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Theoretical Analysis Read chapter Read first chapter

2019 | OriginalPaper | Chapter

3. Generalization to Variable Bending Moment

Authors : Angelo Marcello Tarantino, Luca Lanzoni, Federico Oyedeji Falope

Published in: The Bending Theory of Fully Nonlinear Beams

Publisher: Springer International Publishing

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Abstract

In this third Chapter, the theoretical model proposed in Chap. 1 for the bending of fully nonlinear beams is generalized to the case of variable bending moment. Such a generalization focuses on the local determination of curvature and bending moment along the deformed beam axis. Once the moment-curvature relationship has been derived, the equilibrium problem for nonlinear beams subjected to variable bending moment has been formulated. The governing equations assume the form of a coupled system of three equations in integral form. To solve this highly nonlinear system, an iterative numerical procedure has been proposed. Definitively, the analysis developed in this Chapter allows considering a very wide class of equilibrium problems for nonlinear beams. By way of example, the Euler beam and a cantilever beam loaded by a concentrated force of the dead or live (follower) type, applied in its free end, has been studied, showing the shape assumed by the deformed beam axis as the load multiplier increases.

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previous chapter Numerical and Experimental Analyses
Footnotes
1
The dependence on the angle \(\alpha _{0}\), at least in the case of short beams, suggests that the slenderness of a beam in nonlinear theory should be estimated by referring to the deformed configuration. Thus accepting the concept that the slenderness of a beam is also related to the curvature \(R_{0}^{-1}\).
 
2
A curve is called simple if it is injective:
$$ \mathbf {f}(t_{1})\ne \mathbf {f}(t_{2}),\quad \forall (t_{1},\,t_{2}). $$
.
 
3
In the Chap. 1, the bending of a beam with constant curvature was studied. Using the displacement field (1.​11), the spatial coordinates of the points belonging to the deformed axis are obtained (\(X=Y=0\) and \(Z=t\))
$$ {\left\{ \begin{array}{ll} y(t)=v(t)=-R_{0}(1-\cos \frac{t}{R_{0}})\\ z(t)=t+w(t)=R_{0}\sin \frac{t}{R_{0}}. \end{array}\right. } $$
The derivatives of these functions are
$$ {\left\{ \begin{array}{ll} y'(t)=-\sin \frac{t}{R_{0}}\\ z'(t)=\cos \frac{t}{R_{0}}, \end{array}\right. } $$
from which
$$ \Vert \mathbf {f}'(t)\Vert =\sqrt{(\sin \frac{t}{R_{0}})^{2}+(\cos \frac{t}{R_{0}})^{2}}=1. $$
That is, the curve is inextensible (as stated above considering the unit value of the longitudinal stretch) and directly re-parametrizable with respect to the arc length s.
 
4
For the constitutive parameters of the compressible Mooney-Rivlin stored energy function see also [7, 8].
 
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T.M. Wang, S.L. Lee, O.C. Zienkiewicz, Numerical analysis of large deflections of beams. Int. J. Mech. Sci. 3, 219–228 (1961)CrossRef
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T.M. Wang, Non-linear bending of beams with uniformly distributed loads. Int. J. Nonlinear Mech. 4, 389–395 (1969)CrossRef
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J.T. Holden, On the finite deflections of thin beams. Int. J. Solid Struct. 8, 1051–11055 (1972)CrossRef
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A.M. Tarantino, Thin hyperelastic sheets of compressible material: field equations, Airy stress function and an application in fracture mechanics. J. Elast. 44, 37–59 (1996)MathSciNetCrossRef
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A.M. Tarantino, Nonlinear fracture mechanics for an elastic Bell material. Quart. J. Mech. Appl. Math. 50, 435–456 (1997)MathSciNetCrossRef
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A. Pflüger, Stabililatsprobleme der Elastostatik (Springer, Berlin, Heidelberg, 1964)CrossRef
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V.I. Feodos’ev, On a stability problem. Prikl. Matem. Mekhan. 29, 391–392 (1965)
Metadata
Title
Generalization to Variable Bending Moment
Authors
Angelo Marcello Tarantino
Luca Lanzoni
Federico Oyedeji Falope
Copyright Year
2019
Book
The Bending Theory of Fully Nonlinear Beams

Print ISBN: 978-3-030-14675-7

Electronic ISBN: 978-3-030-14676-4

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-14676-4_3

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