Dynamic Response Analysis of Floating Nuclear Power Plant Containment Under Marine Environment

Abstract

Floating nuclear containment is in a harsher environment than conventional onshore nuclear containment. In view of the Marine environment under the condition of floating nuclear power plant containment structure safety, combined water dynamics and structural mechanics, considering the containment response under random movement of hull in the Marine environment, the influence of the containment structure load calculation, thus checking containment when working in pile structure safety, provide theoretical basis for the safe operation of floating nuclear power plants. In this paper, taking a floating nuclear power plant as an example, ANSYS 2021R1, Workbench, Fluent and other software of finite element analysis are used to conduct fatigue simulation of floating nuclear power plant. The time course curve of the 6-dof motion of the ship’s center of gravity is obtained, then, a remote displacement method is adopted to transfer the hull motion to the containment vessel to realize the numerical simulation of the containment vessel movement with the hull, thus to solve maximum normal stress and strain, the maximum load component of containment bearing under the action of Marine environmental load is obtained. The results show that the maximum stress and strain of the vessel increase obviously in the moving state compared with the static state of the vessel, which indicates that the random motion response of the vessel must be considered in the structural safety analysis of the floating nuclear power plant containment.

1 Introduction

With the continuous adjustment and optimization of China’s energy structure and the continuous promotion of the strategy of becoming a maritime power, traditional fossil energy and emerging energy such as wind, wave and solar energy are increasingly difficult to meet the energy needs brought about by China’s coastal oil and gas resources and island development. Therefore, as a clean, efficient and flexible location of offshore nuclear power generation technology, the national government pays more and more attention to it. Marine floating nuclear power station is a mobile floating offshore platform equipped with nuclear reactor and power generation system. It is the organic combination of mobile small nuclear power station technology with ship and ocean engineering technology. In floating nuclear power plants, a sealed steel containment structure is usually installed around the reactor and other auxiliary power generation structures to protect the normal operation of the reactor and the external environment. Compared with the traditional onshore nuclear power plant containment, the environment and load borne by the small steel containment (including support) of floating nuclear power plant at sea are very different, especially the complexity of the Marine environment leads to more complex load borne by the containment (including support). Therefore, in order to ensure the safe operation of floating nuclear power plants and protect the surrounding personnel and the external environment from nuclear radiation damage, it is urgent to carry out researches on mechanical analysis and safety evaluation technology of steel containment of floating nuclear power plants in marine environment.

The floating nuclear power plant containment vessel is not only subjected to huge vertical and horizontal loads, but also inevitably subjected to wind load, wave load and current load. Therefore, it is particularly important to analyze the dynamic response of the floating nuclear power plant containment vessel in the Marine environment. Reissner [1] studied the vibration characteristics of rigid circular foundation plate under vertical load, and proposed and verified the feasibility of elastic half-space theory in vibration research of foundation and foundation. Choprah [2] proposed the dynamic substructure method, which made numerical calculation effectively applied in this field. Lysmer et al. [3] proposed lumped parameter method, which laid the foundation for structural dynamic response analysis. Gazetas [4] and Mrakis et al. [5] proposed the calculation and analysis method of pile-soil-structure dynamic interaction, and provided empirical expressions of stiffness coefficients and damping coefficients. Fan Min et al. [6] conducted nonlinear seismic response analysis and research, and the results showed that the soil-pile-structure interaction system would affect the dynamic characteristics of the structure, resulting in the extension of the natural vibration period and the increase of damping of the system. Wang et al. [7] used hydrodynamic model to simulate the evolution of wind, wave and tide under 32 typhoon events in Bohai Bay from 1985 to 2014, and used two-dimensional Gumbel Logistic model to establish the joint distribution of wave and storm surge in Bohai Sea. De Waal and Van Gelder [8] established the joint distribution of extreme wave height and period through Copula function. Michele et al. [9] used two-dimensional Copula function to analyze the frequency of effective wave height, storm duration, storm direction and storm interval of ocean storms, and established the joint probability distribution between pairs. Xu et al. [10] established the two-dimensional joint distribution of storm surge height and effective wave height. Dong Sheng et al. [11] established the joint distribution of the annual maximum wave height and the corresponding wind speed of a jacket platform for 24 years based on Archimedean Copula function, combined with the response of the offshore platform, and found that considering the joint effect, the response of the offshore platform could be reduced in the same return period. Chen Minglu et al. [12] conducted hydrodynamic analysis and wave load prediction for semi-submersible offshore platforms. Zhou Sulian et al. [13] studied the mooring system design of deep-water semi-submersible platform.

Due to the complex wind, wave and current environment under the action of Marine environment of floating nuclear power plant containment, the research on dynamic response characteristics of floating nuclear power plant containment is insufficient and almost no reports have been reported. Therefore, it is of great significance to carry out dynamic response analysis of floating nuclear power plant containment under Marine environment conditions, and to explore the variability and ultimate load effect of short-term time history analysis, so as to understand dynamic response characteristics of floating nuclear power plant containment under Marine environment.

The main work of this paper is to establish a hydrodynamic analysis model, using Ansys Workbench HD software module for frequency domain analysis of the platform and obtain the hydrodynamic parameters of the platform. The anchor chain model was added to the platform model to control the six degrees of freedom movement of the platform under the action of wave and flow. The HR module of Ansys Workbench software was used to conduct time-domain analysis of the platform to obtain the time-history curve of the platform motion response. The structural dynamic response analysis of the containment vessel of floating nuclear power plant under the action of wind load, wave load and current load is carried out, which provides important reference for the safe operation of floating nuclear power plant.

2 Theoretical Basis of Potential Flow

2.1 Small Scale Member

The stress of offshore floating structures in waves is studied. The stress of offshore structures is the most important topic in the field of offshore engineering, in which the wave force of piles is the basis of the stress of offshore structures. The method proposed by Morison et al. in 1950 is used to calculate wave force for small components, that is, structures whose diameter is smaller than the wavelength of the incident wave. Morison equation is basically an empirical formula, which takes wave particle velocity, acceleration and cylinder diameter as parameters to calculate the wave force in each depth of water, and then obtains the wave force along the length of the column.

Morrison et al. believed that the horizontal wave force acting on any height of the cylinder included two components:

$$ f_{H} = f_{D} + f_{I} $$

(1)

Its magnitude is in the same mode as the drag force exerted on the column by unidirectional steady water flow, that is, it is proportional to the square of the horizontal velocity of the wave water point and the projection area of the unit column height perpendicular to the wave direction. The difference is that the wave water points oscillate periodically, and the horizontal velocity is positive and negative, so the drag force on the cylinder is also positive and negative:

$$ f_{D} = \frac{1}{2}C_{D} \rho Au_{x} \left| {u_{x} } \right| $$

(2)

$$ f_{I} = \rho V_{0} \frac{{du_{x} }}{dt} + C_{m} \rho V_{0} \frac{{du_{x} }}{dt} = C_{M} \rho V_{0} \frac{{du_{x} }}{dt} $$

(3)

In the engineering design of floating buildings and piled offshore platforms, one of the main problems to be solved is to determine the movement, stress and deformation of these structures under the action of external forces such as wave and wind. We can regard these structures as a dynamic system, the wave action is called the input of the system, and the movement, stress and deformation of the structure are called the output of the system.

Remember this transformation as:

$$ y(t) = K\left[ {x(t)} \right] $$

(4)

For different systems and different inputs, the operator K may have different forms. According to the different operators, dynamic systems can be divided into linear systems and nonlinear systems. An operator is a linear system if it has the following properties, and its operator is denoted by L.

a. Superposition property

$$ L\left[ {x_{1} (t) + x_{2} (t)} \right] = L\left[ {x_{1} (t)} \right] + L\left[ {x_{2} (t)} \right] = y_{1} (t) + y_{2} (t) $$

(5)

b. The constant $\alpha$ can be removed from the operator

$$ L\left[ {\alpha x(t)} \right] = \alpha L\left[ {x(t)} \right] = \alpha y(t) $$

(6)

therefore:

$$ L\left[ {\sum\limits_{i} {\alpha_{i} x_{i} (t)} } \right] = \sum\limits_{i} {\alpha_{i} y_{i} (t)} $$

(7)

It’s called the principle of linear superposition, which means that the response of a linear system to inputs is equal to the sum of the responses of the inputs acting independently.

The following operations are linear transformations:

$$ y(t) = \frac{{\text{d}}}{{{\text{d}}t}}x(t),\,y(t) = \int_{0}^{T} {x(t)} {\text{d}}t,\,y(t) = \varphi (t)x(t) $$

(8)

In Formula (1-179), $\varphi (t)$ is A non-random function. The above types are linear secondary operations. If a certain function is added, they are called linear non-secondary operations, such as:

$$ y(t) = \frac{{\text{d}}}{{{\text{d}}t}}x(t) + \varphi (t) $$

(9)

Systems whose operators do not conform to the above conditions are called nonlinear systems. Linear systems are often encountered in practical work, and some nonlinear systems can be linearized within a certain range.

2.2 Large Scale Member

Ship hydrodynamic problems can be solved by frequency domain method. The frequency-domain method is based on the assumption that the wave-ship interaction has lasted for quite a long time, the initial disturbance of the incident wave and the transient influence of the initial rocking of the ship have disappeared, and the fluid motion in the field has reached a steady state. In this case, if the incident wave is harmonic, then the ship’s motion is also harmonic (the encounter frequency must be the changing frequency), and the steady-state solution can be obtained in the frequency domain.

Due to the action of waves, the ship has six degrees of freedom besides constant speed forward motion. Assuming that the motion of the six degrees of freedom is small, the ship’s center of gravity G point can be at. The three linear displacements (swing, roll, and heave) and the three angular displacements (roll, pitch, and yaw) around the G point in the O-XYZ coordinate system are represented. In the stable state, its displacement vector will be regarded as the harmonic quantity with the encounter frequency field as the changing frequency:

$$ \left\{ {\eta \left( t \right)} \right\} = \left\{ \eta \right\}e^{i\omega t} = \left( {\eta_{1} \;\eta_{2} \;\eta_{3} \;\eta_{4} \;\eta_{5} \;\eta_{6} } \right)^{T} e^{i\omega t} $$

(10)

According to rigid body dynamics, the ship motion equation with the center of gravity G as the center of moment can be expressed as:

$$ \left[ M \right]\left\{ {\mathop \eta \limits^{..} \left( t \right)} \right\} = \left\{ {F\left( t \right)} \right\} = \left\{ F \right\}e^{i\omega t} $$

(11)

For the convenience of calculation, the fluid loads acting on the hull are divided into two parts: hydrostatic loads due to changes in the position of the ship’s relative hydrostatic equilibrium and hydrodynamic loads dependent on wave and ship motion. Hydrostatic load comes from the contribution of hydrostatic pressure change caused by ship movement, which can be directly given by ship statics as follows:

$$ \left\{ {F^{S} \left( t \right)} \right\} = - \left[ C \right]\left\{ {\eta \left( t \right)} \right\} $$

(12)

Among them, only 5 items of hydrostatic coefficient $C_{ij} ,\,\,i,j = 1,2, \cdots ,6$ are not zero, they are:

$$ \left\{ {\begin{array}{*{20}c} {C_{33} = \rho gA} \\ {C_{35} = C_{53} = - \rho gS_{y} } \\ {C_{44} = \rho g\forall h_{x} } \\ {C_{55} = \rho g\forall h_{y} } \\ \end{array} } \right. $$

(13)

In Formula (13), A and Sy are the waterplane area of the ship and the static moment to the Y-axis, $\forall$ is the drainage volume of the ship, h_x and h_y are the transverse and longitudinal metacentric heights of the ship respectively. According to the different dimensions of the mathematical model used to solve the flow field, it can be divided into three dimensional method and two dimensional method (slice method).

The above equation shows that the free surface condition under low speed has the same form as that under zero speed.

If zero velocity radiation potential $\phi_{j}^{0}$ and $\phi_{j}^{U}$ additional velocity potential $\phi_{j}^{U}$ are defined, let them satisfy continuity equation $\left[ L \right]$ of the fixed solution, bottom condition $\left[ D \right]$, distant radiation condition $\left[ R \right]$, free surface condition and object surface condition $\left[ S \right]$ defined by the following formula:

$$ \left\{ {\begin{array}{*{20}c} {\frac{\partial }{\partial n}\phi_{j}^{0} = n_{j} ,\left( {j = 1,2,...6} \right)} & {It^{\prime}s \, on \, plane \, S} \\ {\frac{\partial }{\partial n}\phi_{j}^{U} = m_{j} ,\left( {j = 1,2,...6} \right)} & {It^{\prime}s \, on \, plane \, S} \\ \end{array} } \right. $$

(14)

For the above fixed solution problem, the disturbance potential $\phi_{j}$ and its gradient $\nabla \phi_{j} \left( {j = 1\sim 7} \right)$ can be determined by using appropriate numerical solution method. Introducing differential operator:

$$ \frac{d}{dt} \equiv \frac{\partial }{\partial t} - U\frac{\partial }{\partial x} = i\omega - U\frac{\partial }{\partial x} $$

(15)

According to the linearized Bernoulli equation, the hydrodynamic pressure after deducting the change of hydrostatic pressure is as follows:

$$ p\left( {x,y,z,t} \right) = - \rho \frac{d}{dt}\left[ {\phi_{T} \left( {x,y,z} \right)e^{i\omega t} } \right] $$

(16)

3 Wave Load and Structural Response in Frequency Domain

3.1 Wave Load in Frequency Domain

A floating nuclear power plant with a total length of 229.8 m and a total weight of 83,200 tons is taken as the numerical simulation object, and the model is modeled by the APDL module in Ansys 2021R1. The finite element model of hull structure is constructed by Shell181 and BEAM188 elements. The mesh size of the bottom and supporting part of the containment is 0.1 m, the mesh size of the upper part of the containment is 0.2 m, and the mesh size of the rest of the containment is 0.8 m. The total number of the whole ship elements is about 1.56 million. The finite element model of the built floating nuclear power plant and containment vessel is shown in the figure below (Figs. 1 and 2).

Fig. 1.

Overall finite element model of floating nuclear power plant structure

Fig. 2.

Finite element model of floating nuclear power plant containment

In general, the hydrodynamic response of large floating body structure is a linear system. Therefore, when calculating the hydrodynamic response of the containment vessel of floating nuclear power plant under random waves, regular waves can be used to calculate first, and the calculated response can be divided by wave amplitude. In this way, the RAO of the floating nuclear power plant containment vessel can be obtained, which preliminarily reflects the hydrodynamic performance of the containment vessel, and then the HD module in Ansys Workbench is used for frequency domain analysis. As the floating nuclear power plant in this paper is symmetric about X-axis and Y-axis, the dynamic response caused by waves within the range of 0º–180º to the containment vessel of floating nuclear power plant is mainly analyzed. A wave direction is set every 15º, which is divided into 13 wave directions in total. The schematic diagram of wave incidence Angle is shown in the figure below (Fig. 3).

Fig. 3.

Diagram of wave incidence Angle

The wave load calculation and structure analysis in this paper are based on linear theory. Under this condition, if the wave is a stationary random process, so is the alternating stress obtained by transformation. According to random process theory, the power spectral density of the above two stationary random processes has the following relationship:

$$ G_{XX} (\omega ) = |T(\omega )|^{2} G_{NN} (\omega ) $$

(17)

For A linear system composed of ships and waves, the stress response follows the characteristics of the linear system, and the synthetic stress can be written as $\sigma = \sigma_{C} + i\sigma_{S}$. In actual calculation, it is necessary to process the loads generated by regular waves of unit amplitude with different frequencies according to the real and imaginary parts respectively to obtain the corresponding response $\sigma_{c}$ and $\sigma_{s}$, and then synthesize it into $\sigma_{A} (\omega_{e} )$. Thus, the transfer function of stress can be written as:

$$ H_{\sigma } (\omega_{e} ) = \sigma_{A} (\omega_{e} ) $$

(18)

P-m spectrum can be written as the expression of different parameters. If it is expressed by the two parameters of meaningful wave height Hs and mean zero-crossing period Tz, the expression of wave spectrum can be written as follows:

$$ G_{\eta \eta } (\omega ) = \frac{{H_{S}^{2} }}{4\pi }(\frac{2\pi }{{T_{Z} }})^{2} \omega^{ - 5} \exp ( - \frac{1}{\pi }(\frac{2\pi }{{T_{Z} }})^{4} \omega^{ - 4} ) $$

(19)

In the analysis, the actual response frequency should be the encounter frequency A, and its relationship with wave frequency A is as follows:

$$ \omega e = \omega (1 + \frac{2\omega U}{g}\cos \theta ) $$

(20)

Therefore, the response spectrum of stress can be expressed as:

$$ G_{\chi \chi } (\omega_{e} ) = |H_{\sigma } (\omega_{e} )|^{2} G_{\eta \eta } (\omega_{e} ) $$

(21)

3.2 Structural Response Calculation of Wave Load

In the pre-processing of AQWA, the mass information and moment of inertia information of the hull structure need to be obtained through the whole ship finite element analysis. The incident wave direction interval is 15°, the number of wet surface units of the hydrodynamic model is 20403, and the total number of units is 38559, as shown in the figure below (Fig. 4).

Fig. 4.

Hydrodynamic model

The calculated frequencies of waves in the frequency domain were 0.01592 Hz–0.27 Hz, and 48 calculated frequency points were interpolated at equal intervals, totaling 50 calculated frequency points.

In AQWA calculating unit amplitude structural response under the action of amplitude, according to the main control parameters (RY), namely the total longitudinal bending moment, derived the relationship between the frequency and phase, and then according to the wave height (2 m), wave Angle of incidence, frequency and phase extraction wet surface wave pressure and the ship’s hull acceleration, and loads it into structural response of the hull computation, as shown in the figure below (Figs. 5 and 6):

Fig. 5.

Stress cloud of containment at wave frequency 0.06139 Hz

Fig. 6.

Stress cloud of containment at wave frequency 0.06688 Hz

4 Wind Load and Structural Response

4.1 Wind Load

The wind load rapid loading plug-in currently in use is based on API 4F specification: “Drilling and Workover Structure Specification”, 2008 edition. Typical structural analysis is derrick, fan pile leg and jacket platform. The API Wind load Quick loading plugin is limited to API 4F specification profiles and methods and is not intended for general use. Other wind codes (like ASCE 7-05/7-10) are not included in this plugin.

This plug-in has the following advantages:

• (1) Suitable for different geometric types. No matter solid, shell or beam structure can take advantage of this plug-in.

• (2) Enable load step selection, allowing multiple wind load conditions.

• (3) Directly implement API 4F specification, allowing factor coverage.

• (4) Wind load can be applied to the leeward side.

• (5) The actual windward surface can be detected.

The total wind force on the structure is estimated by the vector sum of the wind force acting on individual components and accessories, as shown in the following formula:

$$ F_{{\text{m}}} = 0.00338 \times K_{i} \times V^{2}_{z} \times C_{s} \times A $$

(22)

$$ F_{t} = G_{f} \times K_{sh} \times \Sigma F_{m} $$

(23)

In Eqs. (3-1) and (3-2):

Fm – The force of the wind perpendicular to the vertical axis of a single member, or to the surface of the wind wall, or to the projected area of the appendage.

Ki – a factor of the inclination Angle φ between the longitudinal axis of a single member and the wind.

Vz – Local wind speed at altitude Z.

Cs – shape coefficient.

A – The projected area of A single member is equal to the length of the member multiplied by its projected width with respect to the normal wind component.

Gf – Gust effect factor, used to explain spatial coherence.

Ksh – the conversion factor for the total shielding of a member or accessory and the variation of airflow around the end of the member or accessory.

Ft – the vector sum of wind forces acting on each individual member or accessory throughout the drilling structure.

4.2 Structural Response Calculation of Wind Load

In this paper, the wind-loading plug-in in ANSYS Workbench is used. Firstly, the structural model of floating nuclear power plant is fixed rigidly and released inertia, and then constant Wind load is carried out on the Wind receiving surface of floating nuclear power plant (Wind speed 50.7 = 98.56 knot, Wind direction: 13, 0° to 180° with wind direction set at 15° intervals), at the same time, then export the result file, and perform post-processing in classic ANSYS.

The wind speed is 50.7 m/s, and the maximum equivalent stress of the containment is 8.15 MPa. The maximum stress occurs at the junction between the upper support and the bulkhead, as shown in the figure below (Figs. 7, 8 and 9).

Fig. 7.

Ship - wide stress cloud

Fig. 8.

Containment stress cloud map

Fig. 9.

Maximum stress and position of containment under wind load

5 Flow Load and Structural Response

5.1 Current Loading

Ocean currents can be caused by many factors, such as local stationary currents caused by ocean circulation, tidal currents caused by periodic changes in the gravitational pull of the sun and moon on the Earth, differences in the density of ocean water, and the action of wind. It should be pointed out that the speed of wind on the sea surface is about 3% of the speed at 10 m above the sea surface. Tidal currents have an important influence on the flow field in some restricted waters. Tidal currents in restricted waters generally have a speed of 2–3 m/s and a maximum of 10 m/s.

For floating bodies, the flow of the ocean surface is of greatest concern to us. However, for the mooring system at sea, the distribution of water flow along the water depth is also our concern. For the designers, the maximum limit flow encountered during the operation of floating body is the most important factor affecting the design, so the actual measurement and monitoring of water flow velocity is essential. Since the velocity and direction of the water flow change slowly, we can approximately consider the water flow to be steady.

The action of water flow on floating body can be divided into the following two parts:

(1) Viscosity effect. Viscosity resistance due to frictional effects, and differential pressure resistance. For blunt body, the friction resistance can be ignored, and the pressure difference resistance is mainly.

(2) Influence of potential flow. The lift effect caused by the ring volume and the drag effect caused by the free surface effect are small in comparison.

Using flow force coefficient to estimate the flow load on the surface ship floating body:

The flow force/moment can be calculated by the following formula, in which the flow force coefficients are usually determined by model test methods.

$$ \left. \begin{gathered} X_{c} = \frac{1}{2}\rho v_{c}^{2} C_{{X_{c} }} (\alpha_{c} )A_{TS} \hfill \\ Y_{c} = \frac{1}{2}\rho v_{c}^{2} C_{{Y_{c} }} (\alpha_{c} )A_{LS} \hfill \\ N_{c} = \frac{1}{2}\rho v_{c}^{2} C_{{N_{c} }} (\alpha_{c} )A_{LS} L \hfill \\ \end{gathered} \right\} $$

(24)

Similar to the wind load calculation of ship type floating body, the key to the flow load of ship type floating body is how to obtain the flow load coefficient, which is mainly obtained by model test.

For the calculation of flow load of large oil tankers, OCIMF based on model test data gives flow force coefficient curves of two different bow forms, full load and ballast, which have good reference value and are widely used in engineering design and analysis of mooring ships.

Remery and Van Oortmerssen carried out an experimental study in the MARIN Tank to test the flow loads on tanker models of different profiles and sizes. Since ships are mostly slender bodies, the axial flow load is mainly caused by frictional resistance. If the axial flow velocity is small, this resistance is difficult to measure, and it is not accurate to predict the axial resistance of real ships from model tests, because the scale effect is very obvious.

The axial flow load is important for anchored vessels. The force can be estimated based on the frictional resistance of the plate. The following formula is recommended by ITTC:

$$ X_{c} = \frac{0.075}{{\left( {\log_{10} \left( {R_{N} - 2} \right)} \right)}} \cdot \frac{1}{2}\rho V_{{\text{c}}}^{2} \cos \alpha_{c} \cdot \left| {\cos \alpha_{c} } \right| \cdot S $$

(25)

$$ {\text{Re}} = \frac{{\left| {\cos \alpha_{c} } \right|V_{c} \cdot L}}{\upsilon } $$

(26)

For tanker, it is generally not a problem to estimate the transverse force and yawing moment of the real ship. For the transverse flow of ocean current in an oil tanker, the ship can be regarded as a blunt body. Because the bilge radius of the hull is relatively small, it can be considered that the flow separation situation in the model and the real ship is consistent, and the transverse force and the bow rolling moment can be considered independent of the Reynolds number.

In MARIN pool, the lateral force and bow rolling moment are expanded into Fourier series through experiments.

$$ C_{{Y_{c} }} \left( {\alpha_{c} } \right) = \sum\limits_{1}^{n} {b_{n} \cdot \sin \left( {n \cdot \alpha_{c} } \right)} $$

(27)

$$ C_{{N_{c} }} \left( {\alpha_{c} } \right) = \sum\limits_{1}^{n} {c_{n} \cdot \sin \left( {n \cdot \alpha_{c} } \right)} $$

(28)

The above formula can be applied in deep water, but for shallow water, the lateral force and bow torque coefficients need to be multiplied by a correction factor. In practice, the free surface effect is small for deep water with flow rates of 3kn.

5.2 Structural Response Calculation of Flow Load

In this paper, Workbench Fluent and classical ANSYS 2021R1 are used to realize flow field analysis and structural response calculation. The viscous flow model, namely K-Omega (2EQN) SST model, is selected for flow field analysis. A THREE-DIMENSIONAL flow field model is established in Workbench DM module, and the flow field size is $500 \times 600 \times 27.6$ (m), as shown in the figure below (Fig. 10):

Fig. 10.

3d model of flow field

The mesh of fluid domain is divided in Workbench mesh module. The mesh size of near-field fluid domain is 1.5 m, and that of far-field fluid domain is 3.5 m. The flow field and structural stress (75° flow direction as an example) corresponding to flow velocity of 0.83 m/s are shown as follows (Fig. 11):

Fig. 11.

Calculation results of 75° flow direction

At a flow rate of 0.83 m/s and a flow direction of 75°, the maximum equivalent stress of the containment fourth strength is 1.15 MPa. The maximum stress occurs at the arc transition of the equipment base inside the containment vessel, as shown below (Fig. 12).

Fig. 12.

Maximum stress position

6 Conclusions

• (1) Under the action of wave load in the frequency domain, the maximum load component (absolute value) of the containment support is as follows: F_X = ${2}{\text{.68}} \times {10}^{{5}} N$ (wave direction 60°), F_Y = $5.89 \times {10}^{{6}} N$ (wave direction 90°), F_Z = $1.16 \times {10}^{{6}} N$ (wave direction 90°), M_X = $3.93 \times {10}^{7} N \cdot m$ (wave direction 90°), M_Y = $1.93 \times {10}^{6} N \cdot m$ (wave direction 75°), M_Z = $1.13 \times {10}^{5} N \cdot m$ (wave direction 105°). This part of the resultant force only includes the structural response caused by waves, not the structural response caused by static equilibrium.

• (2) Under the action of wave load in the frequency domain, the maximum equivalent stress of the containment vessel appears when the wave direction is 90° and the frequency is 0.09433 Hz, and the size is 98.3 MPa. The maximum stress occurs at the junction between the upper support and bulkhead.

• (3) When the wind speed is 50.7 m/s and the wind direction is 75°, the equivalent stress of the fourth strength of the containment vessel is the largest, which is 8.15 MPa. The maximum stress occurs at the junction between the upper support and bulkhead.

• (4) When the flow velocity is 0.83 m/s and the flow direction is 75°, the equivalent stress of the fourth strength of the containment vessel is the largest, which is 1.15 MPa. The maximum stress occurs at the arc transition of the equipment base inside the containment vessel.

• (5) The wind speed and flow velocity are 50.7 m/s and 0.83 m/s respectively, and the maximum structural stress caused by wind load and flow load is 8.15 MPa and 1.15 MPa respectively, which can be almost ignored. Therefore, wind load and flow load can be ignored in the analysis of ultimate load.