A nonlinear quadrilateral shell element with drilling degrees of freedom

SummaryA bending theory for thin shells undergoing finite deformations is presented, and its associated finite element model is described. The kinematic assumptions are of Reissner-Mindlin type. The formulation is based on the introduction of a mixed functional with independent in-plane rotation field and skew-symmetric part of membrane forces. The resulting Euler-Lagrangian equations yield the equilibrium of stress resultants and the couple resultant with respect to the surface normal. Furthermore, the equality of the independent rotation field with the displacement dependent rotation field is enforced. Hence, the symmetry of the stress resultants is fulfilled in a weak sence. Naturally, the development of a quadrilateral finite element includes drilling degrees of freedom. The displacement field is approximated using an Allman-type interpolation.ÜbersichtEs wird eine Biegetheorie dünner Schalen bei finiten Deformationen sowie die zugehörige Finite-Element-Formulierung vorgestellt. Die kinematischen Annahmen sind vom Reissner-Mindlin-Typ. Die Darstellung basiert auf der Einführung eines gemischten Funktionals mit unabhängigen Drehfreiheitsgraden in der Schalenmmittelfläche und einem schiefsymmetrischen Anteil der Membrankräfte. Die resultierenden Euler-Lagrange-Gleichungen ergeben das Gleichgewicht und die Symmetrie der Schnittgrößen. Weiterhin ergibt sich die Gleichheit des unabhängigen mit dem verschiebungsabhängigen Rotationsfeld. Die Entwicklung des Vier-Knoten-Schalenelementes enthält somit auch den Drehfreiheitsgrad um die Schalenormale. Für das Verschiebungsfeld wird ein Allman-Ansatz gewählt.

[1]  R. Taylor,et al.  Theory and finite element formulation of rubberlike membrane shells using principal stretches , 1992 .

[2]  J. C. Simo,et al.  On stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization , 1989 .

[3]  E. Stein,et al.  Elastic-plastic analysis of thin shells and folded plate structures with finite rotations , 1990 .

[4]  Edward L. Wilson,et al.  A unified formulation for triangular and quadrilateral flat shell finite elements with six nodal degrees of freedom , 1991 .

[5]  W. Pietraszkiewicz,et al.  On Geometrically Non-Linear Theory of Elastic Shells Derived from Pseudo-Cosserat Continuum with Constrained Micro-Rotations , 1986 .

[6]  François Frey,et al.  A four node Marguerre element for non‐linear shell analysis , 1986 .

[7]  J. Altenbach.  Zienkiewicz, O. C., The Finite Element Method. 3. Edition. London. McGraw‐Hill Book Company (UK) Limited. 1977. XV, 787 S. , 1980 .

[8]  Peter Wriggers,et al.  Theory and numerics of thin elastic shells with finite rotations , 1989 .

[9]  J. Z. Zhu,et al.  The finite element method , 1977 .

[10]  D. Allman.  A quadrilateral finite element including vertex rotations for plane elasticity analysis , 1988 .

[11]  D. Allman.  A compatible triangular element including vertex rotations for plane elasticity analysis , 1984 .

[12]  E. Reissner.  A note on variational principles in elasticity , 1965 .

[13]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory , 1990 .

[14]  Robert L. Harder,et al.  A refined four-noded membrane element with rotational degrees of freedom , 1988 .

[15]  Peter Wriggers,et al.  Thin shells with finite rotations formulated in biot stresses : theory and finite element formulation , 1993 .

[16]  Edward L. Wilson,et al.  A robust quadrilateral membrane finite element with drilling degrees of freedom , 1990 .

[17]  Richard H. Macneal,et al.  A theorem regarding the locking of tapered four‐noded membrane elements , 1987 .

[18]  Carlos A. Felippa,et al.  A triangular membrane element with rotational degrees of freedom , 1985 .

[19]  F. Brezzi,et al.  On drilling degrees of freedom , 1989 .

[20]  A. B. Sabir,et al.  The applications of finite elements to large deflection geometrically nonlinear behaviour of cylindrical shells , 1972 .