First contact
only Start of
first uplift only
Second or subsequent
contact
Start repenetrating
when still in contact
Start of second or subsequent
uplift Break
contact
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where
su(z) = undrained shear strength at penetration z. This is given by su(z) = su0 + ρz, where su0 is the undrained shear strength at the mudline and ρ is the undrained shear strength gradient, both of which are specified in the Seabed Soil Properties data.
D = penetrator contact diameter. For 3D Buoys and 6D Buoys the contact diameter is taken to be the square root of the contact area (see 3D Buoy contact area and 6D Buoy Theory). For Lines the contact diameter is as specified in the Line Type Contact Data.
Nc(z/D) = bearing factor. For z/D ≥ 0.1 this is modelled using the power law formula Nc(z/D) = a(z/D)b, where a and b are the non-dimensional Penetration Resistance Parameters of the model, as specified in the Soil Model Parameters. For z/D < 0.1 the formula Nc = Nc(0.1)(10z/D)½ is used instead, which gives a good approximation to the theoretical bearing factor for shallow penetration.
fsuc = non-dimensional Suction Resistance Ratio parameter of the model, as specified in the Soil Model Parameters.
Penetration Resistance Formulae
In Not In Contact mode the penetration resistance P(z) is zero.
In the other three modes the resistance P(z) is modelled using formulae that involve the following variables:
ζ = z / (D/Kmax). This is the penetration, but non-dimensionalised to be in units of D/Kmax, where Kmax is the Normalised Maximum Stiffness parameter of the model, as specified in the Soil Model Parameters.
z0 = penetration z at which the latest episode of this contact mode started, i.e. the value at the time the latest transition into this contact mode occurred.
ζ0 = z0 / (D/Kmax) = non-dimensionalised penetration at which the latest episode of this contact mode started.
P0 = resistance P(z) at which the latest episode of this contact mode started.
Initial Penetration Mode
For Initial Penetration mode the starting penetration and resistance values, z0 and P0, are both zero. The penetration resistance is then given by
P(z) = HIP(ζ).Pu(z) (1)
where
HIP(ζ) = ζ / [1 + ζ]
The term HIP(ζ) is a hyperbolic factor that equals 0 when ζ = 0 when initial penetration starts, equals ½ when ζ = 1, i.e. when z = D/Kmax, and asymptotically approaches 1 as penetration gets large compared to D/Kmax. The purpose of this factor is to provide a high initial stiffness while ensuring that the penetration resistance P(z) rises smoothly from zero when contact first starts (when ζ and z are both 0) and asymptotically approaches the ultimate penetration resistance, Pu(z), if ζ gets large (i.e. if z gets large compared to D/Kmax). This is illustrated by the blue curve in the model characteristics diagram below, which approaches the ultimate penetration resistance limit (upper grey curve) as penetration gets large compared to D/Kmax.
Uplift Mode
For Uplift mode the penetration resistance is given by
P(z) = P0 - HUL(ζ0 - ζ)(P0 - Pu-suc(z)) (2a)
but subject to a suction limit – see below. Here:
HUL(ζ0 - ζ) = ( ζ0 - ζ ) / [ AUL(z) + (ζ0 - ζ) ]
AUL(z) = [P0 - Pu-suc(z)] / Pu(z0)
The term HUL(ζ0 - ζ) is a hyperbolic factor that equals 0 when ζ = ζ0 at the start of this uplift, and asymptotically approaches 1 if the non-dimensional uplift (ζ0 - ζ) gets large compared to AUL(z). So in uplift mode the resistance given by equation (2a) drops from its value P0 when this uplift started, and asymptotically approaches the (negative) ultimate suction resistance Pu-suc(z) if the non-dimensional uplift (ζ0 - ζ) gets large compared to AUL(z).
See the green curve in the model characteristics diagram below.
Suction Limit
Experiments (Bridge et al) have found that suction resistance can only be sustained for a limited displacement past the point where the net resistance becomes negative, and suction then decays as uplift continues. To model this the
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Theory, Environment Theory resistance given by equation (2a) is limited to be no less than (i.e. no more suction than) a negative lower bound Pmin(z), given by:Pmin(z) = EUL(z)Pu-suc(z) (2b)
where
EUL(z) = exp[Min{0, (z - zP=0) / (λsuc zmax)}]
zmax = largest ever penetration z for this penetrator
zP=0 = largest penetration z at which suction has started during any uplift
λsuc = non-dimensional Normalised Suction Decay Distance parameter of the model, as specified in the Soil Model Parameters.
The exponent in the expression for EUL(z) is zero or negative, so EUL(z) ≤ 1.EUL(z) equals 1 when z ≥ zP=0, but decays towards zero if the penetration z is less than the largest penetration, zP=0, at which suction has ever occurred during uplift. The effect of this is that the term Pmin(z) limits suction to be no more than Pu-suc(z) when the first uplift starts, but as the penetrator lifts up higher (relative to the maximum penetration at which suction has ever occurred during uplift) then the suction is limited more. This models the suction decay effect that experimental evidence has found.
Repenetration Mode
For Repenetration mode the penetration resistance is given by
P(z) = P0 + HRP(ζ - ζ0)(Pu(z) - P0) (3a)
but subject to a repenetration resistance upper bound – see below. Here
ζ0 and P0 = non-dimensional penetration and resistance at the start of this repenetration
HRP(ζ - ζ0) = (ζ - ζ0) / [ ARP(z) + (ζ - ζ0) ]
ARP(z) = ( Pu(z)-P0 ) / Pu*
Pu* = Pu(z) if P0 ≤ 0, i.e. if this repenetration started from a zero or negative resistance
Pu* = Pu(z*) if P0 > 0, where z* is the penetration when the preceding episode of uplift started
The term HRP(ζ - ζ0) in equation (3a) is a hyperbolic factor that equals 0 when ζ = ζ0 at the start of this repenetration, and asymptotically approaches 1 if the non-dimensional repenetration (ζ - ζ0) gets large compared to ARP(z). So the repenetration mode resistance given by equation (3a) rises from its value P0 when this repenetration starts, and asymptotically approaches the ultimate penetration resistance Pu(z) if the non-dimensional repenetration (ζ - ζ0) gets large compared to ARP(z). See the purple curve in the model characteristics diagram below.
Repenetration Resistance Reduction After Uplift
Experiments (Bridge et al) have found that when repenetration occurs following large uplift movement the repenetration resistance is reduced until the previous maximum penetration is approached. To model this the repenetration resistance given by equation (3a) is limited to be no more than an upper limit Pmax(z) given by:
Pmax(z) = ERP(z)PIP(z) (3b)
where
PIP(z) = penetration resistance that initial penetration mode would give at this penetration, as given by equation (1)
ERP(z) = exp[min{0, - λrep + (z - zP=0) / (λsuc zmax)}]
zmax = largest ever penetration z for this penetrator
zP=0 = largest penetration z at which suction has started during any uplift
λsuc = non-dimensional Normalised Suction Decay Distance parameter of the model, as specified in the Soil Model Parameters.
λrep = non-dimensional Repenetration Offset After Uplift parameter of the model, as specified in the Soil Model Parameters.
The exponent in the expression for ERP(z) is zero or negative, so ERP(z) ≤ 1. The expression for ERP(z) gives a value <
1, and so limits the repenetration resistance to be less than the ultimate penetration resistance Pu(z), until the penetration z exceeds zP=0 by a certain amount, quantified by λrep. This models the effect that repenetration following large uplift movement shows reduced resistance until the previous maximum penetration is approached.
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Model Characteristics
The following diagram illustrates the effect of the above equations as penetration changes, for a catenary line moving up and down on the seabed.
Figure: Soil Model Characteristics
The model starts in Initial Penetration mode and gives a resistance (blue curve, see note (1) in diagram) that increases as the pipe sinks into the seabed, and asymptotically approaches the ultimate penetration resistance Pu
(upper dashed grey curve).
Then, when the pipe starts to lift up again the model enters Uplift mode and the resistance falls (green curve, see note (2) in diagram) and asymptotically approaches the ultimate suction resistance Pu-suc (lower dashed grey curve).
In this case the uplift is enough that the resistance becomes negative – i.e. suction (note (3) in diagram).
If the uplift continues and the pipe lifts off the seabed then the model stays in Uplift mode and the model follows the green curve further (note (4) in diagram). The suction reduces as the uplift continues, and drops to zero when the penetration drops to zero.
If, however, the uplift ends and repenetration starts, then the model enters Repenetration mode (purple curve, note (5) in diagram) and the suction rapidly falls and soon instead becomes +ve resistance. As repenetration continues increases the resistance rises (note (6) in diagram) and again asymptotically approaches the ultimate penetration resistance.
Further cycles of uplift and repenetration would give further episodes of Uplift and Repenetration modes and so give hysteresis loops of seabed resistance.
Soil Extra Buoyancy Force
The seabed resistance formulae given above model the resistance P(z) due to the soil shear strength. In addition to this there is an extra buoyancy force due to the fact that the penetrator displaces soil that has a higher saturated density than the water. To model this the following extra buoyancy force is applied, vertically upwards, in addition to the resistance P(z).
Extra Soil Buoyancy Force = fbVdisp(ρsoil - ρsea)g where
fb is a non-dimensional soil buoyancy factor, as specified on in Soil Model Parameters data.
(5) suction releases if repenetrates ..
5
Ultimate penetration resistance, Pu Normal
seabed reaction force
(3) Further uplift is resisted by suction
Non-dimensional Penetration, ζ (2) Uplift
(6) Further repenetration (1) Initial
penetration
(4) .. or if uplift continues O
-ve reaction
(ie suction) Ultimate suction resistance, Pu-suc
1 2 3 4 10 15 20 25
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Theory, Environment Theory Vdisp = displacement volume = volume of the part of the penetrating object that is below the seabed tangent plane.
ρsoil = saturated soil density, as specified in the Soil Properties data.
ρsea = sea water density at the seabed origin, as specified in the sea density data.
g = acceleration due to gravity for the units being used
The factor fb is normally greater than 1. This models the fact that when seabed soil is displaced it does not disperse thinly across the seabed plane, but instead tends to heave locally around the penetrating object. The effect of this is that the extra buoyancy is greater than the standard theoretical buoyancy force Vdisp(ρsoil - ρsea)g that would apply if the soil was fully fluid.
Soil Model Parameters
Several non-dimensional constants are used in the formulae given above for the seabed normal reaction force. These are parameters that control how the soil response is modelled by the non-linear soil model. Their values can be edited on the Seabed Soil Model page of the Environment data form, but we recommend that these parameters are normally left at their default values. The effects of the parameters are now described.
Penetration Resistance Parameters
The parameters a and b control how the bearing factor Nc(z), and hence the ultimate penetration and suction resistance limits Pu(z) and Pu-suc(z), vary with penetration z. See Ultimate Resistance Limits above.
Soil Buoyancy Factor
This is the factor fb that controls the modelling of the extra buoyancy effect that occurs when a penetrating object displaces soil. See Soil Extra Buoyancy Force above. The buoyancy factor, fb, should normally be greater than 1, to model the fact that the displaced soil tends to heave locally around the penetrating object.
Normalised Maximum Stiffness
This is the factor Kmax that determines the reference penetration, D/Kmax, that is used to calculate the non-dimensional penetration values, ζ and ζ0, that are used in the hyperbolic factors in the Penetration Resistance Formulae above. It therefore controls the maximum stiffness during initial penetration or on reversal of motion, and also how fast the penetration resistance asymptotically approaches its limiting value. A higher value means the resistance more rapidly approaches the limit as penetration changes, and so gives a stiffer seabed model.
Suction Resistance Ratio
This is the factor fsuc that controls the ultimate suction resistance Pu-suc(z). See Ultimate Resistance Limits above. A lower value gives less suction, a higher value gives more.
Normalised Suction Decay Distance
This is the factor λsuc that controls the suction decay limit term Pmin(z) in equation (2b) in Uplift mode. A lower value gives less suction effect, by causing suction to decay after less uplift. A higher value causes suction to persist over greater uplift distances. This parameter also affects the repenetration limit term Pmax(z) in equation (3b) in Repenetration mode.
Repenetration Offset After Uplift
This is the parameter λrep that controls the penetration at which the repenetration resistance limit Pmax(z) in equation (3b) in Repenetration mode merges with the bounding curve for initial penetration resistance, PIP(z). A smaller value results in less penetration past zP=0 before the repenetration resistance after uplift merges with the bounding curve of initial penetration resistance. A higher value leads to greater penetration before the bounding curve is reached.
5.10.5 Morison's Equation
OrcaFlex calculates hydrodynamic loads on lines, 3D Buoys and 6D Buoys using an extended form of Morison's Equation. See Morison, O'Brien, Johnson and Schaaf.
Morison's equation was originally formulated for calculating the wave loads on fixed vertical cylinders. There are two force components, one related to water particle acceleration, the inertia force, and one related to water particle velocity, the drag force. For moving objects, the same principle is applied, but the force equation is modified to take account of the movement of the body.
The extended form of Morison's equation used in OrcaFlex is:
Fw = (Δ.aw + Ca.Δ.ar) + ½.ρ.Cd.A.Vr|Vr| where
Theory, Environment Theory
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Δ is the mass of fluid displaced by the body aw is the fluid acceleration relative to earth Ca is the added mass coefficient for the body ar is the fluid acceleration relative to the body ρ is the density of water
Vr is the fluid velocity relative to the body Cd is the drag coefficient for the body A is the drag area
The term in parentheses is the inertia force, the other term is the drag force. The drag force is familiar to most engineers, but the inertia force can cause confusion.
The inertia force consists of two parts, one proportional to fluid acceleration relative to earth (the Froude-Krylov component), and one proportional to fluid acceleration relative to the body (the added mass component).
To understand the Froude-Krylov component, imagine the body being removed and replaced with an equivalent volume of water. This water would have mass Δ and be undergoing an acceleration aw. It must therefore be experiencing a force Δ.aw.
Now remove the water and put the body back: the same force must now act on the body. This is equivalent to saying that the Froude-Krylov force is the integral over the surface of the body of the pressure in the incident wave, undisturbed by the presence of the body. (Note the parallel with Archimedes' Principle: in still water, the integral of the fluid pressure over the wetted surface must exactly balance the weight of the water displaced by the body.) The added mass component is due to the distortion of the fluid flow by the presence of the body. A simple way to understand it is to consider a body accelerating through a stationary fluid. The force required to sustain the acceleration may be shown to be proportional to the body acceleration and can be written:
F = (m + Ca.Δ).a where
F is the total force on the body m is the mass of the body
(Ca.Δ) is a constant related to the shape of the body and its displacement a is the acceleration of the body.
Another way of looking at the problem is in terms of energy. The total energy required to accelerate a body in a stationary fluid is the sum of the kinetic energy of the body itself, and the kinetic energy of the flow field about the body. These energies correspond to the terms (m.a) and Ca.Δ.a respectively.
Trapped Water
The term (Ca.Δ) has the dimensions of mass and has become known as the added mass. This is an unfortunate name which has caused much confusion over the years. It should not be viewed as a body of fluid trapped by and moving with the body. Some bodies are so shaped that this does occur, but this trapped water is a completely different matter. Trapped water occurs when the body contains a closed flooded space, or where a space is sufficiently closely surrounded to prevent free flow in and out. Trapped water should be treated as part of the body: the mass of the trapped water should be included in the body mass, and its volume should be included in the body volume.
For a more complete description of Morison's equation and a detailed derivation of the added mass component see Barltrop and Adams, 1991 and Faltinsen, 1990.
5.10.6 Waves Wave Theory
Each wave train can be a regular wave, a random wave or specified by a time history file.
Regular Waves
OrcaFlex offers a choice of a long-crested, regular, linear Airy wave (including seabed influence on wave length) or non-linear waves using Dean, Stokes' 5th or Cnoidal wave theories (see Non-linear Wave Theories). Waves are specified in terms of height and period, and direction of propagation.
Random Waves
OrcaFlex offers five standard frequency spectra: JONSWAP, ISSC (also known as Bretschneider or modified Pierson-Moskowitz), Ochi-Hubble, Torsethaugen and Gaussian Swell.
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Theory, Environment Theory The program synthesises a wave time history from a user-determined number of linear wave components. The wave component frequencies are chosen using an equal energy approach – see below. The phases associated with each wave component are pseudo-random: a random number generator is used to assign phases, but the sequence is repeatable, so the same user data will always give the same train of waves. Different wave component phasing for the same spectrum can be obtained by shifting the simulation time origin relative to the wave time origin, or by specifying a different random number seed.OrcaFlex provides special facilities to assist in selecting an appropriate section of random sea. These are available on the Waves Preview page of the Environment data form. The facilities include:
A profile graph plotting the wave elevation for a selected period and
A table listing all the waves in a selected time interval whose height or steepness is large by comparison with the reference wave Hs, Tz.
Wave components
An irregular wave train is constructed by linear superposition of a number of linear wave components. OrcaFlex creates the components using an equal area approach, over a user-specified range of the frequency spectrum.
However this approach can result in some components (e.g. near the tails of the spectrum where the spectral energy is low) representing a wide range of frequencies. Such components can result in poor modelling of system responses, since a wide frequency range of spectral energy is then concentrated at a single frequency. To solve this the user can specify a maximum component frequency range, and any component that covers a wider frequency range is then subdivided into multiple components (which then have lower energy, so they are no longer have equal energy) until all the components satisfy the specified maximum frequency range.
This method of allocating wave components is now described in more detail. We denote by rmin and rmax the minimum and maximum relative frequencies, and by δfmax the maximum component frequency range.
The wave components are created as follows:
1. We define fm- to be the frequency of the spectral peak with the lowest frequency. Likewise define fm+ to be the frequency of the spectral peak with the highest frequency. For single peaked spectra fm- = fm+ = fm. For the Ochi-Hubble spectrum, the spectral peak frequencies are data items named fm1 and fm2. For the Torsethaugen spectrum, the spectral peak frequencies are calculated internally by the program as described in the Torsethaugen and Haver paper.
2. The overall frequency range considered is [rminfm-, rmaxfm+]. The nature of wave spectra means that the energy outside the range is negligible, at least for the default values of rmin and rmax.
3. This overall frequency range is then broken into n component frequency bands [fi-, fi+] (i = 1 to n), such that each band contains the same amount of spectral energy. Here f1- = rminfm-, fn+ = rmaxfm+ and fi+ = f(i+1)-, and n is the user-specified number of components. See the illustration below, which for clarity is for only n=10 components (the default value of n is much larger).
4. Any frequency band [fi-, fi+] for which fi+ - fi- > δfmax is then recursively subdivided into multiple bands, until the frequency width of each band is less than the specified maximum δfmax. Any such subdivision will result in the number of components, n, increasing, and the components resulting from subdivision will no longer have the same energy as the non-subdivided components.
5. A wave component is then created for each resulting frequency band. The wave component frequency, fi, is chosen so that there is equal spectral energy either side of it in the frequency band represented by that component. In other words there is equal spectral energy in the ranges [fi-, fi] and [fi, fi+].
Note: When the spectrum discretisation method is set to Legacy or 9.3a, the values of fm- and fm+ are defined differently. Both fm- and fm+ are set to the nominal value of fm. For single peaked spectra this
Note: When the spectrum discretisation method is set to Legacy or 9.3a, the values of fm- and fm+ are defined differently. Both fm- and fm+ are set to the nominal value of fm. For single peaked spectra this