Matrix or Vector | Geometry / Midside Nodes [1] | Shape Functions | Integration Points |
---|---|---|---|
Stiffness and Damping Matrices, and Pressure Load Vector | Quad with midside nodes | Equation 11–86 | 3 x 3 |
Quad without midside nodes | Equation 11–71 | 2 x 2 | |
Triangle with midside nodes | Equation 11–114 | 6 | |
Triangle without midside nodes | Equation 11–68 | 3 | |
Mass and Stress Stiffness Matrices | Quad with midside nodes | Equation 11–84, Equation 11–85 and Equation 11–86 | 3 x 3 |
Quad without midside nodes | Equation 11–69, Equation 11–70 and Equation 11–71 | 2 x 2 | |
Triangle with midside nodes | Equation 11–114 | 6 | |
Triangle without midside nodes | Equation 11–66, Equation 11–67 and Equation 11–68 | 3 | |
Surface Tension Load Vector | Quad with midside nodes | Equation 11–84 and Equation 11–85 | 3 x 3 |
Quad without midside nodes | Equation 11–69 and Equation 11–70 | 2 x 2 | |
Triangle with midside nodes | Equation 11–112 and Equation 11–113 | 6 | |
Triangle without midside nodes | Equation 11–66 and Equation 11–67 | 3 |
Load Type | Distribution |
---|---|
All Loads | Same as shape functions |
The stiffness matrix is:
(13–254) |
where:
k^{f} = foundation stiffness (input as EFS on R command) |
A = area of element |
{N_{z}} = vector of shape functions representing motions normal to the surface |
The mass matrix is:
(13–255) |
where:
t_{h} = thickness (input as TKI, TKJ, TKK, TKL on RMORE command) |
ρ = density (input as DENS on MP command) |
{N} = vector of shape functions |
A_{d} = added mass per unit area (input as ADMSUA on R command) |
If the command LUMPM,ON is used, [M_{e}] is diagonalized as described in Lumped Matrices.
The element damping matrix is:
(13–256) |
where:
μ = dissipation (input as VISC on MP command) |
The element stress stiffness matrix is:
(13–257) |
where:
[S_{g}] = derivatives of shape functions of normal motions |
s = in-plane force per unit length (input as SURT on R command) |
If pressure is applied to face 1, the pressure load stiffness matrix is computed as described in Pressure Load Stiffness.
The element load vector is:
(13–258) |
where:
{N_{p}} = vector of shape functions representing in-plane motions normal to the edge |
E = edge of element |
{N_{x}} = vector of shape functions representing motion in element x direction |
{N_{y}} = vector of shape functions representing motion in element y direction |
P_{v} = uniform pressure magnitude |
P_{1} = input (VAL1 with LKEY = 5 on SFE command) |
θ = angle between element normal and applied load direction |
D_{x}, D_{y}, D_{z} = vector directions (input as VAL2 thru VAL4 with LKEY = 5 on SFE command) |
{N_{X}}, {N_{Y}}, {N_{Z}} = vectors of shape functions in global Cartesian coordinates |
The integration used to arrive at is the usual numerical integration, even if KEYOPT(6) ≠ 0. The output quantities “average face pressures” are the average of the pressure values at the integration points.