Euler-Bernoulli beam equation

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Euler-Bernoulli beam theory or just beam theory is a simplification of the linear isotropic theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris Wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution.

Additional analysis tools have been developed such as plate theory and finite element analysis, but the simplicity of beam theory makes it an important tool in the sciences, especially civil and mechanical engineering.

1 History
2 The Beam Equation
3 Boundary Considerations
4 Loading Considerations
5 Extensions
6 See also
7 References
8 External links

[edit] History

The prevailing consensus is that Galileo Galilei made the first attempts at developing a theory of beams, but recent studies argue that Leonardo da Vinci was the first to make the crucial observations. Da Vinci lacked Hooke's law and calculus to complete the theory, whereas Galileo was held back by an incorrect assumption he made. ^[1]

Leonhard Euler and Daniel Bernoulli were the first to put together a useful theory circa 1750. At the time, science and industrial art 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 the Eiffel Tower and Ferris wheel demonstrated the validity of the theory on large scales.

[edit] The Beam Equation

For a long, slender, one dimensional beam made of isotropic material, it can be shown that the elastic curve of the beam must satisfy:

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

This is the Euler-Bernoulli equation. 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); it may be a function of x, u, or other variables.

Note that E is the elastic modulus and that I is the second moment of area. I must be calculated with respect to the centroidal axis perpendicular to the applied loading, for an Euler-Bernoulli beam this axis is called the neutral 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 is very common in engineering practice: it describes the deflection of a uniform, static beam.

Successive derivatives of u have important meaning:

$\textstyle{u}\,$ is the deflection.

$\textstyle{\frac{\partial u}{\partial x}}\,$ is the slope of the beam.

$\textstyle{EI\frac{\partial^2 u}{\partial x^2}}\,$ is the bending moment in the beam.

$\textstyle{-\frac{\partial}{\partial x}\left(EI\frac{\partial^2 u}{\partial x^2}\right)}\,$ is the shear force in the beam.

That the beam is a one dimensional object is an important assumption. Alongside that, the beam must be straight, the distributed load must be contained in one plane, and there must be no torsion.

It can be shown that the stress experienced by the beam may be expressed as:

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

Here, c is a position along u, it is the distance from the neutral axis to a point of interest; 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; this 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.

[edit] 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, they can also model point loads, moments, or other effects.

A cantilever beam.

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):

$u|_{x = 0} = 0 \quad ; \quad \frac{\partial u}{\partial x}\bigg|_{x = 0} = 0 \qquad \mbox{(fixed end)}\,$

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

Some commonly encountered boundary conditions include:

$\textstyle{u = \frac{\partial u}{\partial x} = 0}\,$ represents a fixed support.

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

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

$\textstyle{-\frac{\partial}{\partial x}\left(EI\frac{\partial^2 u}{\partial x^2}\right)} = F\,$ represents the application of a point load F.

[edit] 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, the practice is 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:

$EI \frac{d^4 u}{d x^4} = 0\,$

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

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

Using the dirac function,

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

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

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

Note that shear force boundary condition (third derivative) is removed, otherwise there would be a contradiction. These are equivalent boundary value problems, and both yield the following solution:

$u = \frac{2 F}{EI}(L x^2 - 3 x^3)\,$

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 secions, each with four boundary conditions solved separately.

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 using the load function:

$w(x, t) = -\mu \frac{\partial^2 u}{\partial t^2}\,$

Where μ is the linear density of the beam, not necessarily a constant. With this time dependent loading, the beam equation will be a partial differential equation. Another interesting example describes the deflection of a beam rotating with a constant angular velocity of ω:

$w(u) = \mu \omega^2 u\,$

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 autonomous ordinary differential equation.

[edit] Extensions

Using different assumptions and derivations, the theory can be extended in a number of ways. A simple superposition allows 3D loading of the beam, although still without torsion. Other "versions" allow for plastic bending, curved beams, beam buckling, and orthotropic materials (wood, for example).

[edit] See also

[edit] References

^ Ballarini, Roberto (April 18 2003). "The Da Vinci-Euler-Bernoulli Beam Theory?". Mechanical Engineering Magazine Online. Retrieved on July 22, 2006.

2. E.A. Witmer (1991-1992). "Elementary Bernoulli-Euler Beam Theory". MIT Unified Engineering Course Notes: pp. 5-114 to 5-164.