Euler–Bernoulli beam theory

Euler–Bernoulli beam equationbeam theoryEuler-Bernoulli beam equationbeamEuler–Bernoulli theorybeam equationbeam-theorybeamsEuler-Bernoulli beam theoryEuler-Bernoulli beams
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 and deflection characteristics of beams.wikipedia
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Structural engineering

structuralstructural designstructural engineer
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 structural and mechanical engineering.

Timoshenko beam theory

TimoshenkoTimoshenko beams
It is thus a special case of Timoshenko beam theory.
The resulting equation is of 4th order but, unlike Euler–Bernoulli beam theory, there is also a second-order partial derivative present.

Deflection (engineering)

deflectiondeflectionsdeflect
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 and deflection characteristics of beams.
The deflection of beam elements is usually calculated on the basis of the Euler–Bernoulli beam equation while that of a plate or shell element is calculated using plate or shell theory.

Finite element method

finite element analysisfinite elementfinite elements
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 structural and mechanical engineering.
In applying FEA, the complex problem is usually a physical system with the underlying physics such as the Euler-Bernoulli beam equation, the heat equation, or the Navier-Stokes equations expressed in either PDE or integral equations, while the divided small elements of the complex problem represent different areas in the physical system.

Beam (structure)

beambeamscrossbeam
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 and deflection characteristics of beams.
In the beam equation I is used to represent the second moment of area.

Plate theory

plateplates and shellstheory of plates and shells
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 structural and mechanical engineering.
The Kirchhoff–Love theory is an extension of Euler–Bernoulli beam theory to thin plates.

Macaulay's method

technique
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".
Macaulay’s method (the double integration method) is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams.

Second moment of area

area moment of inertiaMoment of Inertiasecond moments of area
E is the elastic modulus and I is the second moment of area of the beam's cross-section.
The second moment of the area is crucial in Euler–Bernoulli theory of slender beams.

Leonhard Euler

EulerLeonard EulerEuler, Leonhard
Leonhard Euler and Daniel Bernoulli were the first to put together a useful theory circa 1750.
Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering.

Daniel Bernoulli

BernoulliDanielBernoulli, Daniel
Leonhard Euler and Daniel Bernoulli were the first to put together a useful theory circa 1750.
He worked with Euler on elasticity and the development of the Euler–Bernoulli beam equation.

Curvature

curvednegative curvatureextrinsic curvature
:where \kappa is the curvature of the beam.
It is common in physics and engineering to approximate the curvature with the second derivative, for example, in beam theory or for deriving wave equation of a tense string, and other applications where small slopes are involved.

Direct integration of a beam

direct integration
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".

Dirichlet boundary condition

Dirichlet boundary conditionsDirichletDirichlet condition
Such boundary conditions are also called Dirichlet boundary conditions.

Sandwich theory

sandwichsandwich compositestwo thin layers of material (skins) on either side of a lightweight core
Sandwich theory describes the behaviour of a beam, plate, or shell which consists of three layers—two facesheets and one core.

Slope deflection method

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".
*Beam theory

Stiffness

flexibilityrigidityrigid
:The quantity A_{xx} is the extensional stiffness,B_{xx} is the coupled extensional-bending stiffness, and D_{xx} is the bending stiffness.
For example, a point on a horizontal beam can undergo both a vertical displacement and a rotation relative to its undeformed axis.

Bending

flexurebendbeam
In the Euler–Bernoulli theory of slender beams, a major assumption is that 'plane sections remain plane'.

Generalised beam theory

It is a generalization of classical Euler–Bernoulli beam theory that approximates a beam as an assembly of thin-walled plates that are constrained to deform as a linear combination of specified deformation modes.

Section modulus

plastic section modulussection moduli
:The quantities S_1,S_2 are the section moduli and are defined as

Singularity function

A well organized family of functions called Singularity functions are often used as a shorthand for the Dirac function, its derivative, and its antiderivatives.
The deflection of a simply supported beam as shown in the diagram, with constant cross-section and elastic modulus, can be found using Euler-Bernoulli beam theory.

Bending stiffness

bending modulus
:The quantity A_{xx} is the extensional stiffness,B_{xx} is the coupled extensional-bending stiffness, and D_{xx} is the bending stiffness.
According to elementary beam theory, the relationship between the applied bending moment M and the resulting curvature \kappa of the beam is:

Cantilever

cantileveredcantilever wingcantilevers
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.

Dirac delta function

Dirac deltadelta functionimpulse
Point loads can be modeled with help of the Dirac delta function.
As another example, the equation governing the static deflection of a slender beam is, according to Euler–Bernoulli theory,

Three-point flexural test

three point bending testThree point flexural testSENB
The three point bending test is a classical experiment in mechanics.

Shear and moment diagram

bending moment diagramShear and moment diagramsbeam sign convention
We now use the Euler-Bernoulli beam theory to compute the deflections of the four segments.