PIPE59 is similar to PIPE16. The principal differences are that the mass matrix includes the:

The origin for any problem containing PIPE59 must be at the free surface (mean sea level). Further, the Z axis is always the vertical axis, pointing away from the center of the earth.

The element may be located in the fluid, above the fluid, or in both regimes simultaneously. There is a tolerance of only below the mud line, for which

(1–64)

where:

The mud line is located at distance d below the origin (input as DEPTH with TB,WATER (water motion table)). This condition is checked with:

(1–65)

(1–66)

where Z(N) is the vertical location of node N. If it is desired to generate a structure below the mud line, one can set up a second material property for those elements using a greater d and deleting hydrodynamic effects. Alternatively, a second element type such as PIPE288 can be used.

If the problem is a large deflection problem, greater tolerances apply for second and subsequent iterations:

(1–67)

(1–68)

(1–69)

where Z(N) is the present vertical location of node N. In other words, the element is allowed to sink into the mud for 10 diameters before generating a warning message. If a node sinks into the mud a distance equal to the water depth, the run is terminated. If the element is supposed to lie on the ocean floor, gap elements must be provided.

The element stiffness matrix for the pipe option (KEYOPT(1) ≠ 1) is the same as for a 3D elastic beam, except that:

where:

GT = twist-tension stiffness constant, which is a function of the helical winding of the armoring (input as TWISTEN on RMORE command, may be negative)

Di = inside diameter of pipe = Do - 2 tw

tw = wall thickness (input as TWALL on R command)

L = element length

J = 2I

The element mass matrix for the pipe option (KEYOPT(1) ≠ 1) and KEYOPT(2) = 0) is the same as for a 3D elastic beam, except that (1,1), (7,7), (1,7), and (7,1), as well as M(4,4), M(10,10), M(4,10), and M(10,4), are multiplied by the factor (M_a /M_t).

where:

Mt = (mw + mint + mins + madd) L = mass/unit length for motion normal to axis of element

Ma = (mw + mint + mins) L= mass/unit length for motion parallel to axis of element

ρ = density of the pipe wall (input as DENS on MP command)

εin = initial strain (input as ISTR on RMORE command)

mint = mass/unit length of the internal fluid and additional hardware (input as CENMPL on RMORE command)

ρi = density of external insulation (input as DENSIN on RMORE command)

CI = coefficient of added mass of the external fluid (input as CI on RMORE command)

ρw = fluid density (input as DENSW with TB,WATER)

The element load vector consists of two parts:

The hydrostatic and hydrodynamic effects work with the original diameter and length, that is, initial strain and large deflection effects are not considered.

Hydrostatic effects may affect the outside and the inside of the pipe. Pressure on the outside crushes the pipe and buoyant forces on the outside tend to raise the pipe to the water surface. Pressure on the inside tends to stabilize the pipe cross-section.

The buoyant force for a totally submerged element acting in the positive z direction is:

(1–70)

Also, an adjustment for the added mass term is made.

The crushing pressure at a node is:

(1–71)

where:

= crushing pressure due to hydrostatic effects

g = acceleration due to gravity

z = vertical coordinate of the node

= input external pressure (input on SFE command)

The internal (bursting) pressure is:

(1–72)

where:

Pi = internal pressure

ρo = internal fluid density (input as DENSO on R command)

Sfo = z coordinate of free surface of fluid (input as FSO on R command)

= input internal pressure (input as SFE command)

To ensure that the problem is physically possible as input, a check is made at the element midpoint to see if the cross-section collapses under the hydrostatic effects. The cross-section is assumed to be unstable if:

(1–73)

where:

The axial force correction term (F_x) is computed as

(1–74)

(1–75)

where:

The axial stress is:

(1–76)

and using the Lamé stress distribution,

(1–77)

(1–78)

where:

= hydrodynamic pressure, described below

D = diameter being studied

P_i and P_o are taken as average values along each element. Combining Equation 1–75 thru Equation 1–78.

(1–79)

Note:

Note that if the cross-section is solid (D_i = 0.), Equation 1–77 reduces to:

(1–80)

See Hydrodynamic Loads in the Element Tools section of this document for information about this subject.

The below two equations are specialized either to end I or to end J.

The stress output for the pipe format (KEYOPT(1) ≠ 1), is similar to PIPE16. The average axial stress is:

(1–81)

where:

σx = average axial stress (output as SAXL)

Fn = axial element reaction force (output as FX, adjusted for sign)

Pi = internal pressure (output as the first term of ELEMENT PRESSURES)

Po = external pressure = (output as the fifth term of the ELEMENT PRESSURES)

and the hoop stress is:

(1–82)

where:

Equation 1–82 is a specialization of Equation 1–77. The outside surface is chosen as the bending stresses usually dominate over pressure induced stresses.

All stress results are given at the nodes of the element. However, the hydrodynamic pressure had been computed only at the two integration points. These two values are then used to compute hydrodynamic pressures at the two nodes of the element by extrapolation.

For the stress output for the cable format (KEYOPT(1) = 1 with D_i = 0.0), the stress is given with and without the external pressure applied:

(1–83)

(1–84)

(1–85)

where:

σxI = axial stress (output as SAXL)

σeI = equivalent stress (output as SEQV)

= axial force on node (output as FX)

Fa = axial force in the element (output as FAXL)