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2013 | OriginalPaper | Chapter

5. Bending Element

Authors : Andreas Öchsner, Markus Merkel

Published in: One-Dimensional Finite Elements

Publisher: Springer Berlin Heidelberg

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Abstract

By this element the basic deformation bending will be described. First, several elementary assumptions for modeling will be introduced and the element used in this chapter will be outlined towards other 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 bar with constant cross-section and load on the nodes also variable cross-sections, generalized loads between the nodes and orientation in the plane and the space will be analyzed.

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previous chapter Torsion Bar
next chapter General 1D Element
Footnotes
1
More precisely this is the neutral fiber or the bending line.
 
2
Consequently the width \(b\) and the height \(h\) of a, for example, rectangular cross-section remain the same.
 
3
For this see the explanations in Chap. 8.
 
4
The sum of all points with \(\sigma =0\) along the beam axis is called neutral fiber.
 
5
Note that according to the precondition regarding the Bernoulli beam the length 01 and \(0^{\prime }1^{\prime }\) remain unchanged.
 
6
The positive cutting surface 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 cutting surface the surface normal and the positive \(x\)-axis are oriented antiparallel.
 
7
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.
 
8
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.
 
9
In English-speaking countries this mathematical notation is mostly named after the British mathematician and engineer W.H. Macaulay (1853–1936). However this notation was originally proposed by the German mathematician Rudolf Friedrich [11] Alfred Clebsch (1833–1872) [12].
 
10
Alternatively the expression interpolation or form function is used.
 
11
In the general three-dimensional case the form \(\varPi _\text{int}=\frac{1}{2}\int _\varOmega \varvec{\varepsilon }^\text{T} \varvec{\sigma }\mathrm{d} \varOmega \) can be applied, whereat \(\varvec{\sigma }\) and \(\varvec{\varepsilon }\) represent the column matrix with the stress and strain components.
 
12
The principle of virtual work encompasses the principle of the virtual displacements and the principle of the virtual forces [13].
 
13
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.
 
14
A common representation of the partial integration of two functions \(f(x)\) and \(g(x)\) is: \(\int f^{\prime }g\,\mathrm{d} x=fg -\int fg^{\prime } \mathrm{d} x\).
 
15
See Sect. 5.2.2 with the executions for the internal reactions and external loads.
 
16
Also see Table 5.6 and supplementary problems 5.6.
 
17
Since the transformation matrix \(\varvec{T}\) is an orthogonal matrix, the following applies: \(\varvec{T}^\text{T}=\varvec{T}^{-1}\).
 
18
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 within this chapter the case of no further subdivision of the mesh ought to be regarded.
 
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Macaulay WH (1919) A note on the deflections of beams. Messenger Math 48:129–130
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Clebsch RFA (1862) Theorie der Elasticität fester Körper. B.G, Teubner, Leipzig
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Szabó I (1996) Geschichte der mechanischen Prinzipien und ihrer wichtigsten Anwendungen. Birkhäuser Verlag, Basel
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Szabó I (2001) Höhere Technische Mechanik: Nach Vorlesungen István Szabó. Springer, Berlin
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Betten J (2004) Finite Elemente für Ingenieure 2: Variationsrechnung, Energiemethoden, Näherungsverfahren, Nichtlinearitäten, Numerische Integrationen, Springer, Berlin
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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
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Buchanan GR (1995) Schaum’s outline of theory and problems of finite element analysis. McGraw-Hill Book, New York
Metadata
Title
Bending Element
Authors
Andreas Öchsner
Markus Merkel
Copyright Year
2013
Publisher
Springer Berlin Heidelberg
Book
One-Dimensional Finite Elements

Print ISBN: 978-3-642-31796-5

Electronic ISBN: 978-3-642-31797-2

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-3-642-31797-2_5

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