Universität Karlsruhe (TH) - am Institut für Baustatik

A robust nonlinear mixed hybrid quadrilateral shell element
W. Wagner F. Gruttmann
Institut für Baustatik
Universität Karlsruhe (TH)
Kaiserstraße 12
76131 Karlsruhe
Germany
Institut für Werkstoffe und Mechanik im Bauwesen
Technische Universität Darmstadt
Petersenstraße 12
64287 Darmstadt
Germany
Contents
1 Introduction 2
2 Kinematics and Variational Formulation 4
3 Finite Element Equations 6
3.1 Interpolation of the initial and current reference surface ............. 6
3.2 Interpolation of the Green–Lagrangean strains and associated variations .... 8
3.3 Second variation of the functional ......................... 10
3.4 Interpolation of the stress resultants and shell strains .............. 11
3.5 Linearized variational formulation ......................... 12
4 Examples 15
4.1 Membrane and bending patch test ......................... 15
4.2 Linear test problem: Twisted beam ........................ 15
4.3 Hemispherical shell with a 18 ◦ hole ........................ 17
4.4 L–shaped beam ................................... 18
4.5 Channel–section beam ............................... 19
4.6 Steel girder with holes ............................... 22
5 Conclusions 24
A Second variation of the current director vector 24
B Remarks on patch test and stability 25
B.1 The patch test .................................... 25
B.2 Stability of the discrete problem .......................... 25
C Numerical integration of the stress resultants 26
D Analytical integration of some matrices 28
E J 2 -plasticity model for small strains 29
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