Axial Sand Methods

There are various methods available that can be used to calculate the axial capacity of a pile, either for generation of TZ & QZ curves or in ALLCAP.

Method

Soil Parameter Input in Main Table

Additional Parameters

1

2

3

4

5

6

7

8

9

10

11

SANDAPI

X

X

X

X

K sand compression & tension

SANDAPI_RP2GEO

X

X

X

X

X

SANDUSER

X

X

SANDICP

X

X

X

CPT diameter, Pile roughness

SANDICP_API

X

X

X

CPT diameter

SANDUWA

X

X

X

CPT diameter

SANDUWAFULL

X

X

X

CPT diameter, Pile roughness

SANDNGI

X

X

SANDF05

X

X

CPT diameter

SANDUNCPT1

X

X

CPT diameter

List of soil parameters used in the axial methods:

No.

Symbol

Unit

Soil Parameter

1

\(f_{lim}\)

(kPa)

limiting skin friction

2

\(q_{lim}\)

(kPa)

limiting end bearing

3

\(q_{c,top}\)

(kPa)

CPT cone resistance at the top of the layer

4

\(q_{c,bottom}\)

(kPa)

CPT cone resistance at the bottom of the layer

5

\(\delta _{f}\)

(deg)

ultimate interface friction angle (Ring shear test)

6

\(\delta _{API}\)

(deg)

interface friction angle

7

\(N_{q}\)

(-)

bearing capacity factor

8

\(f_{u}\)

(kPa)

user defined ultimate shaft resistance

9

\(q_{u}\)

(kPa)

user defined ultimate end bearing resistance

10

\(\beta _{API}\)

(-)

shaft friction factor in compression

11

\(\beta _{API,Tens}\)

(-)

shaft friction factor in tension

Note

Where possible the notation used in the explanation of each method is the same as used in the original source paper, rather than translate many different methods and styles of notation to a common one to be used in OPILE. It is always advised to consult the original reference sources of these methods if the user is not familiar with the implementation of a particular method. The references cited within the OPILE documentation are by no means exhaustive. Note that easy unit conversions can be made when inputting data such as dimensions and shear strengths, see units convention section for details.

SANDAPI

Skin Friction

The calculation of skin friction for cohesionless soils according to API (2000) is as follows:

\[f = K \cdot p_0 \cdot \tan{\delta_{API}}\]

Where:

\(K\)

is the coefficient of lateral earth pressure (ratio of horizontal to vertical normal stress).

For open ended piles driven unplugged it is usually appropriate to assume \(K = 0.8\) for tension and compression loadings.

\(P_0\)

is the effective overburden pressure [kPa] at the point in question.

\(\delta_{API}\)

is the friction angle between the soil and pile wall.

The skin friction f is subject to a limiting skin friction \(f_{Lim}\). Typical values of \(\delta_{API}\) and \(f_{Lim}\) (the limiting skin friction value) are shown in the table below.

End bearing

The unit end bearing [kPa] is calculated using the equation:

\[q = p_0 \cdot N_q\]

Where:

\(N_q\)

is the dimensionless bearing capacity factor, as specified in the table below.

The end bearing pressure q is subject to a limiting end bearing value \(q_{lim}\). It is recommended that for further information on this method API (2000) should be referred to.

Density

Soil Description

Soil-Pile Friction Angle \(\delta\) [degrees]

Limiting Skin Friction [kPa]

N q

Limiting End Bearing [MPa]

Very Loose

Loose

Medium

Sand

Sand-Silt

Silt

15

47.8

8

1.9

Loose

Medium

Sand

Silt

20

67

Loose

Medium

Dense

Sand

Sand-Silt

Silt

20

67

12

2.9

Medium

Dense

Sand

Sand-Silt

25

81.3

20

4.8

Dense

Very Dense

Sand

Sand-Silt

30

95.7

40

9.6

Dense

Very Dense

Gravel

Sand

35

114.8

50

12

The following soil density categories by Bentley & Carter (1991) are provided for guidance:

Sands & Gravels

Bulk Density [kN/m3]

Very Loose

17-18

Loose

18-19

Medium Dense

19-21

Dense

20-22

Very Dense

22-23


SANDAPI_RP2GEO

Skin Friction

The calculation of skin friction for cohesionless soils according to API RP2GEO (2011) is as follows:

\[f(z) = \beta \cdot p'_0(z)\]

Where:

\(\beta\)

is the dimensionless shaft friction factor for sands, for compression and tension.

\(p'_0(z)\)

is the effective vertical stress at depth z.

The skin friction, \(f\), is subject to a limiting skin friction \(f_{Lim}\).

Typical values of \(\beta\) and \(f_{Lim}\) (the limiting skin friction value) are shown in the table below.

Relative Density

Soil Description

Shaft Friction Factor [-]

Limiting Shaft Friction Values [kPa]

Nq [-]

Limiting Unit End Bearing Values [MPa]

Very Loose

Loose

Loose

Medium dense

Dense

Sand

Sand

Sand-Silt

Silt

Silt

Not applicable

Not applicable

Not applicable

Not applicable

Medium dense

Sand-Silt

0.29

67

12

3

Medium dense

Dense

Sand

Sand-Silt

0.37

81

20

5

Dense

Very Dense

Sand

Sand-Silt

0.46

96

40

10

Very Dense

Sand

0.56

115

50

12

The following relative density percentage categories are provided by API RP2GEO (2011):

Very Loose

0-15

Loose

15-35

Medium Dense

35-65

Dense

65-85

Very Dense

85-100

End bearing

The unit end bearing [kPa] is calculated using the equation:

\[q = N_q \cdot p'_{0, tip}\]

Where:

\(N_q\)

is the dimensionless bearing capacity factor, as specified in the table below.

\(p'_{0,tip}\)

is the effective vertical stress at the pile tip.

The end bearing pressure q is subject to a limiting end bearing value \(q_{lim}\).

It is recommended that for further information on this method API RP2GEO (2011) should be referred to. Design parameters for cohesionless siliceous soil, ref API RP2GEO (2011).


CPT Methods

Skin Friction

The CPT based methods SANDUWA, SANDICP_API and SANDF05 are implemented in OPILE using the equations provided in API RP2GEO (2011). The SANDICP_API method is a variation of the original SANDICP method which ignores the factors accounting for sand dilation in the original method. The SANDUWA method is the simplified offshore version which, similar to the presented SANDICP_API, does not account for sand dilation in the original method SANDUWAFULL. It is recommended that for further guidance API RP2GEO (2011) or some of the other references are referred to.

End bearing for the methods is covered in the other relevant sections, SANDUWA, SANDICP and SANDF05.

The unit skin friction (f) formulae for open ended steel pipe piles for the first three recommended CPT-based methods (Simplified SANDICP_API, SANDUWA and SANDF05) can all be considered as special cases of the general formula:

\[f = u \cdot q_c \cdot \left(\frac{\sigma'_{v0}}{p_a}\right)^a \cdot {A_r}^b \cdot \textrm{max}\left(\frac{L-z}{D},v\right)^{-c} \cdot \tan({\delta_{cv}})^d \cdot \textrm{min}\left(\frac{L-z}{D} \cdot \frac{1}{v},1\right)^e\]

Where:

\(A_p\)

Pile gross end area = \(\large\frac{\pi . D_o^2}{4}\)

\(A_r\)

Pile displacement ratio = \(1-\large\frac{D_i}{D_o}^2\)

\(D_o\)

Pile outer diameter

\(D_i\)

Pile inner diameter = \(D_o - 2WT\)

\(WT\)

is the wall thickness of the pile

\(e\)

Base natural logarithms = 2.718

\(L\)

Pile embedded length (below original seabed)

\(pa\)

Atmospheric pressure = 100kPa

\(q_{c,z}\)

CPT cone tip resistance qc at depth z

\(\sigma'_{vo}\)

is the overburden pressure at depth z

Values for parameters a, b, c, d, e, u and v for compression and tension (extracted from API RP2GEO (2011)):

Method

Parameter

a

b

c

d

e

u

v

SANDICP_API

Compression

0.1

0.2

0.4

1

0

0.023

\(\sqrt[4]{A_r}\)

Tension

0.1

0.2

0.4

1

0

0.016

\(\sqrt[4]{A_r}\)

SANDUWA

Compression

0

0.3

0.5

1

0

0.030

2

Tension

0

0.3

0.5

1

0

0.022

2

SANDF05

Compression

0.05

0.45

0.90

0

1

0.043

\(\sqrt[2]{A_r}\)

Tension

0.15

0.42

0.85

0

0

0.025

\(\sqrt[2]{A_r}\)


SANDNGI

Skin Friction

The skin friction in compression loading is calculated by:

\[f_{cz} = 1.3 \cdot \frac{z}{L} \cdot p_a \cdot F_{sig} \cdot F_{Dr}\]

And the skin friction in tension loading is calculated by:

\[f_{tz} = \frac{z}{L} \cdot p_a \cdot F_{sig} \cdot F_{Dr}\]

Where:

\(F_{sig}\)

\(F_{sig} = \left(\Large\frac{p_o}{100\textrm{kPa}}\right)^{0.25}\)

\(F_{Dr}\)

\(2.1 \cdot (D_r - 0.1)^{1.7}\)

\(D_r = 0.4 \cdot log\left[\Large\frac{q_c}{22(\sigma'_{vo} \cdot p_a)^{0.5}}, \small 2.718 \right]\)

And

\(D_r = max(D_r, 0.1)\)

\(p_a\)

Atmospheric pressure = 100kPa

\(\sigma'_{vo}\)

is the effective overburden pressure [kPa] at the point in question.

\(z\)

is the depth in question [m]

\(L\)

is the pile penetration below the original sea floor

\(q_c\)

is the cone resistance at the depth in question

End Bearing

The base pressure is determined by:

\[\begin{split}q_p = \left\|\begin{array}{c} \Large\frac{0.7 \cdot q_c}{1 + 3 \cdot D_r^2} \quad & \text{For Plugged condition} \\ q_c \quad & \text{For Unplugged condition} \end{array}\right.\end{split}\]

Where:

\(q_c\)

Is the cone tip resistance at the specified depth.


SANDICP

Skin Friction

There is a modified ICP method, SANDICP_API which is presented in the CPT methods section. For the unmodified ICP method SANDICP the equations are as presented in Jardine et al (2005) and the API variant ignores the pile roughness term.

If the pile roughness input in OPILE is set to zero the two methods give the same results in OPILE, even though the equations are implemented as they were presented in the cited references.

In compression the skin fricton is calculated from:

\[\tau_f = \sigma_{rf} \cdot \tan(\delta_{cv})\]

And in tension the skin fricton is calculated from:

\[\tau_{tf} = 0.9 \cdot [0.8 (sigma_{rc} + \Delta\sigma_{rd})]\]

Where:

\(\sigma_{rf}\)

\(\sigma_{rf} = (\sigma_{rc} + \Delta\sigma_{rd})\)

\(\sigma_{rc}\)

\(\sigma_{rc} = 0.029 \cdot q_c \cdot \Big(\frac{p_o}{100\textrm{kPa}}\Big)^{0.13} \cdot \Big(\frac{h}{R^*}\Big)^{-0.38}\)

A lower limit of \(\frac{h}{R^*} \geq 8\) applies

\(\Delta\sigma_{rd}\)

\(\Delta\sigma_{rd} = 2 G \cdot \frac{\Delta r}{R}\)

\(G\)

\(G = q_c \cdot (A + B \cdot \eta - C \cdot \eta^2)^{-1}\)

with:

\(A = 0.0203\)

\(B = 0.00125\)

\(C = 0.000001216\)

\(\eta = [q_c \cdot (100\textrm{kPa} \cdot p_o)^{-0.5}]\)

\(p_o\)

is the overburden pressure at the depth in question

\(h\)

is the distance of the depth in question above the pile tip

\(R^*\)

\(R^* = \big[R^2-(R-t)^2]^{0.5}\big]\)

\(t\)

is the pile wall thickness

\(R\)

is the pile radius

End Bearing

If the pile is plugged both of the following conditions must be met, otherwise the pile is unplugged:

\[\begin{split}Plug \ state = \left\| \begin{array} PPlugged & \quad & \text{if } \large\frac{D_i}{1m} \small < (D_r - 0.3) \\ Unplugged & \quad & \text{otherwise } \end{array}\right.\end{split}\]

And:

\[\begin{split}Plug \ state = \left\| \begin{array} UUnplugged & \quad & \text{if Plug state = Unplugged} \\ Plugged & \quad & \text{if } \large\frac{D_i}{D_{CPT}} \small < 0.083 \cdot \large \frac{q_c}{p_a} \\ Unplugged & \quad & \text{otherwise } \end{array}\right.\end{split}\]

After it has been determined whether or not the pile is plugged or unplugged the end bearing pressure is calculated by:

\[\begin{split}q_b = \left\| \begin{array} qq_c \cdot \left( 0.5 - 0.25 \cdot \textrm{log}\Big( \large \frac{D}{D_{CPT}},10 \Big) \right) & \quad & \text{if Plug state = Plugged} \\ q_c & \quad & \text{otherwise} \end{array}\right.\end{split}\]

Where:

\(D_r\)

\(D_r = 0.4 \cdot \textrm{log}\left[\Large\frac{q_c}{22 \cdot (\sigma'_{vo} . p_a)^{0.5}}, \small 2.718 \right]\)

\(D_{CPT}\)

is the diameter of the cone penetrometer, commonly 0.036m


SANDUWA

Skin Friction

This is a modified UWA method for offshore piles, which account for full pile fill ratio (IFR = 1) and ignores the effect of sand dilation.

Skin friction is calculated according to the equations which are published in API RP2GEO (2011), presented in the CPT methods section.

End Bearing

The UWA method, as presented in API RP2GEO (2011), only considers piles to behave in a plugged manner.

End bearing pressure is calculated using the following equation:

\[q_p = q_c \cdot (0.15 + 0.45 \cdot A_r)\]

Where:

\(q_c\)

is the cone tip resistance at the specified depth

\(A_r\)

Pile displacement ratio = \(1-\large\frac{D_i}{D_o}^2\)

For further information see Lehane et al (2005).


SANDUWAFULL

Skin Friction

Skin friction is calculated according to the equations published in Lehane et al (2005):

\[\tau_f = \sigma^{'}_{rf} \tan(\delta_{cv}) = \frac{f}{f_{c}} (\sigma^{'}_{rc} + \Delta\sigma^{'}_{rd}) \tan(\delta_{cv})\]
\[\sigma^{'}_{rc} = 0.03 \cdot q_{c} (A^{*}_{rs})^{0.3} \left[ \textrm{max}\left( \frac{h}{D},2 \right) \right]^{-0.5}\]
\[A^{*}_{rs} = 1 - \textrm{IFR} \left(\frac{D_{i}^{2}}{D^{2}}\right)\]
\[\textrm{IFR} = \textrm{min}\left[1, \left(\frac{D_{i}(m)}{1.5m}\right)^{0.2}\right]\]
\[\Delta\sigma^{'}_{rd} = 4 G \cdot \Delta r/D\]

Where:

\(\delta_{cv}\)

constant volume interface friction angle, with upper limit \(\tan \delta \leq 0.55\)

\(\sigma^{'}_{rf}\)

radial effective stress at failure

\(\sigma^{'}_{rc}\)

radial effective stress after installation and equalization

\(\Delta\sigma^{'}_{rd}\)

change in radial stress due to loading stress path (dilation)

\(\frac{f}{f_{c}}\)

1 for compression and 0.75 for tension

\(\frac{G}{q_{c}}\)

\(185 \cdot (q_{c1N})^{-0.75}\) with \(q_{c1N}=(q_{c}/p_{a}) / (\sigma^{'}_{v0} / p_{a})^{0.5}\)

\(p_{a}\)

a reference stress equal to 100kPa

\(\sigma^{'}_{v0}\)

in situ vertical effective stress

\(\Delta r\)

dilation, inputted as ‘Pile Roughness [m]’

End Bearing

End bearing capacity for driven pipe piles is calculated using the following equation:

\[Q_{b} = q_{b0.1} \frac{\pi}{4} D^{2}\]
\[q_{b01} / \overline{q_{c}} = 0.15 + 0.45 A_{rb}^{*}\]
\[A_{rb}^{*} = 1 - \textrm{FFR} \left(\frac{D_{i}^{2}}{D^{2}}\right)\]
\[\textrm{FFR} = \textrm{min} \left[1, (\frac{D_{i}(m)}{1.5m})^{0.2}\right]\]

Where:

\(D_i\)

is the inner pile diameter

For further information see Lehane et al (2005).


SANDF05

Skin Friction

Skin friction is calculated according to the equations which are published in API (2006) and presented in the CPT methods section.

End Bearing

The SANDF05 method only considers that piles can behave in a plugged manner and end bearing pressure is calculated using the following equation:

\[q_p = 8.5 \cdot \left[ p_a \cdot \left( \frac{q_c}{p_a} \right)^{0.5} \right] \cdot (A_r)^{0.25}\]

Where:

\(q_c\)

Is the cone tip resistance at the specified depth

\(A_r\)

Pile displacement ratio = \(1-\large\frac{D_i}{D_o}^2\)

\(p_a\)

Atmospheric pressure, introduced for units purposes = 100kPa

For further information see CUR (2001) and Lehane et al (2005).

Note

Note that only the equations presented in Lehane et al (2005) and API (2006) are intended by the authors for use in design. The earlier versions have been subjected to modifications.


SANDUNCPT1

The unified CPT method is a CPT based pile capacity method for assessing the capacity in sands. The equations are recommended for the evaluation of medium-term (two weeks after driving) static capacity of steel tubular piles. The method can be applied to sands with a fines content of less than 12% (up to 20% in the case of non-plastic fines). The method should not be used for sands with unusually weak grains or compressible structure, including those sands containing amounts of mica, volcanic grains or calcium carbonate sufficient to lead to a mechanical response that differs from that of a silica sand. This method should only be applied to driven piles and is not directly applicable to piles installed by vibration or jacking. The reliability of the method has been evaluated and the method was shown, when used with the partial load and resistance factors of ISO 19902 1st Edition, to provide pile foundations that are more reliable than those obtained using the former main text method. Further information on the method and equations below is provided in Lehane et al (2020) and Nadim et al (2020).

Skin Friction

The external unit skin friction for capacity two weeks after driving, \(\tau_f\), in stress units, at depth, \(z\), can be calculated using:

\[\tau_f = f_{L} \cdot (\sigma^{'}_{rc} + \Delta\sigma^{'}_{rd}) \cdot \tan(29)\]

Where:

\(\sigma^{'}_{rc}\)

\(\sigma^{'}_{rc} = \Big(\frac{q_{c}}{44}\Big) \cdot A_{re}^{0.3} \cdot \Big(\textrm{max}(1,\frac{h}{D})\Big)^{-0.4}\)

\(\Delta\sigma^{'}_{rd}\)

\(\Delta\sigma^{'}_{rd} = \Big(\frac{q_{c}}{10}\Big) \cdot \Big(\frac{q_{c}}{\sigma^{'}_{v}}\Big)^{-0.33} \cdot \Big(\frac{d_{ref}}{D}\Big)\)

\(A_{re}\)

\(1 - PLR \cdot \Big(\frac{D_{i}}{D}\Big)^{2}\) where PLR = 1 for typical offshore pile

\(f_{L}\)

is a loading coeffcient taken as 0.75 for tension loading and 1.0 for compression loading

\(29degrees\)

is the angle of interface friction used for calibration of the method.

\(\sigma^{'}_{rc}\)

is the horizontal effective stress acting on a driven pile at a depth, z, about two weeks after driving.

\(\sigma^{'}_{v}\)

is the vertical effective stress at a depth, \(z\).

\(q_{c}\)

is the cone resistance at a depth, \(z\).

\(A_{re}\)

the effective area ratio, defined above, is a measure of the soil displacement induced by the driven pile and expressed as a fraction of the soil displacement induced by a closed-ended pile (for which Are=1).

\(D\)

is the outer pile diameter.

\(D_{i}\)

is the inner pile diameter.

\(PLR\)

is the plug length ratio, with a maximum value of 1.0, defined as the ratio of the plug length (\(L_{p}\)) to the pile embedment (\(L\)).

\(\Delta\sigma^{'}_{rd}\)

is the change in horizontal stress, acting a depth, z, arising due to interface shear dilation as the pile is loaded.

\(d_{ref}\)

0.0356m.

\(h\)

is the distance above the pile tip at which \(\tau_f\) acts (\(= L-z\)) .

Note

It should be ensured that other factors such as paint, coatings or mill-scale varnish that are likely to reduce pile roughness below that usually expected (typically around 10 microns centre line average roughness for lightly rusted steel used for offshore piles) do not negatively affect interface friction angle that can be mobilised.

End Bearing

The end bearing method assumes a plugged base and is applicable for piles with a length to diameter ratio greater than five. Caution should be applied for plugs with low permeability within two pile diameters of the pile tip, such as those comprising interbedded clay layers and, in such cases, it should be confirmed that unplugged end bearing is not less than plugged end bearing.

End bearing pressure, in case of plugged behaviour, is calculated using the following equation:

\[q = (0.12 + 0.38 \cdot A_{re}) \cdot q_{p}\]

Where:

\(A_{re}\)

\(1 - \textrm{PLR} \cdot \Big(\frac{D_{i}}{D}\Big)^{2}\)

where PLR = 1 for typical offshore pile

\(q_{p}\)

is the average qc value within a zone 1.5D above and below the pile tip.

In the absence of more definitive criteria, a cautious estimate of unplugged end bearing an be taken as follows:

\[q(unplugged) = A_{re} \cdot q_{p}\]

Caution

The cone resistance in sand can vary considerably horizontally. If measured cone resistance over a limited area is used to obtain pile capacity over large areas, the user should exercise caution with the extrapolation of cone resistance away from cone test data and consider the effect of lateral variability on the soil resistance. Where cone resistance varies significantly, a cautious estimate should be used. It should be noted that end bearing capacity can be more sensitive to local variations in cone resistance than shaft resistance.


SANDUSER

Skin Friction and End Bearing

The SANDUSER method allows the user to enter skin friction and end bearing pressures for use in the calculation of capacity and TZ and QZ curves.

The frictions and end bearings are entered as shown in the figure below:

../_images/image1817831563.jpg

If zero end bearing is required then “0” should be entered. It is not possible to leave the end bearing box blank.

SANDUSER differs from CLAYUSER only in the shape of the TZ curves which are generated.