Techniques for modeling large diameter monopiles
In example embodiments, a new model for modeling monopiles in is provided that, in addition to distributed lateral load along the monopile, considers distributed moment along the length of the pile, base moment at the pile tip, and base shear force at the pile tip. The new model may avoid the overly conservative designs for large diameter piles (e.g., 10 m+) with small length-to-diameter ratios (e.g., <6), while using standardized reaction curves (i.e., p-y curves and t-z curves) and considering axial and combined loading.
1. A method for modeling monopiles, comprising:
accessing, by structural analysis and design software executing on one or more computing devices, a p-y curve and a t-z curve for a type of soil for a monopile at each of one or more layers, the monopile including a plurality of nodes in the one or more layers, wherein each p-y curve is a nonlinear function that relates lateral soil reaction force (p) to lateral displacement (y), and each t-z curve is a nonlinear function that relates skin friction (t) to vertical displacement (z);
determining, by the structural analysis and design software, a distributed moment along the monopile and based thereon a rotational stiffness of a given node, wherein the rotational stiffness is calculated as a function of both force per unit area from a respective t-z curve for the layer of the given node and slope of the respective t-z curve for the layer of the given node;
determining, by the structural analysis and design software, a base moment at a pile tip of the monopile and based thereon a base rotational stiffness, wherein the base rotational stiffness is calculated as a function of both force per unit area from a respective p-y curve for the layer of the pile tip and slope of the respective p-y curve for the layer of the pile tip;
determining, by the structural analysis and design software, a base shear force at the pile tip and based thereon base shear stiffness, wherein the base shear stiffness is calculated as a function of slope of a respective t-z curve for the layer of the pile tip and pile tip area;
iteratively solving, by the structural analysis and design software, a system of non-linear equations that uses a structural stiffness matrix that includes the rotational stiffness, the base rotational stiffness and the base shear stiffness in addition to pile material structural stiffness to model the monopile; and
outputting, by the structural analysis and design software, a description of the modeled monopile.
2. The method of claim 1 , wherein the monopile is a monopile of an offshore wind turbine.
3. The method of claim 1 , wherein the monopile has a diameter of 10 meters or more.
4. The method of claim 1 , wherein the monopile has a length-to-diameter ratio between 2 and 6 inclusive.
5. The method of claim 1 , wherein rotational stiffness (k θ i ) for a node (i) corresponding to an element (j) is calculated as:
k
θ
i
=
∫
0
2
π
(
d
M
j
d
θ
)
d
α
=
4
R
[
∫
0
π
/
2
T
j
d
j
2
d
α
+
θ
∫
0
π
/
2
(
d
T
j
d
Z
j
)
d
j
3
d
α
]
where θ is an angle of a rotated horizonal plane from an x-axis, α is an angle to the element j in the horizonal plane from the x-axis, R is radius of the monopile, d j is a distance in the horizonal plane from the y-axis, M j is total moment, T j is force per unit area from the respective t-z curve for the layer of the node (i), and dT j /dZ j is slope of the respective t-z curve for the layer of the node (i).
6. The method of claim 1 , wherein base rotational stiffness (K M b ) for base (b) given an element (j) is calculated as:
K
M
b
=
∫
-
π
2
π
2
(
d
M
b
d
θ
b
)
d
α
=
2
R
2
[
∫
0
π
2
P
j
d
j
(
Sin
2
α
)
d
α
+
θ
b
∫
0
π
2
(
d
P
j
d
Z
j
)
d
j
2
(
Sin
2
α
)
d
α
where θ b is pile tip rotation, α is an angle to the element j in the horizonal plane from the x-axis, R is radius of the monopile, d j is a distance in the pile tip surface from the y-axis, M b is base moment, P j is force per unit area from the respective p-y curve for the layer at the pile tip and dP j /dZ j is slope of the respective p-y curve for the layer at the pile tip.
7. The method of claim 1 , wherein base shear stiffness (K V b ), for base (b) is calculated as:
K
V
b
=
π
R
2
d
T
b
d
X
b
where R is radius of the monopile, and dT b /dX b is slope of the respective t-z curve for the layer at the pile tip.
8. A computing device comprising:
a processor; and
a memory coupled to the processor and configured to store structural analysis and design software, the structural analysis and design software when executed on the processor configured to:
access a p-y curve and a t-z curve for a type of soil for a monopile at each of one or more layers, the monopile including a plurality of nodes in the one or more layers, wherein each p-y curve is a nonlinear function that relates lateral soil reaction force (p) to lateral displacement (y), and each t-z curve is a nonlinear function that relates skin friction (t) to vertical displacement (z),
determine a rotational stiffness of a given node as a function of both force per unit area from a respective t-z curve for the layer of the given node and slope of the respective t-z curve for the layer of the given node,
determine a base rotational stiffness as a function of both force per unit area from a respective p-y curve for the layer of a pile tip and slope of the respective p-y curve for the layer of the pile tip,
determine a base shear stiffness as a function of slope of a respective t-z curve for the layer of the pile tip and pile tip area, and
iteratively solve a system of non-linear equations that uses a structural stiffness matrix that includes the rotational stiffness, the base rotational stiffness and the base shear stiffness in addition to pile material structural stiffness to model the monopile.
9. The computing device of claim 8 , wherein the monopile is a monopile of an offshore wind turbine.
10. The computing device of claim 8 , wherein the monopile has a diameter of 10 meters or more or a length-to-diameter ratio between 2 and 6, inclusive.
11. The computing device of claim 8 , wherein rotational stiffness (k θ i ) for a node (i) corresponding to an element (j) is calculated as:
k
θ
i
=
∫
0
2
π
(
d
M
j
d
θ
)
d
α
=
4
R
[
∫
0
π
/
2
T
j
d
j
2
d
α
+
θ
∫
0
π
/
2
(
d
T
j
d
Z
j
)
d
j
3
d
α
]
where θ is an angle of a rotated horizonal plane from an x-axis, α is an angle to the element j in the horizonal plane from the x-axis, R is radius of the monopile, d j is a distance in the horizonal plane from the y-axis, M j is total moment, T j is force per unit area from the respective t-z curve for the layer of the node (i), and dT j /dZ j is slope of the respective t-z curve for the layer of the node (i).
12. The computing device of claim 8 , wherein base rotational stiffness (K M b ) for base (b) given an element (j) is calculated as:
K
M
b
=
∫
-
π
2
π
2
(
d
M
b
d
θ
b
)
d
α
=
2
R
2
[
∫
0
π
2
P
j
d
j
(
Sin
2
α
)
d
α
+
θ
b
∫
0
π
2
(
d
P
j
d
Z
j
)
d
j
2
(
Sin
2
α
)
d
α
where θ b is pile tip rotation, α is an angle to the element j in the horizonal plane from the x-axis, R is radius of the monopile, d j is a distance in the pile tip surface from the y-axis, M b is base moment, P j is force per unit area from the respective p-y curve for the layer of the pile tip and dP j /dZ j is slope of the respective p-y curve for the layer of the pile tip.
13. The computing device of claim 8 , wherein base shear stiffness (K V b ) for base (b) is calculated as:
K
V
b
=
π
R
2
d
T
b
d
X
b
where R is radius of the monopile, and dT b /dX b is slope of the respective t-z curve for the layer of at the pile tip.
14. A non-transitory computing device readable medium having instructions stored thereon, the non-transitory computing device readable medium being a volatile or persistent memory device or a hard-disk, the instructions when executed by one or more computing devices operable to:
access a p-y curve and a t-z curve for a type of soil for a monopile at each of one or more layers, the monopile including a plurality of nodes in the one or more layers, wherein each p-y curve is a nonlinear function that relates lateral soil reaction force (p) to lateral displacement (y), and each t-z curve is a nonlinear function that relates skin friction (t) to vertical displacement (z);
determine a distributed moment along the monopile and based thereon a rotational stiffness of a given node, wherein the rotational stiffness is calculated as a function of both force per unit area from a respective t-z curve for the layer of the given node and slope of the t-z curve for the layer of the given node;
determine a base moment at a pile tip of the monopile and based thereon a base rotational stiffness, wherein the base rotational stiffness is calculated as a function of both force per unit area from a respective p-y curve for the layer of the pile tip and slope of the respective p-y curve for the layer of the pile tip;
determine a base shear force at the pile tip from the t-z curve and based thereon base shear stiffness, wherein the base shear stiffness is calculated as a function of slope of a respective t-z curve for the layer of the pile tip and pile tip area;
iteratively solve a system of non-linear equations that uses a structural stiffness matrix that includes the rotational stiffness, the base rotational stiffness and the base shear stiffness in addition to pile material structural stiffness to model the monopile; and
output a description of the modeled monopile.
15. The non-transitory computing device readable medium of claim 14 , wherein the monopile is a monopile of an offshore wind turbine.
16. The non-transitory computing device readable medium of claim 14 , wherein the monopile has a diameter of 10 meters or more.
17. The non-transitory computing device readable medium of claim 14 , wherein the monopile has a length-to-diameter ratio between 2 and 6, inclusive.
18. The non-transitory computing device readable medium of claim 14 , wherein rotational stiffness (k θ i ) for a node (i) corresponding to an element (j) is calculated as:
k
θ
i
=
∫
0
2
π
(
d
M
j
d
θ
)
d
α
=
4
R
[
∫
0
π
/
2
T
j
d
j
2
d
α
+
θ
∫
0
π
/
2
(
d
T
j
d
Z
j
)
d
j
3
d
α
]
where θ is an angle of a rotated horizonal plane from an x-axis, α is an angle to the element j in the horizonal plane from the x-axis, R is radius of the monopile, d j is a distance in the horizonal plane from the y-axis, M j is total moment, T j is force per unit area from the respective t-z curve for the layer of the node (i), and dT j /dZ j is slope of the respective t-z curve for the layer of the node (i).
19. The non-transitory computing device readable medium of claim 14 , wherein base rotational stiffness (K M b ) for base (b) given an element (j) is calculated as:
K
M
b
=
∫
-
π
2
π
2
(
d
M
b
d
θ
b
)
d
α
=
2
R
2
[
∫
0
π
2
P
j
d
j
(
Sin
2
α
)
d
α
+
θ
b
∫
0
π
2
(
d
P
j
d
Z
j
)
d
j
2
(
Sin
2
α
)
d
α
where θ b is pile tip rotation, α is an angle to the element j in the horizonal plane from the x-axis, R is radius of the monopile, d j is a distance in the pile tip surface from the y-axis, M b is base moment, P j is force per unit area from the respective p-y curve for the layer of at the pile tip and dP j /dZ j is slope of the respective p-y curve for the layer of at the pile tip.
20. The non-transitory computing device readable medium of claim 14 , wherein base shear stiffness (K V b ), for base (b) is calculated as:
K
V
b
=
π
R
2
d
T
b
d
X
b
where R is radius of the monopile, and dT b /dX b is slope of the respective t-z curve for the layer of at the pile tip.