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Shells

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A shell is a structure or part of a structure which has a relatively small thickness in comparison with the other two dimensions.  A general shell forms a curved surface in space.  When it forms a flat surface, it is also called a plate.  In this program, when the term “plate” is used, it refers to flat shell in the out-of-plane bending action.  

 

The shell element in the program is a four-node (quadrilateral) element that combines the in-plane membrane action and the out-of-plane bending action.  The in-plane membrane action is a standard isoparametric compatible formulation with the option to add incompatible modes.  The out-of-plane bending action is a thick-plate formulation, with the option to use the thin-plate formulation when the element is rectangular.  The element can be used to model both flat-surface plates and curved-surface general shells.  Applications of shell elements in structures are wide and far-reaching.  Examples are concrete floors, mat foundations, shear wall, folded plates, barrel vaults, cooling towers, spherical domes, water tanks, etc.  The program provides powerful commands to generate these and other commonly used plate and shell structures.  These commands include Geometry > Generate Shells, Edit > Extrude, Revolve, etc.

 

For many years, a great number of papers have been published on the subject of plate and shell structures.  Although the membrane action of a shell element is relatively simple, the (plate) bending action is much more complex.  Many plate elements have been proposed, some of which have been implemented in commercial programs.  However, most of these proposed plate elements are either ineffective or unreliable.  One of the main hurdles is known as transverse “shear locking”, that is, elements behave too stiff with respect to shear deformation especially when elements are thin or geometrically distorted.

 

One of the few reliable plate elements is a rectangular thin plate element developed by O.C. Zienkiewicz [Ref. 2].  It is based on the Kirchhoff thin plate bending theory in which a line straight and normal to the mid-surface of the plate before loading is assumed to remain straight and normal to the deformed mid-surface after loading.  The transverse shear strain is therefore assumed to be zero.  This plate element is important in that it is the first plate element that can be applied reliably in engineering practice.  Prior to this, plate analysis depended mainly on very few “closed-form” solutions of simple geometry and boundary conditions, and other very approximate methods such as equivalent frame method of ACI [Ref. 12].  The Kirchhoff rectangular thin plate is implemented in the program.  It produces results that converge to “closed-form” solutions as finite element meshes are refined.  The element, however, has to be rectangular in shape and does not account for shear deformation.

 

A much more reliable and effective plate bending element is the MITC4 developed by K.J. Bathe and others [Ref. 1].  It is a thick plate that is based on Mindlin plate theory in which a line straight and normal to the mid-surface of the plate before loading is assumed to remain straight but not necessarily normal to the deformed mid-surface after loading.  The element considers shear as well as bending deformations and may be used for both thick and thin plates.  This plate element differs from earlier Mindlin theory based plate elements in that different (mixed) interpolations are used to account for the bending and transverse shear strains.  The MITC4 plate bending element is implemented in the program.  It is free from “shear locking” and performs well even when element meshes are distorted.  The shape of the element may be any general quadrilateral as long as the aspect ratio is within a reasonable range (say 0.2 to 5.0).

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