Corotational02

Corotational02#

The corotational coordinate transformation allows small-strain frame elements to be employed in a large deformation analysis. [1] [2] Corotational02 superceeds the original Corotational transformation, which is now deprecated.

Model.geomTransf("Corotational02", tag, vecxz[, offi, offj])

Define a corotational geometric transformation for frame elements.

Parameters:

tag: integer: integer tag identifying transformation
vecxz: tuple of floats: X, Y, and Z components of vecxz, the vector used to define the local x-z plane of the local-coordinate system, required in 3D. The local y-axis is defined by taking the cross product of the vecxz vector and the x-axis.
offi: tuple of floats: joint offset values – offsets specified with respect to the global coordinate system for element-end node i (optional, the number of arguments depends on the dimensions of the current model).
offj: tuple of floats: joint offset values – offsets specified with respect to the global coordinate system for element-end node j (optional, the number of arguments depends on the dimensions of the current model).

geomTransf Corotational02 $tag < $vecxzX $vecxzY $vecxzZ > <-jntOffset $dXi $dYi $dZi $dXj $dYj $dZj>

Argument	Type	Description
$tag	integer	integer tag identifying transformation
$vecxzX $vecxzY $vecxzZ	float	X, Y, and Z components of vecxz, the vector used to define the local x-z plane of the local-coordinate system. The local y-axis is defined by taking the cross product of the vecxz vector and the x-axis. These components are specified in the global-coordinate system X,Y,Z and define a vector that is in a plane parallel to the x-z plane of the local-coordinate system. These items need to be specified for the three-dimensional problem.
$dXi $dYi $dZi	float	joint offset values – offsets specified with respect to the global coordinate system for element-end node i (optional, the number of arguments depends on the dimensions of the current model).
$dXj $dYj $dZj	float	joint offset values – offsets specified with respect to the global coordinate system for element-end node j (optional, the number of arguments depends on the dimensions of the current model).

Note

The element coordinate system and joint offsets are the same as that documented for the Linear transformation.

Examples#

Orienting Frames

../../../../../examples/frames/frame-0059.html

Theory#

../../../../../_images/directors1.png — Corotational transformation of a two-node frame element.#

Under a corotational transformation, an element’s state determination is performed in a transformed configuration space represented by director fields $\left\{\bar{\mathbf{d}}_k\right\}$, and $\left\{\bar{\mathbf{D}}_k\right\}$ with the expressions:

\[\begin{split}\left.\begin{aligned} \mathbf{d}_k &\triangleq \boldsymbol{\Lambda}\mathbf{D}_k \\ \bar{\mathbf{d}}_k &\triangleq \boldsymbol{R}\mathbf{D}_k \\ \bar{\mathbf{D}}_k &\triangleq \bar{\boldsymbol{\Lambda}}\mathbf{D}_k \\ \end{aligned}\right., \quad\text{ implying }\qquad \begin{aligned} \boldsymbol{\Lambda} &= \mathbf{d}_k\otimes\mathbf{D}_k \\ \boldsymbol{R} &= \bar{\mathbf{d}}_k\otimes\mathbf{D}_k \\ \bar{\boldsymbol{\Lambda}} &= \bar{\mathbf{D}}_k\otimes\mathbf{D}_k \\ \end{aligned}\end{split}\]

Note

It is more appropriate to think of the corotational transformation as a family of transformations.

References#

Code Developed by: Remo Magalhaes de Souza, Claudio M. Perez