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Universität Karlsruhe (TH) Institut für Baustatik A robust nonlinear mixed hybrid quadrilateral shell element W. Wagner, F. Gruttmann Mitteilung 3(2004) BAUSTATIK c○ Prof. Dr.–Ing. W. Wagner Telefon: (0721) 608–2280 Institut für Baustatik Telefax: (0721) 608–6015 Universität Karlsruhe E–mail: bs@.uni-karlsruhe.de Postfach 6980 Internet: http://www.bs.uni-karlsruhe.de 76128 Karlsruhe
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 1
Page 1: Universität Karlsruhe (TH) Institu
Page 5 and 6: the strain field and the non-consta
Page 7 and 8: with membrane strains ε αβ , cur
Page 9 and 10: as close as possible to the coordin
Page 11 and 12: with δγξ M = [δx, ξ ·d + x,
Page 13 and 14: To avoid numerical difficulties the
Page 15 and 16: where numel denotes the total numbe
Page 17 and 18: 4 Examples The derived element form
Page 19 and 20: 4.3 Hemispherical shell with a 18
Page 21 and 22: Load P [N] 3,5 3,0 2,5 2,0 1,5 1,0
Page 23 and 24: Load P in kN 20,0 18,0 16,0 14,0 12
Page 25 and 26: 200 175 150 Load P [kN] 125 100 u_z
Page 27 and 28: B Remarks on patch test and stabili
Page 29 and 30: The plane stress condition S 33 (E
Page 31 and 32: E J 2 -plasticity model for small s
Page 33 and 34: [16] Betsch P, Gruttmann F, Stein,

UniversitÃ¤t Karlsruhe (TH) - am Institut fÃ¼r Baustatik

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