User:Dhollm/beam theory (de)

Beam theory describes the behavior of beams under stress. It is a branch of applied mechanics, especially strength of materials, elasticity and statics (mechanics).

Beam theory is used in many engineering disciplines, for example:
 * Civil engineering
 * Mechanical engineering
 * Shipbuilding
 * Aerospace engineering and aircraft.

Requirements
Beam theory deals with the calculation of the behaviour of components with the following characteristics:
 * A bar is a rod-shaped support member that exerts force can be charged to its axis by longitudinal and transverse loads. The response of the beam to the pressures associated longitudinal, bending, shear, slumping, drill-and lateral deformation associated with each cut sizes at which the internal stresses are combined in an appropriate manner.
 * As long as the deformation is seen inadirection ($$ y $$ as a function of $$ x $$) is the dimension in the third dimension ($$z$$) is irrelevant: the theory applies in this particular case for a plate and comprises an important application, the shelf.
 * When can a bar in the narrow sense is the axis in the unloaded condition straight, although you calculate bends with a suitably extended form of the beam theory.
 * A bar in the narrow sense consists of elastic material, such as steel, wood, or (the elastic limit) of reinforced concrete, although many other materials can be approximated shall calculate, as if they were elastic. A beam behaves rigidly. Approximate bending slack rope act and are therefore no bar.
 * The load of the beam is perpendicular to its axis, so that it bends itself. If the device will be charged at only along its axis (train / compression, torsion) and non-buckles, it is called no bar, but bar. If the component is indeed loaded only longitudinally, but for stability failure buckles on side, although it is called Knickstab and not buckling beams, but it is calculated with an extended form of the beam theory (theory Higher order).
 * Strictly speaking we mean by a bar one Euler-Bernoulli beam. This is the hypothesis: cross-sections, which are originally perpendicular to the zero line, stay with the same deformation. For pure bending ($$ M $$ = const) the cross-sections remain perpendicular to the zero line and also because the bending line is then a circle and the cross-sectional level coincides with the circle radius. In all other cases, the cross-sectional plane is rotated by the shear angle. This is for example to capture a broader and more complex beam theory tries, namely the theory of Timoshenko beam. This takes into account the shear deformation of the cross section level.

The beam theory also applies to components that are assembled from individual bars.

Next conversion
Generally we distinguish
 * First order beam theory: It is considered to approximate the undeformed beam and a beam element accounted for the forces and moments. They almost always sufficient.
 * Second order beam theory: It is considered the deformed bar, a bar element, however, the linearized mathematical model. It is needed for stability problems, as well as large deflections for angles up to about 20 °.
 * Third order beam theory: It is considered the deformed beams a beam element, and the mathematical model is not linearized. It is required in special cases, for very large deflections and angles of 20 °.

Determined statically
At statically determined stored beam can the support reactions and internal forces determined from the equilibrium conditions. Statically determinate beam in the longitudinal direction have a fixed bearing and a longitudinally moving support or are clamped to a beam end. As a "solid" refers to a support when it is held horizontal and vertical and thus can transmit horizontal and vertical forces. A movable bearing can move against horizontal or vertical, and thus pay off any forces in that direction.



Statically indeterminate (or overdetermined)
For statically indeterminate beams have to be met in addition to the conditions of equilibrium and compatibility conditions in order to determine the support reactions and internal forces can. Statically indeterminate beams have any number of supports or clampings.

In the simplest case, a beam from the equation of the elastic line, a linear inhomogeneous differential equation is calculated. It provides a link between the deflection $$ w $$ (in $$ y $$-direction) and the line load (weight per leg) $$ q $$ as a function of the coordinate x along the beam axis ago.

$$ EI \ w( x) = q (x) $$.

Bending rigidity and bending stress
Thebending rigidity $$ EI $$ is composed of the modulus $$ E $$ of the material and the moment of inertia $$ I $$ of the given geometric cross section. The latter is calculated as $$ I = \ int y ^ 2 (\ rm d) y (\ rm d) z $$. For a beam with rectangular cross-section $$ h \ cdot b $$ (in $$ y $$ - respectively $$ z $$-direction) is $$ I = (h ^ 3 b \ over 12) $$.

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

For the dynamic boundary conditions is relevant, what is the connection between the deflection and the average expense arises, namely

Bending moment $$ M (x) =-EI \ w''(x) $$

Shear force: $$ Q (x) =-EI \ w'''(x) $$

The bending moment is made up of bending stresses, these are in the axial direction acting voltages with a linear pressure distribution between the fiber and Zugfaser: $$ \ Sigma_b (z) = \ frac (M) (I) z $$ It $$ I $$ is the area moment of inertia of the cross section at the axis around which revolves the bending moment. The characteristic $$ I / z $$ on the maximum $$ z $$ (at the extreme fiber of cross section) is also called modulus $$ W $$. It follows a fairly well-known result: the carrying capacity of a beam is proportional to $$ I / h = bh ^ 2 $$.



can be rotated in the case of unsymmetrical cross-sections must coordinate in direction of the principal axes of inertia, so that separated the bend in both directions from each charge. For example, if an L-profile is loaded from above, it can also bend forward or backward. Only in the direction of a principal axis of inertia is a beam bends in the direction of the load and not across it.

How strong a beam bends depends also strongly on the position of the supports, at constant load $$ q (x) $$ = const is obtained from the differential equation as the optimal stock positions, the Bessel-point e.

The bending stress in particular describes the force on the cross section (eg a beam) acts, which is loaded perpendicular to its extension direction.

The normal stress in the beam cross section is:

$$ \ Sigma \ = \ frac (M) (I) \ cdot z $$

If the moment is positive, enter for $$ z $$> to 0 and train for $$ z $$ <0 compressive stresses. The greatest stress occurs in amount, therefore, in the extreme fiber $$ z_ \ mathrm (max) $$.

The modulus$$ W $$ is to counteract the tension $$ \ sigma \, $$ to

$$ W = \ frac (I) (| z | _ \ mathrm (max)) $$ ($$ I $$ describes the moment of inertia)

Accordingly, results for the maximum bending stress:

$$ \ Sigma_ \ mathrm (max) = \ frac (M) (W) $$

The larger the modulus, the smaller is thus the bending stress.

First order theory: Dynamics
Until only the static treated. The beam dynamics, for example to calculate beam vibrations, based on the equation $$ EI \ w( x, t) b \, \ dot (w) (x, t) m \, \ mbox (w) (x, t) = q (x, t) $$ The problem is not only the location $$ x $$, but also on the time $$ t $$. There are two further parameters of the beam, namely the mass distribution $$ m $$ (in kg / m) and the structural damping $$ b $$. If the component oscillates under water, including $$ m $$ and the hydrodynamic mass, and in $$ b $$ can you include a linearized form of the hydrodynamic damping, see Morison's equation.

Second order theory: Knickstab
While so far the forces and moments have been reported at approximately undeformed component, it is in the case of bend bars necessary to consider a beam element in the deformed state. Knickstab calculations are based on the equation $$ EI \ w( x) N \, f''(x) = q (x) $$ namely, in the simplest case with q = $$ 0 $$. In addition, the axial compressive force acting in Knickstab $$ N $$, which may not exceed, depending on the conditions buckling load not to allow the rod buckles not.

Third order theory
An application in which third order beam theory is needed is, for example, the laying of Offshore Pipelines of a vessel into deep water, reproduced here as planar static case. A very long pipe string hangs down from the vehicle to the seafloor, is curved like a rope, but rigid. The nonlinear differential equation is here $$ EI \, \ varphi''(s) - H \, \ sin \ varphi (s) (ws-V) \ cos \ varphi (s) = 0 $$ The coordinate is not more $$ x $$, but $$ s $$. This is the arc length along the pipeline. $$ H $$ is the constant horizontal component of the pipeline along the cutting force (Horizontalzug) and is influenced by how much the vehicle moves with its anchors and the tensioner on the pipeline, so they do not durchsackt and breaks. The tensioner is a device composed of two caterpillar tracks, which harnesses the pipeline on board and keep it under tension. $$ w $$ is the weight per length minus buoyancy. $$ V $$ is a math computation size, which can be thought of as small Bodenauflagerkraft. The geometry is described by the angle $$ \ varphi $$, the $$ with Horizontalkoordinate x (s) $$ and the vertical coordinates $$ z (s) $$ in the following context is : $$ \ Partial x (s) / \ partial s = \ cos \ varphi (s) \ Quad \ quad \ quad \ Partial z (s) / \ partial s = \ sin \ varphi (s) $$

History
After qualitative work done by Leonardo da Vinci, the beam theory of Galileo Galilei was founded. He ordered the neutral sheet, however, errors at the base of the beam. buckling rods have been considered by Leonhard Euler.

"Fathers" of the modern theory of bending of Leonardo da Vinci to Navier:
 * Leonardo da Vinci (1452-1519) - Qualitative statements about the sustainability
 * Galileo Galilei (1564-1642) - Discorsi ... - Galilean problem
 * Edme Mariotte (1620-1684) - power distribution - "axis of balance"
 * Robert Hooke (1635-1703) - Hooke's law, proportionality strain / stress
 * Isaac Newton (1643-1727) - balance of power, calculus
 * Gottfried Wilhelm Leibniz (1646-1716) - Moments of resistance, Calculus
 * Jakob Bernoulli (1655-1705) - the relationship between load and bending
 * Antoine Parent (1666-1716) - Triangular distribution of tensile stress
 * Jakob Leupold (1674-1727) - deflection and load capacity
 * Leonhard Euler (1707-1783) - Investigation of elastic lines, buckling equation
 * Charles Augustin de Coulomb (1736-1806) - Final resolution of the bending problem
 * Claude Louis Marie Henri Navier (1785-1836) - bending theory, elasticity, elastomechanics, Structural Analysis

Literature

 * Gross, Hauger, Schröder, Wall:Mechanics. Volume 1-3. Springer, Berlin 2006 / 2007.
 * István Szabó:Introduction to Engineering Mechanics.Springer, Berlin 2001, ISBN 3-540-67653-8
 * Peter Gummert, Karl-August Reckling:mechanism.Vieweg, Braunschweig 1994, ISBN 3-528-28904-X