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2016 | OriginalPaper | Buchkapitel

7. Structural Elements

verfasst von : Jože Korelc, Peter Wriggers

Erschienen in: Automation of Finite Element Methods

Verlag: Springer International Publishing

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Abstract

Trusses, beams and shells belong to the most important structural elements in engineering practice. Many structures in civil engineering—like masts, domes, frames or cooling towers—consist of such structural elements.

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Vorheriges Kapitel Continuum Elements

Nächstes Kapitel Automation of Sensitivity Analysis

Since in the global configuration of the truss is already taken into account in (7.2) and (7.3) it is not necessary to introduce a transformation between local and global truss coordinates.

In case that the stresses are needed for post-processing purposes they simply can be computed from (5.1) by \(S_X = \partial W^{SV,O} \, / \,\partial E_X = 1 \, / \,\lambda _X \,\partial W^{SV,O} \, / \,\partial \lambda _X\). Cauchy stresses then follow from (1.71) with \(J_F=1\): \(\sigma _x = S_X\,\lambda _X^2\).

Quadratic or higher order interpolations could be introduced as well, but for most applications linear elements are sufficient since the linear shape functions are solution of the homogeneous differential equations of the truss in the geometrically linear case when linear elastic constitutive behaviour is assumed.

The second term in (7.13) can be considered at global level after assembly of the element residuals, as is well known from the linear finite element method.

This unexpected feature leads to a linear solution for special cases, like rolling up a beam under an end moment where the normal and shear forces are zero.

Note that the stress \({\varvec{T}}_B\) does not follow from the polar decomposition of the deformation gradient like the Biot stress, see Table 1.2.

Note that a plane isoparametric surface assumes in general a hyper surface after deformation and thus the cross section does not remain plane in general.

The classical Kirchhoff–Love hypothesis would, of course, be a natural assumption for kinematics of thin shells. However this additional constraint requires \(C^1\)-continuous interpolation functions that cannot be constructed by interpolations using only primary variables for triangular and quadrilateral finite elements. In this context, a new approach which combines interpolations of the deformations with the CAD description of the shell surfaces are of interest. These discretization employ Bezier or other \(C^1\)-continuous polynomials, see e.g. Cirak et al. (2000), Onate and Cervera (1993) and lately Hughes et al. (2005) who introduced the notion of isogeometric analysis.

This formulation is contrary to classical shell theories where stress resultants are introduced with respect to the shell midsurface and related strain energies are constructed. The latter is especially complicated when finite strain constitutive models have to be considered.

In general this basis does not coincidence with the global cartesian coordinate system \(\{\varvec{E}_i\}\) of the reference configuration. Thus a transformation of vectors and tensors to the global coordinate system has to be performed in order to assemble the shell elements correctly since the FE displacements and rotations are defined in the global system.

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Titel: Structural Elements
verfasst von: Jože Korelc
Peter Wriggers
Verlag: Springer International Publishing
Buch: Automation of Finite Element Methods
Print ISBN: 978-3-319-39003-1

Electronic ISBN: 978-3-319-39005-5

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-319-39005-5_7

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