Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Introduction Kapitel lesen Erstes Kapitel lesen

2013 | OriginalPaper | Buchkapitel

8. Beam with Shear Contribution

verfasst von : Andreas Öchsner, Markus Merkel

Erschienen in: One-Dimensional Finite Elements

Verlag: Springer Berlin Heidelberg

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

By this element the basic deformation bending under consideration of the shear influence will be described. First, several basic assumptions for the modeling of the Timoshenko beam will be introduced and the element used in this chapter will be distinguished from other formulations. The basic equations from the strength of materials, meaning kinematics, the equilibrium as well as the constitutive equation will be introduced and used for the derivation of a system of coupled differential equations. The section about the basics is ended with analytical solutions. Subsequently the Timoshenko bending element will be introduced with the definitions for load and deformation parameters which are commonly accepted at the handling via the FE method. The derivation of the stiffness matrix at this point also takes place via various methods and will be described in detail. Besides linear shape functions a general concept for an arbitrary arrangement of the shape functions will be introduced.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Plane and Spatial Frame Structures
Nächstes Kapitel Beams of Composite Materials
Fußnoten
1
For a function \(f(x,y)\) of two variables usually a TAYLOR’ s series expansion of first order is assessed around the point \((x_0, y_0)\) as follows: \(f(x,y)=f(x_0+\text{d} x,y_0 +\text{d} x)\approx f(x_0, y_0)+\left(\displaystyle \frac{\partial f}{\partial x}\right)_{x_0, y_0}\times (x-x_0)+\left(\displaystyle \frac{\partial f}{\partial y}\right)_{x_0, y_0}\times (y-y_0)\).
 
2
A closer analysis of the shear stress distribution in the cross-sectional area shows that the shear stress does not just alter through the height of the beam but also through the width of the beam. If the width of the beam is small compared to the height, only a small change along the width occurs and one can assume in the first approximation a constant shear stress throughout the width: \(\tau _{xy}(y,z) \rightarrow \tau _{xy}(y)\). See for example [1, 2].
 
3
One notes that in the English literature often the so-called form factor for shear is stated. This results as the reciprocal of the shear correction factor.
 
4
Maple\(^{\textregistered} \), Mathematica\(^{\textregistered} \) and Matlab\(^{\textregistered} \) can be listed at this point as commercial examples.
 
5
A numerical Gauss integration with two integration points yields the same results as the exact analytical integration.
 
6
For this see Fig. 8.6 and the supplementary problem 8.6.
 
7
One considers the definition of \(I_z\) and \(A\) in Eq. (8.111) and divides the fraction by \(h^3\).
 
8
For this see Fig. 8.6 and the supplementary problem 8.6.
 
9
The numerical integration according to the Gauss–Legendre method with \(n\) integration points integrates a polynomial, which degree is at most \(2n-1\), exactly.
 
10
MacNeal hereford uses the expression ‘residual bending flexibility’ [16, 17].
 
11
For this see the supplementary problem 8.5.
 
12
At the so-called Lagrange interpolation, \(m\) points are approximated via the ordinate values with the help of a polynomial of the order \(m-1\). In the case of the Hermite interpolation the slope of the regarded points is considered in addition to the ordinate value. For this see Chap. 6.
 
13
It needs to be remarked that the influence of distributed loads is disregarded in the derivation. If distributed loads occur, the equivalent nodal loads have to be distributed on the remaining nodes.
 
14
A similar example is presented in [19].
 
15
For this see the supplementary problem 8.6.
 
Literatur
1.
Timoshenko SP, Goodier JN (1970) Theory of elasticity. McGraw-Hill, New YorkMATH
2.
Beer FP, Johnston ER Jr, DeWolf JT, Mazurek DF (2009) Mechanics of materials. McGraw-Hill, Singapore
3.
Cowper GR (1966) The shear coefficient in Timoshenko’s beam theory. J Appl Mech 33:335–340MATHCrossRef
4.
5.
Weaver W Jr, Gere JM (1980) Matrix analysis of framed structures. Van Nostrand Reinhold Company, New York
6.
Gere JM, Timoshenko SP (1991) Mechanics of materials. PWS-KENT Publishing Company, Boston
7.
Gruttmann F, Wagner W (2001) Shear correction factors in Timoshenko’s beam theory for arbitrary shaped cross-sections. Comput Mech 27:199–207MATHCrossRef
8.
9.
Reddy JN (1984) A simple higher-order theory for laminated composite plate. J Appl Mech 51:745–752MATHCrossRef
10.
Reddy JN (1997) Mechanics of laminated composite plates: theory and analysis. CRC Press, Boca RatonMATH
11.
Reddy JN (1997) On locking-free shear deformable beam finite elements. Comput Method Appl Mech Eng 149:113–132MATHCrossRef
12.
Wang CM (1995) Timoshenko beam-bending solutions in terms of Euler-Bernoulli solutions. J Eng Mech-ASCE 121:763–765CrossRef
13.
Cook RD, Malkus DS, Plesha ME, Witt RJ (2002) Concepts and applications of finite element analysis. Wiley, New York
14.
Reddy JN (2006) An introduction to the finite element method. McGraw Hill, Singapore
15.
MacNeal RH (1994) Finite elements: their design and performance. Marcel Dekker, New York
16.
Russel WT, MacNeal RH (1953) An improved electrical analogy for the analysis of beams in bending. J Appl Mech 20:349–355
17.
18.
Reddy JN (1999) On the dynamic behaviour of the Timoshenko beam finite elements. Sadhana-Acad Proc Eng Sci 24:175–198MATH
19.
Steinke P (2010) Finite-Elemente-Methode–Rechnergestützte Einführung. Springer, Berlin
20.
Hibbeler RC (2008) Mechanics of materials. Prentice Hall, Singapore
Metadaten
Titel
Beam with Shear Contribution
verfasst von
Andreas Öchsner
Markus Merkel
Copyright-Jahr
2013
Verlag
Springer Berlin Heidelberg
Buch
One-Dimensional Finite Elements

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

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

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-3-642-31797-2_8

Premium Partner

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen. 

    Zur Marktübersicht
    Bildnachweise