Cantilever

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The cantilevered beam (green) projects out into space from its supports (blue). In this case the beam is balanced by a load over the structure (red block) which counteracts the force of gravity (red arrow). The orange arrow indicates the location of the maximum bending and shear forces on the cantilever at the support.

A cantilever is a beam anchored at one end and projecting into space. This beam may be fixed at the support, or extend to another support as illustrated. The beam carries the load to the support where it is resisted by bending moment and shear. Cantilever construction allows for long structures without external bracing.

This is in contrast to a post and lintel system where the beam is supported at both ends and loads applied between them.

The Forth Bridge, a cantilever railway bridge with three balanced (double) cantilevers

1 In bridges, towers, and buildings
2 In aircraft
3 In MEMS
4 See also
5 References

[edit] In bridges, towers, and buildings

Less obvious examples are free-standing radio towers without cable stays and chimneys, which resist being blown over by the wind through cantilever action at their base.

Arguably the most famous cantilever in architecture, a balcony at Fallingwater.

[edit] In aircraft

Another use of the cantilever is in aircraft design, pioneered by Hugo Junkers in 1915. Early aircraft wings bore their loads by building two (or more) wings, and bracing them with wires. They were similar to truss bridges in some aspects, the wings on each side of the plane were braced with crossed wires both along their length, so they would stay parallel, as well as front-to-back to resist twisting. The cables generated considerable drag however, and there was constant experimentation on ways to eliminate them.

A British Hawker Hurricane from World War II with cantilever wings

It was also desirable to build a monoplane aircraft, as additional drag is formed by having a stack of wings. Early monoplanes used either struts (as do some modern personal aircraft), or cables (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, which reduces maximum speed (for a given power) and increases fuel consumption (for a given speed).

The most successful wing design was the cantilever. A single large beam, referred to as the spar, runs through the wing, and often right through the aircraft. Looking at a plane from the front, the wings are both trying to rotate up at the tips, a force that is resisted either by mounting the two spars to each other (each one is twisting in the opposite direction) or to a strong box-like structure in the middle, or by a shell like structure forward of the spar that forms the aerodynamic shape and resists twisting (this is called a D tube).

Cantilever wings require a much heavier spar than would otherwise be needed in cable-stayed designs. However as the size of aircraft grew, this additional weight dropped in comparison to the overall weight, as well as the growing weight of the cables needed to brace larger wings. 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.

[edit] In MEMS

Cantilevered beams are the most ubiquitous structures in the field of microelectromechanical systems (MEMS). MEMS cantilevers are commonly fabricated from Si, SiN or polymers. The fabrication process typically involves undercutting the cantilever structure to release it, often with an anisotropic wet or dry 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 biosensors for medical diagnostic applications. MEMS cantilevers are also finding application as radio frequency filters and resonators.

Two equations are key to understanding the behavior of MEMS cantilevers. The first is Stoney's formula, which relates cantilever end deflection δ to applied stress σ:

$\delta = \frac{3\sigma\left(1 - \nu \right)}{E} \left(\frac{L}{t}\right)^2$

where ν is Poisson's ratio, $E$ is Young's modulus, $L$ is the beam length and $t$ 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 $k$ to the cantilever dimensions and material constants:

$k = \frac{F}{\delta} = \frac{Ewt^3}{4L^3}$

where $F$ is force and $w$ is the cantilever width. The spring constant is related to the cantilever resonant frequency $ω 0$ by the usual harmonic oscillator formula $\omega_0 = \sqrt{k/m}$ . A change in the force applied to a cantilever can shift the resonant frequency. The frequency shift can be measured with exquisite accuracy using heterodyne 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 can also be difficult.

[edit] See also

[edit] References

Roth, Leland M (1993). Understanding Architecture: Its Elements History and Meaning. Oxford, UK: Westview Press. ISBN 0-06-430158-3. pp. 23-4
Madou, Marc J (2002). Fundamentals of Microfabrication. Taylor & Francis. ISBN 0-8493-0826-7.
Sarid, Dror (1994). Scanning Force Microscopy. Oxford University Press. ISBN 0-19-509204-X.