Euler–Bernoulli beam theory

The beam theory describes the behavior of the beam under load, and in particular its bending. It is a branch of engineering mechanics, especially the strength of materials, theory of elasticity and the static. Often, one also speaks of the bending theory of beam.

For use, the beam theory is used in many engineering disciplines, for example,

  • Civil Engineering
  • Engineering
  • Shipbuilding
  • Air and space technology and aerospace.

Be calculated V.A., the longitudinal, transverse and bending deformation, and the bending line, the bending moment of the forces or load distribution follows. This requires precise knowledge of the support points of the beam or support, or whether they are static un - or overdetermined. Necessary for calculating the modulus of elasticity of the material and geometrical moment of inertia of the beam cross-section, from which it follows the bending stiffness.

Requirements

See also: cantilever

The beam theory is concerned with the calculation of components with the following features, which are called bar:

  • A bar is a rod-like support member which can be loaded by loads along and transverse to its axis. The response of the beam to the longitudinal loads, bending, shear, camber, drill and transverse deformations associated with each corresponding section sizes at which the internal stresses are combined in a suitable manner. As long as the deformation in one direction is considered (as a function of ) the dimension in the third dimension ( ) is irrelevant: the theory applicable in this particular case also comprises a disk and thereby an important application, the shelf.

The beam theory also applies to components that are composed of individual beams.

Basic features of the theory

Approximation steps

Generally, a distinction is

  • Beam theory to the first order: Consider a beam element approximation on the undeformed beam and accounted for the forces and moments. You almost always sufficient.
  • Second order beam theory: Consider a beam element on the deformed beams, however, the mathematical model is linearized. It is required for stability problems, as well as for large deflections at tilt angles up to 20 °.
  • Beam theory Third order: Consider a beam element on the deformed beam, and the mathematical model is not linearized. It is needed in special cases, for very large deflections and angles of inclination over 20 °.

First order theory: statics

Statically determined

For statically determinate supported beam, the support reactions and internal forces from the equilibrium conditions can be determined. Statically determinate beams have longitudinally a fixed support and a longitudinally movable supports or clamped to a beam end. As a "firm" refers to a bearing when it is held horizontally and vertically, and thus can transmit horizontal and vertical forces. A movable supports can, however, move horizontally and thus erode no forces in this direction.

Statically indeterminate (or overdetermined )

In statically indeterminate supported beam and compatibility conditions are in addition to the equilibrium conditions to fulfill in order to determine the support reactions and internal forces can. Statically indeterminate beams have any number of supports or restraints.

In the simplest case, a beam is calculated using the equation of the elastic line, a linear inhomogeneous differential equation is calculated. It provides a link between the deflection ( in direction) and the line load (weight per route) ago as a function of the coordinate along the beam axis.

Bending stiffness and bending stress

The bending stiffness indicates how large is the bending moment in relation to the curvature. For homogeneous cross-sections she is the product of the modulus of elasticity of the material and the geometric area moment of inertia of the given cross section. The latter is calculated as

Is - For a beam of rectangular cross section (or per - direction )

Boundary and transition conditions arising from the nature of the supports and consist of kinematic boundary conditions and from dynamic (forces and moments in question ) boundary conditions.

For the dynamic boundary conditions is relevant, what is the connection between the deflection and the load averages over there, namely

Bending moment:

Transverse force:

The bending moment is composed of bending stresses, it is acting in the axial direction of tension to a linear distribution between the pressure and the fiber tension fiber:

It is the area moment of inertia of the cross section around the axis about which rotates the bending moment. The characteristic value at the maximum ( at the outermost fibers of the cross section ) is also called modulus. It follows a fairly well-known result: the carrying capacity of a beam is proportional to.

In the case of unsymmetrical cross-sections, the coordinate system in the direction of the principal axes must be rotated so that you separated from each other in both directions can calculate the deflection. For example, if an L-profile from the top is charged, it can bend well forward or backward. Only in the direction of a principal axis of inertia, a beam bends in the direction of the load and not across it.

How much bend a bar, also depends strongly on the position of the supports; at constant load = const is obtained from the differential equation as the optimal bearing positions, the Bessel points.

The bending stress in particular describes the force (e.g., a bar), acting on the cross section which is loaded perpendicular to its extension direction.

The normal stress in the beam cross-section is:

This is the torque positive, occur for > 0 train and for <0 compressive stresses. The maximum absolute voltage therefore occurs on the outermost fiber.

The section modulus is counteracting to the voltage at

It describes the area moment of inertia. For the maximum bending stress results in:

The larger the moment of resistance, so the smaller the bending stress.

First order theory: dynamics

Until here only the static was treated. The beam dynamics to calculate about to beam vibrations, based on the equation

The problem here depends not only on the position but also on the time. It added two more parameters of the beam, namely the mass distribution and the structural damping. If the component oscillates under water, also includes the hydrodynamic mass, and can include a linearized form of the hydrodynamic damping, see Morison equation.

Second order theory: Buckling

While so far the forces and moments were approximately accounted for the undeformed component, it is in the case of articulated rods necessary to consider a beam element in the deformed state. Buckling calculations are based on the equation

And in the simplest case. Add to that the axially acting in Buckling pressure force, which shall not exceed, depending on the boundary conditions, the buckling load, so that the rod does not buckling.

Third order theory

A use case in which beam theory third order is necessary, for example, the laying of offshore pipelines from a vessel into deep water, are listed here only as a planar static case. A very long pipe section depends on the vehicle down to the ocean floor, is curved like a rope, but rigid. The non-linear differential equation is here

The coordinate is no longer called here, but. Is the arc length along the pipeline. is along the pipeline constant horizontal component of the cutting force ( horizontal draft ) and is influenced by how much the vehicle with its anchors and the tensioner on the pipeline draws, so they do not sag and break. The tensioner is a device consisting of two crawlers, which clamps the pipeline on board and keeps them under tension. is the weight per unit length less buoyancy. is an operand that can be thought of as small Bodenauflagerkraft. The geometry is described by the angle of inclination is consistent with the horizontal coordinate and vertical coordinate in the following context:

History

After qualitative work done by Leonardo da Vinci, the beam theory of Galileo Galilei was founded. He ordered the neutral surface, however flawed at the bottom of the beam at. Buckling rods were considered by Leonhard Euler.

" Fathers " of the modern theory of bending of Leonardo da Vinci to Navier:

  • Leonardo da Vinci (1452-1519) - Qualitative statements on the sustainability
  • Galileo Galilei (1564-1642) - Discourses ... - Galilean problem
  • Edme Mariotte (1620-1684) - Power distribution - "axis of equilibrium "
  • Robert Hooke (1635-1703) - Hooke's law, proportionality strain / stress
  • Isaac Newton (1643-1727) - Balance of Power, Calculus
  • Gottfried Wilhelm Leibniz (1646-1716) - resistance moments, Calculus
  • Jacob Bernoulli (1655-1705) - relationship between load and deflection
  • Antoine Parent (1666-1716) - Triangular distribution of the tensile stress
  • Jacob Leupold (1674-1727) - deflection and load capacity
  • Leonhard Euler (1707-1783) - Study of elastic lines, buckling equation
  • Charles Augustin de Coulomb (1736-1806) - Final solution of the bending problem
  • Claude Louis Marie Henri Navier (1785-1836) - bending theory, elasticity, elasto, Structural

See also cantilever

Pictures of Euler–Bernoulli beam theory

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