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An extension of a discrete-module-beam-bending-based hydroelasticity method for a flexible structure with complex geometric features
Ocean Engineering 163 (2018) 22–28
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Short communication
An extension of a discrete-module-beam-bending-based hydroelasticity method for a flexible structure with complex geometric features
T
Xiantao Zhanga,b, Da Lua,∗ a b
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, China School of Civil, Environmental and Mining Engineering, The University of Western Australia, Australia
A R T I C LE I N FO
A B S T R A C T
Keywords: Hydroelasticity Waves Flexible structure Finite element method
The discrete-module-beam-bending based hydroelasticity method proposed by Lu et al. (2016) deals with the hydroelastic response of a flexible structure by discretising it into several rigid submodules which are connected by Euler-Bernoulli beam elements. A stiffness matrix is determined in terms of the geometrical and physical properties of the flexible structure for each beam element and it has an analytical form for simple geometric features such as a rectangular cross section unchanged along the longitudinal direction. However, the analytical stiffness matrix may not exist for a flexible structure with complex geometric features. In the present study, with the aid of finite element method, the discrete-module-beam-bending-based hydroelasticity method proposed by Lu et al. (2016) is extended to be applicable for a flexible structure with complex geometric features.
1. Introduction Hydroelasticity is concerned with the deformation of flexible structures responding to hydrodynamic excitations and simultaneously the modification of the excitations due to the structural deformation. Traditional three-dimensional hydroelasticity theory based on mode superposition approach has been widely adopted for the dynamic response of flexible structures in waves (Senjanović et al., 2008). This method contains three steps: (1) Evaluation of the natural oscillation modes of the flexible structure; (2) hydrodynamic analysis for each mode and (3) Superposition of all modes together and solution of the coupled modal equation to obtain the hydroelastic response of the flexible structure. Unlike the traditional mode superposition approach, Lu et al. (2016) proposed a discrete-module-beam-bending based hydroelasticity method. The underlying idea is outlined as follow. First, a continuous flexible structure is discretised into several rigid submodules. Multirigid-body hydrodynamics is adopted to obtain the hydrodynamic forces (wave excitation force, added mass force and radiation damping force) on each rigid submodule, which, together with the hydrostatic force and inertia force, comprises the total external force on each submodule. Then each submodule is simplified as a lumped mass at its center of gravity and adjacent lumped masses are connected by a beam element to account for effects of structural deformation. Finally, the equations of motion for a flexible structure in waves are established by considering the equilibrium of total external forces and structural
∗
Corresponding author. E-mail address: [email protected] (D. Lu).
https://doi.org/10.1016/j.oceaneng.2018.05.050 Received 17 February 2018; Received in revised form 18 April 2018; Accepted 27 May 2018 0029-8018/ © 2018 Elsevier Ltd. All rights reserved.
deformation induced forces on each lumped mass. Some recent researches on application of this hydroelasticity method can refer to Sun et al. (2018), Wei et al. (2017), Xu et al. (2017), Zhang et al. (2018a, 2018b). The flexible structures investigated in the above-mentioned researches have simple shapes with unchanged cross section in the longitudinal direction (a rectangular plate in Sun et al. (2018), Wei et al. (2017), Xu et al. (2017) and Zhang et al. (2018b). And an elliptic cylinder in Zhang et al. (2018a)). Thus for each beam element added between two adjacent lumped masses, an analytical stiffness matrix exists (see Appendix A). However, in reality, large ships or very large floating structures may have complicated geometric features and thus no analytical stiffness matrix exists for the beam element. The aim of the present study is to extend the discrete-module-beambending-based hydroelasticity method proposed by Lu et al. (2016) to be applicable for a flexible structure with complex geometric features with the aid of finite element method. An outline of the underlying idea for the extension is given and some validations are provided. 2. Extension of the discrete-module-beam-bending based hydroelasticity method 2.1. Revisitation of the hydroelasticity method The discrete-module-beam-bending-based hydroelasticity method proposed by Lu et al. (2016) is revisited here. For Lu's hydroelasticity
Ocean Engineering 163 (2018) 22–28
X. Zhang, D. Lu
Fig. 1. Schematic of the discrete-module-beam-bending based hydroelasticity method.
2.2. Extension of the hydroelasticity method If the flexible structure has a simple shape, an analytical expression of the beam element stiffness matrix KE (the dimension is 12×12) can be given. For example, for a uniform beam element with a rectangular cross section, the expression of KE is given in Appendix A. However, for a flexible structure with complex geometric features, we can never obtain such an analytical expression for the beam element stiffness matrix. As a result, the finite element approach will be adopted to extend the present hydroelasticity method. When it comes to the finite element approach, we mean that the beam between two adjacent lumped masses is discretised into a large number of small elements (a standard discretization process used in the finite element analysis). We consider a beam element with arbitrary shape of cross section and varied geometric features along the longitudinal direction shown in Fig. 2. This beam element is bounded by the center of gravity of the ith and jth submodules. The extension of the approach actually includes two steps: (1) to represent the physical variables (displacement and forces) of all the nodes of the beam element by the physical variables of nodes on the cross section of the beam element; and (2) to represent the physical variables of nodes on the cross section by the ones at the center of cross section of the beam element. The finite element method (FEM) is applied to the beam element for calculation of the overall stiffness of the structure. Suppose that the beam element is discretised into a number of small elements with the number of nodes n. Each node has three components of displacement
Fig. 2. A beam element with arbitrary shape of cross section and varied geometric features along the longitudinal direction.
method, a linear hydrodynamic approach is used and the solution is obtained in frequency domain. As shown in Fig. 1, a continuous flexible structure is first discretised into N rigid submodules. Then multi-rigidbody hydrodynamics theory can be used to obtain the wave excitation force FE , added mass force FA = ω2 A (ω) ξ , radiation damping force FRd = iωB (ω) ξ for all submodules. Here, ω is the wave frequency; ξ is the total displacement vector at the center of each submodule with the dimension 6N×1 (each submodule has 6 degrees of freedom); A(ω) and B(ω) are the added mass and radiation damping, respectively (with the dimension 6N×6N). The hydrostatic restoring force is FHs = −Cξ with C (whose dimension is 6N×6N) being the hydrostatic restoring coefficients. The inertia force of all submodules is FIn = ω2 Mξ , where M (whose dimension is 6N×6N) is the mass matrix. The hydrodynamic formulations related to the calculation of the above-mentioned forces can be referred to Lu et al. (2016) and Zhang et al. (2018b). The total external force exerted on the center of all submodules is
FT = FE + FA + FRd + FIn + FHs
T
ξ∗p = ⎜⎛ξ∗xp ξ∗yp ξ∗zp⎟⎞ and ⎝ ⎠ ∗ ∗ ∗ ∗ T F p = (Fxp F yp Fzp) (p = 1, 2, …, n). The force is related with the displacement by the spatial stiffness matrix K∗(the dimension is 3n×3n) as and
(1)
Subsequently, each submodule is simplified as a lumped mass at its center of gravity. And the adjacent lumped masses are connected by a beam element, the geometrical and physical properties of which are derived from the flexible structure. The force on lumped masses caused by structural deformation is FSt = −K St ξ , where K St (6N×6N) is the stiffness matrix of the entire structure and it is given by overlaying the beam element stiffness matrix KE according to the standard process of finite element method. More details can be referred to Zhang et al. (2018a, 2018b). Finally, the equations of motion of a flexible structure in waves are established in frequency domain by considering the equilibrium of forces for each lumped mass,
{ −ω2 (M + A (ω)) − iωB (ω) + (C + KSt)} ξ = FE
force,
∗
which
is
⎡ ξ1 ⎤ ⎢ ξ∗ ⎥ = F∗ = K∗ξ∗ = K∗ ⎢ 2 ⎥ ⎢ ⋮⎥ ⎢ ξ∗ ⎥ ⎣ n ⎦3n × 1
defined
as
∗
⎡ F1 ⎤ ⎢ F∗2 ⎥ ⎢ ⎥ ⎢ ⋮⎥ ⎥3n × 1 ⎢ F∗n ⎦ ⎣
(3)
where the spatial stiffness matrix K∗ can be obtained using the software ANSYS (2013). In Eq. (3), the nodes are spatially distributed within the beam element. This means that there are nodes both on the two end cross sections of the beam (two cross sections where the centers i and j are located) and in the inner space of the beam. So the first step is to represent the physical variables (i.e. displacements and forces) of all the nodes of the beam element by the physical variables of nodes on two end cross sections of the beam element, i.e. to establish the stiffness matrix of the nodes on two end cross sections, K B* . This process is quite similar to the static condensation process in finite element method (Wilson, 1974). Eq. (3) can be modified as
(2)
By solving Eq. (2), the displacement at the center of gravity of each submodule is obtained. Then the beam bending theory can be applied to solve the structural deflection, shear force and bending moment.
23
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X. Zhang, D. Lu
* * * * ⎡ KBB KBI ⎤ ⎡ ξ B ⎤ ⎡ FB ⎤ ⎢ ⎢ K* K* ⎥ ξ * ⎥ = ⎢ F* ⎥ II ⎦ ⎣ I ⎦ ⎣ IB ⎣ I⎦
(i)
⎡ T1 ⎤ T (i) ( i ) T = ⎢ 2 ⎥ and ⎢ ⋮⎥ ⎢ Tn(ii) ⎥ ⎣ ⎦
(4)
where ξ B* and ξ *I are the displacement of the nodes on two end cross sections and in the inner space of the beam element, respectively. FB* and F*I are the external forces of the nodes on two end cross sections and * * , KBI , K∗IB , K*II in the inner space of the beam element, respectively. KBB ∗ are the partitioned matrix of the matrix K . It should be noted that F*I = 0 because these nodes are inner nodes of the beam element and no external forces are exerted on inner nodes (for the discrete-modulebeam-bending method, all forces have been assumed to be exerted on the center of gravity of each submodule). From Eq. (4), we can obtain the following expressions, −1
* * K B* ξ B* = (KBB − KBI [K*II] K*IB) ξ B* = FB*
(8)
where T (pi) (T (Pj) ) is the partitioned matrix of T (i) (T (j) ) corresponding to the transformation of the displacements from the center i (j) to the pth node. Assuming that the relative position of the pth node (which belongs to the left end cross section) to center i of the cross section is rip = [x ip yip z ip]T , then the expression of T (pi) is given as follows,
⎡1 T (pi) = ⎢ 0 ⎢ 0 ⎢ ⎣
(5)
0
0
1 0
0 −z ip 1 yip
0
z ip 0 − xip
− yip x ip 0
⎤ ⎥ ⎥ ⎥ ⎦3 × 6
(9)
Similarly, we can obtain T (pj) as follows,
The second step is to represent the physical variables of nodes on two end cross sections by the ones at the center of each cross section of the beam element (center i and j in Fig. 2). The equivalent displacements and forces on the centers of two cross sections at the end of the beam are defined as ξ CG and FCG , respectively. If the equivalent stiffness is defined as K CG , then
K CGξCG = FCG
(j )
⎡ T1 ⎤ T (j) T (j) = ⎢ 2 ⎥ ⎢ ⋮⎥ ⎢ Tn(jj) ⎥ ⎣ ⎦
T (pj)
⎡1 = ⎢0 ⎢ 0 ⎢ ⎣
0
0
0
1 0
0 1
− zjp 0 − x jp yjp
zjp
− yjp x jp 0
⎤ ⎥ ⎥ ⎥ ⎦3 × 6
(10)
where rjp = [x jp yjp zjp]T is the relative position of the pth node (which belongs to the right end cross section) to center j of the cross section. As the forces on the nodes of two end cross sections and the relative coordinates of nodes to the centers of two cross sections are both known, it can be easy to transform the forces on the nodes FB* to the forces on the centers of two end cross sections FCG by the following expression,
(6)
The problem now is to seek the relationship between the equivalent stiffness matrix K CG and the stiffness matrix of nodes on two end cross sections, K B* , which will make use of the rigid-body-motion transformation matrix linking between the motion of center i and j and the motion of other nodes on two end cross sections. The displacements of nodes on two end cross sections can be related to the displacements at the centers (i and j in Fig. 2) of two cross sections as follows,
FCG = TT FB*
(11)
Combining Eqs. 5–11, we have
K CG = T TK B*T T (i) ξ B* = TξCG = ⎡ ⎢ 0 ⎣
(i) 0 ⎤ ⎡ ξ CG ⎤ ⎢ (j) ⎥ j ( ) ⎥ T ⎦ ξ CG ⎦ ⎣
−1
* * K B* = KBB − KBI [K*II] K*IB * * KBI ⎤ ⎡K K∗ = ⎢ BB * * ⎥ ⎣ KIB KII ⎦
(7)
where T is a transform matrix. If the number of nodes on two end cross sections is n1, then the dimension for ξ B* , T and ξ CG is 3n1×1, 3n1×12 and 12×1, respectively. It should be noted that the dimension is 6×1 for (i) (j ) (i) (j ) and ξ CG . This is because ξ CG and ξ CG represent the displaceboth ξ CG ments of two rigid submodules used in hydroelastic analysis, which have six components including surge, sway and heave (linear displacement along x, y and z direction, respectively) and roll, pitch and yaw (angular displacement along x, y and z axis, respectively). The number of nodes at the left and right end cross section is ni and nj (ni + nj = n1), respectively. Then
(12)
Eq. (12) shows the relationship between the equivalent stiffness at the centers of two end cross sections K CG and the stiffness matrix of all nodes of a beam element K∗. And K∗ can be obtained with the aid of a finite element software such as ANSYS. Using Eq. (12), the hydroelastic response of a floating flexible structure with complex geometric features can be solved. 2.3. Validation of the extension of the hydroelasticity method First, we choose a cubic steel model, the physical properties and
Fig. 3. A cubic steel model used for the purpose of validation. (a) The geometrical and physical properties; (b) The discretization of the cubic model used in ANSYS software. 24
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X. Zhang, D. Lu
Table 1 The stiffness matrix KAnal calculated by analytical expression (× 1010 ).
1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
5
6
7
8
9
10
11
12
8.40 0 0 0 0 0 −8.40 0 0 0 0 0
0 0.291 0 0 0 1.45 0 −0.290 0 0 0 1.45
0 0 0.291 0 −1.50 0 0 0 −0.290 0 −1.50 0
0 0 0 1.82 0 0 0 0 0 −1.80 0 0
0 0 −1.50 0 10.1 0 0 0 1.45 0 4.47 0
0 1.45 0 0 0 10.1 0 −1.50 0 0 0 4.47
−8.40 0 0 0 0 0 8.40 0 0 0 0 0
0 −0.290 0 0 0 −1.50 0 0.291 0 0 0 −1.50
0 0 −0.290 0 1.45 0 0 0 0.291 0 1.45 0
0 0 0 −1.80 0 0 0 0 0 1.82 0 0
0 0 −1.50 0 4.47 0 0 0 1.45 0 10.1 0
0 1.45 0 0 0 4.47 0 −1.50 0 0 0 10.1
Table 2 The stiffness matrix K CG calculated by finite element method (× 1010 , from Eq. (12)).
1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
5
6
7
8
9
10
11
12
8.40 0 0 0 0 0 −8.40 0 0 0 0 0
0 0.299 0 0 0 1.49 0 −0.30 0 0 0 1.49
0 0 0.299 0 −1.50 0 0 0 −0.30 0 −1.50 0
0 0 0 1.82 0 0 0 0 0 −1.80 0 0
0 0 −1.50 0 10.3 0 0 0 1.49 0 4.67 0
0 1.49 0 0 0 10.3 0 −1.50 0 0 0 4.67
−8.40 0 0 0 0 0 8.40 0 0 0 0 0
0 −0.30 0 0 0 −1.50 0 0.299 0 0 0 −1.50
0 0 −0.30 0 1.49 0 0 0 0.299 0 1.49 0
0 0 0 −1.80 0 0 0 0 0 1.82 0 0
0 0 −1.50 0 4.67 0 0 0 1.49 0 10.3 0
0 1.49 0 0 0 4.67 0 −1.50 0 0 0 10.3
hydroelastic response of the model is calculated for heading waves of different wave length. The continuous structure is discretised into 8 submodules to perform the hydroelastic analysis (refer to Fig. 1). The structural stiffness matrix for each beam element is calculated using both analytical expression in Appendix A (denoted as ‘Approach 1’) and FEM method with Eq. (12) (denoted as ‘Approach 2’). The finite element model of a beam element to connect the centers of gravity of two adjacent submodules is shown in Fig. 5. The length, width and height of this macro-beam element are 60, 60 and 2 m, respectively. The number of elements in these three directions are 20, 20 and 4, respectively. The experimental results obtained by Yago and Endo (1996) is denoted as ‘Experiment’. The numerical results calculated using three-dimensional hydroelasticity method based on mode-superposition approach (Fu et al., 2007) is denoted as “Mode-superposition”. As shown in Fig. 6, the dynamic responses of the continuous VLFS in waves are almost identical for two approaches, both of which agree well with experimental results. This further validates the correctness of extension of the hydroelasticity theory proposed by Lu et al. (2016). A detailed comparison of numerical results calculated by different numerical methods with experimental results (expressed as the relative differences) is listed in Table 4.
Table 3 The difference between KAnal and K CG (%).
1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
5
6
7
8
9
10
11
12
0 0 0 0 0 0 0 0 0 0 0 0
0 2.79 0 0 0 2.79 0 2.79 0 0 0 2.79
0 0 2.79 0 2.79 0 0 0 2.79 0 2.79 0
0 0 0 0.29 0 0 0 0 0 0.29 0 0
0 0 2.79 0 2.01 0 0 0 2.79 0 4.53 0
0 2.79 0 0 0 2.01 0 2.79 0 0 0 4.53
0 0 0 0 0 0 0 0 0 0 0 0
0 2.79 0 0 0 2.79 0 2.79 0 0 0 2.79
0 0 2.79 0 2.79 0 0 0 2.79 0 2.79 0
0 0 0 0.29 0 0 0 0 0 0.29 0 0
0 0 2.79 0 4.53 0 0 0 2.79 0 2.01 0
0 2.79 0 0 0 4.53 0 2.79 0 0 0 2.01
dimension of which are given in Fig. 3(a). For this simple shape, the stiffness matrix can be calculated using the expression given in Appendix A. The result is noted as KAnal and it is given in Table 1. The stiffness matrix for this cubic model is also calculated from Eq. (12) with the aid of finite element software ANSYS. The steel model is discretised with the number of nodes 20, 8 and 6 in the direction of length, width and height, respectively (see Fig. 3(b)). The ANSYS software is used to extract the stiffness matrix K∗, after which the equivalent stiffness matrix for the centers of two end cross sections K CG is calculated using Eq. (12). The result is given in Table 2. And the difference between KAnal and K CG is given in Table 3. It can be seen that the relative error of all components of the stiffness matrix calculated by two approaches is within 10%, which validates the correctness of Eq. (12). Next we consider a model adopted in the experiment conducted by Yago and Endo (1996) to further evaluate the above-mentioned methodology. As shown in Fig. 4, the model is a continuous pontoon-type very large floating structure (VLFS). The physical properties and dimension shown in Fig. 4 all correspond to the prototype scale. The
3. Conclusions The discrete-module-beam-bending-based hydroelasticity method proposed by Lu et al. (2016) is extended to be applicable for a flexible structure with complex geometric features with the aid of finite element method. First, the elastic beam (used to consider the structural deformation effects) is considered as a macro-element and it is discretised following the standard finite element method, after which the structural stiffness matrix of this macro-element can be obtained. Using the static condensation technique and rigid-body-motion transformation matrix, the equivalent stiffness matrix corresponding to the displacements at the centers of two end cross sections of this beam can then be derived 25
Ocean Engineering 163 (2018) 22–28
X. Zhang, D. Lu
Fig. 4. Schematic of a very large floating structure (VLFS).
hydroelastic response of a flexible structure with complex geometric features can be solved. The extension of the hydroelasticity method is validated by (Ⅰ) a small difference (within 5%) of the stiffness matrix for a cubic steel model calculated using the analytical formulation and the proposed approach and (Ⅱ) a good agreement between the experimentally measured vertical response of a VLFS and that calculated using the present method. The extension of the discrete-module-beam-bending based hydroelasticity method is a big step for its application in practical engineering problems. In the future work, relevant experiments will be carried out on the hydroelastic response of a flexible structure with complex geometric features in waves to further validate the present study.
Acknowledgement The first author would like to acknowledge the support of the IPRS, APA and Shell-UWA offshore engineering PhD research top-up scholarships.
Fig. 5. The finite element model of a beam element to connect the centers of gravity of two adjacent submodules.
from
the
macro-element's
stiffness
matrix.
Subsequently,
the
Appendix A The stiffness matrix KE for a uniform beam element with unchanged cross section is given as follows, EA
⎡ l ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 KE = ⎢ −EA ⎢ ⎢ l ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣
12EIz l3 (1 + βy )
0
12EIy
0
0
0
−6EIy
6EIz
Symmetry
l3 (1 + βz )
l2 (1 + βz )
Gκh3b l
0
(4 + βz ) EIy l (1 + βz )
0
0
0
0
0
0
0
−12EIz
0
0
0
−12EIy
0
l2 (1 + βy )
l3 (1 + βy )
0 0 0 6EIz l2 (1 + βy )
l3 (1 + βz )
0 −6EIy l2 (1 + βz )
0
−Gκh3b l
0 0
6EIy l2 (1 + βz )
0 (2 − βz ) EIy l (1 + βz )
0
(4 + βy ) EIz l (1 + βy )
0
EA l
−6EIz
0
l2 (1 + βy )
12EIz l3 (1 + βy )
0
0
0
0
0
0
0
0
0
0
−6EIz
(2 − βy ) EIz l (1 + βy )
l2 (1 + βy )
12EIy l3 (1 + βz )
0 6EIy l2 (1 + βz )
0
Gκh3b l
0 0
(4 + βz ) EIy l (1 + βz )
0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (4 + βy ) EIz ⎥ l (1 + βy ) ⎥ ⎦
(A1)
The size of the symmetric matrix is 12 × 12. l is the length of beam element (the dimension along X axis). b is width of the beam element (the dimension along Y axis). h is the height (the dimension along Z axis). κ is a constant related to b and h. A is cross-sectional area with A = bh. E is Young modulus. βy is the correction factor considering shear deformation for the beam element bending in XOY plane. βz is the correction factor considering shear deformation for the beam element bending in XOZ plane. Iy is the moment of inertia around the Y axis. Iz is the moment of inertia around the Z axis. G is the shear modulus.
26
Ocean Engineering 163 (2018) 22–28
X. Zhang, D. Lu
Fig. 6. Proof of correctness of extension of hydroelasticity theory proposed by Lu et al. (2016). Table 4 The difference between numerical and experimental results for vertical response of a VLFS (%). λ/L = 60 m x/L 0 0.12 0.25 0.38 0.50 0.63 0.75 0.87 1
A.1 14.65 2.52 16.48 18.42 8.76 10.50 9.39 31.49 10.58
λ/L = 120 m A.2 14.54 2.44 17.17 18.68 9.06 10.03 9.91 30.24 10.28
A.3 3.63 0.60 5.51 15.48 8.08 4.28 17.18 34.35 4.88
A.1 5.89 8.43 8.28 14.98 22.84 25.58 35.78 23.44 12.91
λ/L = 180 m A.2 5.77 8.38 8.67 15.63 23.03 26.10 35.05 22.93 12.45
A.3 5.12 3.63 6.17 7.86 14.16 3.17 2.81 11.05 5.55
A.1 3.00 6.52 1.17 11.60 7.77 11.92 29.14 17.91 11.65
λ/L = 300 m A.2 3.08 6.46 1.40 11.02 7.40 11.37 28.41 18.05 11.38
A.3 2.35 6.79 1.08 15.08 6.43 17.48 33.24 8.75 15.60
A.1 6.56 5.61 4.57 4.68 8.58 7.29 5.36 6.17 8.22
A.2 6.54 5.65 4.63 4.80 8.49 7.10 5.46 6.23 8.22
A.3 0.91 5.91 4.46 3.31 10.07 9.04 4.96 6.29 4.56
Note: A.1, A.2 and A.3 represent the relative difference between experimental results and (1) the numerical results with analytical stiffness matrix; (2) the numerical results calculated by the proposed method; and (3) the numerical results calculated using mode-superposition method (Fu et al., 2007). Senjanović, I., Malenica, Š., Tomas, S., 2008. Investigation of ship hydroelasticity. Ocean. Eng. 35 (5), 523–535. Sun, Y., Lu, D., Xu, J., Zhang, X., 2018. A study of hydroelastic behavior of hinged VLFS. Int. J. Nav. Architect. Ocean. Eng. and Ocean Engineering 10 (2), 170–179. Wei, W., Fu, S., Moan, T., Lu, Z., Deng, S., 2017. A discrete-modules-based frequency domain hydroelasticity method for floating structures in inhomogeneous sea conditions. J. Fluid Struct. 74, 321–339. Wilson, E.L., 1974. The static condensation algorithm. Int. J. Numer. Meth. Eng. 8 (1), 198–203.
References ANSYS, A., 2013. Version 15.0; ANSYS. Inc.: Canonsburg, PA, USA November, 2013. Fu, S., Moan, T., Chen, X., Cui, W., 2007. Hydroelastic analysis of flexible floating interconnected structures. Ocean. Eng. 34 (11–12), 1516–1531. Lu, D., Fu, S., Zhang, X., Guo, F., Gao, Y., 2016. A method to estimate the hydroelastic behaviour of VLFS based on multi-rigid-body dynamics and beam bending. Ships Offshore Struct. 1–9.
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by two interconnected floaters: effects of structural flexibility. Appl. Ocean Res. 71, 34–47. Zhang, X., Lu, D., Gao, Y., Chen, L., 2018b. A time domain discrete-module-beam-bending based hydroelasticity method for the transient response of very large floating structures under unsteady external loads. Ocean. Eng., OE-D-17-01474.
Xu, J., Sun, Y., Li, Z., Zhang, X., Lu, D., 2017. Analysis of the hydroelastic performance of very large floating structures based on multimodules beam theory. Math. Probl Eng. 2017. Yago, K., Endo, H., 1996. Model experiment and numerical calculation of the hydroelastic behavior of matlike VLFS. VLFS 96, 209–214. Zhang, X., Lu, D., Guo, F., Gao, Y., Sun, Y., 2018a. The maximum wave energy conversion
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Short communication
An extension of a discrete-module-beam-bending-based hydroelasticity method for a flexible structure with complex geometric features
T
Xiantao Zhanga,b, Da Lua,∗ a b
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, China School of Civil, Environmental and Mining Engineering, The University of Western Australia, Australia
A R T I C LE I N FO
A B S T R A C T
Keywords: Hydroelasticity Waves Flexible structure Finite element method
The discrete-module-beam-bending based hydroelasticity method proposed by Lu et al. (2016) deals with the hydroelastic response of a flexible structure by discretising it into several rigid submodules which are connected by Euler-Bernoulli beam elements. A stiffness matrix is determined in terms of the geometrical and physical properties of the flexible structure for each beam element and it has an analytical form for simple geometric features such as a rectangular cross section unchanged along the longitudinal direction. However, the analytical stiffness matrix may not exist for a flexible structure with complex geometric features. In the present study, with the aid of finite element method, the discrete-module-beam-bending-based hydroelasticity method proposed by Lu et al. (2016) is extended to be applicable for a flexible structure with complex geometric features.
1. Introduction Hydroelasticity is concerned with the deformation of flexible structures responding to hydrodynamic excitations and simultaneously the modification of the excitations due to the structural deformation. Traditional three-dimensional hydroelasticity theory based on mode superposition approach has been widely adopted for the dynamic response of flexible structures in waves (Senjanović et al., 2008). This method contains three steps: (1) Evaluation of the natural oscillation modes of the flexible structure; (2) hydrodynamic analysis for each mode and (3) Superposition of all modes together and solution of the coupled modal equation to obtain the hydroelastic response of the flexible structure. Unlike the traditional mode superposition approach, Lu et al. (2016) proposed a discrete-module-beam-bending based hydroelasticity method. The underlying idea is outlined as follow. First, a continuous flexible structure is discretised into several rigid submodules. Multirigid-body hydrodynamics is adopted to obtain the hydrodynamic forces (wave excitation force, added mass force and radiation damping force) on each rigid submodule, which, together with the hydrostatic force and inertia force, comprises the total external force on each submodule. Then each submodule is simplified as a lumped mass at its center of gravity and adjacent lumped masses are connected by a beam element to account for effects of structural deformation. Finally, the equations of motion for a flexible structure in waves are established by considering the equilibrium of total external forces and structural
∗
Corresponding author. E-mail address: [email protected] (D. Lu).
https://doi.org/10.1016/j.oceaneng.2018.05.050 Received 17 February 2018; Received in revised form 18 April 2018; Accepted 27 May 2018 0029-8018/ © 2018 Elsevier Ltd. All rights reserved.
deformation induced forces on each lumped mass. Some recent researches on application of this hydroelasticity method can refer to Sun et al. (2018), Wei et al. (2017), Xu et al. (2017), Zhang et al. (2018a, 2018b). The flexible structures investigated in the above-mentioned researches have simple shapes with unchanged cross section in the longitudinal direction (a rectangular plate in Sun et al. (2018), Wei et al. (2017), Xu et al. (2017) and Zhang et al. (2018b). And an elliptic cylinder in Zhang et al. (2018a)). Thus for each beam element added between two adjacent lumped masses, an analytical stiffness matrix exists (see Appendix A). However, in reality, large ships or very large floating structures may have complicated geometric features and thus no analytical stiffness matrix exists for the beam element. The aim of the present study is to extend the discrete-module-beambending-based hydroelasticity method proposed by Lu et al. (2016) to be applicable for a flexible structure with complex geometric features with the aid of finite element method. An outline of the underlying idea for the extension is given and some validations are provided. 2. Extension of the discrete-module-beam-bending based hydroelasticity method 2.1. Revisitation of the hydroelasticity method The discrete-module-beam-bending-based hydroelasticity method proposed by Lu et al. (2016) is revisited here. For Lu's hydroelasticity
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Fig. 1. Schematic of the discrete-module-beam-bending based hydroelasticity method.
2.2. Extension of the hydroelasticity method If the flexible structure has a simple shape, an analytical expression of the beam element stiffness matrix KE (the dimension is 12×12) can be given. For example, for a uniform beam element with a rectangular cross section, the expression of KE is given in Appendix A. However, for a flexible structure with complex geometric features, we can never obtain such an analytical expression for the beam element stiffness matrix. As a result, the finite element approach will be adopted to extend the present hydroelasticity method. When it comes to the finite element approach, we mean that the beam between two adjacent lumped masses is discretised into a large number of small elements (a standard discretization process used in the finite element analysis). We consider a beam element with arbitrary shape of cross section and varied geometric features along the longitudinal direction shown in Fig. 2. This beam element is bounded by the center of gravity of the ith and jth submodules. The extension of the approach actually includes two steps: (1) to represent the physical variables (displacement and forces) of all the nodes of the beam element by the physical variables of nodes on the cross section of the beam element; and (2) to represent the physical variables of nodes on the cross section by the ones at the center of cross section of the beam element. The finite element method (FEM) is applied to the beam element for calculation of the overall stiffness of the structure. Suppose that the beam element is discretised into a number of small elements with the number of nodes n. Each node has three components of displacement
Fig. 2. A beam element with arbitrary shape of cross section and varied geometric features along the longitudinal direction.
method, a linear hydrodynamic approach is used and the solution is obtained in frequency domain. As shown in Fig. 1, a continuous flexible structure is first discretised into N rigid submodules. Then multi-rigidbody hydrodynamics theory can be used to obtain the wave excitation force FE , added mass force FA = ω2 A (ω) ξ , radiation damping force FRd = iωB (ω) ξ for all submodules. Here, ω is the wave frequency; ξ is the total displacement vector at the center of each submodule with the dimension 6N×1 (each submodule has 6 degrees of freedom); A(ω) and B(ω) are the added mass and radiation damping, respectively (with the dimension 6N×6N). The hydrostatic restoring force is FHs = −Cξ with C (whose dimension is 6N×6N) being the hydrostatic restoring coefficients. The inertia force of all submodules is FIn = ω2 Mξ , where M (whose dimension is 6N×6N) is the mass matrix. The hydrodynamic formulations related to the calculation of the above-mentioned forces can be referred to Lu et al. (2016) and Zhang et al. (2018b). The total external force exerted on the center of all submodules is
FT = FE + FA + FRd + FIn + FHs
T
ξ∗p = ⎜⎛ξ∗xp ξ∗yp ξ∗zp⎟⎞ and ⎝ ⎠ ∗ ∗ ∗ ∗ T F p = (Fxp F yp Fzp) (p = 1, 2, …, n). The force is related with the displacement by the spatial stiffness matrix K∗(the dimension is 3n×3n) as and
(1)
Subsequently, each submodule is simplified as a lumped mass at its center of gravity. And the adjacent lumped masses are connected by a beam element, the geometrical and physical properties of which are derived from the flexible structure. The force on lumped masses caused by structural deformation is FSt = −K St ξ , where K St (6N×6N) is the stiffness matrix of the entire structure and it is given by overlaying the beam element stiffness matrix KE according to the standard process of finite element method. More details can be referred to Zhang et al. (2018a, 2018b). Finally, the equations of motion of a flexible structure in waves are established in frequency domain by considering the equilibrium of forces for each lumped mass,
{ −ω2 (M + A (ω)) − iωB (ω) + (C + KSt)} ξ = FE
force,
∗
which
is
⎡ ξ1 ⎤ ⎢ ξ∗ ⎥ = F∗ = K∗ξ∗ = K∗ ⎢ 2 ⎥ ⎢ ⋮⎥ ⎢ ξ∗ ⎥ ⎣ n ⎦3n × 1
defined
as
∗
⎡ F1 ⎤ ⎢ F∗2 ⎥ ⎢ ⎥ ⎢ ⋮⎥ ⎥3n × 1 ⎢ F∗n ⎦ ⎣
(3)
where the spatial stiffness matrix K∗ can be obtained using the software ANSYS (2013). In Eq. (3), the nodes are spatially distributed within the beam element. This means that there are nodes both on the two end cross sections of the beam (two cross sections where the centers i and j are located) and in the inner space of the beam. So the first step is to represent the physical variables (i.e. displacements and forces) of all the nodes of the beam element by the physical variables of nodes on two end cross sections of the beam element, i.e. to establish the stiffness matrix of the nodes on two end cross sections, K B* . This process is quite similar to the static condensation process in finite element method (Wilson, 1974). Eq. (3) can be modified as
(2)
By solving Eq. (2), the displacement at the center of gravity of each submodule is obtained. Then the beam bending theory can be applied to solve the structural deflection, shear force and bending moment.
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* * * * ⎡ KBB KBI ⎤ ⎡ ξ B ⎤ ⎡ FB ⎤ ⎢ ⎢ K* K* ⎥ ξ * ⎥ = ⎢ F* ⎥ II ⎦ ⎣ I ⎦ ⎣ IB ⎣ I⎦
(i)
⎡ T1 ⎤ T (i) ( i ) T = ⎢ 2 ⎥ and ⎢ ⋮⎥ ⎢ Tn(ii) ⎥ ⎣ ⎦
(4)
where ξ B* and ξ *I are the displacement of the nodes on two end cross sections and in the inner space of the beam element, respectively. FB* and F*I are the external forces of the nodes on two end cross sections and * * , KBI , K∗IB , K*II in the inner space of the beam element, respectively. KBB ∗ are the partitioned matrix of the matrix K . It should be noted that F*I = 0 because these nodes are inner nodes of the beam element and no external forces are exerted on inner nodes (for the discrete-modulebeam-bending method, all forces have been assumed to be exerted on the center of gravity of each submodule). From Eq. (4), we can obtain the following expressions, −1
* * K B* ξ B* = (KBB − KBI [K*II] K*IB) ξ B* = FB*
(8)
where T (pi) (T (Pj) ) is the partitioned matrix of T (i) (T (j) ) corresponding to the transformation of the displacements from the center i (j) to the pth node. Assuming that the relative position of the pth node (which belongs to the left end cross section) to center i of the cross section is rip = [x ip yip z ip]T , then the expression of T (pi) is given as follows,
⎡1 T (pi) = ⎢ 0 ⎢ 0 ⎢ ⎣
(5)
0
0
1 0
0 −z ip 1 yip
0
z ip 0 − xip
− yip x ip 0
⎤ ⎥ ⎥ ⎥ ⎦3 × 6
(9)
Similarly, we can obtain T (pj) as follows,
The second step is to represent the physical variables of nodes on two end cross sections by the ones at the center of each cross section of the beam element (center i and j in Fig. 2). The equivalent displacements and forces on the centers of two cross sections at the end of the beam are defined as ξ CG and FCG , respectively. If the equivalent stiffness is defined as K CG , then
K CGξCG = FCG
(j )
⎡ T1 ⎤ T (j) T (j) = ⎢ 2 ⎥ ⎢ ⋮⎥ ⎢ Tn(jj) ⎥ ⎣ ⎦
T (pj)
⎡1 = ⎢0 ⎢ 0 ⎢ ⎣
0
0
0
1 0
0 1
− zjp 0 − x jp yjp
zjp
− yjp x jp 0
⎤ ⎥ ⎥ ⎥ ⎦3 × 6
(10)
where rjp = [x jp yjp zjp]T is the relative position of the pth node (which belongs to the right end cross section) to center j of the cross section. As the forces on the nodes of two end cross sections and the relative coordinates of nodes to the centers of two cross sections are both known, it can be easy to transform the forces on the nodes FB* to the forces on the centers of two end cross sections FCG by the following expression,
(6)
The problem now is to seek the relationship between the equivalent stiffness matrix K CG and the stiffness matrix of nodes on two end cross sections, K B* , which will make use of the rigid-body-motion transformation matrix linking between the motion of center i and j and the motion of other nodes on two end cross sections. The displacements of nodes on two end cross sections can be related to the displacements at the centers (i and j in Fig. 2) of two cross sections as follows,
FCG = TT FB*
(11)
Combining Eqs. 5–11, we have
K CG = T TK B*T T (i) ξ B* = TξCG = ⎡ ⎢ 0 ⎣
(i) 0 ⎤ ⎡ ξ CG ⎤ ⎢ (j) ⎥ j ( ) ⎥ T ⎦ ξ CG ⎦ ⎣
−1
* * K B* = KBB − KBI [K*II] K*IB * * KBI ⎤ ⎡K K∗ = ⎢ BB * * ⎥ ⎣ KIB KII ⎦
(7)
where T is a transform matrix. If the number of nodes on two end cross sections is n1, then the dimension for ξ B* , T and ξ CG is 3n1×1, 3n1×12 and 12×1, respectively. It should be noted that the dimension is 6×1 for (i) (j ) (i) (j ) and ξ CG . This is because ξ CG and ξ CG represent the displaceboth ξ CG ments of two rigid submodules used in hydroelastic analysis, which have six components including surge, sway and heave (linear displacement along x, y and z direction, respectively) and roll, pitch and yaw (angular displacement along x, y and z axis, respectively). The number of nodes at the left and right end cross section is ni and nj (ni + nj = n1), respectively. Then
(12)
Eq. (12) shows the relationship between the equivalent stiffness at the centers of two end cross sections K CG and the stiffness matrix of all nodes of a beam element K∗. And K∗ can be obtained with the aid of a finite element software such as ANSYS. Using Eq. (12), the hydroelastic response of a floating flexible structure with complex geometric features can be solved. 2.3. Validation of the extension of the hydroelasticity method First, we choose a cubic steel model, the physical properties and
Fig. 3. A cubic steel model used for the purpose of validation. (a) The geometrical and physical properties; (b) The discretization of the cubic model used in ANSYS software. 24
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Table 1 The stiffness matrix KAnal calculated by analytical expression (× 1010 ).
1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
5
6
7
8
9
10
11
12
8.40 0 0 0 0 0 −8.40 0 0 0 0 0
0 0.291 0 0 0 1.45 0 −0.290 0 0 0 1.45
0 0 0.291 0 −1.50 0 0 0 −0.290 0 −1.50 0
0 0 0 1.82 0 0 0 0 0 −1.80 0 0
0 0 −1.50 0 10.1 0 0 0 1.45 0 4.47 0
0 1.45 0 0 0 10.1 0 −1.50 0 0 0 4.47
−8.40 0 0 0 0 0 8.40 0 0 0 0 0
0 −0.290 0 0 0 −1.50 0 0.291 0 0 0 −1.50
0 0 −0.290 0 1.45 0 0 0 0.291 0 1.45 0
0 0 0 −1.80 0 0 0 0 0 1.82 0 0
0 0 −1.50 0 4.47 0 0 0 1.45 0 10.1 0
0 1.45 0 0 0 4.47 0 −1.50 0 0 0 10.1
Table 2 The stiffness matrix K CG calculated by finite element method (× 1010 , from Eq. (12)).
1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
5
6
7
8
9
10
11
12
8.40 0 0 0 0 0 −8.40 0 0 0 0 0
0 0.299 0 0 0 1.49 0 −0.30 0 0 0 1.49
0 0 0.299 0 −1.50 0 0 0 −0.30 0 −1.50 0
0 0 0 1.82 0 0 0 0 0 −1.80 0 0
0 0 −1.50 0 10.3 0 0 0 1.49 0 4.67 0
0 1.49 0 0 0 10.3 0 −1.50 0 0 0 4.67
−8.40 0 0 0 0 0 8.40 0 0 0 0 0
0 −0.30 0 0 0 −1.50 0 0.299 0 0 0 −1.50
0 0 −0.30 0 1.49 0 0 0 0.299 0 1.49 0
0 0 0 −1.80 0 0 0 0 0 1.82 0 0
0 0 −1.50 0 4.67 0 0 0 1.49 0 10.3 0
0 1.49 0 0 0 4.67 0 −1.50 0 0 0 10.3
hydroelastic response of the model is calculated for heading waves of different wave length. The continuous structure is discretised into 8 submodules to perform the hydroelastic analysis (refer to Fig. 1). The structural stiffness matrix for each beam element is calculated using both analytical expression in Appendix A (denoted as ‘Approach 1’) and FEM method with Eq. (12) (denoted as ‘Approach 2’). The finite element model of a beam element to connect the centers of gravity of two adjacent submodules is shown in Fig. 5. The length, width and height of this macro-beam element are 60, 60 and 2 m, respectively. The number of elements in these three directions are 20, 20 and 4, respectively. The experimental results obtained by Yago and Endo (1996) is denoted as ‘Experiment’. The numerical results calculated using three-dimensional hydroelasticity method based on mode-superposition approach (Fu et al., 2007) is denoted as “Mode-superposition”. As shown in Fig. 6, the dynamic responses of the continuous VLFS in waves are almost identical for two approaches, both of which agree well with experimental results. This further validates the correctness of extension of the hydroelasticity theory proposed by Lu et al. (2016). A detailed comparison of numerical results calculated by different numerical methods with experimental results (expressed as the relative differences) is listed in Table 4.
Table 3 The difference between KAnal and K CG (%).
1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
5
6
7
8
9
10
11
12
0 0 0 0 0 0 0 0 0 0 0 0
0 2.79 0 0 0 2.79 0 2.79 0 0 0 2.79
0 0 2.79 0 2.79 0 0 0 2.79 0 2.79 0
0 0 0 0.29 0 0 0 0 0 0.29 0 0
0 0 2.79 0 2.01 0 0 0 2.79 0 4.53 0
0 2.79 0 0 0 2.01 0 2.79 0 0 0 4.53
0 0 0 0 0 0 0 0 0 0 0 0
0 2.79 0 0 0 2.79 0 2.79 0 0 0 2.79
0 0 2.79 0 2.79 0 0 0 2.79 0 2.79 0
0 0 0 0.29 0 0 0 0 0 0.29 0 0
0 0 2.79 0 4.53 0 0 0 2.79 0 2.01 0
0 2.79 0 0 0 4.53 0 2.79 0 0 0 2.01
dimension of which are given in Fig. 3(a). For this simple shape, the stiffness matrix can be calculated using the expression given in Appendix A. The result is noted as KAnal and it is given in Table 1. The stiffness matrix for this cubic model is also calculated from Eq. (12) with the aid of finite element software ANSYS. The steel model is discretised with the number of nodes 20, 8 and 6 in the direction of length, width and height, respectively (see Fig. 3(b)). The ANSYS software is used to extract the stiffness matrix K∗, after which the equivalent stiffness matrix for the centers of two end cross sections K CG is calculated using Eq. (12). The result is given in Table 2. And the difference between KAnal and K CG is given in Table 3. It can be seen that the relative error of all components of the stiffness matrix calculated by two approaches is within 10%, which validates the correctness of Eq. (12). Next we consider a model adopted in the experiment conducted by Yago and Endo (1996) to further evaluate the above-mentioned methodology. As shown in Fig. 4, the model is a continuous pontoon-type very large floating structure (VLFS). The physical properties and dimension shown in Fig. 4 all correspond to the prototype scale. The
3. Conclusions The discrete-module-beam-bending-based hydroelasticity method proposed by Lu et al. (2016) is extended to be applicable for a flexible structure with complex geometric features with the aid of finite element method. First, the elastic beam (used to consider the structural deformation effects) is considered as a macro-element and it is discretised following the standard finite element method, after which the structural stiffness matrix of this macro-element can be obtained. Using the static condensation technique and rigid-body-motion transformation matrix, the equivalent stiffness matrix corresponding to the displacements at the centers of two end cross sections of this beam can then be derived 25
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Fig. 4. Schematic of a very large floating structure (VLFS).
hydroelastic response of a flexible structure with complex geometric features can be solved. The extension of the hydroelasticity method is validated by (Ⅰ) a small difference (within 5%) of the stiffness matrix for a cubic steel model calculated using the analytical formulation and the proposed approach and (Ⅱ) a good agreement between the experimentally measured vertical response of a VLFS and that calculated using the present method. The extension of the discrete-module-beam-bending based hydroelasticity method is a big step for its application in practical engineering problems. In the future work, relevant experiments will be carried out on the hydroelastic response of a flexible structure with complex geometric features in waves to further validate the present study.
Acknowledgement The first author would like to acknowledge the support of the IPRS, APA and Shell-UWA offshore engineering PhD research top-up scholarships.
Fig. 5. The finite element model of a beam element to connect the centers of gravity of two adjacent submodules.
from
the
macro-element's
stiffness
matrix.
Subsequently,
the
Appendix A The stiffness matrix KE for a uniform beam element with unchanged cross section is given as follows, EA
⎡ l ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 KE = ⎢ −EA ⎢ ⎢ l ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣
12EIz l3 (1 + βy )
0
12EIy
0
0
0
−6EIy
6EIz
Symmetry
l3 (1 + βz )
l2 (1 + βz )
Gκh3b l
0
(4 + βz ) EIy l (1 + βz )
0
0
0
0
0
0
0
−12EIz
0
0
0
−12EIy
0
l2 (1 + βy )
l3 (1 + βy )
0 0 0 6EIz l2 (1 + βy )
l3 (1 + βz )
0 −6EIy l2 (1 + βz )
0
−Gκh3b l
0 0
6EIy l2 (1 + βz )
0 (2 − βz ) EIy l (1 + βz )
0
(4 + βy ) EIz l (1 + βy )
0
EA l
−6EIz
0
l2 (1 + βy )
12EIz l3 (1 + βy )
0
0
0
0
0
0
0
0
0
0
−6EIz
(2 − βy ) EIz l (1 + βy )
l2 (1 + βy )
12EIy l3 (1 + βz )
0 6EIy l2 (1 + βz )
0
Gκh3b l
0 0
(4 + βz ) EIy l (1 + βz )
0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (4 + βy ) EIz ⎥ l (1 + βy ) ⎥ ⎦
(A1)
The size of the symmetric matrix is 12 × 12. l is the length of beam element (the dimension along X axis). b is width of the beam element (the dimension along Y axis). h is the height (the dimension along Z axis). κ is a constant related to b and h. A is cross-sectional area with A = bh. E is Young modulus. βy is the correction factor considering shear deformation for the beam element bending in XOY plane. βz is the correction factor considering shear deformation for the beam element bending in XOZ plane. Iy is the moment of inertia around the Y axis. Iz is the moment of inertia around the Z axis. G is the shear modulus.
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Fig. 6. Proof of correctness of extension of hydroelasticity theory proposed by Lu et al. (2016). Table 4 The difference between numerical and experimental results for vertical response of a VLFS (%). λ/L = 60 m x/L 0 0.12 0.25 0.38 0.50 0.63 0.75 0.87 1
A.1 14.65 2.52 16.48 18.42 8.76 10.50 9.39 31.49 10.58
λ/L = 120 m A.2 14.54 2.44 17.17 18.68 9.06 10.03 9.91 30.24 10.28
A.3 3.63 0.60 5.51 15.48 8.08 4.28 17.18 34.35 4.88
A.1 5.89 8.43 8.28 14.98 22.84 25.58 35.78 23.44 12.91
λ/L = 180 m A.2 5.77 8.38 8.67 15.63 23.03 26.10 35.05 22.93 12.45
A.3 5.12 3.63 6.17 7.86 14.16 3.17 2.81 11.05 5.55
A.1 3.00 6.52 1.17 11.60 7.77 11.92 29.14 17.91 11.65
λ/L = 300 m A.2 3.08 6.46 1.40 11.02 7.40 11.37 28.41 18.05 11.38
A.3 2.35 6.79 1.08 15.08 6.43 17.48 33.24 8.75 15.60
A.1 6.56 5.61 4.57 4.68 8.58 7.29 5.36 6.17 8.22
A.2 6.54 5.65 4.63 4.80 8.49 7.10 5.46 6.23 8.22
A.3 0.91 5.91 4.46 3.31 10.07 9.04 4.96 6.29 4.56
Note: A.1, A.2 and A.3 represent the relative difference between experimental results and (1) the numerical results with analytical stiffness matrix; (2) the numerical results calculated by the proposed method; and (3) the numerical results calculated using mode-superposition method (Fu et al., 2007). Senjanović, I., Malenica, Š., Tomas, S., 2008. Investigation of ship hydroelasticity. Ocean. Eng. 35 (5), 523–535. Sun, Y., Lu, D., Xu, J., Zhang, X., 2018. A study of hydroelastic behavior of hinged VLFS. Int. J. Nav. Architect. Ocean. Eng. and Ocean Engineering 10 (2), 170–179. Wei, W., Fu, S., Moan, T., Lu, Z., Deng, S., 2017. A discrete-modules-based frequency domain hydroelasticity method for floating structures in inhomogeneous sea conditions. J. Fluid Struct. 74, 321–339. Wilson, E.L., 1974. The static condensation algorithm. Int. J. Numer. Meth. Eng. 8 (1), 198–203.
References ANSYS, A., 2013. Version 15.0; ANSYS. Inc.: Canonsburg, PA, USA November, 2013. Fu, S., Moan, T., Chen, X., Cui, W., 2007. Hydroelastic analysis of flexible floating interconnected structures. Ocean. Eng. 34 (11–12), 1516–1531. Lu, D., Fu, S., Zhang, X., Guo, F., Gao, Y., 2016. A method to estimate the hydroelastic behaviour of VLFS based on multi-rigid-body dynamics and beam bending. Ships Offshore Struct. 1–9.
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by two interconnected floaters: effects of structural flexibility. Appl. Ocean Res. 71, 34–47. Zhang, X., Lu, D., Gao, Y., Chen, L., 2018b. A time domain discrete-module-beam-bending based hydroelasticity method for the transient response of very large floating structures under unsteady external loads. Ocean. Eng., OE-D-17-01474.
Xu, J., Sun, Y., Li, Z., Zhang, X., Lu, D., 2017. Analysis of the hydroelastic performance of very large floating structures based on multimodules beam theory. Math. Probl Eng. 2017. Yago, K., Endo, H., 1996. Model experiment and numerical calculation of the hydroelastic behavior of matlike VLFS. VLFS 96, 209–214. Zhang, X., Lu, D., Guo, F., Gao, Y., Sun, Y., 2018a. The maximum wave energy conversion
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