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Erschienen in:
Introduction Kapitel lesen Erstes Kapitel lesen

2018 | OriginalPaper | Buchkapitel

8. Beam with Shear Contribution

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 under the consideration of the shear influence. First, several basic assumptions for the modeling of the Timoshenko beam will be introduced and the element used in this chapter will be differentiated 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, 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 linear shape functions, a general concept for an arbitrary arrangement of the shape functions will be reviewed.

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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 Taylors’ series expansion of first order is assessed around the point \((x_0,y_0)\) as follows: \(f(x,y)=f(x_0+\mathrm {d}x,y_0 +\mathrm {d}x)\approx f(x_0,y_0)+\left( \tfrac{\partial f}{\partial x}\right) _{x_0,y_0}\times (x-x_0)+\left( \tfrac{\partial f}{\partial y}\right) _{x_0,y_0}\times (y-y_0)\).
 
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\(^\circledR \), Mathematica\(^\circledR \) and Matlab\(^\circledR \) can be listed at this point as commercial examples.
 
5
Consider also the supplementary Problem 8.4.
 
6
A numerical Gauss integration with two integration points yields the same results as the exact analytical integration.
 
7
For this see Fig. 8.6 and the supplementary Problem 8.6.
 
8
One considers the definition of \(I_z\) and A in Eq. (8.123) and divides the fraction by \(h^3\).
 
9
For this see Fig. 8.6 and the supplementary Problem 8.6.
 
10
The numerical integration according to the Gauss–Legendre method with n integration points integrates a polynomial, which degree is at most \(2n-1\), exactly.
 
11
MacNeal therefore uses the expression ‘residual bending flexibility’ [16, 17].
 
12
For this see the supplementary Problem 8.5.
 
13
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.
 
14
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.
 
15
A similar example is presented in [19].
 
16
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 33:335–340CrossRef
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, BostonCrossRef
7.
Gruttmann F, Wagner W (2001) Shear correction factors in Timoshenko’s beam theory for arbitrary shaped cross-sections. Comput Mech 27:199–207CrossRef
8.
Levinson M (1981) A new rectangular beam theory. J Sound Vib 74:81–87CrossRef
9.
Reddy JN (1984) A simple higher-order theory for laminated composite plate. J Appl Mech 51:745–752CrossRef
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 M 149:113–132CrossRef
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
17.
MacNeal RH (1978) A simple quadrilateral shell element. Comput Struct 8:175–183CrossRef
18.
Reddy JN (1999) On the dynamic behaviour of the Timoshenko beam finite elements. Sadhana-Acad P Eng S 24:175–198MathSciNetCrossRef
19.
Steinke P (2010) Finite-Elemente-Methode - Rechnergestützte Einführung. Springer, BerlinCrossRef
20.
Hibbeler RC (2008) Mechanics of materials. Prentice Hall, Singapore
Metadaten
Titel
Beam with Shear Contribution
verfasst von
Andreas Öchsner
Markus Merkel
Copyright-Jahr
2018
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_8

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