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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 Deflection or deflexion may refer to: Board games * Deflection (chess), a tactic that forces an opposing chess piece to leave a square * Khet (game), formerly ''Deflexion'', an Egyptian-themed chess-like game using lasers Mechanics * Deflection ...
characteristics of beams. It covers the case corresponding to small deflections of a
beam Beam may refer to: Streams of particles or energy *Light beam, or beam of light, a directional projection of light energy **Laser beam *Particle beam, a stream of charged or neutral particles **Charged particle beam, a spatially localized grou ...
that is subjected to lateral loads only. By ignoring the effects of shear deformation and rotatory inertia, it is thus a special case of Timoshenko–Ehrenfest beam theory. It was first enunciated circa 1750, but was not applied on a large scale until the development of the
Eiffel Tower The Eiffel Tower ( ; french: links=yes, tour Eiffel ) is a wrought-iron lattice tower on the Champ de Mars in Paris, France. It is named after the engineer Gustave Eiffel, whose company designed and built the tower. Locally nicknamed "'' ...
and the
Ferris wheel A Ferris wheel (also called a Giant Wheel or an observation wheel) is an amusement ride consisting of a rotating upright wheel with multiple passenger-carrying components (commonly referred to as passenger cars, cabins, tubs, gondolas, capsules ...
in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the
Second Industrial Revolution The Second Industrial Revolution, also known as the Technological Revolution, was a phase of rapid scientific discovery, standardization, mass production and industrialization from the late 19th century into the early 20th century. The Firs ...
. Additional
mathematical models A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
have been developed, such as
plate theory In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions ...
, but the simplicity of beam theory makes it an important tool in the sciences, especially
structural A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such ...
and
mechanical engineering Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and ...
.


History

Prevailing consensus is that
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
made the first attempts at developing a theory of beams, but recent studies argue that
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
was the first to make the crucial observations. Da Vinci lacked
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
and
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
to complete the theory, whereas Galileo was held back by an incorrect assumption he made. The Bernoulli beam is named after
Jacob Bernoulli Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
, who made the significant discoveries.
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and Daniel Bernoulli were the first to put together a useful theory circa 1750.


Static beam equation

The Euler–Bernoulli equation describes the relationship between the beam's
deflection Deflection or deflexion may refer to: Board games * Deflection (chess), a tactic that forces an opposing chess piece to leave a square * Khet (game), formerly ''Deflexion'', an Egyptian-themed chess-like game using lasers Mechanics * Deflection ...
and the applied load: :\frac\left(EI \frac\right) = q\, The curve w(x) describes the deflection of the beam in the z direction at some position x (recall that the beam is modeled as a one-dimensional object). q is a distributed load, in other words a force per unit length (analogous to
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
being a force per area); it may be a function of x, w, or other variables. E is the
elastic modulus An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
and I is the
second moment of area The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...
of the beam's cross section. I must be calculated with respect to the axis which is perpendicular to the applied loading and which passes through the centroid of the cross section.For an Euler–Bernoulli beam not under any axial loading this axis is called the
neutral axis The neutral axis is an axis in the cross section of a beam (a member resisting bending) or shaft along which there are no longitudinal stresses or strains. If the section is symmetric, isotropic and is not curved before a bend occurs, then the ne ...
.
Explicitly, for a beam whose axis is oriented along x with a loading along z, the beam's cross section is in the yz plane, and the relevant second moment of area is : I = \iint z^2\; dy\; dz, where it is assumed that the centroid of the cross section occurs at y=z=0. Often, the product EI (known as the
flexural rigidity Flexural rigidity is defined as the force couple required to bend a fixed non- rigid structure by one unit of curvature, or as the resistance offered by a structure while undergoing bending. Flexural rigidity of a beam Although the moment M(x) an ...
) is a constant, so that :EI \frac = q(x).\, This equation, describing the deflection of a uniform, static beam, is used widely in engineering practice. Tabulated expressions for the deflection w 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 Macaulay’s method (the double integration method) is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams. Use of Macaulay’s technique is very convenient for cases of discontinuous and/or discrete load ...
", " moment area method, "
conjugate beam method The conjugate-beam methods is an engineering method to derive the slope and displacement of a beam. A conjugate beam is defined as an imaginary beam with the same dimensions (length) as that of the original beam but load at any point on the conjug ...
", " the principle of virtual work", "
Castigliano's method Castigliano's method, named after Carlo Alberto Castigliano, is a method for determining the displacements of a linear-elastic system based on the partial derivatives of the energy. He is known for his two theorems. The basic concept may be eas ...
", "
flexibility method In structural engineering, the flexibility method, also called the method of consistent deformations, is the traditional method for computing member forces and displacements in structural systems. Its modern version formulated in terms of the me ...
", "
slope deflection method The slope deflection method is a structural analysis method for beams and frames introduced in 1914 by George A. Maney. The slope deflection method was widely used for more than a decade until the moment distribution method was developed. In the ...
", "
moment distribution method The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal. The method only accounts for flexural effects and ignores axia ...
", or "
direct stiffness method As one of the methods of structural analysis, the direct stiffness method, also known as the matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type. It is a ...
". Sign conventions are defined here since different conventions can be found in the literature. In this article, a
right-handed In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjecti ...
coordinate system is used with the x axis to the right, the z axis pointing upwards, and the y axis pointing into the figure. The sign of the bending moment M is taken as positive when the torque vector associated with the bending moment on the right hand side of the section is in the positive y direction, that is, a positive value of M produces compressive stress at the bottom surface. With this choice of bending moment sign convention, in order to have dM = Qdx , it is necessary that the shear force Q acting on the right side of the section be positive in the z direction so as to achieve static equilibrium of moments. If the loading intensity q is taken positive in the positive z direction, then dQ = -qdx is necessary for force equilibrium. Successive derivatives of the deflection w have important physical meanings: dw/dx is the slope of the beam, which is the anti-clockwise angle of rotation about the y-axis in the limit of small displacements; : M = -EI \frac is the
bending moment In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending mome ...
in the beam; and :Q = -\frac\left(EI\frac\right) is the
shear force In solid mechanics, shearing forces are unaligned forces acting on one part of a body in a specific direction, and another part of the body in the opposite direction. When the forces are collinear (aligned with each other), they are called t ...
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 The neutral axis is an axis in the cross section of a beam (a member resisting bending) or shaft along which there are no longitudinal stresses or strains. If the section is symmetric, isotropic and is not curved before a bend occurs, then the ne ...
in the figure is \rho d \theta. The length of a fiber with a radial distance z below the neutral axis is (\rho + z)d \theta. Therefore, the strain of this fiber is : \frac = \frac . The stress of this fiber is E\dfrac where E is the
elastic modulus An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
in accordance with
Hooke's Law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
. The differential force vector, d\mathbf, resulting from this stress is given by, : d\mathbf = E \frac dA \mathbf. This is the differential force vector exerted on the right hand side of the section shown in the figure. We know that it is in the \mathbf direction since the figure clearly shows that the fibers in the lower half are in tension. dA is the differential element of area at the location of the fiber. The differential bending moment vector, d\mathbf associated with d\mathbf is given by : d\mathbf = -z\mathbf \times d\mathbf = -\mathbf E \frac dA. This expression is valid for the fibers in the lower half of the beam. The expression for the fibers in the upper half of the beam will be similar except that the moment arm vector will be in the positive z direction and the force vector will be in the -x direction since the upper fibers are in compression. But the resulting bending moment vector will still be in the -y direction since \mathbf \times -\mathbf = -\mathbf. Therefore, we integrate over the entire cross section of the beam and get for \mathbf the bending moment vector exerted on the right cross section of the beam the expression :\mathbf =\int d\mathbf = -\mathbf \frac \int \ dA = -\mathbf \frac , where I is the
second moment of area The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...
. From calculus, we know that when \dfrac is small, as it is for an Euler–Bernoulli beam, we can make the approximation \dfrac \simeq \dfrac, where \rho is the
radius of curvature In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
. Therefore, :\mathbf = -\mathbf EI . This vector equation can be separated in the bending unit vector definition (M is oriented as \mathbf), and in the bending equation: :M = - EI .


Dynamic beam equation

The dynamic beam equation is the
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
for the following action : S = \int_^\int_0^L \left \frac \mu \left( \frac \right)^2 - \frac EI \left( \frac \right)^2 + q(x) w(x,t)\rightdx dt. The first term represents the kinetic energy where \mu is the mass per unit length, the second term represents the potential energy due to internal forces (when considered with a negative sign), and the third term represents the potential energy due to the external load q(x) . The
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
is used to determine the function that minimizes the functional S. For a dynamic Euler–Bernoulli beam, the Euler–Lagrange equation is : \cfrac\left(EI\cfrac\right) = - \mu\cfrac + q(x). When the beam is homogeneous, E and I are independent of x, and the beam equation is simpler: : EI\cfrac = - \mu\cfrac + q \,.


Free vibration

In the absence of a transverse load, q, 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 : w(x,t) = \text
hat(x)~e^ A hat is a head covering which is worn for various reasons, including protection against weather conditions, ceremonial reasons such as university graduation, religious reasons, safety, or as a fashion accessory. Hats which incorporate mecha ...
where \omega is the frequency of vibration. Then, for each value of frequency, we can solve an ordinary differential equation : EI~\cfrac - \mu\omega^2\hat = 0 \,. The general solution of the above equation is : \hat = A_1\cosh(\beta x) + A_2\sinh(\beta x) + A_3\cos(\beta x) + A_4\sin(\beta x) \quad \text \quad \beta := \left(\frac\right)^ where A_1,A_2,A_3,A_4 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 : \hat_n = A_1\cosh(\beta_n x) + A_2\sinh(\beta_n x) + A_3\cos(\beta_n x) + A_4\sin(\beta_n x) \quad \text \quad \beta_n := \left(\frac\right)^\,. The quantities \omega_n 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

The boundary conditions for a cantilevered beam of length L (fixed at x = 0) are : \begin &\hat_n = 0 ~,~~ \frac = 0 \quad \text ~~ x = 0 \\ &\frac = 0 ~,~~ \frac = 0 \quad \text ~~ x = L \,. \end If we apply these conditions, non-trivial solutions are found to exist only if \cosh(\beta_n L)\,\cos(\beta_n L) + 1 = 0 \,. This nonlinear equation can be solved numerically. The first four roots are \beta_1 L = 0.596864\pi, \beta_2 L = 1.49418\pi, \beta_3 L = 2.50025\pi, and \beta_4 L = 3.49999\pi. The corresponding natural frequencies of vibration are : \omega_1 = \beta_1^2 \sqrt = \frac\sqrt ~,~~ \dots The boundary conditions can also be used to determine the mode shapes from the solution for the displacement: : \hat_n = A_1 \left \cosh\beta_n x - \cos\beta_n x) + \frac(\sin\beta_n x - \sinh\beta_n x)\right The unknown constant (actually constants as there is one for each n), A_1, which in general is complex, is determined by the initial conditions at t = 0 on the velocity and displacements of the beam. Typically a value of A_1 = 1 is used when plotting mode shapes. Solutions to the undamped forced problem have unbounded displacements when the driving frequency matches a natural frequency \omega_n, 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. The boundary conditions for a free–free beam of length L extending from x=0 to x=L are given by: : \frac = 0 ~,~~ \frac = 0 \quad \text ~~ x=0 \,\text \, x=L \,. If we apply these conditions, non-trivial solutions are found to exist only if \cosh(\beta_n L)\,\cos(\beta_n L) - 1 = 0 \,. This nonlinear equation can be solved numerically. The first four roots are \beta_1 L= 1.50562\pi, \beta_2 L = 2.49975\pi, \beta_3 L = 3.50001\pi, and \beta_4 L = 4.50000\pi. The corresponding natural frequencies of vibration are: : \omega_1 = \beta_1^2 \sqrt = \frac\sqrt ~,~~ \dots The boundary conditions can also be used to determine the mode shapes from the solution for the displacement: : \hat_n = A_1 \Bigl (\cos\beta_n x + \cosh\beta_n x) - \frac(\sin\beta_n x + \sinh\beta_n x)\Bigr As with the cantilevered beam, the unknown constants are determined by the initial conditions at t = 0 on the velocity and displacements of the beam. Also, solutions to the undamped forced problem have unbounded displacements when the driving frequency matches a natural frequency \omega_n.


Stress

Besides deflection, the beam equation describes
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s and moments and can thus be used to describe stresses. For this reason, the Euler–Bernoulli beam equation is widely used in
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, especially civil and mechanical, to determine the strength (as well as deflection) of beams under bending. Both the
bending moment In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending mome ...
and the
shear force In solid mechanics, shearing forces are unaligned forces acting on one part of a body in a specific direction, and another part of the body in the opposite direction. When the forces are collinear (aligned with each other), they are called t ...
cause stresses in the beam. The stress due to shear force is maximum along the
neutral axis The neutral axis is an axis in the cross section of a beam (a member resisting bending) or shaft along which there are no longitudinal stresses or strains. If the section is symmetric, isotropic and is not curved before a bend occurs, then the ne ...
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 continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
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: :\sigma = \frac = -zE ~ \frac.\, Here, z 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 Pure bending ( Theory of simple bending) is a condition of stress where a bending moment is applied to a beam without the simultaneous presence of axial, shear, or torsional forces. Pure bending occurs only under a constant bending moment (M) sinc ...
(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

The maximum tensile stress at a cross-section is at the location z = c_1 and the maximum compressive stress is at the location z = -c_2 where the height of the cross-section is h = c_1 + c_2 . These stresses are : \sigma_1 = \cfrac = \cfrac ~;~~ \sigma_2 = -\cfrac = -\cfrac The quantities S_1,S_2 are the section moduli and are defined as : S_1 = \cfrac ~;~~ S_2 = \cfrac The section modulus combines all the important geometric information about a beam's section into one quantity. For the case where a beam is doubly symmetric, c_1 = c_2 and we have one section modulus S = I/c.


Strain in an Euler–Bernoulli beam

We need an expression for the
strain Strain may refer to: Science and technology * Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes * Strain (chemistry), a chemical stress of a molecule * Strain (injury), an injury to a mu ...
in terms of the deflection of the neutral surface to relate the stresses in an Euler–Bernoulli beam to the deflection. To obtain that expression we use the assumption that normals to the neutral surface remain normal during the deformation and that deflections are small. These assumptions imply that the beam bends into an arc of a circle of radius \rho (see Figure 1) and that the neutral surface does not change in length during the deformation. Let \mathrmx be the length of an element of the neutral surface in the undeformed state. For small deflections, the element does not change its length after bending but deforms into an arc of a circle of radius \rho. If \mathrm\theta is the angle subtended by this arc, then \mathrmx = \rho~\mathrm\theta. Let us now consider another segment of the element at a distance z above the neutral surface. The initial length of this element is \mathrmx. However, after bending, the length of the element becomes \mathrmx' = (\rho-z)~\mathrm\theta = \mathrmx - z~\mathrm\theta. The strain in that segment of the beam is given by : \varepsilon_x = \cfrac = -\cfrac = -\kappa~z where \kappa is the
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...
of the beam. This gives us the axial strain in the beam as a function of distance from the neutral surface. However, we still need to find a relation between the radius of curvature and the beam deflection w.


Relation between curvature and beam deflection

Let P be a point on the neutral surface of the beam at a distance x from the origin of the (x,z) coordinate system. The slope of the beam is approximately equal to the angle made by the neutral surface with the x-axis for the small angles encountered in beam theory. Therefore, with this approximation, : \theta(x) = \cfrac Therefore, for an infinitesimal element \mathrmx, the relation \mathrmx = \rho~\mathrm\theta can be written as : \cfrac = \cfrac = \cfrac = \kappa Hence the strain in the beam may be expressed as : \varepsilon_ = -z\kappa


Stress-strain relations

For a homogeneous
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
linear elastic material, the stress is related to the strain by \sigma = E\varepsilon, where E is the
Young's modulus Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
. Hence the stress in an Euler–Bernoulli beam is given by : \sigma_x = -zE\cfrac Note that the above relation, when compared with the relation between the axial stress and the bending moment, leads to : M = -EI\cfrac Since the shear force is given by Q = \mathrmM/\mathrmx, we also have : Q = -EI\cfrac


Boundary considerations

The beam equation contains a fourth-order derivative in x. To find a unique solution w(x,t) we need four boundary conditions. The boundary conditions usually model ''supports'', but they can also model point loads, distributed loads and moments. The ''support'' or displacement boundary conditions are used to fix values of displacement (w) and rotations (\mathrmw/\mathrmx) on the boundary. Such boundary conditions are also called
Dirichlet boundary conditions In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differentia ...
. Load and moment boundary conditions involve higher derivatives of w and represent momentum flux. Flux boundary conditions are also called
Neumann boundary condition In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann. When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative appl ...
s. As an example consider a
cantilever A cantilever is a rigid structural element that extends horizontally and is supported at only one end. Typically it extends from a flat vertical surface such as a wall, to which it must be firmly attached. Like other structural elements, a canti ...
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. 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): :w, _ = 0 \quad ; \quad \frac\bigg, _ = 0 \qquad \mbox\, :\frac\bigg, _ = 0 \quad ; \quad \frac\bigg, _ = 0 \qquad \mbox\, 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'' is a constant, and defining \lambda=F/EI where ''F'' is the magnitude of a point force, and \tau=M/EI 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'' increases is denoted by \Delta followed by that derivative. For example, \Delta w''=w''(x+)-w''(x-) where w''(x+) is the value of w'' at the lower boundary of the upper segment, while w''(x-) is the value of w'' at the upper boundary of the lower segment. When the values of the particular derivative are not only continuous across the boundary, but fixed as well, the boundary condition is written e.g., \Delta w''=0^* which actually constitutes two separate equations (e.g., w''(x-) = w''(x+) = fixed). : 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

Applied loads may be represented either through boundary conditions or through the function q(x,t) 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 In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
. 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 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: : \begin & EI \frac = 0 \\ & w, _ = 0 \quad ; \quad \frac\bigg, _ = 0 \quad ; \quad \frac\bigg, _ = 0 \quad ; \quad -EI \frac\bigg, _ = F\, \end Alternatively we can represent the point load as a distribution using the Dirac function. In that case the equation and boundary conditions are : \begin & EI \frac = F \delta(x - L) \\ & w, _ = 0 \quad ; \quad \frac\bigg, _ = 0 \quad; \quad \frac\bigg, _ = 0\, \end 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 :w = \frac(3 L x^2 - x^3)\,~. The application of several point loads at different locations will lead to w(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 Singularity functions are a class of discontinuous functions that contain singularities, i.e. they are discontinuous at their singular points. Singularity functions have been heavily studied in the field of mathematics under the alternative names ...
s are often used as a shorthand for the Dirac function, its
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
, and its
antiderivatives In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
. 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 Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic function, periodic, such as the motion of a pendulum ...
of a beam can be accounted for by using the load function: :q(x, t) = \mu \frac\, where \mu is the
linear mass density Linear density is the measure of a quantity of any characteristic value per unit of length. Linear mass density (titer in textile engineering, the amount of mass per unit length) and linear charge density (the amount of electric charge per unit ...
of the beam, not necessarily a constant. With this time-dependent loading, the beam equation will be a
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
: : \frac \left( EI \frac \right) = -\mu \frac. Another interesting example describes the deflection of a beam rotating with a constant
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of \omega: :q(x) = \mu \omega^2 w(x)\, This is a
centripetal force A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
distribution. Note that in this case, q is a function of the displacement (the dependent variable), and the beam equation will be an autonomous
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
.


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). The bending moments (M), shear forces (Q), and deflections (w) 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 A cantilever is a rigid structural element that extends horizontally and is supported at only one end. Typically it extends from a flat vertical surface such as a wall, to which it must be firmly attached. Like other structural elements, a canti ...
beams. The bending moments (M), shear forces (Q), and deflections (w) 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 moment In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending mome ...
s and
shear force In solid mechanics, shearing forces are unaligned forces acting on one part of a body in a specific direction, and another part of the body in the opposite direction. When the forces are collinear (aligned with each other), they are called t ...
s in Euler–Bernoulli beams can often be determined directly using static balance of
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s 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 In statics and structural mechanics, a structure is statically indeterminate when the static equilibrium equations force and moment equilibrium conditions are insufficient for determining the internal forces and reactions on that structure. Math ...
''. 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 equation In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approx ...
s can allow for
viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly ...
or
plastic Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptab ...
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 Shear may refer to: Textile production *Animal shearing, the collection of wool from various species **Sheep shearing *The removal of nap during wool cloth production Science and technology Engineering *Shear strength (soil), the shear strength ...
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 Tymoshenko ( uk, Тимошенко, translit=Tymošenko), Timoshenko (russian: Тимошенко), or Tsimashenka/Cimašenka ( be, Цімашэнка) is a surname of Ukrainian origin. It derives from the Christian name Timothy, and its Ukrainian ...
(developed by the Russian-born scientist
Stephen Timoshenko Stepan Prokofyevich Timoshenko (russian: Степан Прокофьевич Тимошенко, p=sʲtʲɪˈpan prɐˈkofʲjɪvʲɪtɕ tʲɪmɐˈʂɛnkə; uk, Степан Прокопович Тимошенко, Stepan Prokopovych Tymoshenko; ...
) 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 : u_1 = u_0(x) - z \cfrac ~;~~u_2 = 0 ~;~~ u_3 = w_0(x) Using the definition of the Lagrangian Green strain from
finite strain theory In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...
, we can find the von Kármán strains for the beam that are valid for large rotations but small strains by discarding all the higher-order terms (which contain more than two fields) except \frac \frac. The resulting strains take the form: : \begin \varepsilon_ & = \cfrac - z\cfrac + \frac\left \left(\cfrac - z\cfrac\right)^2 + \left(\cfrac \right)^2 \right \approx \cfrac - z\cfrac + \frac\left( \frac \right)^2 \\ .25 em \varepsilon_ & = 0 \\ .25 em \varepsilon_ & = \frac\left(\frac\right)^2 \\ .25 em \varepsilon_ & = 0 \\ .25 em \varepsilon_ & = - \frac\left \left(\cfrac-z\cfrac\right) \left(\cfrac\right) \right \approx 0 \\ .25 em \varepsilon_ & = 0. \end From the
principle of virtual work A principle is a proposition or value that is a guide for behavior or evaluation. In law, it is a rule that has to be or usually is to be followed. It can be desirably followed, or it can be an inevitable consequence of something, such as the la ...
, the balance of forces and moments in the beams gives us the equilibrium equations : \begin \cfrac + f(x) & = 0 \\ \cfrac + q(x) + \cfrac\left(N_\cfrac\right) & = 0 \end where f(x) is the axial load, q(x) is the transverse load, and : N_ = \int_A \sigma_~ \mathrmA ~;~~ M_ = \int_A z\sigma_~ \mathrmA To close the system of equations we need the
constitutive equations In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
that relate stresses to strains (and hence stresses to displacements). For large rotations and small strains these relations are : \begin N_ & = A_\left cfrac + \frac\left(\cfrac\right)^2 \right- B_\cfrac \\ M_ & = B_\left cfrac + \frac\left(\cfrac\right)^2 \right- D_\cfrac \end where : A_ = \int_A E~\mathrmA ~;~~ B_ = \int_A zE~\mathrmA ~;~~ D_ = \int_A z^2E~\mathrmA ~. The quantity A_ is the '' extensional stiffness'',B_ is the coupled ''extensional-bending stiffness'', and D_ is the ''
bending stiffness The bending stiffness (K) is the resistance of a member against bending deformation. It is a function of the Young's modulus E, the second moment of area I of the beam cross-section about the axis of interest, length of the beam and beam boundary c ...
''. 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 : EI~\cfrac - \frac~EA~\left(\cfrac\right)^2\left(\cfrac\right) = q(x)


See also

*
Applied mechanics Applied mechanics is the branch of science concerned with the motion of any substance that can be experienced or perceived by humans without the help of instruments. In short, when mechanics concepts surpass being theoretical and are applied and e ...
*
Bending In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to ...
*
Bending moment In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending mome ...
*
Buckling In structural engineering, buckling is the sudden change in shape ( deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a ...
*
Flexural rigidity Flexural rigidity is defined as the force couple required to bend a fixed non- rigid structure by one unit of curvature, or as the resistance offered by a structure while undergoing bending. Flexural rigidity of a beam Although the moment M(x) an ...
*
Generalised beam theory In structural engineering and mechanical engineering, generalised beam theory (GBT) is a one-dimensional theory used to mathematically model how beams bend and twist under various loads. It is a generalization of classical Euler–Bernoulli beam th ...
*
Plate theory In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions ...
*
Sandwich theory Sandwich theoryPlantema, F, J., 1966, Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates, and Shells, Jon Wiley and Sons, New York.Zenkert, D., 1995, An Introduction to Sandwich Construction, Engineering Materials Advisory S ...
*
Shear and moment diagram Shear force and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear forces and bending moments at a given point of a structural element such ...
*
Singularity function Singularity functions are a class of discontinuous functions that contain singularities, i.e. they are discontinuous at their singular points. Singularity functions have been heavily studied in the field of mathematics under the alternative names ...
*
Strain (materials science) In physics, deformation is the continuum mechanics transformation of a body from a ''reference'' configuration to a ''current'' configuration. A configuration is a set containing the positions of all particles of the body. A deformation can ...
*
Timoshenko beam theory Tymoshenko ( uk, Тимошенко, translit=Tymošenko), Timoshenko (russian: Тимошенко), or Tsimashenka/Cimašenka ( be, Цімашэнка) is a surname of Ukrainian origin. It derives from the Christian name Timothy, and its Ukrainian ...
* Theorem of three moments (Clapeyron's theorem) *
Three-point flexural test The three-point bending flexural test provides values for the modulus of elasticity in bending E_f, flexural stress \sigma_f, flexural strain \epsilon_f and the flexural stress–strain response of the material. This test is performed on a un ...


References


Notes


Citations


Further reading

*


External links


Beam stress & deflection, beam deflection tables
{{DEFAULTSORT:Euler-Bernoulli Beam Equation Elasticity (physics) Equations Leonhard Euler Mechanical engineering Solid mechanics Structural analysis