# 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.