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Erschienen in:

2018 | OriginalPaper | Buchkapitel

5. Bending Element

verfasst von : Andreas Öchsner, Markus Merkel

Erschienen in: One-Dimensional Finite Elements

Verlag: Springer International Publishing

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Abstract

This element describes the basic deformation mode of bending. First, several elementary assumptions for modeling will be introduced and the beam element used in this chapter will be differentiated from other element formulations. The basic equations from the strength of materials, meaning kinematics, equilibrium and constitutive equation will be introduced and used for the derivation of the differential equation of the bending line. Analytical solutions will conclude the section of the basic principles. Subsequently, the bending element will be introduced, according to the common definitions for load and deformation parameters, which are used in the handling of the FE method. The derivation of the stiffness matrix is carried out through various methods and will be described in detail. Besides the simple, prismatic beam with constant cross-section also variable cross-sections, generalized loads between the nodes and orientation in the plane and the space will be analyzed.

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Vorheriges Kapitel Equivalences to Tension Bar

Nächstes Kapitel General 1D Element

More precisely this is the neutral fiber or the bending line.

Consequently the width b and the height h of a, for example, rectangular cross-section remain the same.

For this see the explanations in Chap. 8.

The sum of all points with \(\sigma =0\) along the beam axis is called the neutral fiber.

Note that according to the assumption for the Bernoulli beam the lengths 01 and 0’1’ remain unchanged.

The positive cut face is defined by the surface normal on the cutting plane which features the same orientation as the positive x-axis. It should be regarded that the surface normal is always directed outwardly. Regarding the negative cut face the surface normal and the positive x-axis are oriented antiparallel.

If the axis is grasped with the right-hand in a way so that the spread out thumb points in the direction of the positive axis, the bent fingers then show the direction of the positive rotational direction.

Just for the case that an external moment \(M^\text {ext}\) would act at position \(x=L\), the internal moment would result in: \(M_z(x=L)=M^\text {ext}\). Hereby it was assumed that the external moment \(M^\text {ext}\) would be positive in a mathematical sense.

Alternatively the expression interpolation or form function is used.

In the general three-dimensional case the form \(\varPi _\text {int}=\frac{1}{2}\int \limits _\varOmega \varvec{\varepsilon }^\text {T} \varvec{\sigma }\mathrm{d}\varOmega \) can be applied, whereat \(\varvec{\sigma }\) and \(\varvec{\varepsilon }\) represent the column matrix with the stress and strain components.

The principle of virtual work encompasses the principle of the virtual displacements and the principle of the virtual forces [9].

Castigliano’s theorems were formulated by the Italian builder, engineer and scientist Carl Alberto Castigliano (1847–1884). The second theorem signifies: The partial derivative of the stored potential energy in a linear-elastic body with regards to the displacement \(u_i\) yields the force \(F_i\) in the direction of the displacement at the considered point. An analog coherence also applies for the rotation and the moment.

The use of the transposed ‘T’ for the scalar weight function is not obvious at the first glance. However, the following matrix operations will clarify this approach.

A common representation of the partial integration of two functions f(x) and g(x) is: \(\int fg'\,\mathrm{d}x=fg -\int f'g \mathrm{d}x\).

See Sect. 5.2.2 with the executions for the internal reactions and external loads.

Pay attention to the sign: the virtual displacements have the same direction as the external loads.

Also see Table 5.5 and supplementary problems 5.6.

If point loads appear between nodes, the discretization can of course be further sub-divided, so that a new node is positioned on the location of the loading point. However, this chapter considers only the case that no further subdivision is regarded.

Hartman F, Katz C (2007) Structural analysis with finite elements. Springer, Berlin

Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-Hill Book Company, New York

Gould PL (1988) Analysis of shells and plates. Springer, New York

Altenbach H, Altenbach J, Naumenko K (1998) Ebene Flächentragwerke: Grundlagen der Modellierung und Berechnung von Scheiben und Platten. Springer, Berlin

Szabó I (2003) Einführung in die Technische Mechanik: Nach Vorlesungen István Szabó. Springer, Berlin

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Budynas RG (1999) Advanced strength and applied stress analysis. McGraw-Hill Book, Singapore

Hibbeler RC (2008) Mechanics of materials. Prentice Hall, Singapore

Szabó I (1996) Geschichte der mechanischen Prinzipien und ihrer wichtigsten Anwendungen. Birkhäuser, Basel

10.

Szabó I (2001) Höhere Technische Mechanik: Nach Vorlesungen István Szabó. Springer, Berlin

11.

Betten J (2004) Finite Elemente für Ingenieure 2: Variationsrechnung, Energiemethoden, Näherungsverfahren, Nichtlinearitäten, Numerische Integrationen. Springer, Berlin

12.

Oden JT, Reddy JN (1976) Variational methods in theoretical mechanics. Springer, Berlin

13.

Betten J (2001) Kontinuumsmechanik: Elastisches und inelastisches Verhalten isotroper und anisotroper Stoffe. Springer, Berlin

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Hutton DV (2004) Fundamentals of finite element analysis. McGraw-Hill Book, Singapore

Titel: Bending Element
verfasst von: Andreas Öchsner
Markus Merkel
Verlag: Springer International Publishing
Buch: One-Dimensional Finite Elements
Print ISBN: 978-3-319-75144-3

Electronic ISBN: 978-3-319-75145-0

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-75145-0_5

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