Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying capacity and deflection of beams.

When external forces are applied to a beam, internal shear forces and bending moments develop causing bending and curvature. Euler-Bernoulli beam theory states that the shear force at any point on a beam is the cumulative sum of the loads applied along the length of the beam up to that point. Similarly, the bending moment at any point is the sum of the shear forces along the beam up to that point. Additionally, the theory states that the deflection at any point on the beam is the fourth integral of the applied loads up to that point, and depends on flexural rigidity. Through the use of calculus, and boundary conditions describing the beam's curvature at its supports, the theory provides a mathematical model to predict the structural behavior of beams.

Euler-Bernoulli beam theory is limited to small deflections of a beam that is subjected to lateral loads only, causing elastic bending. Beam theories were later refined in the 20th century to account for the effects of shear deformation and rotatory inertia; Euler-Bernoulli beam theory is now considered a case of Timoshenko–Ehrenfest beam theory that neglects these effects.

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 models have been developed, such as plate theory, but the simplicity of beam theory makes it an important tool in the sciences, especially structural and mechanical engineering.

History

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.

Jacob Bernoulli postulated that the curvature at any point on a deflected beam is proportional to its bending moment at that point, first published in Histoire de l'Académie royale des sciences in 1705 . He also formulated an approach for calculating the deflection of a cantilevered beam with a concentrated load at its free end, however he erroneously assumed a beam's curvature is due to its cross-section rotating about an axis that intersects the extreme compression fiber.

Leonhard Euler built on Jacob Bernoulli's work, and applied differential calculus to describe the behavior of elastic curves and published Methodus inveniendi lineas curvas in 1744 .

Daniel Bernoulli furthered theories and formulated the differential equation of motion of a vibrating beam.

Static beam equation

The Euler–Bernoulli equation describes the relationship between the beam's deflection and the applied load: is a distributed load, in other words a force per unit length (analogous to pressure is the elastic modulus is the second moment of area must be calculated with respect to the axis which is perpendicular to the applied loading. plane, and the relevant second moment of area is

.

(known as the flexural rigidity) is a constant, so that

for common beam configurations can be found in engineering handbooks. For more complicated situations, the deflection can be determined by solving the Euler–Bernoulli equation using techniques such as "direct integration", "Macaulay's method", "moment area method", "conjugate beam method", "the principle of virtual work", "Castigliano's method", "flexibility method", "slope deflection method", "moment distribution method", or "direct stiffness method".

Sign conventions are defined here since different conventions can be found in the literature. In this article, a right-handed produces compressive stress at the bottom surface. With this choice of bending moment sign convention is necessary for force equilibrium.

-axis in the limit of small displacements;

is the bending moment in the beam; and

is the shear force in the beam.

The stresses in a beam can be calculated from the above expressions after the deflection due to a given load has been determined.

Derivation of the bending equation

Because of the fundamental importance of the bending moment equation in engineering, we will provide a short derivation. We change to polar coordinates. The length of the neutral axis Therefore, the strain of this fiber is

is the elastic modulus in accordance with Hooke's law resulting from this stress, is given by

is given by

the bending moment vector exerted on the right cross section of the beam the expression

is the second moment of area is the radius of curvature. Therefore,

This vector equation can be separated in the bending unit vector), and in the bending equation:

Dynamic beam equation

The dynamic beam equation is the Euler–Lagrange equation for the following action:

The first term represents the kinetic energy. The Euler–Lagrange equation. For a dynamic Euler–Bernoulli beam, the Euler–Lagrange equation is

, and the beam equation is simpler:

Free vibration

, we have the free vibration equation. This equation can be solved using a Fourier decomposition of the displacement into the sum of harmonic vibrations of the form

is the frequency of vibration. Then, for each value of frequency, we can solve an ordinary differential equation

The general solution of the above equation is

are constants. These constants are unique for a given set of boundary conditions. However, the solution for the displacement is not unique and depends on the frequency. These solutions are typically written as

are called the natural frequencies of the beam. Each of the displacement solutions is called a mode, and the shape of the displacement curve is called a mode shape.

Example: Cantilevered beam

) are

.

The corresponding natural frequencies of vibration are

The boundary conditions can also be used to determine the mode shapes from the solution for the displacement:

, i.e., the beam can resonate. The natural frequencies of a beam therefore correspond to the frequencies at which resonance can occur.

Example: free–free (unsupported) beam

A free–free beam is a beam without any supports. are given by:

If we apply these conditions, non-trivial solutions are found to exist only if

.

The corresponding natural frequencies of vibration are:

The boundary conditions can also be used to determine the mode shapes from the solution for the displacement:

.

Example: hinged-hinged beam

) are

enforces this condition. Rearranging for natural frequency gives

Stress

Besides deflection, the beam equation describes forces and moments and can thus be used to describe stresses. For this reason, the Euler–Bernoulli beam equation is widely used in engineering, 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 that stress concentrations commonly occur at surfaces, meaning that the maximum stress in a beam is likely to be at the surface.

Simple or symmetrical bending

For beam cross-sections that are symmetrical about a plane perpendicular to the neutral plane, it can be shown that the tensile stress experienced by the beam may be expressed as:

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.

Maximum stresses at a cross-section

. These stresses are

are the section moduli and are defined as

. === Strain in an Euler–Bernoulli beam === We need an expression for the strain (see Figure 1) and that the neutral surface does not change in length during the deformation.

.

. The strain in that segment of the beam is given by

is the curvature.

Relation between curvature and beam deflection

-axis for the small angles encountered in beam theory. Therefore, with this approximation,

can be written as

Standard deflection behaviour for simply supported beams under uniformly distributed loads can be visualized using well-known shear and moment diagrams and closed-form elastic equations.

Hence the strain in the beam may be expressed as

Stress-strain relations

For a homogeneous isotropic linear elastic is the Young's modulus. Hence the stress in an Euler–Bernoulli beam is given by

Note that the above relation, when compared with the relation between the axial stress and the bending moment, leads to

, we also have

Boundary considerations

we need four boundary conditions. The boundary conditions usually model supports, but they can also model point loads, distributed loads and moments. The support) on the boundary. Such boundary conditions are also called Dirichlet boundary conditions and represent momentum flux. Flux boundary conditions are also called Neumann boundary conditions.

As an example consider a cantilever beam that is built-in at one end and free at the other as shown in the adjacent figure. At the built-in end of the beam there cannot be any displacement or rotation of the beam. This means that at the left end both deflection and slope are zero. Since no external bending moment is applied at the free end of the beam, the bending moment at that location is zero. In addition, if there is no external force applied to the beam, the shear force at the free end is also zero.

is a constant):

A simple support (pin or roller) is equivalent to a point force on the beam which is adjusted in such a way as to fix the position of the beam at that point. A fixed support or clamp, is equivalent to the combination of a point force and a point torque which is adjusted in such a way as to fix both the position and slope of the beam at that point. Point forces and torques, whether from supports or directly applied, will divide a beam into a set of segments, between which the beam equation will yield a continuous solution, given four boundary conditions, two at each end of the segment. Assuming that the product EI where F where M is the magnitude of a point torque, the boundary conditions appropriate for some common cases is given in the table below. The change in a particular derivative of w across the boundary as x = fixed).

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Note that in the first cases, in which the point forces and torques are located between two segments, there are four boundary conditions, two for the lower segment, and two for the upper. When forces and torques are applied to one end of the beam, there are two boundary conditions given which apply at that end. The sign of the point forces and torques at an end will be positive for the lower end, negative for the upper end.

Loading considerations

which represents an external distributed load. 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 continuous function. Point loads can be modeled with help of the Dirac delta function applied at the free end. Using boundary conditions, this may be modeled in two ways. In the first approach, the applied point load is approximated by a shear force applied at the free end. In that case the governing equation and boundary conditions are:

Alternatively we can represent the point load as a distribution using the Dirac function. In that case the equation and boundary conditions are

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 solution

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 functions are often used as a shorthand for the Dirac function, its derivative, and its antiderivatives.

Dynamic phenomena can also be modeled using the static beam equation by choosing appropriate forms of the load distribution. As an example, the free vibration of a beam can be accounted for by using the load function:

is the linear mass 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 frequency:

This is a centripetal force is a function of the displacement (the dependent variable), and the beam equation will be an autonomous ordinary differential equation.

Examples

Three-point bending

The three-point bending test is a classical experiment in mechanics. It represents the case of a beam resting on two roller supports and subjected to a concentrated load applied in the middle of the beam. The shear is constant in absolute value: it is half the central load, P / 2. It changes sign in the middle of the beam. The bending moment varies linearly from one end, where it is 0, and the center where its absolute value is PL / 4, is where the risk of rupture is the most important. The deformation of the beam is described by a polynomial of third degree over a half beam (the other half being symmetrical). ) for a beam subjected to a central point load and an asymmetric point load are given in the table below.

Cantilever beams

Another important class of problems involves cantilever) for a cantilever beam subjected to a point load at the free end and a uniformly distributed load are given in the table below.

Solutions for several other commonly encountered configurations are readily available in textbooks on mechanics of materials and engineering handbooks.

Statically indeterminate beams

The bending moments and shear forces in Euler–Bernoulli beams can often be determined directly using static balance of forces and moments. However, for certain boundary conditions, the number of reactions can exceed the number of independent equilibrium equations. Such beams are called statically indeterminate.

The built-in beams shown in the figure below are statically indeterminate. To determine the stresses and deflections of such beams, the most direct method is to solve the Euler–Bernoulli beam equation with appropriate boundary conditions. But direct analytical solutions of the beam equation are possible only for the simplest cases. Therefore, additional techniques such as linear superposition are often used to solve statically indeterminate beam problems.

The superposition method involves adding the solutions of a number of statically determinate problems which are chosen such that the boundary conditions for the sum of the individual problems add up to those of the original problem.

Another commonly encountered statically indeterminate beam problem is the cantilevered beam with the free end supported on a roller. The bending moments, shear forces, and deflections of such a beam are listed below:

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 equations 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 scientist Stephen Timoshenko) have been developed to account for these effects.

Large deflections

The original Euler–Bernoulli theory is valid only for infinitesimal strains and small rotations. The theory can be extended in a straightforward manner to problems involving moderately large rotations provided that the strain remains small by using the von Kármán strains.

The Euler–Bernoulli hypotheses that plane sections remain plane and normal to the axis of the beam lead to displacements of the form

Using the definition of the Lagrangian Green strain from finite strain theory, we can find the von Kármán strains The resulting strains take the form:

From the principle of virtual work, the balance of forces and moments in the beams gives us the equilibrium equations

is the transverse load, and

To close the system of equations we need the constitutive equations that relate stresses to strains (and hence stresses to displacements). For large rotations and small strains these relations are

where

is the extensional stiffness is the coupled extensional-bending stiffness is the bending stiffness.

For the situation where the beam has a uniform cross-section and no axial load, the governing equation for a large-rotation Euler–Bernoulli beam is

See also

• Applied mechanics

• Bending

• Bending moment

• Buckling

• Flexural rigidity

• Generalised beam theory

• Plate theory

• Sandwich theory

• Shear and moment diagram

• Singularity function

• Strain (materials science)

• Timoshenko beam theory

• Theorem of three moments (Clapeyron's theorem)

• Three-point flexural test

References

Notes

Citations

Further reading

External links

• Beam stress & deflection, beam deflection tables