## 13.154. SURF154 - 3-D Structural Surface Effect Matrix or Vector Geometry / Midside Nodes  Shape Functions Integration Points
Stiffness and Damping Matrices, and Pressure Load VectorQuad 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 MatricesQuad 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
1. Midside node setting is controlled by KEYOPT(4).

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. (13–258) {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.