- Euler-Bernoulli beam equation
**Euler-Bernoulli beam theory**, or just**beam theory**, is a simplification of the lineartheory of elasticity which provides a means of calculating the load-carrying anddeflection characteristics of beams. It was first enunciated circa 1750, but was not applied on a large scale until the development of theEiffel Tower and theFerris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of theSecond Industrial Revolution .Additional analysis tools have been developed such as

plate theory andfinite element analysis , but the simplicity of beam theory makes it an important tool in the sciences, especially structural andmechanical engineering .**History**The prevailing consensus is that

Galileo Galilei made the first attempts at developing a theory of beams, but recent studies argue thatLeonardo da Vinci was the first to make the crucial observations. Da Vinci lackedHooke's law andcalculus to complete the theory, whereas Galileo was held back by an incorrect assumption he made. [*cite journal*]

last = Ballarini

first = Roberto

title = The Da Vinci-Euler-Bernoulli Beam Theory?

journal = Mechanical Engineering Magazine Online

date =April 18 ,2003

url = http://www.memagazine.org/contents/current/webonly/webex418.html

accessdate = 2006-07-22The Bernoulli beam is named after

Jacob Bernoulli , who made the significant discoveries.Leonhard Euler andDaniel Bernoulli were the first to put together a useful theory circa 1750. [*cite paper*] At the time, science and

author = Seon M. Han, Haym Benaroya and Timothy Wei

title = Dynamics of Transversely Vibrating Beams using four Engineering Theories

version = final version

publisher = Academic Press

date =March 22 ,1999

url = http://csxe.rutgers.edu/research/vibration/51.pdf

format = PDF

accessdate = 2007-04-15engineering were generally seen as very distinct fields, and there was considerable doubt that a mathematical product of academia could be trusted for practical safety applications. Bridges and buildings continued to be designed by precedent until the late 19th century, when theEiffel Tower andFerris wheel demonstrated the validity of the theory on large scales.**The beam equation**The Euler-Bernoulli equation describes the relationship between the beam's deflection and the applied load:

:$frac\{partial^2\}\{partial\; x^2\}left(EI\; frac\{partial^2\; u\}\{partial\; x^2\}\; ight)\; =\; w,$

The curve $u(x)$ describes the deflection $u$ of the beam at some position $x$ (recall that the beam is modeled as a one-dimensional object). $w$ is a distributed load, in other words a force per unit length (analogous to

pressure being a force per area); it may be a function of $x$, $u$, or other variables.Note that $E$ is the

elastic modulus and that $I$ is thesecond moment of area . $I$ must be calculated with respect to the centroidal axis perpendicular to the applied loading. For an Euler-Bernoulli beam not under any axial loading this axis is called theneutral axis .Often, $u\; =\; u(x)$, $w\; =\; w(x)$, and EI is a constant, so that:

:$EI\; frac\{d^4\; u\}\{d\; x^4\}\; =\; w(x),$

This equation, describing the deflection of a uniform, static beam, is very common in engineering practice.

Successive derivatives of $u$ have important meanings:

:* $extstyle\{u\},$ is the deflection.

:* $extstyle\{frac\{partial\; u\}\{partial\; x,$ is the slope of the beam.

:* $extstyle\{EIfrac\{partial^2\; u\}\{partial\; x^2,$ is the bending moment in the beam.

:* $extstyle\{-frac\{partial\}\{partial\; x\}left(EIfrac\{partial^2\; u\}\{partial\; x^2\}\; ight)\},$ is the shear force in the beam.

**tress**Besides deflection, the beam equation describes

force s and moments and can thus be used to describe stresses. For this reason, the Euler-Bernoulli beam equation is widely used inengineering , especially civil and mechanical, to determine the strength (as well as deflection) of beams under bending.Both the bending moment and the shear force cause stresses in the beam. The stress due to shear force is maximum along the

neutral axis of the beam (when the width of the beam, t, is constant along the cross section of the beam; otherwise an integral involving the first moment and the beam's width needs to be evaluated for the particular cross section), and the maximum tensile stress is at either the top or bottom surfaces. Thus the maximum principal stress in the beam may be neither at the surface nor at the center but in some general area. However, shear force stresses are negligible in comparison to bending moment stresses in all but the stockiest of beams as well as the fact thatstress concentration s commonly occur at surfaces, meaning that the maximum stress in a beam is likely to be at the surface.It can be shown that the tensile stress experienced by the beam may be expressed as:

:$sigma\; =\; frac\{Mc\}\{I\}\; =\; E\; c\; frac\{partial^2\; u\}\{partial\; x^2\},$

Here, $c$, a position along $u$, is the distance from the neutral axis to a point of interest; and $M$ is the bending moment. Note that this equation implies that "pure" bending (of positive sign) will cause zero stress at the neutral axis, positive (tensile) stress at the "top" of the beam, and negative (compressive) stress at the bottom of the beam; and also implies that the maximum stress will be at the top surface and the minimum at the bottom. This bending stress may be superimposed with axially applied stresses, which will cause a shift in the neutral (zero stress) axis.

**Boundary considerations**The beam equation contains a fourth-order derivative in $x$, hence it mandates at most four conditions, normally boundary conditions. The boundary conditions usually model "supports", but they can also model point loads, moments, or other effects.

An example is a

cantilever beam: a beam that is completely fixed at one end and completely free at the other. "Completely fixed" means that at the left end both deflection and slope are zero; "completely free" implies (though it may or may not be obvious) that at the right end both shear force and bending moment are zero. Taking the $x$ coordinate of the left end as $0$ and the right end as $L$ (the length of the beam), these statements translate to the following set of boundary conditions (assume $EI$ is a constant)::$u|\_\{x\; =\; 0\}\; =\; 0\; quad\; ;\; quad\; frac\{partial\; u\}\{partial\; x\}igg|\_\{x\; =\; 0\}\; =\; 0\; qquad\; mbox\{(fixed\; end)\},$

:$frac\{partial^2\; u\}\{partial\; x^2\}igg|\_\{x\; =\; L\}\; =\; 0\; quad\; ;\; quad\; frac\{partial^3\; u\}\{partial\; x^3\}igg|\_\{x\; =\; L\}\; =\; 0\; qquad\; mbox\{(free\; end)\},$

Some commonly encountered boundary conditions include:

:* $extstyle\{u\; =\; frac\{partial\; u\}\{partial\; x\}\; =\; 0\},$ represents a fixed support.

:* $extstyle\{u\; =\; frac\{partial^2\; u\}\{partial\; x^2\}\; =\; 0\},$ represents a pin connection (deflection and moment fixed to zero).

:* $extstyle\{frac\{partial^2\; u\}\{partial\; x^2\}\; =\; frac\{partial^3\; u\}\{partial\; x^3\}\; =\; 0\},$ represents no connection (no restraint) and no load.

:* $extstyle\{-frac\{partial\}\{partial\; x\}left(EIfrac\{partial^2\; u\}\{partial\; x^2\}\; ight)\}\; =\; F,$ represents the application of a point load "F".

**Loading considerations**Applied loading may be represented either through boundary conditions or through the distributed function $w$. Using distributed loading is often favorable for simplicity. Boundary conditions are, however, often used to model loads depending on context; this practice being especially common in vibration analysis.

By nature, the distributed load is very often represented in a piecewise manner, since in practice a load isn't typically a "nice" continuous function. Point loads can be modeled with help of the

Dirac delta function . For example, consider a static uniform cantilever beam of length $L$ with an upward point load $F$ applied at the free end. Using boundary conditions, this may be modeled through::$EI\; frac\{d^4\; u\}\{d\; x^4\}\; =\; 0,$

:$u|\_\{x\; =\; 0\}\; =\; 0\; quad\; ;\; quad\; frac\{d\; u\}\{d\; x\}igg|\_\{x\; =\; 0\}\; =\; 0,$

:$frac\{d^2\; u\}\{d\; x^2\}igg|\_\{x\; =\; L\}\; =\; 0\; quad\; ;\; quad\; -EI\; frac\{d^3\; u\}\{d\; x^3\}igg|\_\{x\; =\; L\}\; =\; F,$

Using the Dirac function,

:$EI\; frac\{d^4\; u\}\{d\; x^4\}\; =\; F\; delta(x\; -\; L),$

:$u|\_\{x\; =\; 0\}\; =\; 0\; quad\; ;\; quad\; frac\{d\; u\}\{d\; x\}igg|\_\{x\; =\; 0\}\; =\; 0,$

:$frac\{d^2\; u\}\{d\; x^2\}igg|\_\{x\; =\; L\}\; =\; 0,$

Note that shear force boundary condition (third derivative) is removed, otherwise there would be a contradiction. These are equivalent

boundary value problem s, and both yield the following solution::$u\; =\; frac\{F\}\{6\; EI\}(3\; L\; x^2\; -\; x^3),$

The application of several point loads at different locations will lead to $u(x)$ being a piecewise function. Use of the Dirac function greatly simplifies such situations; otherwise the beam would have to be divided into sections, each with four boundary conditions solved separately. A well organized family of functions called

Singularity function s are often used as a shorthand for the Dirac function, itsderivative , and itsantiderivatives .Clever formulation of the load distribution allows for many interesting phenomena to be modeled. As an example, the

vibration of a beam can be accounted for by using the load function::$w(x,\; t)\; =\; -mu\; frac\{partial^2\; u\}\{partial\; t^2\},$

where $mu$ is the

linear mass density of the beam, not necessarily a constant. With this time-dependent loading, the beam equation will be apartial differential equation ::$mu\; frac\{partial^2\; u\}\{partial\; t^2\}\; +\; frac\{partial^2\}\{partial\; x^2\}\; left(\; EI\; frac\{partial^2\; u\}\{partial\; x^2\}\; ight)\; =\; 0.$

Another interesting example describes the deflection of a beam rotating with a constant

angular frequency of $omega$::$w(u)\; =\; mu\; omega^2\; u,$

This is a

centripetal force distribution. Note that in this case, $w$ is a function of the displacement (the dependent variable), and the beam equation will be an autonomousordinary differential equation .**Extensions**The kinematic assumptions upon which the Euler-Bernoulli beam theory is founded allow it to be extended to more advanced analysis. Simple superposition allows for three-dimensional transverse loading. Using alternative

constitutive equation s can allow for viscoelastic or plastic beam deformation. Euler-Bernoulli beam theory can also be extended to the analysis of curved beams, beam buckling, composite beams, and geometrically nonlinear beam deflection.Euler-Bernoulli beam theory does not account for the effects of transverse

shear strain. As a result it underpredicts deflections and overpredicts natural frequencies. For thin beams (beam length to thickness ratios of the order 20 or more) these effects are of minor importance. For thick beams, however, these effects can be significant. More advanced beam theories such as the Timoshenko beam theory (developed by the Russian-born scientistStephen Timoshenko ) have been developed to account for these effects.**See also***

Buckling

* Clapeyron's theorem

* Stress

* Strain

*Applied mechanics

*Singularity function

*Flexural rigidity **Notes****References***

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