Linear Transformation

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This command is used to construct a linear coordinate transformation (LinearCrdTransf) object, which performs a linear geometric transformation of beam stiffness and resisting force from the basic system to the global-coordinate system.

For a two-dimensional problem:

geomTransf Linear $transfTag <-jntOffset $dXi $dYi $dXj $dYj>

For a three-dimensional problem:

geomTransf Linear $transfTag $vecxzX $vecxzY $vecxzZ <-jntOffset $dXi $dYi $dZi $dXj $dYj $dZj>

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$transfTag	integer tag identifying transformation
$vecxzX $vecxzY $vecxzZ	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	joint offset values -- offsets specified with respect to the global coordinate system for element-end node i (the number of arguments depends on the dimensions of the current model). The offset vector is oriented from node i to node j as shown in a figure below. (optional)
$dXj $dYj $dZj	joint offset values -- offsets specified with respect to the global coordinate system for element-end node j (the number of arguments depends on the dimensions of the current model). The offset vector is oriented from node j to node i as shown in a figure below. (optional)

A refresher on Euclidean Geometry and Coordinate Systems:

A single vector may be defined by two points. It has length, direction, and location in space. When this vector is used to define a coordinate axis, only its direction is important. Now any 2 vectors, Vr and Vs, not parallel, define a plane that is parallel to them both. The cross-product of these vectors define a third vector, Vt, that is perpendicular to both Vr and Vs and hence normal to the plane: Vt = Vr X Vs.

The element coordinate system is specified as follows:

The x-axis is a vector given by the two element nodes; The vector vecxz is a vector the user specifies that must not be parallel to the x-axis. The x-axis along with the vecxz Vector define the xz plane. The local y-axis is defined by taking the cross product of the x-axis vector and the vecxz vector (Vy = Vx X Vxz). The local z-axis is then found simply by taking the cross product of the y-axis and x-axis vectors (Vz = Vy X Vx). The section is attached to the element such that the y-z coordinate system used to specify the section corresponds to the y-z axes of the element.

NOTE: When in 2D, local x and y axes are in the X-Y plane, where X and Y are global axes. Local x axis is the axis connecting the two element nodes, and local y and z axes follow the right-hand rule (e.g., if the element is aligned with the positive Y axis, the local y axis is aligned with the negative X axis, and if the element is aligned with the positive X axis, the local y axis is aligned with the positive Y axis). Orientation of local y and z axes is important for definition of the fiber section.

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EXAMPLE:

1. Element 1 : tag 1 : vecxZ = zaxis

geomTransf Linear 1 0 0 -1

1. Element 2 : tag 2 : vecxZ = y axis

geomTransf Linear 2 0 1 0

1. If there was a rigid offset at the top of element 1:

geomTransf Linear 1 0 0 -1 -jntOffset 0.0 0.0 0.0 0.0 -$Offset 0.0

Code Developed by: Remo Magalhaes de Souza

Images Developed by: Silvia Mazzoni