A Review on Membrane Finite Elements with Drilling Degree of Freedom
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ORIGINAL PAPER
A Review on Membrane Finite Elements with Drilling Degree of Freedom Djamel Boutagouga1 Received: 17 January 2020 / Accepted: 27 August 2020 © CIMNE, Barcelona, Spain 2020
Abstract Membrane finite elements have gained increasing importance since the early years of the finite elements method due to their convenience to a wide range of plane and 3D shell problems. Many research works have been focussing on membrane elements with drilling rotation. These elements are plane stress finite elements with a rotational in-plane degree of freedom. The main motivation for introducing the drilling rotation into membrane/shell elements is to improve the accuracy of the model and to avoid the problem of singularity of the stiffness matrix of shell elements in a natural way. In this article, a comprehensive review has been written for both new and expert researchers in the field. By summarizing the basic theories that have been used, new researchers can get clarification on the relevant computational methods and, gradually get familiarized with the advantages and drawbacks of each formulation. Expert researchers in the field can track-back the history of membrane elements with drilling rotation through a comprehensive review of relevant references.
1 Introduction In structural mechanics, membrane components are sheets in space that offer stiffness only in the plane of the element. These structural components have no bending or transverse shear stiffness. Therefore, they can carry membrane forces only, and do not support or transmit any bending moment or transverse load. Membrane finite elements are generally triangular or quadrangular plane surface elements formulated in tow- or three-dimensional space; they are intended for modelling thin surfaces under in-plane loading. Membranes have several basic differences with plates in structural behaviour. Plate elements are different from membrane elements in that they sustain transverse “out of plane” loading using bending stresses while membrane elements sustain only membrane loading using in-plane stresses. Shell elements are different from plate elements in that they combine bending and membrane behaviours so they can carry in-plane and out of plane loadings. Shell finite elements can be plane or curved elements. plane elements, commonly known as flat shell elements, are the simplest ones as they combine a plate-bending element with a membrane element. In this * Djamel Boutagouga [email protected] 1
Civil Engineering Department, Laboratoire des Mines, Larbi Tebessi University – Tebessa, Tébessa, Algeria
case, the plate element provides the out-plane stiffness while the membrane element provides the in-plane stiffness. As the in-plane state of strain can be described in terms of the two in-plane translations (u and v) only, conventional membrane elements require only two degrees of freedom at each node. As this family combines simplicity and efficiency, flat shell elements have been widely used in structural analysis. Several formulations of triangular and
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