Matrix or Vector | Option | Shape Functions[1] | Integration Points |
---|---|---|---|
Stiffness and Damping Matrices | Longitudinal | Equation 11–6 | None |
Torsional | Equation 11–18 | None | |
Stress Stiffening Matrix | Longitudinal | Equation 11–7, and Equation 11–8 | None |
COMBIN14 essentially offers two types of elements, selected with KEYOPT(2).
Single DOF per node (KEYOPT(2) > 0). The orientation is defined by the value of KEYOPT(2) and the two nodes are usually coincident.
Multiple DOFs per node (KEYOPT(2) = 0). The orientation is defined by the location of the two nodes; therefore, the two nodes must not be coincident.
Consider the case of a single DOF per node first. The orientation is selected with KEYOPT(2). If KEYOPT(2) = 7 (pressure) or = 8 (temperature), the concept of orientation does not apply. The form of the element stiffness and damping matrices are:
(13–9) |
(13–10) |
where:
= stiffness (input as K on R command) |
= constant damping coefficient (input as CV1 on R command) |
= linear damping coefficient (input as CV2 on R command) |
= relative velocity between nodes computed from the nodal Newmark velocities |
In full harmonic, full transient, and static analyses, when the stiffness and damping real constants are input as table parameters, the stiffness () and damping () coefficients are interpolated at each frequency or time step.
Next, consider the case of multiple DOFs per node. Only the case with three DOFs per node will be discussed, as the case with two DOFs per node is simply a subset. The stiffness, damping, and stress stiffness matrices in element coordinates are developed as:
(13–11) |
(13–12) |
(13–13) |
where subscript refers to element coordinates.
and where:
= force in element from previous iteration |
= distance between the two nodes |
There are some special notes that apply to the torsion case (KEYOPT(3) = 1):
Rotations are simply treated as a vector quantity. No other effects (including displacements) are implied.
In a large rotation problem (NLGEOM,ON), the coordinates do not get updated, as the nodes only rotate. (They may translate on other elements, but this does not affect COMBIN14 with KEYOPT(3) = 1). Therefore, there are no large rotation effects.
Similarly, as there is no axial force computed, no stress stiffness matrix is computed.
The stretch is computed as:
(13–14) |
where:
= coordinates in global Cartesian coordinates |
= displacements in global Cartesian coordinates |
= displacements in nodal Cartesian coordinates (UX, UY, UZ) |
= rotations in nodal Cartesian coordinates (ROTX, ROTY, ROTZ) |
= pressure (PRES) |
= temperature (TEMP) |
If KEYOPT(3) = 1 (torsion), the expression for has rotation instead of translations, and is output as TWIST. Next, the static force (or torque) is computed:
(13–15) |
where:
= static force (or torque) (output as FORC (TORQ if KEYOPT(3) = 1)) |
Finally, if a nonlinear transient dynamic (ANTYPE,TRANS, with TIMINT,ON) analysis is performed, a damping force is computed:
(13–16) |
where:
= damping force (or torque) (output as DAMPING FORCE (DAMPING TORQUE if KEYOPT(3) = 1)) |
= relative velocity |
relative velocity is computed using Equation 13–14, where the nodal displacements , etc. are replaced with the nodal Newmark velocities , etc.