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Advanced Solver Settings : 2D Element Analysis

Definition​

Linear Element Formulation

Available separately for both in-plane and out-of-plane problems, GSA provides a separate additional method for the formulation of the element’s stiffness. In each case there is an original formulation kept for consistency with existing models, and an improved formulation based on refinements to the standard 2D element isoparametric theory. More information is provided below on the differences between the different options.

In general, both later formulations are driven by a desire to upgrade the performance and/or stability of GSA’s Linear TRI / QUAD elements, admitting their use in a wider spread of applications where a preference for linear elements is of practical concern. Both make use of changes to the element formulation that help alleviate common problems of shear locking that is symptomatic of standard linear TRI and QUAD elements. As a general rule, the improved formulations perform much better in bending-type problems than their original formulations. Note however that the formulations remain as elements based on linear approximation theory and so while improved, do not match the bending performance of quadratic elements (but at greater computational cost).

For out of-of-plane behaviour, the formulations are named:

  • Mindlin – Original formulation consistent with versions of GSA v8.4 and previous. Based on Mindlin-Reissner plate theory.
  • MITC – An improved formulation that both removes the inherent problems of out-of-plane shear locking with the traditional Mindlin formulation while not reducing the number of stiff modes and the consequential problems of hourglassing.

For in-plane behaviour, the formulations are named:

  • Bilinear – Original formulation consistent with versions of GSA v8.5 and previous. Uses a linear interpolation for the dependent variable in both dimensions.
  • Allman-Cook – An improved formulation that makes use of the local zz ‘drilling’ degree of freedom to add four additional stiff modes of deformation. While not required by the basic formulation, these modes both help alleviate in-plane shear locking while also gaining the practical significance of providing access to stiff ‘zz’ degrees of freedom.

By default both options are set to the recommended, later formulations. The default in-plane and out-of pane formulations can be set in the preferences. This preference setting can be found in the miscellaneous tab of the GSA Preferences.

Suppression of Non-stiff Degrees of Freedom

This governs the method by which the non-stiff degrees of freedom on 2D elements are handled. This problem arises because the 2D elements do not have any zz stiffness. The options are

  • Geometry based automatic constraints, Flatness parameter – This approach removes the degrees of freedom with no stiffness based on a geometric or pseudo stiffness criterion. The surface is planar and the degree of freedom removed if the minimum principal pseudo stiffness is less than the flatness parameter. This is the recommended option.
  • Stiffness based automatic constraints, Zero stiffness value – This approach removes the degrees of freedom with no stiffness based on the directions of principal stiffness at the nodes. The degree of freedom surface is removed if the minimum principal stiffness is less than the zero stiffness value.
  • Artificial zz stiffness in shells – This approach retains the degrees of freedom but assigned a small artificial stiffness to shells in the zz direction. This option should be used with caution.

When tied interfaces are used with shell elements with five degrees of freedom (no drilling degree of freedom) there can be issued in setting up the constraint equations used by the tied interface. Assigning a full set of degrees of freedom avoids this problem.

Geometry Checks

Normally when the element geometry is poor the results are likely to be less reliable. The default behaviour of Treat geometry check failures as Errors will cause the analysis to abort when poor geometry is found. Selecting Severe warnings allows the analysis to continue despite the presence of badly shaped elements.

This option affects geometry checks on

  • large internal angles in elements
  • warping of flat shell elements

Considerations when using the Allman-Cook formulation​

The Bilinear formulation associates no stiffness with the local ‘zz’ degrees of freedom in linear 2D elements. In this case, these rotational ‘drilling’ degrees of freedom are not activated in the solution (or are suppressed) such that any topological connection to these local freedoms go unrestrained in the solution.

Choosing the Allman-Cook formulation activates the local ‘zz’ degrees of freedom through refinements to the stiffness formulation that connect the local rotational ‘zz’ freedoms to the in-plane translational stiffness of the element. The results provides a means to connect a moment applied in the ‘zz’ direction to translational strain of the element. While this has the clear advantage in practical problems, consideration must be made to allow for a suitably stable connection. Specifically, ‘zz’ moment connections to 2D elements made over a single node are not recommended as these can cause unwanted local deformations around the individual node and possible imbalances in load transfer. A warning from the solver is provided in this case. Either use additional in-plane beams to distribute the load connection over a local finite area or use a rigid constraint to achieve a similar effect. As an example, if tying a column to a slab, the area under the column itself may be a suitable candidate area to use for the connection.