11.1. Understanding Shape Function Labels

The given functions are related to the nodal quantities by:

Table 11.1:  Shape Function Labels

VariableInput/Output LabelMeaning
uUXTranslation in the x (or s) direction
vUYTranslation in the y (or t) direction
wUZTranslation in the x (or r) direction
θx ROTXRotation about the x direction
θy ROTYRotation about the y direction
θz ROTZRotation about the z direction
Ax AXX-component of vector magnetic potential
Ay AYY-component of vector magnetic potential
Az AZZ-component of vector magnetic potential
CCONCConcentration
PPRESPressure
TTEMP, TBOT, TE2, ... TTOPTemperature
VVOLTElectric potential or source current
φMAGScalar magnetic potential

The vector correspondences are not exact, since, for example, u, v, and w are in the element coordinate system, whereas UX, UY, UZ represent motions in the nodal coordinate system. Generally, the element coordinate system is the same as the global Cartesian system, except for:

1. Line elements (2-D Lines to Axisymmetric Harmonic Shells and General Axisymmetric Surfaces), where u motions are axial motions, and v and w are transverse motions.

2. Shell elements (3-D Shells), where u and v are in-plane motions and w is the out-of-plane motion.

Subscripted variables such as uJ refer to the u motion at node J. When these same variables have numbers for subscripts (e.g. u1), nodeless variables for extra shape functions are being referred to. Coordinates s, t, and r are normalized, going from -1.0 on one side of the element to +1.0 on the other, and are not necessarily orthogonal to one another. L1, L2, L3, and L4 are also normalized coordinates, going from 0.0 at a vertex to 1.0 at the opposite side or face.

Elements with midside nodes allow those midside nodes to be dropped in most cases. A dropped midside node implies that the edge is and remains straight, and that any other effects vary linearly along that edge.

Gaps are left in the equation numbering to allow for additions. Labels given in subsection titles within parentheses are used to relate the given shape functions to their popular names, where applicable.

Some elements in Element Library (notably the 8-node solids) imply that reduced element geometries (e.g., wedge) are not available. However, the tables in Element Library refer only to the available shape functions. In other words, the shape functions used for the 8-node brick is the same as the 6-node wedge.