Theory of Suction Pile Final -...

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Theory of a vertically loaded Suction Pile in SAND

1. Convention

Z Waterdepth d Penetrationdepth D Caisson diameter L Caisson length t1 Caisson wall thickness t2 Top plate thickness W Submerged weight of caisson φi Internal angle of internal friction φe External angle of internal friction (sand-steel interface) c Cohesion coefficient κ Intrinsic flow parameter p Pore pressure ρsoil Soil density γsoil Soil Volumetric weight σ’ v(z) Effective stress in vertical direction at depth z

L

t1

D

d

t2

Z

Water

Soil

COG

zCOG

Figure 1: Overview of main components Figure 2: Overview of main parameters


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σ’h(z) Effective stress in horizontal direction at depth z σ v(z) Total stress in vertical direction at depth z σ h(z) Total stress in vertical direction at depth z

Free body diagram

Figure 3: Free Body Diagram

Possible forces:

• Vertical load force (Fv) • (Submerged) Self weight caisson (Fg) • Pressure differential force (Fp) • Shaft friction outer surface (Fo, τo) • Shaft friction inner surface (Fi, τi) • Tip resistance (Ft, qt)

Total force balance: z p o i tipF W F F F F= + − − −∑ (1.1)


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2. Installation Phase A: Self weight penetration In this phase the suction anchor is lowered onto the seabed, an opened valve on top of the anchor enables water to escape the caisson freely. The anchor’s submerged weight causes the soil at the tip of the anchor to fail allowing penetration into the soil. As the penetration depth increases friction starts to act on the anchors inner and outer surface. At a certain penetration depth the friction forces combined with the tip force will equal the submerged weight of the anchor and penetration will cease. The cylindrical shell that has penetrated into the soil is called the skirt and the final depth is referred to as the self weight penetration depth.

Force balance • (Submerged) Self weight of anchor (W) • Friction outer surface (Fo, τo) • Friction inner surface (Fi, τi) • Tip Force (Ft, qt)

At the self weight penetration depth these forces are in equilibrium resulting in a force balance given by equation(2.1). 0z o i tipF W F F F= − − − =∑ (2.1)

Soil conditions The vertical stresses are assumed to be caused by gravity alone. The pore pressure, total vertical stress and effective vertical stress are therefore given by equation (2.2) through (2.4). ( ) waterp Z z γ= + (2.2)

v water soilZ zσ γ γ= + (2.3)

' ( )v v soil waterp zσ σ γ γ= − = − (2.4)

Past events can cause the horizontal stresses to differ from the expected relation for passive stresses. To determine these horizontal stresses exactly one must perform a survey, preferably in situ. However, it is possible to determine boundaries for the

horizontal stresses. Since soil conditions outside of the passive and active case are in conflict with Mohr-Coulomb criterion the actual stress must be somewhere in between

these two values as shown by (2.5) and visualized by Figure 4. ' ' 'p v h a vK Kσ σ σ≤ ≤ (2.5)


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Figure 4: Circles of Mohr

Shaft friction outer surface As derived by Mohr the value for shear transferred over a steel-sand interface is given by equation(2.6). Since the focus is on installation it is reasonable to look at the case of maximal horizontal soil stress. The active stress relation represents the desired upper bound. The value of the external angle of friction for steel is estimated by (2.7). max ,max' tan ' tanh e a v ec c Kτ σ ϕ σ ϕ= + = + (2.6)

2

3e iϕ ϕ≈ (2.7)

The total friction over the outer surface can be found by integrating the shear stress over the outer surface area of the caisson. Substituting (2.6), (2.4) and solving yields equation (2.8) for maximal friction on the outer surface.

2max max

0

1tan ( )

2

d

o a e soil waterF dA D dz DK dτ π τ π ϕ γ γ= = = −∫∫ ∫ (2.8)

Shaft friction inner surface Since there are virtually no differences between the soil conditions inside and outside of the caisson, the friction on the inner surface is approximately equal to the friction on the outer surface as shown by (2.9).

σh,min σv

σh,max

σ

τ


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i oF F≈ (2.9)

Tip resistance force The maximal tip force is determined with the strip foundation theory by Brinch Hanssen (2.10). In this case q is represented by the effective vertical stress. c qp cN qN= + (2.10)

'vq σ= (2.11)

The parameters for sand are determined by (2.12) and (2.13).

1 sin

exp( tan )1 sin

iq i

i

Nϕ π ϕϕ

+=−

(2.12)

0c = (2.13) Now the tip force can be found by multiplying the tip pressure by the surface area of the tip. This results in equation (2.14) for the tip force. 1 1 1' ( )tip q v q soil waterF Dt p Dt N Dt N dπ π σ π γ γ= = = − (2.14)

Self eight penetration depth Substituting the derived expressions for all active forces into the force balance results in expression(2.15) which is dependent on various system constants and the penetration depth.

2 21 2 1( )( 0.25 ) tan ( ) ( ) 0steel water a e soil water q soil waterDt L D t DK d Dt N dγ γ π π π ϕ γ γ π γ γ− + − − − − =

(2.15) By solving equation (2.15) for d, one can find a lower limit for the penetration depth. If no further data about the soil conditions is available this is the value that should be used for design purposes.


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3. Installation Phase B: Pump out trapped water and penetrate by suction

In this phase the pump is activated, lowering the pressure inside the caisson. The difference between the caisson pressure and the hydrostatic water pressure causes a distributed force over the top of the anchor directed downward. Water starts to flow through the soil inside and outside the skirt into the caisson and out again through the pump. This water flow affects the soil properties greatly and must be taken into account.

Force Balance Active forces:

• (Submerged) Self weight of anchor (W) • Pressure differential force (Fp) • Shaft friction outer surface (Fo, τo) • Shaft friction inner surface (Fi, τi) • Tip resistance force (Ft, qt)

The resulting force balance is given by expression (3.1), if the sum of all forces is positive the suction caisson will penetrate into the soil. z p o i tipF W F F F F= + − − −∑ (3.1)

Pressure differential force The magnitude of the downward directed pressure differential force is given by equation(3.2).

2 21 1( ) (( ) )

4 4p h c w cF A P D p p D Z L d pπ π γ= ∆ = − = − + − (3.2)

There are limits to the pressure difference that can be applied. Firstly the pressure inside the caisson cannot be less than zero; therefore the pressure difference is limited to the hydrostatic pressure. In general this is not the limiting factor, but in some shallow water applications this can be upper limit for the pressure difference. 0;c h c hp P p p p> ∆ = − < (3.3)

Secondly soil failure inside the caisson can occur if the pressure gradient becomes to great. As an estimation the pore pressure at the tip of the caisson is set to be the mean of the caisson pressure and the hydrostatic pressure at the seabed as shown by relation(3.4).


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( ) ( )

2 2h c w c

t

p p Z pp

γ+ += = (3.4)

By assuming a constant pressure gradient inside the skirt a relationship for the pore pressure inside the caisson can be found. It is given by equation (3.5).

t cc c

p ppp p z p z

z d

−∂= + = +∂

(3.5)

Soil failure will occur if the effective vertical stresses become negative since in this case the soil start to flow upwards into the suction caisson. From relationship (3.6) for the vertical effective stress a limit state for the pressure difference is formulated. The pressure difference must be kept within the boundaries given by (3.7).

,2 ,2' ( ) ( )2

t c h cz z s s

p p p pp z z

d dσ σ γ γ− −= − = − = − (3.6)

2h c sp p dγ− ≤ (3.7)

So there are two limiting criteria, (3.3) and (3.7). Naturally the smallest value of the to must be used to find the maximal allowable pressure difference. To be on the save side of things the pressure difference must be kept below 90% of this value as show by(3.8). 0.9 min( ,2 )h c h sp p p dγ− ≤ ⋅ (3.8)

Soil conditions As shown in the former text, the vertical effective stresses will decrease inside the caisson as a result of the water flow. Consequently the upper- and lower limit for the horizontal stresses will decrease as well. This concept is visualized in figure AA and quantified by relationship(3.9). ,2 ,2 ,2' ' 'p v h a vK Kσ σ σ≤ ≤ (3.9)


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Outside the caisson water is attracted from all directions, therefore at the shaft surface there is no significant change in soil stresses and relationship(2.5) can be used.

Shaft friction outer surface Since there is no significant change in soil stresses at the outer surface the same relationship as before can be used to find shaft friction acting on the outer surface.

Shaft friction inner surface The reduction in vertical effective soil stresses effects the shaft friction greatly. Since we are still looking at installation the worst case scenario is given by active soil stresses. Substituting (3.11) into (3.10) and integrating over the inner surface lead to an expression for the shaft friction(3.12). max ,max,2 ,2' tan ' tanh e a v ec c Kτ σ ϕ σ ϕ= + = + (3.10)

2

3e iϕ ϕ≈ (3.11)

2max max

0

1tan ( )

2 2

dh c

i a e s

p pF dA D dz DK d

dτ π τ π ϕ γ −= = = −∫∫ ∫ (3.12)

σh,min σv

σh,max

σ

τ Decreasing vertical stress


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Tip resistance force Water flow around the tip of the anchor causes the soil to weaken due to a ‘jetting’ effect. A measure for this weakening effect is given by equation(3.13).

tan '

,0 1tan

i

i

ϕα αϕ

= ≤ ≤ (3.13)

As was the case for the self weight penetration phase, the tip force is estimated by the Brinch-Hanssen method. In this instance the parameters are somewhat different as shown by (3.14) through(3.18). c qp cN qN= + (3.14)

'vq σ= (3.15)

1 sin '

' exp( tan ' )1 sin '

iq i

i

Nϕ π ϕϕ

+=−

(3.16)

0c = (3.17) 1 1 1' ' ' ( )tip q v q soil waterF Dt p Dt N Dt N dπ π σ π γ γ= = = − (3.18)

Penetration depth With all active forces known the final penetration depth can be calculated. For an anchor to operate correctly it is necessary that is penetrates completely. It is important to note that not all anchors will allow full penetration by suction. Therefore this is a very important design aspect. To solve this problem a numerical method has been chosen by means of an Excel sheet that can be downloaded here.


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4. Operation in sand

Force balance Active forces:

• Vertical load force (Fv) • (Submerged) Self weight caisson (Fg) • Shaft friction outer surface (Fo, τo) • Shaft friction inner surface (Fi, τi)

The resulting force balance is given by expression (3.1), as long as the sum of all forces remains positive the anchor will hold. z o i loadF W F F F= + + −∑ (3.19)

Soil conditions It is assumed that all changes in the soil stresses caused by installation of the anchor gradually go away and the soil returns to its initial state. The vertical stresses are assumed to be caused by gravity alone. The pore pressure, total vertical stress and effective vertical stress are therefore given by equation (2.2) through (2.4). ( ) waterp Z z γ= + (3.20)

v water soilZ zσ γ γ= + (3.21)

' ( )v v soil waterp zσ σ γ γ= − = − (3.22)

Past events can cause the horizontal stresses to differ from the expected relation for passive stresses. To determine these horizontal stresses exactly one must perform a survey, preferably in situ. However, it is possible to determine boundaries for the horizontal stresses. Since soil conditions outside of the passive and active case are in conflict with Mohr-Coulomb criterion the actual stress must be somewhere in between these two values as shown by (2.5). ' ' 'p v h a vK Kσ σ σ≤ ≤ (3.23)


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Shaft friction outer surface During operation the worst case scenario is given by the case in which the relationship between vertical and horizontal effective stresses is the passive one. As derived by Mohr the value for shear transferred over a steel-sand interface is given by equation(2.6). The value of the external angle of friction for steel is estimated by (2.7). min ,min' tan ' tanh e p v ec c Kτ σ ϕ σ ϕ= + = + (3.24)

2

3e iϕ ϕ≈ (3.25)

The total friction over the outer surface can be found by integrating the shear stress over the outer surface area of the caisson. Substituting (2.6), (2.4) and solving yields equation (2.8) for maximal friction on the outer surface.

2min min

0

1tan ( )

2

d

o p e soil waterF dA D dz DK dτ π τ π ϕ γ γ= = = −∫∫ ∫ (3.26)

Shaft friction inner surface Since there are virtually no differences between the soil conditions inside and outside of the caisson, the friction on the inner surface is approximately equal to the friction on the outer surface as shown by (2.9).

σh,min σv

σh,max

σ

τ


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i oF F≈ (3.27)

There is an upper limit to the shaft friction acting on the inner surface. Since there is no cohesion in sand the friction force can never exceed the weight of the soil plug inside the anchor as shown by (3.28). 20.25i soil soilF W Dπ γ≤ = (3.28)

Load capacity With all forces known except for the load force, it is now easy to determine the load capacity of the suction caisson. The relationship follows directly from force balance (3.19) and is given by(3.29). load o iF W F F≤ + + (3.29)

Theory of Suction Pile Final -...

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