Structural Analysis and Design

The structural modelling of WECs is a rather complicated process through which few devices have undergone in a detailed manner. There are a number of reasons for this, primarily the fact that numerical modelling of survival conditions is beyond the state of the art in numerical modelling methods used in the estimation of power absorption. BIEM codes rely on the assumption of linear wave theory, i.e. small amplitude waves relative to the wavelength. Survival conditions typically involve highly nonlinear wave forms and the reaction of the device following slamming and

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splashing events. Emerging numerical techniques including CFD and meshless methods may provide the answer to this. In the past and for the near future, the analysis of device responses and loads in survival conditions have been carried out in physical model tests at a scale typically larger than that used in functionality tests.

This process is time consuming and quite expensive due to the requirement for large basins to accommodate large models etc. This section will investigate the possible simplified methods of assessment of the structural loads in WECs from a number of perspectives and determine the most appropriate governing loads to use in the estimation of structural materials for its construction.

2.3.5.1 Breaking Wave Loads Estimations and Local Strength Design

One of the primary concerns for many WEC structures is the hydrodynamic pressures imposed on the structure during breaking wave events. While the occurrence of these events is typically seldom, the structure nonetheless must be designed to resist the large impact loads. One of the primary challenges faced by designers of conceptual structures, is the estimation of structural materials required for survivability of offshore structures. In many cases, this estimate is guided by experience of similar structures etc., however in the case of innovative MRE platforms, simplified estimates of the hydrodynamic pressures imposed on the structure are required, typically referred to as “basic scantlings”. Reference [102]

suggests a formula for the estimation of breaking wave loads on vertical piles in Equation 2-1,

𝐹_𝑏𝑟 = 1

2 𝐶^𝑑𝑏𝜌𝑔𝐷𝐻_𝑏² Equation 2-1

where F_br is the breaking wave force acting at the stillwater line (N), C_db is the drag coefficient, ρ is the density of seawater (kg/m³), g is the acceleration due to gravity (m/s²), D is the pile diameter (m) and Hb is the breaking wave height (m). The force estimated by this formula is a concentrated force acting at the stillwater line. To estimate the thickness of steel plate required for a small WEC, this force would be distributed over an area of the WEC structure as shown in Equation 2-2.

𝑝 = 𝐹𝑏𝑟

𝐴

Equation 2-2

where p is the hydrodynamic pressure imposed on the WEC structure (N/m²) and A is the area affected by the breaking wave (m²). Furthermore, if numerical modelling, even in the frequency domain, is carried out, the linearised response amplitude operator (RAO) of hydrodynamic pressure on each panel of the discretised wetted hull is produced. This can also be used to produce a local design of the hull.

The hydrodynamic pressure estimated from these methods may be used as an input into Equation 2-3 from [103] Subsection 6,

89 𝑡 = 15.8 𝑠 𝑘_𝑎 𝑘_𝑟 �𝑝

�𝜎𝑝𝑘_𝑝 + 𝑡𝑘 Equation 2-3

where t is the estimated plate thickness (mm), s is the stiffner spacing (mm), p is the hydrodynamic pressure (kN/m²), k_a is an aspect ratio, k_p represents the long side (stiffner) boundary condition, k_r is a curvature factor, σp is the nominal yield based allowable stress (kN/m²) and tk is the added plate thickness to resist corrosion (mm).

For a conservative estimate, ka, kr and kp may be assumed to be one, while assuming a suitable corrosion prevention system is in place on the structure, t_k may be assumed to be zero. This plate thickness design method assumes that, e.g. column diameters are greater than 12m. Smaller members would most likely be ring stiffened and the equation is inapplicable. To estimate the stiffner mass, reference [103] Subsection 6 uses Equation 2-4,

𝑍 = 1000 𝑙² 𝑠 𝑝

𝑚 𝜎_𝑝 𝑘_𝑠 + 𝑍_𝑘 Equation 2-4

where Z is the section modulus (cm³), l is the plate length (m), s is the stiffner spacing (mm), p is the hydrodynamic pressure (kN/m²), m is the denominator of the beam equation, σp is the nominal yield based allowable stress (kN/m²), k_s is a stiffner factor and Zk is the additional steel to account for corrosion. This equation is coupled with a minimum requirement for Z to be 15cm³.

These equations provide a preliminary estimate of the required structural material required for local strength checks. This methodology may be used for smaller WEC structures, while a more global strength approach is appropriate for larger WEC structures.

2.3.5.2 Global Strength Design of Large Structures

The global strength of a large structure depends on the structural configuration. It may be a long slender vertical beam, such as a spar foundation, or a horizontal beam, such as a barge or ship, or a space frame structure such as semi-submersibles and tension leg platforms. Each of these structure types would require specific consideration depending on the fabrication methods, installation methods etc., to determine what the global loads distribution would be throughout the lifecycle.

These load cases are required to determine the governing load case and the design would be such to satisfy the worst case scenario. To address all possibilities of worst case loading for each structure would result in an exhaustive report, therefore as an example, consider the global loading of a ship in extreme wave conditions. The vessel may be supported only at the bow and stern by a large wave with a wavelength equal to the length of the ship. This scenario results in the majority of the hull being unsupported. The ship essentially would be pinned at either end, and a simple static structural analysis may be carried out to give a preliminary estimate of the basic scantlings of the hull. The problem may be simplified further by considering the maximum load of the vessel, i.e. including ballast for stability and

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containers full, and transform this into an equivalent uniformly distributed load (UDL) by dividing by the length between the bow and stern. The maximum longitudinal bending moment may then be estimated from Equation 2-5.

𝐵𝑀_𝑚𝑎𝑥 = 𝜔𝑙²

8 Equation 2-5

where BM_max is the maximum longitudinal bending moment (kNm), ω is the UDL (kN/m) and l is the ship length (m). This preliminary estimate of the bending moment may be used to estimate the steel plate thickness required for the hull to resist global loading.

Figure 2-66: Pinned oil tanker

Reference [104] carried out physical model testing of the WEPTOS WEC at the Cantabria Coastal and Ocean Basin (CCOB) in Spain at a scale of 1:15. The WEPTOS is an innovative triangular shaped device with a number of “Salter Duck”

type WECs mounted on each side of the “V” shape. The internal angle of the “V”

can be changed dynamically through a sliding rail joined between the beams. The link between the two outer main beams is of a constant length and therefore as it slides towards the bow of the device it reduces the angle in the “V” shape. This is to reduce the loads on the mooring lines during survival sea states. The structural bending moments (BM) in the main beams have been monitored by strain gauges mounted on a measuring flange incorporated into the main beams. The BM was measured during both operational and survival sea states. It was found that the vertical BM in the main beams reduced to a value below the BM during operational conditions due to a reduction in the enclosed angle between the main beams from 90° in operational conditions to 30° in survival conditions. The longitudinal BM continued to increase significantly however due to the larger wave conditions. The largest recorded value of the longitudinal BM was 2151Nm. Reference [105] report that the 1:15 model weighed 1150kg with beam lengths of 7.4m, therefore each beam would have a uniform load of 762N/m. Treating the beam as simply supported at each end under its own self weight load, the maximum longitudinal BM would be 5216Nm. Therefore this ‘simply supported’ method of approximation provides an additional factor of safety of c. 2.5.

91 2.3.6 Mooring Loads

Many of the numerical methods mentioned in the previous section are capable of modelling sufficiently accurately the behaviour of moorings and the associated loads in nonlinear wave conditions.

For a preliminary design of the mooring lines and anchors, a simplified method of estimation of the mooring loads is required. The predominant method used in literature is the force estimated by Morison’s Equation (Equation 2-6).

𝐹𝑥(𝑡) = 𝜌𝐶𝑚V𝑈̇ + 1

2 𝜌𝐶^𝑑𝐴𝑈|𝑈| Equation 2-6

where Fx is the total horizontal force (N), ρ is the fluid density (kg/m³), Cm is the inertia coefficient, V is the structure volume (m³), and U is the fluid velocity (m/s), Cd is the drag coefficient and A is the structure area perpendicular to the flow direction (m²). As an example, consider the case of the Sevan FPSO facility. This structure has a diameter of 60m and a draft of 18m. Considering an inertial coefficient of 2, and a drag coefficient of 1.2, maximum wave height of 22m and period of 14s, the total load is approximately 628MN, as illustrated by Figure 2-67.

Figure 2-67: Morison’s Equation Example

Based on the inputs noted above, the inertial forces clearly dominate. This would vary however depending on the structure type and design. The determination of the inertial and drag coefficients for new platform concepts will inevitably present a problem for designers at an early stage. Drag coefficients may be taken from [106].