2D Finite Elements, An Overview by Kotur Raghavan

1.       Broadly, there are two categories of two-dimensional finite elements. They are Flexure Elements and Continuum Elements. Flexure elements, usually referred to as shell elements, have all three translations and all three rotations as displacement fields. On the other hand, continuum elements have only in-plane translations as displacement fields. The present article will focus only on the continuum elements. 

2.       All finite element packages include 2D continuum elements in the element library. The elements are often called 2D solid elements (for example the element PLANE182 in ANSYS). The formulation aspects of these elements are identical to those pertaining to three dimensional elements and are based on the Theory of Elasticity. The only difference is that the 2D elements have only two displacement fields and the nodes have only two degrees of freedom. 

3.       Everything in the world is three-dimensional. But geometrical aspects and relative dimensions permit the usage of 2D elements without significant loss of accuracy. To facilitate this, users can choose one of the four theoretical bases on which the elements are developed. The options are 

a.        Axisymmetric

b.       Plane Stress

c.       Plane Strain

d.       Generalized Plane Strain. 

4.       The axisymmetric option is exclusively for solids of revolution. Only the generator plane is modelled using 2D elements. The degrees of freedom are radial and axial displacements. The necessary condition is that the loads and displacement constraints are also axisymmetric, that is invariant in the tangential direction. 

5.       The other three options are to be used for prismatic objects. For these elements to be valid, it is necessary that the following conditions are satisfied. 

a.       The objects are prismatic. The shape and size are invariant along the prism axis.

b.       All loads act normal to the prism axis.

c.       The loads are invariant along the prism axis.

d.       Any slice of the object will be in a state of self-equilibrium. 

6.       If the above conditions are satisfied, a cross section of the component can be analysed. The option to be used is dependent of the relative dimensions and end conditions. 

7.       Plane Stress formulation is to be used when the cross section dimensions are large as compared to the axial dimension (thickness). Examples: Beams, Thin rotating disks of uniform thickness. 

8.       Plane Strain formulation is appropriate for prismatic objects whose ends are held so that the object will not have axial deformation and hence there is no normal axial strain. Gravity dams fall under this category. Long heat exchanger tubes which are subjected to internal or external pressure and are held between tube sheets also can be analysed using plane strain elements. 

9.       Generalized Plane Strain (GPS) elements are appropriate for long tubes and pipes whose ends are free. In such structural components, there will be a central core portion along which the state of stress remains constant. GPS elements yield results which agree with the core portion results obtained from a 3D model. The element is useful for analyzing piping system branches which are normally designed to have free thermal expansion. 

In articles to follow a few case studies will be presented and discussed. 

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