PIPE59 is similar to PIPE16 (or LINK8 if the cable option (KEYOPT(1) = 1) is selected). 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

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:

where Z(N) is the vertical location of node N. If it is desired to generate a structure below the mud line, the user can set up a second material property for those elements using a greater d and deleting hydrodynamic effects. Alternatively, the user can use a second element type such as PIPE16, the elastic straight pipe element.

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

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 BEAM4 ((Equation 14–10)), 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 stiffness matrix for the cable option (KEYOPT(1) = 1) is the same as for LINK8.

The element mass matrix for the pipe option (KEYOPT(1) ≠ 1) and KEYOPT(2) = 0) is the same as for BEAM4 ((Equation 14–11)), 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 mass matrix for the cable option (KEYOPT(1) = 1) or the reduced mass matrix option (KEYOPT(2) ≠ 0) is the same form as for LINK8 except that (1,1), (4,4), (1,4) and (4,1) are also multiplied by the factor (M_a/M_t).

The element load vector consists of two parts:

The hydrostatic and hydrodynamic effects work with the original diameter and length, i.e., 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:

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

The crushing pressure at a node is:

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:

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:

where:

The axial force correction term (F_x) is computed as

where:

The axial stress is:

and using the Lamé stress distribution,

where:

= hydrodynamic pressure, described below

D = diameter being studied

P_i and P_o are taken as average values along each element. Combining (Equation 14–397) thru (Equation 14–400).

Note:

Note that if the cross-section is solid (D_i = 0.), (Equation 14–399) reduces to:

Hydrodynamic effects may occur because the structure moves in a motionless fluid, the structure is fixed but there is fluid motion, or both the structure and fluid are moving. The fluid motion consists of two parts: current and wave motions. The current is input by giving the current velocity and direction (input as W(i) and θ(i)) at up to eight different vertical stations (input as Z(i)). (All input quantities referred to in this section not otherwise identified comes from the TBDATA commands used with TB,WATER). The velocity and direction are interpolated linearly between stations. The current is assumed to flow horizontally only. The wave may be input using one of four wave theories in Table 14.5: "Wave Theory Table" (input as KWAVE with TB,WATER).

The free surface of the wave is defined by

where:

ηs = total wave height

Kw = wave theory key (input as KWAVE with TB,WATER)

ηi = wave height of component i

R = radial distance to point on element from origin in the X-Y plane in the direction of the wave

t = time elapsed (input as TIME on TIME command) (Note that the default value of TIME is usually not desired. If zero is desired, 10-12 can be used).

φi = phase shift = input as φ(i)

If λ_i is not input (set to zero) and K_w < 2, λ_i is computed iteratively from:

where:

λi = output quantity small amplitude wave length

g = acceleration due to gravity (Z direction) (input on ACEL command)

d = water depth (input as DEPTH with TB,WATER)

Each component of wave height is checked that it satisfies the “Miche criterion” if Kw ≠3. This is to ensure that the wave is not a breaking wave, which the included wave theories do not cover. A breaking wave is one that spills over its crest, normally in shallow water. A warning message is issued if:

where:

When using wave loading, there is an error check to ensure that the input acceleration does not change after the first load step, as this would imply a change in the wave behavior between load steps.

For K_w = 0 or 1, the particle velocities at integration points are computed as a function of depth from:

where:

= radial particle velocity

= vertical particle velocity

ki = 2π/λi

= height of integration point above the ocean floor = d+Z

= time derivative of ηi

= drift velocity (input as W with TB,WATER)

The particle accelerations are computed by differentiating and with respect to time. Thus:

where:

Expanding equation 2.29 of the Shore Protection Manual(43) for a multiple component wave, the wave hydrodynamic pressure is:

However, use of this equation leads to nonzero total pressure at the surface at the crest or trough of the wave. Thus, (Equation 14–410) is modified to be:

which does result in a total pressure of zero at all points of the free surface. This dynamic pressure, which is calculated at the integration points during the stiffness pass, is extrapolated to the nodes for the stress pass. The hydrodynamic pressure for Stokes fifth order wave theory is:

Other aspects of the Stokes fifth order wave theory are discussed by Skjelbreia et al. (31). The modification as suggested by Nishimura et al.(143) has been included. The stream function wave theory is described by Dean(59).

If both waves and current are present, the question of wave-current interaction must be dealt with. Three options are made available thru K_cr (input as KCRC with TB,WATER):

For K_cr = 0, the current velocity at all points above the mean sea level is simply set equal to W_o, where W_o is the input current velocity at Z = 0.0. All points below the mean sea level have velocities selected as though there were no wave.

For K_cr = 1, the current velocity profile is “stretched” or “compressed” to fit the wave. In equation form, the Z coordinate location of current measurement is adjusted by

where:

For K_cr = 2, the same adjustment as for K_cr = 1 is used, as well as a second change that accounts for “continuity.” That is,

where:

These three options are shown pictorially in Figure 14.38: "Velocity Profiles for Wave-Current Interactions".

To compute the relative velocities ( ,), both the fluid particle velocity and the structure velocity must be available so that one can be subtracted from the other. The fluid particle velocity is computed using relationships such as (Equation 14–406) and (Equation 14–407) as well as current effects. The structure velocity is available through the Newmark time integration logic (see Transient Analysis).

Finally, a generalized Morison's equation is used to compute a distributed load on the element to account for the hydrodynamic effects:

where:

{F/L}d = vector of loads per unit length due to hydrodynamic effects

CD = coefficient of normal drag (see below)

ρw = water density (mass/length3) (input as DENSW on TB,WATER)

De = outside diameter of the pipe with insulation (length)

= normal relative particle velocity vector (length/time)

CM = coefficient of inertia (input as CM on R command)

= normal particle acceleration vector (length/time2)

CT = coefficient of tangential drag (see below)

= tangential relative particle velocity vector (length/time)

Two integration points along the length of the element are used to generate the load vector. Integration points below the mud line are simply bypassed. For elements intersecting the free surface, the integration points are distributed along the wet length only. If the reduced load vector option is requested (KEYOPT(2) = 2), the moment terms are set equal to zero.

The coefficients of drag (C_D,C_T) may be defined in one of two ways:

The dependency on Reynolds number (Re) may be expressed as:

where:

fD = functional relationship (input on the water motion table as RE and CD with TB,WATER)

μ = viscosity (input as VISC on MP command)

and

where:

fT = functional relationship (input on the water motion table as RE and CT with TB,WATER)

Temperature-dependent quantity may be input as μ, where the temperatures used are those given by input quantities T(i) of the water motion table.

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 (PIPE16 - Elastic Straight Pipe). The average axial stress is:

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:

where:

(Equation 14–419) is a specialization of (Equation 14–399). 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.

The stress output for the cable format (KEYOPT(1) = 1 with D_i = 0.0) is similar to that for LINK8 (LINK8 - 3-D Spar (or Truss)), except that the stress is given with and without the external pressure applied:

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)