| 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:
| kf = foundation stiffness (input as EFS on R command) |
| A = area of element |
| {Nz} = vector of shape functions representing motions normal to the surface |
The mass matrix is:
 | (13–255) |
where:
| th = thickness (input as TKI, TKJ, TKK, TKL on RMORE command) |
| ρ = density (input as DENS on MP command) |
| {N} = vector of shape functions |
| Ad = added mass per unit area (input as ADMSUA on R command) |
If the command LUMPM,ON is used, [Me] 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:
| [Sg] = 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:
 |
| {Np} = vector of shape functions representing in-plane motions normal to the edge |
| E = edge of element |
 |
|
|
|
| {Nx} = vector of shape functions representing motion in element x direction |
| {Ny} = vector of shape functions representing motion in element y direction |
|
|
|
|
| Pv = uniform pressure magnitude |
|
| P1 = input (VAL1 with LKEY = 5 on SFE command) |
| θ = angle between element normal and applied load direction |
|
|
|
 |
| Dx, Dy, Dz = vector directions (input as VAL2 thru VAL4 with LKEY = 5 on SFE command) |
| {NX}, {NY}, {NZ} = 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.