Deflection (engineering)From Wikipedia, the free encyclopedia

In engineering, deflection is the degree to which a structural element is displaced under a load. Itmay refer to an angle or a distance.

The deflection distance of a member under a load is directly related to the slope of the deflectedshape of the member under that load and can be calculated by integrating the function thatmathematically describes the slope of the member under that load. Deflection can be calculated bystandard formula (will only give the deflection of common beam configurations and load cases atdiscrete locations), or by methods such as virtual work, direct integration, Castigliano's method,Macaulay's method or the direct stiffness method, amongst others. The deflection of beam elementsis usually calculated on the basis of the Euler–Bernoulli beam equation while that of a plate or shellelement is calculated using plate or shell theory.

An example of the use of deflection in this context is in building construction. Architects andengineers select materials for various applications. The beams used for frame work are selected onthe basis of deflection, amongst other factors.

Contents

1 Beam deflection for various loads and supports1.1 End loaded cantilever beams1.2 Uniformly loaded cantilever beam1.3 Center loaded beam1.4 Intermediately loaded beam1.5 Uniformly loaded beam

2 Structural deflection3 See also4 References5 External links

Beam deflection for various loads and supports

End loaded cantilever beams

The elastic deflection and angle of deflection (in radians) in the example image, a (weightless)cantilever beam, with an end load, can be calculated (at the free end B) using:[1]

where

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Deflection of a cantilevered beam

= Force acting on the tip of the beam = Length of the beam (span) = Modulus of elasticity

= Area moment of inertia

Note that if the span doubles, the deflection increaseseightfold. The deflection at any point, , along the span of anend loaded cantilevered beam can be calculated using:[1]

Note that at (the end of the beam), the and equations are identical to the and equations above.

Uniformly loaded cantilever beam

The deflection, at the free end B, of a cantilevered beam under a uniform load is given by:[1]

where

= Uniform load on the beam (force per unit length) = Length of the beam = Modulus of elasticity

= Area moment of inertia

The deflection at any point, , along the span of a uniformly loaded cantilevered beam can becalculated using:[1]

Center loaded beam

The elastic deflection (at the midpoint C) of a beam, loaded at its center, supported by two simplesupports is given by:[1]

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where

= Force acting on the center of the beam = Length of the beam between the supports = Modulus of elasticity

= Area moment of inertia

The deflection at any point, , along the span of a center loaded simply supported beam can becalculated using:[1]

for

Intermediately loaded beam

The maximum elastic deflection on a beam supported by two simple supports, loaded at a distance from the closest support, is given by:[1]

where

= Force acting on the beam = Length of the beam between the supports = Modulus of elasticity

= Area moment of inertia = Distance from the load to the closest support (i.e. )

This maximum deflection occurs at a distance from the closest support and is given by:[1]

Uniformly loaded beam

The elastic deflection (at the midpoint C) on a beam supported by two simple supports, under auniform load (as pictured) is given by:[1]

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Shows a statically determinate beam,deflecting under an evenly distributedload.

where

= Uniform load on the beam = Length of the beam = Modulus of elasticity

= Area moment of inertia

The deflection at any point, , along the span of a uniformlyloaded simply supported beam can be calculated using:[1]

Structural deflection

Building codes determine the maximum deflection, usually as a fraction of the span e.g. 1/400 or1/600. Either the strength limit state (allowable stress) or the serviceability limit state (deflectionconsiderations amongst others) may govern the minimum dimensions of the member required.

The deflection must be considered for the purpose of the structure. When designing a steel frame tohold a glazed panel, one allows only minimal deflection to prevent fracture of the glass.

The deflected shape of a beam can be represented by the moment diagram, integrated (twice, rotatedand translated to enforce support conditions).

See also

BendingSlope deflection methodVirtual workDirect integrationCastigliano's methodMacaulay's methodDirect stiffness method

References

^ a b c d e f g h i j Gere, James M.; Goodno, Barry J. Mechanics of Materials (Eighth ed.). p. 1083-1087.ISBN 978-1-111-57773-5.

1.

External links

Deflection & stress of beams Calculators (http://www.engineersedge.com/beam_calc_menu.shtml)Deflection of beams (http://www.mathalino.com/reviewer/mechanics-and-strength-of-materials/chapter-6-beam-deflections)Online Calculator for Deflection and slope of beams (http://civilengineer.webinfolist.com/str/sdcalc.htm)Beam Deflections (http://www.clag.org.uk/beam.html)Beam Deflections (Tabulated) (http://www.advancepipeliner.com/Resources/Others/Beams

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/Beam_Deflection_Formulae.pdf)

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