and the arch bridge. The actual resolution of forces will depend upon the design. If no horizontal thrusting forces are generated this becomes an arch-shaped truss, essentially a bent beam — see moon bridge for an example...

s (see Navajo Bridge
Navajo Bridge
Navajo Bridge crosses the Colorado River's Marble Canyon near Lee's Ferry in the US state of Arizona. Apart from the Glen Canyon Bridge a few miles upstream at Page, Arizona, it is the only roadway crossing of the river and the Grand Canyon for nearly...

) are built from each side as cantilevers until the spans reach each other and are then jacked apart to stress them in compression before final joining. Nearly all cable-stayed bridges are built using cantilevers as this is one of their chief advantages. Many box girder bridges are built segmentally
Segmental bridge
As its name implies, a segmental bridge is a bridge built in short sections , i.e., one piece at a time, as opposed to traditional methods that build a bridge in very large sections...

, or in short pieces. This type of construction lends itself well to balanced cantilever construction where the bridge is built in both directions from a single support.

These structures are highly based on torque and rotational equilibrium.

In an architectural application, Frank Lloyd Wright
Frank Lloyd Wright
Frank Lloyd Wright was an American architect, interior designer, writer and educator, who designed more than 1,000 structures and completed 500 works. Wright believed in designing structures which were in harmony with humanity and its environment, a philosophy he called organic architecture...

's Fallingwater
Fallingwater
Fallingwater or Kaufmann Residence is a house designed by architect Frank Lloyd Wright in 1935 in rural southwestern Pennsylvania, 50 miles southeast of Pittsburgh...

used cantilevers to project large balconies. The East Stand at Elland Road
Elland Road is an all-seater football stadium in Beeston, Leeds, West Yorkshire, England. It has been the permanent residence of Leeds United A.F.C...

Stadium in Leeds was, when completed, the largest cantilever stand in the world holding 17,000 spectators. The roof
Roof
A roof is the covering on the uppermost part of a building. A roof protects the building and its contents from the effects of weather. Structures that require roofs range from a letter box to a cathedral or stadium, dwellings being the most numerous....

built over the stands at Old Trafford Football Ground uses a cantilever so that no supports will block views of the field. The old, now demolished Miami Stadium
Miami Stadium was a baseball stadium in Miami, Florida. It was primarily used for baseball, and was the home field of the Miami Marlins minor league baseball team, as well as other minor league teams. It opened in 1949 and held 13,500 people...

had a similar roof over the spectator area. The largest cantilever in Europe is located at St James' Park
St James' Park
St James' Park, known for sponsorship reasons as the Sports Direct Arena, is an all-seater stadium in Newcastle upon Tyne, England. It is the home of Newcastle United Football Club and is the sixth largest football stadium in the United Kingdom with a capacity of between 52,387 and 52,409.St James'...

in Newcastle-Upon-Tyne, the home stadium of Newcastle United F.C.
Newcastle United F.C.
Newcastle United Football Club is an English professional association football club based in Newcastle upon Tyne, Tyne and Wear. The club was founded in 1892 by the merger of Newcastle East End and Newcastle West End, and has played at its current home ground, St James' Park, since the merger...

Less obvious examples of cantilevers are free-standing (vertical) radio towers
Radio masts and towers are, typically, tall structures designed to support antennas for telecommunications and broadcasting, including television. They are among the tallest man-made structures...

without guy-wire
Guy-wire
A guy-wire or guy-rope, also known as simply a guy, is a tensioned cable designed to add stability to structures . One end of the cable is attached to the structure, and the other is anchored to the ground at a distance from the structure's base...

s, and chimneys, which resist being blown over by the wind through cantilever action at their base.

## Aircraft

Another use of the cantilever is in fixed-wing aircraft
Fixed-wing aircraft
A fixed-wing aircraft is an aircraft capable of flight using wings that generate lift due to the vehicle's forward airspeed. Fixed-wing aircraft are distinct from rotary-wing aircraft in which wings rotate about a fixed mast and ornithopters in which lift is generated by flapping wings.A powered...

design, pioneered by Hugo Junkers
Hugo Junkers
Hugo Junkers was an innovative German engineer, as his many patents in varied areas show...

in 1915. Early aircraft wings typically bore their loads by using two (or more) wings in a biplane
Biplane
A biplane is a fixed-wing aircraft with two superimposed main wings. The Wright brothers' Wright Flyer used a biplane design, as did most aircraft in the early years of aviation. While a biplane wing structure has a structural advantage, it produces more drag than a similar monoplane wing...

configuration braced with wires and struts. They were similar to truss bridge
Truss bridge
A truss bridge is a bridge composed of connected elements which may be stressed from tension, compression, or sometimes both in response to dynamic loads. Truss bridges are one of the oldest types of modern bridges...

s, having been developed by Octave Chanute
Octave Chanute
Octave Chanute was a French-born American railway engineer and aviation pioneer. He provided the Wright brothers with help and advice, and helped to publicize their flying experiments. At his death he was hailed as the father of aviation and the heavier-than-air flying machine...

, a railroad bridge engineer. The wings were braced with crossed wires so they would stay parallel, as well as front-to-back to resist twisting, running diagonally between adjacent strut anchorages. The cables and struts generated considerable drag, and there was constant experimentation on ways to eliminate them.

It was also desirable to build a monoplane
Monoplane
A monoplane is a fixed-wing aircraft with one main set of wing surfaces, in contrast to a biplane or triplane. Since the late 1930s it has been the most common form for a fixed wing aircraft.-Types of monoplane:...

aircraft, as the airflow around one wing negatively affects the other in a biplane's airframe design. Early monoplanes used either strut
Strut
A strut is a structural component designed to resist longitudinal compression. Struts provide outwards-facing support in their lengthwise direction, which can be used to keep two other components separate, performing the opposite function of a tie...

s (as do some current light aircraft), or cables like the 1909 Bleriot XI
Blériot XI
The Blériot XI is the aircraft in which, on 25 July 1909, Louis Blériot made the first flight across the English Channel made in a heavier-than-air aircraft . This achievement is one of the most famous accomplishments of the early years of aviation, and not only won Blériot a lasting place in...

(as do some modern home-built aircraft). The advantage in using struts or cables is a reduction in weight for a given strength, but with the penalty of additional drag. This reduces maximum speed, and increases fuel consumption.
Hugo Junkers endeavored to eliminate all external bracing, only a dozen years after the Wright Brothers
Wright brothers
The Wright brothers, Orville and Wilbur , were two Americans credited with inventing and building the world's first successful airplane and making the first controlled, powered and sustained heavier-than-air human flight, on December 17, 1903...

' initial flights, to decrease airframe drag in flight, with the result being the Junkers J 1
Junkers J 1
The Junkers J 1, nicknamed the Blechesel , was the world's first practical all-metal aircraft. Built early in World War I, when aircraft designers relied largely on fabric-covered wooden structures, the Junkers J 1 was a revolutionary development in aircraft design, being built and flown only 12...

pioneering all-metal monoplane of late 1915, designed from the start with all-metal cantilever wing panels.

The most common current wing design is the cantilever. A single large beam, called the main spar
Spar (aviation)
In a fixed-wing aircraft, the spar is often the main structural member of the wing, running spanwise at right angles to the fuselage. The spar carries flight loads and the weight of the wings whilst on the ground...

, runs through the wing, typically nearer the leading edge
The leading edge is the part of the wing that first contacts the air; alternatively it is the foremost edge of an airfoil section. The first is an aerodynamic definition, the second a structural one....

at about 25 percent of the total chord
Chord (aircraft)
In aeronautics, chord refers to the imaginary straight line joining the trailing edge and the center of curvature of the leading edge of the cross-section of an airfoil...

. In flight, the wings generate lift
Lift (force)
A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...

, and the wing spars are designed to carry this load through the fuselage to the other wing. To resist fore and aft movement, the wing will usually be fitted with a second smaller drag-spar nearer the trailing edge
Trailing edge
The trailing edge of an aerodynamic surface such as a wing is its rear edge, where the airflow separated by the leading edge rejoins. Essential control surfaces are attached here to redirect the air flow and exert a controlling force by changing its momentum...

, tied to the main spar with structural elements or a stressed skin. The wing must also resist twisting forces, done either by a monocoque
Monocoque
Monocoque is a construction technique that supports structural load by using an object's external skin, as opposed to using an internal frame or truss that is then covered with a non-load-bearing skin or coachwork...

"D" tube structure forming the leading edge, or by the aforementioned linking two spars in some form of box beam or lattice girder
Lattice girder
A lattice girder is a girder where the flanges are connected by a lattice web. This type of design has been supplanted in modern construction with welded or bolted plate girders, which use more material but have lower fabrication and maintenance costs...

structure.

Cantilever wings require a much heavier spar than would otherwise be needed in cable-stayed designs. However, as the size of an aircraft increases, the additional weight penalty decreases. Eventually a line was crossed in the 1920s, and designs increasingly turned to the cantilever design. By the 1940s almost all larger aircraft used the cantilever exclusively, even on smaller surfaces such as the horizontal stabilizer.

## In microelectromechanical systems

Cantilevered beams are the most ubiquitous structures in the field of microelectromechanical systems
Microelectromechanical systems
Microelectromechanical systems is the technology of very small mechanical devices driven by electricity; it merges at the nano-scale into nanoelectromechanical systems and nanotechnology...

(MEMS). An early example of a MEMS cantilever is the Resonistor, an electromechanical monolithic resonator. MEMS cantilevers are commonly fabricated from silicon
Silicon
Silicon is a chemical element with the symbol Si and atomic number 14. A tetravalent metalloid, it is less reactive than its chemical analog carbon, the nonmetal directly above it in the periodic table, but more reactive than germanium, the metalloid directly below it in the table...

(Si), silicon nitride
Silicon nitride
Silicon nitride is a chemical compound of silicon and nitrogen. If powdered silicon is heated between 1300° and 1400°C in an atmosphere of nitrogen, trisilicon tetranitride, Si3N4, is formed. The silicon sample weight increases progressively due to the chemical combination of silicon and nitrogen...

(Si3N4), or polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...

s. The fabrication process typically involves undercutting the cantilever structure to release it, often with an anisotropic wet or dry
Reactive ion etching
Reactive-ion etching is an etching technology used in microfabrication. It uses chemically reactive plasma to remove material deposited on wafers. The plasma is generated under low pressure by an electromagnetic field...

etching technique. Without cantilever transducers, atomic force microscopy would not be possible. A large number of research groups are attempting to develop cantilever arrays as biosensor
Biosensor
A biosensor is an analytical device for the detection of an analyte that combines a biological component with a physicochemical detector component.It consists of 3 parts:* the sensitive biological element A biosensor is an analytical device for the detection of an analyte that combines a biological...

s for medical diagnostic applications. MEMS cantilevers are also finding application as radio frequency
Radio frequency is a rate of oscillation in the range of about 3 kHz to 300 GHz, which corresponds to the frequency of radio waves, and the alternating currents which carry radio signals...

filter
Mechanical filter
A mechanical filter is a signal processing filter usually used in place of an electronic filter at radio frequencies. Its purpose is the same as that of a normal electronic filter: to pass a range of signal frequencies, but to block others. The filter acts on mechanical vibrations which are the...

s and resonator
Resonator
A resonator is a device or system that exhibits resonance or resonant behavior, that is, it naturally oscillates at some frequencies, called its resonant frequencies, with greater amplitude than at others. The oscillations in a resonator can be either electromagnetic or mechanical...

s. The MEMS cantilevers are commonly made as unimorph
Unimorph
A unimorph is a cantilever that consists of one active layer and one inactive layer. In the case where active layer is piezoelectric, deformation in that layer may be induced by the application of an electric field. This deformation induces a bending displacement in the cantilever. The inactive...

s or bimorph
Bimorph
A bimorph is a cantilever that consists of two active layers: piezoelectric and metal. These layers produce a displacement via:* thermal activation ....

s.

Two equations are key to understanding the behavior of MEMS cantilevers. The first is Stoney's formula, which relates cantilever end deflection
Deflection
Deflection or deflexion may refer to:* Deflection , the displacement of a structural element under load* Deflection , a technique of shooting ahead of a moving target so that the target and projectile will collide...

δ to applied stress σ:

where ν is Poisson's ratio
Poisson's ratio
Poisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain , to the extension or axial strain ....

, is Young's modulus
Young's modulus
Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...

, is the beam length and is the cantilever thickness. Very sensitive optical and capacitive methods have been developed to measure changes in the static deflection of cantilever beams used in dc-coupled sensors.

The second is the formula relating the cantilever spring constant  to the cantilever dimensions and material constants:

where is force and is the cantilever width. The spring constant is related to the cantilever resonance frequency by the usual harmonic oscillator
Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....

formula . A change in the force applied to a cantilever can shift the resonance frequency. The frequency shift can be measured with exquisite accuracy using heterodyne
Heterodyne
Heterodyning is a radio signal processing technique invented in 1901 by Canadian inventor-engineer Reginald Fessenden where high frequency signals are converted to lower frequencies by combining two frequencies. Heterodyning is useful for frequency shifting information of interest into a useful...

techniques and is the basis of ac-coupled cantilever sensors.

The principal advantage of MEMS cantilevers is their cheapness and ease of fabrication in large arrays. The challenge for their practical application lies in the square and cubic dependences of cantilever performance specifications on dimensions. These superlinear dependences mean that cantilevers are quite sensitive to variation in process parameters. Controlling residual stress
Residual stress
Residual stresses are stresses that remain after the original cause of the stresses has been removed. They remain along a cross section of the component, even without the external cause. Residual stresses occur for a variety of reasons, including inelastic deformations and heat treatment...

can also be difficult.

### Warehouse storage

A cantilever rack is a type of warehouse
Warehouse
A warehouse is a commercial building for storage of goods. Warehouses are used by manufacturers, importers, exporters, wholesalers, transport businesses, customs, etc. They are usually large plain buildings in industrial areas of cities and towns. They usually have loading docks to load and unload...

storage system consisting of the vertical column, the base, the arms, and the horizontal and/or cross bracing. These components are fabricated from both roll formed and structural steel. The horizontal and/or cross bracing are used to connect two or more columns together. They are commonly found in lumber yard
Lumber yard
A lumber yard is a retail location for lumber and wood related products used in construction and/or home improvement projects. Lumber yards can also provide services such as the use of a planer and other large machines....

s, woodworking shops, and plumbing supply warehouses.

### Portable storage

A folding cantilever tray is a type of stacked shelf that can be unfolded to allow convenient access to items on multiple tiers simultaneously. The shelves can be collapsed when not in use for more compact storage. Because of these properties folding cantilever trays are often used in luggage
Luggage
Baggage is any number of bags, cases and containers which hold a traveller's articles during transit.Luggage is more or less the same concept as "baggage", but is normally used in relation to the personal luggage of a specific person or persons Baggage is any number of bags, cases and containers...

and toolboxes.

• Grand Canyon Skywalk
Grand Canyon Skywalk
The Grand Canyon Skywalk is a transparent horseshoe-shaped cantilever bridge and tourist attraction in Arizona near the Colorado River on the edge of a side canyon in the Grand Canyon West area of the main canyon...

• Applied mechanics
Applied mechanics
Applied mechanics is a branch of the physical sciences and the practical application of mechanics. Applied mechanics examines the response of bodies or systems of bodies to external forces...

• Moment (physics)
Moment (physics)
In physics, the term moment can refer to many different concepts:*Moment of force is the tendency of a force to twist or rotate an object; see the article torque for details. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the...

• Statics
Statics
Statics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity...

• Beam theory
• Cantilever bicycle brakes
• Cantilever bicycle frame
• Cantilever bridge
Cantilever bridge
A cantilever bridge is a bridge built using cantilevers, structures that project horizontally into space, supported on only one end. For small footbridges, the cantilevers may be simple beams; however, large cantilever bridges designed to handle road or rail traffic use trusses built from...

• Cantilever chair
Cantilever chair
A cantilever chair is a chair with no back legs, relying for support on the properties of the material from which it is made. This famous form was designed by Mart Stam in 1926, and remains an important example of 20th century design....

• Cantilever mechanics (orthodontics)
Cantilever mechanics (orthodontics)
A cantilever is, in principle, any piece of wire, whose end is inserted, on one side, into a bracket or a tube, or included in the acrylic of a removable appliance, while the other one is tied to another unit, with only a one-point contact...