ExactFrame

Fig. 1: Torsional instability captured by the ExactFrame formulation and rendered with veux.

The ExactFrame element is a geometrically exact beam element based on Simo’s parameterization [1] of Antman’s Special Cosserat Rod [2]. This element is formulated without any geometric simplifications, and consequently, is capable of modeling highly nonlinear geometric phenomena with extreme accuracy. The implementation closely follows the treatment by Perez and Filippou (2024) [3].

Python (RT)Tcl

Model.element("ExactFrame", tag, nodes, section, transform)

Parameters:

• tag (integer) – unique Element tag

• nodes (tuple) – tuple of integer node tags (see Node)

• section (integer) – section tag (see Section)

• transform (integer) – identifier for previously-defined coordinate-transformation (see Transformations)

The valid eleResponse queries to this element are "force".

Geometrically exact elements often exhibit undesirable features. These are rectified as follows:

• Path Dependence The geometrically exact Cosserat rod theory is posed over a non-vectorial configuration space, and consequently may exhibit minor path-dependence.

• Rotation parameters for post-processing purposes, rotations should be obtained using the nodeRotation. Recall that in a finite rotation analysis, “rotational” components of the nodal displacement vector (ie, the vector returned by nodeDisp) is meaningless.

The ExactFrame formulation is appropriate for a wide variety of structural members including thin-walled sections, asymmetric sections, and inelasticity.

As of xara version 0.1.15, the ExactFrame element can be used to model cross-sectional warping through an additional seventh degree of freedom. This feature is implemented through template metaprogramming and consequently incurs absolutely no overhead on standard six-degree-of-freedom simulations.

Note

This element always employs a Gauss-Legendre quadrature of order nen-1 for an element with nen nodes, and does not accept user-defined quadrature schemes.

Example

The following example demonstrates the command to create an ExactFrame element.

TclPython

element ExactFrame 1 1 2 -section 1 -transform 1

References

[1]

Simo, J.C. (1985) ‘A finite strain beam formulation. The three-dimensional dynamic problem. Part I’, Computer Methods in Applied Mechanics and Engineering, 49(1), pp. 55–70. Available at: https://doi.org/10.1016/0045-7825(85)90050-7.

[2]

Antman, S.S. (2005) Nonlinear problems of elasticity. 2nd ed. New York: Springer (Applied mathematical sciences, v. 107).

[3]

Perez, C.M. and Filippou, F.C. (2024) ‘On nonlinear geometric transformations of finite elements’, International Journal for Numerical Methods in Engineering, p. e7506. Available at: https://doi.org/10.1002/nme.7506.

Code developed by: Claudio M. Perez