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An accurate frequency-domain model of water interaction with cylinders of arbitrary shape during earthquakes

Abstract

An accurate frequency domain model is proposed to analyze the seismic response of uniform vertical cylinders with arbitrary cross section surrounded by water. According to the boundary conditions and using the variables separation method, the vertical modes of the hydrodynamic pressure are firstly obtained. Secondly, the three-dimensional wave equation can be simplified to a two-dimensional Helmholtz equation. Introducing the scaled boundary coordinate, a scaled boundary finite element (SBFE) equation which is a linear non-homogeneous second-order ordinary equation is derived by weighted residual method. The dynamic-stiffness matrix equation for the problem is furtherly derived. The continued fraction is acted as the solution of the dynamic-stiffness matrix for cylinder dynamic interaction of cylinder with infinite water. The coefficient matrices of the continued fraction are derived recursively from the SBFE equation of dynamic-stiffness. The accuracy of the present method is verified by comparing the hydrodynamic force on circular, elliptical and rectangle cylinders with the analytical or numerical solutions. Finally, the proposed model is used to analyze the natural frequency and seismic response of cylinders.

Introduction

With the development of the economy and transportation, more and more offshore and coastal structures, such as bridge piers and offshore wind turbines, have been constructed in China in recent years. These structures may be under threat of earthquakes in areas of active seismicity, for instance, the Eastern coast of China. It is well known that water-structure interaction causes hydrodynamic pressure on the structure during earthquakes and the additional hydrodynamic pressure has significant effect on the dynamic responses and properties of the structure (Liaw and Chopra 1974; Han and Xu 1996; Wei et al. 2013). Therefore, it is necessary to investigate earthquake-induced hydrodynamic pressure on the offshore structures. The purpose of this study is mainly to develop an accurate frequency-domain model of water interaction with cylinders of arbitrary shape during earthquakes.

The study on earthquake-induced hydrodynamic pressure has begun with the gravity dams (Westergaard 1933) and the cantilever circular cylinders (Jacobsen 1949). For the simple cross-section cylinder, it has been well studied by analytical method, such as circular cylinder (Liaw and Chopra 1974; Tanaka and Hudspeth 1988), elliptical cylinder (Wang et al. 2018a, 2019a), and arbitrary smooth cross-section (Wang et al. 2018b). Liaw and Chopra (1974) demonstrated that water compressibility was negligible for slender cylinders but important for squat cylinders vibrating at high frequency. Additionally, effect of the hydrodynamic pressure can be simply modeled as an ‘added mass’ when the surface wave and water compressibility are ignored. Assuming the structure rigid, the simplified formulas of the added mass for the earthquake induced hydrodynamic pressure on circular cylinder and elliptical cylinders were given by Jiang et al. (2017), Li and Yang (2013), and Wang et al. (2018a). Considering the flexible of the structure, a simple and accurate added mass representation was also presented by Han and Xu (1996). Later, Wang et al. extended this model to calculate the added mass representation for a flexible vertical elliptical cylinder vibrating in water (Wang et al. 2019a). Considering water compressibility, Du et al. (2014) presented a simplified formula in time domain for hydrodynamic pressure on rigid circular cylinders by introducing three dimensionless parameters including frequency ratio, wide-depth ratio and relative height. Recently, considering the flexible of the circular cylinder, an accurate and efficient time-domain model was proposed to analyze water-cylinder interaction during earthquake (Wang et al. 2018c). In addition, the boundary integral method is also can be used to analyze the earthquake responses of submerged circular cylinder (Williams 1986).

The earthquake-induced hydrodynamic pressure on complex structures should be solved by numerical method, such as the finite element method (FEM) (Liao 1986, Liaw and Chopra 1974), finite-difference scheme (Chen 1997), and the finite element technique incorporating the infinite element (Woo-Sun et al. 1991). A special boundary method based on the use of a complete and non-singular set of Trefftz functions was presented for hydrodynamic pressure on axisymmetric offshore structures (Sun and Nogami 2010; Avilés and Li 2001). The free surface wave effect on the water-cylinder interaction is significant in wave force (Ti et al. 2020), but it has been validated that free surface waves have less effect on the earthquake-induced hydrodynamic force on cylinders by Liaw and Chopra (1974) and Li and Yang (2013).

Recently, coupled the FEM and artificial boundary condition was used to simulated the water-fluid interaction system (Zhao et al. 2018). Wang et al. (2019b) present an accurate and efficient numerical model to calculate the earthquake-induced hydrodynamic pressure on uniform vertical cylinders with an arbitrary cross section surrounded by water and the simplified formulas for the hydrodynamic pressure on round-ended and rectangular cylinder were also given. In addition, the simplified methods for efficient seismic design and analysis of water-surrounded circular tapered cylinders and composite axisymmetric structures were also presented (Wang et al. 2018d, 2019c).

Recently, the scaled boundary finite element method (SBFEM), originally developed to solve soil-structure interaction problems (Song and Wolf 1996, 1997), has been successfully applied to many fields, such as stress singularities as occurring in cracks (Song and Wolf 2002), elastostatics (Deeks and Wolf 2002), potential flow (Deeks and Cheng 2003), and non-linear analysis of unbounded media (Doherty and Deeks 2005). The SBFEM is a novel semi-analytical, combining the advantages of the finite element method (FEM) and the boundary element method (BEM), as reduction of the spatial dimension by one but no fundamental solution required and eliminates singular integrals. The SBFEM is also used to analyze wave interaction with a bottom-mounted uniform porous cylinder of arbitrary shape (Meng and Zou 2012), a single and multiple cylindrical structures of different cross sectional shape (Song et al. 2010).

In this paper, a substructure method in frequency domain is proposed for the analysis of seismic response of the water-cylinder interaction system, where the section of cylinder can be general shape. Proposed approach is efficient in the water-cylinder dynamic interaction. The SBFEM is used to simulate the earthquake-induced hydrodynamic pressure on the uniform vertical cylinder with arbitrary cross section. Firstly, utilizing the variables separation method, the three-dimensional wave equation governing the compressible water is transformed into a two-dimensional (2D) Helmholtz equation. Secondly, a dynamic-stiffness equation is obtained by utilizing the SBFEM. Thirdly, a continued fraction is used to solve the dynamic-stiffness equation. Finally, the finite element equation of the water-cylinder interaction system is obtained.

Mathematical formulation

The water-cylinder interaction system during earthquake is shown in Fig. 1. The structure can be assumed to be linearly elastic. The cylinder varies uniformly from bedrock above water surface along z axis. The water is a horizontally infinite layer of the constant depth h. The rigid bedrock has a horizontal earthquake motion of the displacement time history along the direction of paralleling x axis. The cylinder is fully submerged in water and treated simply as a one-dimensional structure governed by beam theory. The water-cylinder interaction system is initially at rest.

Fig. 1
figure1

The water-cylinder interaction system during earthquake

The governing equation of the water can be expressed in terms of its hydrodynamic pressure, which is controlled by the wave equation, can be written as

$$ \frac{\partial^2P}{\partial {x}^2}+\frac{\partial^2P}{\partial {y}^2}+\frac{\partial^2P}{\partial {z}^2}+\frac{\omega^2}{c^2}P=0 $$
(1)

where P(x, y, z, ω) is the hydrodynamic pressure expressed in frequency domain; ω is the circle frequency; c = 1438m/s is the velocity of the wave propagate in water.

The boundary conditions are as follows

  1. (1)

    At the bottom of the water, no vertical motion

$$ \frac{\partial P}{\partial z}=0 $$
(2a)
  1. (2)

    At the free surface of the water, the hydrodynamic pressure is zero

$$ P=0 $$
(2b)
  1. (3)

    At the water-cylinder interface, the outward normal acceleration of the water in contact with the structure is equal the cylinder surface, which in frequency can be written as

$$ \frac{\partial P}{\partial n}={\rho}_w{\omega}^2{U}_x{n}_x $$
(2c)
  1. (4)

    At the infinity of the water, the hydrodynamic pressure is decayed to zero

$$ P\left|{}_{r=\infty}\right.=0 $$
(2d)

where nx is the outward normal direction of the cylinder surface at the x axis component; Ux is the displacement of the cylinder surface in x-direction; ρw = 1000 kg/m3 is the mass density of water; and r is the distance from the cylinder.

Applying separation of the variables to the hydrodynamic pressure, the vertical modes of the hydrodynamic pressure are obtained. Then, the hydrodynamic pressure P(x, y, z, ω) can be separated out as (Liaw and Chopra 1974)

$$ P\left(x,y,z,\omega \right)=\sum \limits_{j=1}^{\infty }{P}_j{U}_j\cos {\lambda}_jz $$
(3)

where Pj = Pj(x, y, ω) is the modal hydrodynamic pressure in xy-plane; λj = (2j − 1)π/2h; and \( {U}_j=\frac{2}{h}{\int}_0^h{U}_x{n}_x\cos {\lambda}_jz\mathrm{d}z \) is the corresponding modal displacement of the cylinder.

Then, the problem of three-dimensional water-cylinder interaction is reduced to solve a two-dimensional Helmholtz equation with the boundary conditions (Liaw and Chopra 1974) as follows

$$ \frac{\partial^2{P}_j}{\partial {x}^2}+\frac{\partial^2{P}_j}{\partial {y}^2}+\left(\frac{\omega^2}{c^2}-{\lambda}_j^2\right){P}_j=0 $$
(4)
$$ \frac{\partial {P}_j}{\partial n}={\omega}^2{\rho}_w{n}_x $$
(5a)
$$ {P}_j\left|{}_{r=\infty}\right.=0 $$
(5b)

The SBFEM equation

Scaled boundary finite element transformation

SBFEM defines the domain V by scaling a defining curve Si relative to a scaling center (x0y0), which is chosen at the center of the cylinder in this study. As shown in Fig. 2, the circumferential coordinate η is anticlockwise along the defining curve Si, which is closed in this case. The normalized radial coordinate ξ is a scaling factor with 1 ≤ ξ ≤ ∞ for unbounded domain. The coordinate of a point on the straight line element are denote as xb and yb, it can be expressed with mapping function as

$$ {x}_b=\mathbf{NX} $$
(6a)
$$ {y}_b=\mathbf{NY} $$
(6b)
Fig. 2
figure2

The transform of Cartesian coordinate and scaled boundary

with \( \mathbf{N}=\left[\frac{1-\eta }{2}\kern0.5em \frac{1+\eta }{2}\right] \) is the mapping function; \( \mathbf{X}={\left\{{x}_1\kern0.5em {x}_2\right\}}^T \), \( \mathbf{Y}={\left\{{y}_1\kern0.5em {y}_2\right\}}^T \) are the nodes coordinates vector in Cartesian coordinate.

Therefore, the Cartesian coordinates are transformed to the scaled boundary coordinate ξ and η with the scaling equations

$$ x=\xi {x}_b $$
(7a)
$$ y=\xi {y}_b $$
(7b)

The spatial derivatives in the two coordinate systems are related as

$$ \left\{\begin{array}{c}\frac{\partial }{\partial \xi}\\ {}\frac{\partial }{\partial \eta}\end{array}\right\}=\hat{\mathbf{J}}\left(\xi, \eta \right)\left\{\begin{array}{c}\frac{\partial }{\partial x}\\ {}\frac{\partial }{\partial y}\end{array}\right\} $$
(8)

with

$$ \hat{\mathbf{J}}\left(\xi, \eta \right)=\left[\begin{array}{cc}{x}_b& {y}_b\\ {}\xi {x}_b,\eta & \xi {y}_b,\eta \end{array}\right]=\left[\begin{array}{cc}1& \\ {}& \xi \end{array}\right]\mathbf{J} $$
(9)

where \( \mathbf{J}=\left[\begin{array}{cc}{x}_b& {y}_b\\ {}{x}_b,\eta & {y}_b,\eta \end{array}\right] \) is Jacobi matrix.

Then the derivatives with respect to x, y can be obtained

$$ \left\{\begin{array}{c}\frac{\partial }{\partial x}\\ {}\frac{\partial }{\partial y}\end{array}\right\}=\hat{\mathbf{J}}{\left(\xi, \eta \right)}^{-1}\left\{\begin{array}{c}\frac{\partial }{\partial \xi}\\ {}\frac{\partial }{\partial \eta}\end{array}\right\}={\mathbf{b}}_1\frac{\partial }{\partial \xi }+\frac{1}{\xi }{\mathbf{b}}_2\frac{\partial }{\partial \eta } $$
(10)

with

$$ {\mathbf{b}}_1=\frac{1}{\left|\mathbf{J}\right|}\left\{\begin{array}{c}{y}_{b,\eta}\\ {}-{x}_{b,\eta}\end{array}\right\} $$
(11a)
$$ {\mathbf{b}}_2=\frac{1}{\left|\mathbf{J}\right|}\left\{\begin{array}{c}-{y}_b\\ {}{x}_b\end{array}\right\} $$
(11b)

For later use, note that (b2|J|),η =  − b1|J|.

The infinitesimal area dV of the domain is calculated as

$$ \mathrm{d}V=\left|\hat{\mathbf{J}}\left(\xi, \eta \right)\right|\mathrm{d}\xi \mathrm{d}\eta =\xi \left|\mathbf{J}\right|\mathrm{d}\xi \mathrm{d}\eta $$
(12a)
$$ \mathrm{d}S=\sqrt{{x_{,\eta}}^2+{y_{,\eta}}^2}\mathrm{d}\eta $$
(12b)

Scaled boundary finite element equation

To derive the finite element approximation, Eq. (4) is multiplied by a weighting function w and integrating over the domain, it can be obtained as

$$ \int w\left(\frac{\partial^2{P}_j}{\partial {x}^2}+\frac{\partial^2{P}_j}{\partial {y}^2}+\left(\frac{\omega^2}{c^2}-{\lambda}^2\right){P}_j\right)\mathrm{d}V=0 $$
(13)

Substituting Eq. (10) into Eq. (13), we get

$$ \int \left(w{{\mathbf{b}}_1}^T\frac{\partial }{\partial \xi}\left({\mathbf{b}}_1\frac{\partial {P}_j}{\partial \xi }+{\mathbf{b}}_2\frac{\partial {P}_j}{\partial \eta}\right)+\frac{1}{\xi }w{{\mathbf{b}}_2}^T\frac{\partial }{\partial \eta}\left({\mathbf{b}}_1\frac{\partial {P}_j}{\partial \xi }+{\mathbf{b}}_2\frac{\partial {P}_j}{\partial \eta}\right)+w\left(\frac{\omega^2}{c^2}-{\lambda}^2\right){P}_j\right)\mathrm{d}V=0 $$
(14)

Substituting the transformation of the scaled boundary coordinate into Eq. (14), the SBFEM equation can be obtained as

$$ {\mathbf{E}}_0{\xi}^2{\mathbf{P}}_{j,\xi \xi}+\left({\mathbf{E}}_0-{\mathbf{E}}_1+{{\mathbf{E}}_1}^T\right)\xi {\mathbf{P}}_{j,\xi }-{\mathbf{E}}_2{\mathbf{P}}_j+\left(\frac{\omega^2}{c^2}-{\lambda}^2\right){\mathbf{M}}_0{\xi}^2{\mathbf{P}}_j=\mathbf{0} $$
(15)

with the introducing coefficient matrixes are

$$ {\mathbf{E}}_0={\int}_{-1}^1{{\mathbf{B}}_1}^T{\mathbf{B}}_1\left|\mathbf{J}\right| d\eta =\frac{{\varDelta_x}^2+{\varDelta_y}^2}{12\left|\mathbf{J}\right|}\left[\begin{array}{cc}2& 1\\ {}1& 2\end{array}\right] $$
(16a)
$$ {\mathbf{E}}_1={\int}_{-1}^1{{\mathbf{B}}_2}^T{\mathbf{B}}_1\left|\mathbf{J}\right| d\eta =\frac{{\varDelta_x}^2+{\varDelta_y}^2}{24\left|\mathbf{J}\right|}\left[\begin{array}{cc}-1& 1\\ {}1& -1\end{array}\right]-\frac{\varDelta_y\left({y}_1+{y}_2\right)+{\varDelta}_x\left({x}_1+{x}_2\right)}{8\left|\mathbf{J}\right|}\left[\begin{array}{cc}-1& -1\\ {}1& 1\end{array}\right] $$
(16b)
$$ {\mathbf{E}}_2={\int}_{-1}^1{{\mathbf{B}}_2}^T{\mathbf{B}}_2\left|\mathbf{J}\right| d\eta =\frac{3{\left({y}_1+{y}_2\right)}^2+3{\left({x}_1+{x}_2\right)}^2+{\varDelta_x}^2+{\varDelta_y}^2}{24\left|\mathbf{J}\right|}\left[\begin{array}{cc}1& -1\\ {}-1& 1\end{array}\right] $$
(16c)
$$ {\mathbf{M}}_0={\int}_{-1}^1{\mathbf{N}}^T\mathbf{N}\left|\mathbf{J}\right| d\eta =\frac{\left|\mathbf{J}\right|}{3}\left[\begin{array}{cc}2& 1\\ {}1& 2\end{array}\right] $$
(16d)

where B1 = b1N and B2 = b2N,η.

To simply the nomenclature, the same symbols are used for the assembled coefficient matrices.

Dynamic-stiffness equation

The amplitude of the internal nodal forces Q(ξ), which are equal to the normal derivatives of Pj, on a line with a constant ξ are addressed. Applying the principle of virtual work yields

$$ {\mathbf{w}}^T\mathbf{Q}\left(\xi \right)=\underset{S_i}{\int }{\mathbf{w}}^T\frac{\partial {P}_j}{\partial n}\mathrm{d}S $$
(17)

The nodal forces can be obtained by substituting relevant formulas mentioned above

$$ \mathbf{Q}\left(\xi \right)={\mathbf{E}}_0\xi {\mathbf{P}}_{j,\xi }+{{\mathbf{E}}_1}^T{\mathbf{P}}_j $$
(18)

For an unbounded medium, the amplitudes of the modal nodal forces R(ξ) are equal to the opposite of the internal nodal forces Q(ξ), which can be expressed as

$$ \mathbf{R}\left(\xi \right)=-\mathbf{Q}\left(\xi \right) $$
(19)

The same signs of Eqs. (18) and (19) applied to the assembled system. In the frequency domain, the amplitudes of the modal hydrodynamic pressures Pj(ξ) are related to those of the nodal forces R(ξ) as

$$ \mathbf{R}\left(\xi \right)=\mathbf{S}\left(\omega, \xi \right){\mathbf{P}}_j\left(\xi \right) $$
(20)

where S(ω, ξ) is the dynamic-stiffness matrix.

The SBFE equation in dynamic-stiffness is derived from Eqs. (15), (18), (19) and (20) (Song and Wolf 1996). On the boundary (ξ = 1) the dynamic stiffness matrix for the unbounded medium is expressed as

$$ \left(\mathbf{S}\left(\omega \right)+{\mathbf{E}}_1\right){{\mathbf{E}}_0}^{-1}\left(\mathbf{S}\left(\omega \right)+{{\mathbf{E}}_1}^T\right)-\omega \mathbf{S}{\left(\omega \right)}_{,\omega }-{\mathbf{E}}_2+\left(\frac{\omega^2}{c^2}-{\lambda}^2\right){\mathbf{M}}_0=\mathbf{0} $$
(21)

Introducing a dimensionless coefficient \( \varpi =- i\lambda \sqrt{1-{\left(\omega /\lambda c\right)}^2} \), Eq. (21) can be simplified into the general form of the SBFEM as (Song and Wolf 1996)

$$ \left(\mathbf{S}\left(\varpi \right)+{\mathbf{E}}_1\right){{\mathbf{E}}_0}^{-1}\left(\mathbf{S}\left(\varpi \right)+{{\mathbf{E}}_1}^T\right)-\varpi \mathbf{S}{\left(\varpi \right)}_{,\mathrm{a}}-{\mathbf{E}}_2+{\varpi}^2{\mathbf{M}}_0=\mathbf{0} $$
(22)

Solution of the dynamic-stiffness

It can be seen that Eq. (22) is a system of non-linear first-order ordinary differential equations with the independent variable ϖ. To avoid the computationally expensive task, a continued fraction solution for the dynamic-stiffness matrix is developed in (Bazyar and Song 2008) directly from the scaled boundary finite element equation.

The continued fraction solution of the dynamic-stiffness matrix is derived in this section. The solution is assumed as

$$ \mathbf{S}\left(\varpi \right)={\mathbf{g}}_0+ i\varpi {\mathbf{h}}_0-{{\mathbf{S}}_1}^{-1}\left(\varpi \right) $$
(23a)
$$ {{\mathbf{S}}_j}^{-1}\left(\varpi \right)={\mathbf{g}}_j+ i\varpi {\mathbf{h}}_j-{{\mathbf{S}}_{j+1}}^{-1}\left(\varpi \right) $$
(23b)

Substituting Eq. (23a) into (22), three terms in descending order of the power of () are expressed as

$$ {\displaystyle \begin{array}{l}{\left( i\varpi \right)}^2\left({\mathbf{h}}_0{\mathbf{E}}_0^{-1}{\mathbf{h}}_0-{{\mathbf{M}}_0}^{\prime}\right)+\left( i\varpi \right)\left[{\mathbf{h}}_0{{\mathbf{E}}_0}^{-1}\left({\mathbf{g}}_0+{\mathbf{E}}_1^T\right)+\left({\mathbf{g}}_0+{\mathbf{E}}_1\right){\mathbf{E}}_0^{-1}{\mathbf{h}}_0-{\mathbf{h}}_0\right]\\ {}+\left({\mathbf{g}}_0+{\mathbf{E}}_1\right){\mathbf{E}}_0^{-1}\left({\mathbf{g}}_0+{\mathbf{E}}_1^T\right)+{\mathbf{E}}_2-\left({\mathbf{g}}_0+{\mathbf{E}}_1\right){\mathbf{E}}_0^{-1}{\mathbf{S}}_1^{-1}-\left( i\varpi \right){\mathbf{h}}_0{\mathbf{E}}_0^{-1}{\mathbf{S}}_1^{-1}\\ {}-{{\mathbf{S}}_1}^{-1}{{\mathbf{E}}_0}^{-1}\left({\mathbf{g}}_0+{\mathbf{E}}_1^T\right)-{\mathbf{S}}_1^{-1}{\mathbf{E}}_0^{-1}\left( i\varpi \right){\mathbf{h}}_0+{\mathbf{S}}_1^{-1}{\mathbf{E}}_0^{-1}{\mathbf{S}}_1^{-1}-\varpi {\mathbf{S}}_{1,\varpi}^{-1}=\mathbf{0}\end{array}} $$
(24)

It can be seen that Eq. (24) is satisfied only when all the three terms are equal to zero, setting the coefficient of the ()2 and () terms to zero results in

$$ {\mathbf{h}}_0{\mathbf{E}}_0^{-1}{\mathbf{h}}_0-{{\mathbf{M}}_0}^{\prime }=\mathbf{0} $$
(25a)
$$ {\mathbf{h}}_0{{\mathbf{E}}_0}^{-1}\left({\mathbf{g}}_0+{\mathbf{E}}_1^T\right)+\left({\mathbf{g}}_0+{\mathbf{E}}_1\right){\mathbf{E}}_0^{-1}{\mathbf{h}}_0-{\mathbf{h}}_0=\mathbf{0} $$
(25b)

where Eq. (25a) can be solved by the function ‘care’ in MATLAB, and Eq. (25b) can be solved by the function ‘lyap’ in MATLAB.

The remaining part of Eq. (24) which is an equation for S1 can be written as

$$ {\mathbf{S}}_1{{\mathbf{V}}_1}^1{\mathbf{S}}_1+\left({{\mathbf{V}}_2}^1+ i\varpi {{\mathbf{V}}_3}^1\right){\mathbf{S}}_1+{\mathbf{S}}_1\left({{\mathbf{V}}_4}^1+ i\varpi {{\mathbf{V}}_5}^1\right)+{\mathbf{V}}_6^1-\varpi {\mathbf{S}}_{1,\varpi }=\mathbf{0} $$
(26)

with

$$ {{\mathbf{V}}_1}^1=\left({\mathbf{g}}_0+{\mathbf{E}}_1\right){\mathbf{E}}_0^{-1}\left({\mathbf{g}}_0+{\mathbf{E}}_1^T\right)+{\mathbf{E}}_2 $$
(27a)
$$ {{\mathbf{V}}_2}^1=-{{\mathbf{E}}_0}^{-1}\left({\mathbf{g}}_0+{\mathbf{E}}_1^T\right) $$
(27b)
$$ {{\mathbf{V}}_3}^1=-{{\mathbf{E}}_0}^{-1}{\mathbf{h}}_0 $$
(27c)
$$ {{\mathbf{V}}_4}^1=-\left({\mathbf{g}}_0+{\mathbf{E}}_1\right){\mathbf{E}}_0^{-1} $$
(27d)
$$ {{\mathbf{V}}_5}^1=-{\mathbf{h}}_0{\mathbf{E}}_0^{-1} $$
(27e)
$$ {{\mathbf{V}}_6}^1={\mathbf{E}}_0^{-1} $$
(27f)

Substituting Eq. (23b) into Eq. (26), three terms in descending order of the power of () are expressed as

$$ {\displaystyle \begin{array}{l}{\left( i\varpi \right)}^2\left({\mathbf{h}}_j{{\mathbf{V}}_1}^j{\mathbf{h}}_j+{{\mathbf{V}}_3}^j{\mathbf{h}}_j+{\mathbf{h}}_j{{\mathbf{V}}_5}^j\right)+\left( i\varpi \right)\left({\mathbf{h}}_j{{\mathbf{V}}_1}^j{\mathbf{g}}_j+{\mathbf{g}}_j{{\mathbf{V}}_1}^j{\mathbf{h}}_j+{{\mathbf{V}}_2}^j{\mathbf{h}}_j+{{\mathbf{V}}_3}^j{\mathbf{g}}_j+{\mathbf{h}}_j{{\mathbf{V}}_4}^j+{\mathbf{g}}_j{{\mathbf{V}}_5}^j-{\mathbf{h}}_j\right)\\ {}+{\mathbf{g}}_j{{\mathbf{V}}_1}^j\mathbf{g}+{\mathbf{V}}_6^j+{{\mathbf{V}}_2}^j{\mathbf{g}}_j+{\mathbf{g}}_j{{\mathbf{V}}_4}^j-{{\mathbf{S}}_{j+1}}^{-1}\left[{{\mathbf{V}}_1}^j\left({\mathbf{g}}_j+ ia{\mathbf{h}}_j\right)+{{\mathbf{V}}_4}^j+ ia{{\mathbf{V}}_5}^j\right]\\ {}-\left[\left({\mathbf{g}}_j+ ia{\mathbf{h}}_j\right){{\mathbf{V}}_1}^j+{{\mathbf{V}}_2}^j+ ia{{\mathbf{V}}_3}^j\right]{{\mathbf{S}}_{j+1}}^{-1}+{{\mathbf{S}}_{j+1}}^{-1}{{\mathbf{V}}_1}^j{{\mathbf{S}}_{j+1}}^{-1}+\varpi {{\mathbf{S}}_{j+1,\varpi}}^{-1}=0\end{array}} $$
(28)

Equation (28) is satisfied when all the three terms are equal to zero, setting the coefficient of the ()2 and () terms to zero results in

$$ {\mathbf{h}}_j{{\mathbf{V}}_1}^j{\mathbf{h}}_j+{{\mathbf{V}}_3}^j{\mathbf{h}}_j+{\mathbf{h}}_j{{\mathbf{V}}_5}^j=\mathbf{0} $$
(29a)
$$ {\mathbf{h}}_j{{\mathbf{V}}_1}^j{\mathbf{g}}_j+{\mathbf{g}}_j{{\mathbf{V}}_1}^j{\mathbf{h}}_j+{{\mathbf{V}}_2}^j{\mathbf{h}}_j+{{\mathbf{V}}_3}^j{\mathbf{g}}_j+{\mathbf{h}}_j{{\mathbf{V}}_4}^j+{\mathbf{g}}_j{{\mathbf{V}}_5}^j-{\mathbf{h}}_j=\mathbf{0} $$
(29b)

Pre- and post-multiplying Eq. (29a) with \( {\mathbf{h}}_j^{-1} \) respectively yields

$$ {{\mathbf{V}}_1}^j+{\mathbf{h}}_j^{-1}{{\mathbf{V}}_3}^j+{{\mathbf{V}}_5}^j{\mathbf{h}}_j^{-1}=\mathbf{0} $$
(30)

Equations (29b) and (30) can be solved by the function ‘layp’ in MATLAB. Then the remaining part of Eq. (28) is an equation for Sj + 1 (Bazyar and Song 2008)

$$ {\mathbf{S}}_{j+1}{{\mathbf{V}}_1}^{j+1}{\mathbf{S}}_{j+1}+\left({{\mathbf{V}}_2}^{j+1}+ i\varpi {{\mathbf{V}}_3}^{j+1}\right){\mathbf{S}}_{j+1}+{\mathbf{S}}_{j+1}\left({{\mathbf{V}}_4}^{j+1}+ i\varpi {{\mathbf{V}}_5}^{j+1}\right)+{\mathbf{V}}_6^{j+1}-\varpi {\mathbf{S}}_{j+1,\varpi }=\mathbf{0} $$
(31)

with

$$ {{\mathbf{V}}_1}^{j+1}={\mathbf{g}}_j{{\mathbf{V}}_1}^j\mathbf{g}+{\mathbf{V}}_6^j+{{\mathbf{V}}_2}^j{\mathbf{g}}_j+{\mathbf{g}}_j{{\mathbf{V}}_4}^j $$
(32a)
$$ {{\mathbf{V}}_2}^{j+1}=-\left({{\mathbf{V}}_1}^j{\mathbf{g}}_j+{{\mathbf{V}}_4}^j\right) $$
(32b)
$$ {{\mathbf{V}}_3}^{j+1}=-\left({{\mathbf{V}}_1}^j{\mathbf{h}}_j+{{\mathbf{V}}_5}^j\right) $$
(32c)
$$ {{\mathbf{V}}_4}^{j+1}=-\left({\mathbf{g}}_j{{\mathbf{V}}_1}^j+{{\mathbf{V}}_2}^j\right) $$
(32d)
$$ {{\mathbf{V}}_5}^{j+1}=-\left({\mathbf{h}}_j{{\mathbf{V}}_1}^j+{{\mathbf{V}}_3}^j\right) $$
(32e)
$$ {{\mathbf{V}}_6}^{j+1}={{\mathbf{V}}_1}^j $$
(32f)

The continued fraction solution of Eq. (23) is constructed from the coefficient matrix g0, h0, gj, hj, and Sj−1, which can be solved by Eqs. (27), (29b) and (30).

Scaled boundary finite element transformation

The same weight residual method used to the boundary condition Eq. (5a), and transformed to the scaled boundary coordinates, it can be obtained that

$$ {\mathbf{E}}_0\xi {\mathbf{P}}_{\mathrm{j},\xi }+{{\mathbf{E}}_1}^T{\mathbf{P}}_j=-{\omega}^2{\mathbf{M}}_1{\mathbf{n}}_x $$
(33)

with

$$ {\mathbf{n}}_x={\left({n}_{x1}\kern0.5em {n}_{x2}\right)}^{\mathrm{T}} $$
(34a)
$$ {\mathbf{M}}_1={\rho}_w{\int}_{-1}^1{\overline{\mathbf{N}}}^T\mathbf{N} d\eta $$
(34b)

where \( \overline{\mathbf{N}} \) is the mapping function of the structure, and M1 is lumped as

$$ {\mathbf{M}}_1=\frac{\rho_w{l}_e}{2}\left[\begin{array}{cc}1& 0\\ {}0& 1\end{array}\right] $$
(35)

It is obvious that the left of Eq. (33) is the internal nodal force Q(ξ). The same signs in Eq. (33) applied to the assembled system. Substituting Eqs. (19) and (20) into Eq. (33), the relation of the hydrodynamic pressure and the displacement of the cylinder at the interface can be obtained as

$$ {\mathbf{P}}_j={\omega}^2{\left[\mathbf{S}\left(\varpi \right)\right]}^{-1}{\mathbf{M}}_1\mathbf{n} $$
(36)

Coupled finite element equation of water-cylinder interaction system

The cylinder is assumed as a cantilever only the lateral deformation is considered, which can be solved by the finite element method (Chandrupatla and Belegundu 2013). After the spatial discretization to the cylinder, the finite element equation can be written as the partitioned matrix form as follow

$$ \left[\begin{array}{cc}{\mathbf{M}}_{\mathrm{O}}& \boldsymbol{0}\\ {}\boldsymbol{0}& {\mathbf{M}}_{\mathrm{I}}\end{array}\right]\left\{\begin{array}{c}{\ddot{\mathbf{u}}}_{\mathrm{O}}\\ {}{\ddot{\mathbf{u}}}_{\mathrm{I}}\end{array}\right\}+\left[\begin{array}{cc}{\mathbf{C}}_O& {\mathbf{C}}_{OI}\\ {}{\mathbf{C}}_{IO}& {\mathbf{C}}_I\end{array}\right]\left\{\begin{array}{c}{\ddot{\mathbf{u}}}_{\mathrm{O}}\\ {}{\ddot{\mathbf{u}}}_{\mathrm{I}}\end{array}\right\}+\left[\begin{array}{cc}{\mathbf{K}}_O& {\mathbf{K}}_{OI}\\ {}{\mathbf{K}}_{IO}& {\mathbf{K}}_I\end{array}\right]\left\{\begin{array}{c}{\mathbf{u}}_O\\ {}{\mathbf{u}}_I\end{array}\right\}=\left\{\begin{array}{c}\boldsymbol{0}\\ {}{\mathbf{f}}_I\end{array}\right\} $$
(37)

where the subscripts I denotes the nodes of the cylinder immersed in water and O denotes the nodes of the cylinder in air, respectively; u is the absolute motion vector with the given bedrock motion ug; the dot over variable represents the derivative to time; and M, C and K are the lumped mass, damping and stiffness matrices, respectively, and fI is the hydrodynamic force vector caused by the water-cylinder interaction. The element stiffness matrix is obtained as Re. (Chandrupatla and Belegundu 2013).

By using Fourier transform, Eq. (37) can be rewritten as

$$ -{\omega}^2\left[\begin{array}{cc}{\mathbf{M}}_{\boldsymbol{O}}& \mathtt{0}\\ {}\mathtt{0}& {\mathbf{M}}_{\mathrm{I}}\end{array}\right]\left\{\begin{array}{c}{\mathbf{U}}_{\boldsymbol{O}}\\ {}{\mathbf{U}}_{\boldsymbol{I}}\end{array}\right\}+\left( i\omega \right)\left[\begin{array}{cc}{\mathbf{C}}_{\boldsymbol{O}}& {\mathbf{C}}_{\boldsymbol{O}\boldsymbol{I}}\\ {}{\mathbf{C}}_{\boldsymbol{I}\boldsymbol{O}}& {\mathbf{C}}_{\boldsymbol{I}}\end{array}\right]\left\{\begin{array}{c}{\mathbf{U}}_{\boldsymbol{O}}\\ {}{\mathbf{U}}_{\boldsymbol{I}}\end{array}\right\}+\left[\begin{array}{cc}{\mathbf{K}}_{\boldsymbol{O}}& {\mathbf{K}}_{\boldsymbol{O}\boldsymbol{I}}\\ {}{\mathbf{K}}_{\boldsymbol{I}\boldsymbol{O}}& {\mathbf{K}}_{\boldsymbol{I}}\end{array}\right]\left\{\begin{array}{c}{\mathbf{U}}_{\boldsymbol{O}}\\ {}{\mathbf{U}}_{\boldsymbol{I}}\end{array}\right\}=\left\{\begin{array}{c}\mathtt{0}\\ {}{\mathbf{F}}_{\mathrm{I}}\end{array}\right\} $$
(38)

The cantilever immersed in water is separated into N nodes, the corresponding z-coordinates, lateral deformations and mapping functions are

$$ {\mathbf{z}}_{\mathrm{I}}={\left\{{z}_1\kern0.5em {z}_2\kern0.5em \cdots \kern0.5em {z}_{\mathrm{N}}\right\}}^{\mathrm{T}} $$
(39a)
$$ {\mathbf{U}}_{\mathrm{I}}={\left\{{U}_{\mathrm{I}1}\kern0.5em {U}_{\mathrm{I}2}\kern0.5em \cdots \kern0.5em {U}_{\mathrm{I}\mathrm{N}}\right\}}^{\mathrm{T}} $$
(39b)
$$ {\mathbf{N}}_{\mathrm{Z}}(z)=\left\{{N}_1(z)\kern0.5em {N}_2(z)\kern0.5em \cdots \kern0.5em {N}_{\mathrm{N}}(z)\right\} $$
(39c)

Then arbitrary node coordinate and deformation can be expressed as

$$ z={\mathbf{N}}_{\mathrm{z}}(z){\mathbf{z}}_{\mathrm{I}} $$
(40a)
$$ U={\mathbf{N}}_{\mathrm{z}}{\mathbf{U}}_{\mathrm{I}} $$
(40b)

The shape function is defined as

$$ \mathbf{W}={\int}_0^h{\left[{\mathbf{N}}_{\mathrm{z}}(z)\right]}^{\mathrm{T}}{\mathbf{N}}_{\mathrm{z}}(z)\mathrm{dz} $$
(41)

The corresponding modal vector is written as

$$ {\boldsymbol{\upphi}}_j={\left\{\cos \left({\lambda}_j{z}_1\right)\kern0.5em \cos \left({\lambda}_j{z}_2\right)\kern0.5em \cdots \kern0.5em \cos \left({\lambda}_j{z}_{\mathrm{N}}\right)\right\}}^{\mathrm{T}} $$
(42)

Substituting Eq. (41) into the modal displacement, we get

$$ {U}_j=\frac{2}{h}\ {\left[{\boldsymbol{\upphi}}_j\right]}^{\mathrm{T}}{\mathbf{WU}}_{\mathrm{I}} $$
(43)

The interaction force in Eq. (38) is

$$ {\mathbf{F}}_{\mathrm{I}}={\int}_0^h{\mathbf{N}}^{\mathrm{T}}F\mathrm{d}z $$
(44)

with the continuous hydrodynamic force equal to the integral along interface Si

$$ F=\sum \limits_{j=1}^{\infty }-\underset{S}{\int }{\mathbf{P}}_j.{\mathbf{n}}_x\mathrm{ds}{U}_j\cos {\lambda}_jz=\sum \limits_{j=1}^{\infty }{f}_j{U}_j\cos {\lambda}_jz $$
(45)

where \( {f}_j=-\underset{S}{\int }{\mathbf{P}}_j.{\mathbf{n}}_x\mathrm{d}s \), where the dot between two vector is the inner product of the vectors.

Substituting Eqs. (43) and (45) into Eq. (44), the hydrodynamic force can be expressed as follows

$$ {\mathbf{F}}_{\mathrm{I}}=-\tilde{\mathbf{S}}{\mathbf{U}}_{\mathrm{I}} $$
(46)
$$ \tilde{\mathbf{S}}=\frac{2}{h}\sum \limits_{j=1}^{\infty }{f}_j\mathbf{W}{\boldsymbol{\upphi}}_j{{\boldsymbol{\upphi}}_j}^{\mathrm{T}}\mathbf{W} $$
(47)

Verification and application

The definition of section parameters is shown in Fig. 3. Then, the seismic responses of circular, elliptical and rectangle cylinders are investigated, where the density, Yang’s modulus and damping ratio of the cylinder is 2500 kg/m3, 30,000 MPa and 0.05.

Fig. 3
figure3

The cross-section of the cylinders

Verification

The accuracy of the present method is verified by comparing the hydrodynamic force on rigid circular, elliptical and rectangle cylinders with the analytical or numerical solutions (Wang et al. 2018a, 2019a; Du et al. 2014), where the numerical solution is calculated by finite element method.

Figure 4 shows the real and the imaginary part of the hydrodynamic force on circular cylinder, where the horizontal axis is r0 = ϖ, ‘SBFEM’, and ‘Analytical’ represent the solution calculated by the proposed method and the analytical solution. Figures 5 and 6 shows the real and the imaginary part of the hydrodynamic force on elliptical and rectangle cylinders, where ‘Numerical’ represents the numerical solution. It can be seen that the proposed method is in good agreement with the reference solution.

Fig. 4
figure4

The hydrodynamic pressure on the rigid circular model in frequency

Fig. 5
figure5

The hydrodynamic pressure on the rigid elliptical cylinder in frequency

Fig. 6
figure6

The hydrodynamic pressure on the rigid rectangle cylinder in frequency

Application

The seismic responses and natural frequencies of circular, elliptical and rectangle cylinders are further investigated. The seismic responses of cylinders surrounded by water in frequency domain are analyzed by using ground excitation ug = eiωt. Two dimensionless parameters including width-depth ratio (L = 2b/h) and aspect ratio (Rs = a/b) are introduced. Two dimensionless variables Ehωn and ECOMωn are introduced to represent the effect of hydrodynamic pressure and water compressibility on the natural frequency of the cylinders surrounded by water, which can be expressed as

$$ {E}_{h\omega n}=\frac{\left|{\omega}_c-{\omega}_0\right|}{\omega_c}\times 100\% $$
(48)
$$ {ECOM}_{\omega n}=\frac{\left|{\omega}_c-{\omega}_i\right|}{\omega_c}\times 100\% $$
(49)

where, ωc, ωi and ω0 mean the fundamental frequency of the cylinder surrounding by compressible water, incompressible water, and air, respectively.

The frequency responses of circular cylinders are firstly calculated, where the maximum displacement response on the top of the cylinder is selected as the observe object. Figure 7 is the frequency response of the circular cylinder surrounded by incompressible water, compressible water and in air. Figure 8 shows the variation of the ratios ωc/ω0 and ωc/ωi with respect to L. It can be seen from Figs. 7 and 8 that water-cylinder interaction reduces the natural frequency of the cylinder, and this influence decreases as width-depth ratio increasing. It is obvious that water compressibility has little influence on the natural frequency of cylinder. Figure 9 shows the variation of the ratios Ac/An and Ac/Ai with respect to L, where Ac, Ai and An are the amplitude displacement on the top of cylinder at fundamental frequency in the case of compressible water, incompressible water and air. Compared Figs. 7 and 9, it can be seen that water-cylinder interaction can increase the seismic response of slender cylinders (circular cylinder L ≤ 0.8). However, water-cylinder interaction can significantly decrease the seismic response of squat cylinders (circular cylinder L ≥ 1), and this trend increases as width-depth ratio increasing.

Fig. 7
figure7

Frequency response of the circular cylinder surrounded by incompressible water, compressible water and in air

Fig. 8
figure8

Variation of the ratios ωc/ω0 and ωc/ωi with respect to L of circular cylinder

Fig. 9
figure9

Variation of the ratios Ac/An and Ac/Ai with respect to L of circular cylinders

Figure 10 shows the variation of Ehωn with respect to L for the elliptical and rectangle cylinders with different Rs. It is obvious that the effect of hydrodynamic pressure on the natural frequency decreases as L and Rs increase. Figure 11 shows the variation of ECOMωn with respect to L for the elliptical and rectangle cylinders with different Rs. It can be seen that the effect of water compressibility decreases as L increases and this influence is limited in 6% for elliptical and rectangle cylinders with L ≤ 2 and Rs ≤ 4. In case of L = 1.2 and Rs = 2 (L = 1.0 and Rs = 2), the natural frequency of structure and surrounded by water is nearly resonance. Therefore, the effect of hydrodynamic pressure is significant, as shown in Fig. 10.

Fig. 10
figure10

Variation of Ehωn with L for elliptical and rectangle cylinders

Fig. 11
figure11

Variation of ECOMωn with L for elliptical and rectangle cylinder

Conclusion

Based on the scaled boundary finite element, an accurate frequency domain model is proposed to simulate the seismic response of cylinders with arbitrary shapes cross-section surrounded by water. The numerical examples show that the proposed model has high-accuracy. The proposed model is also used to investigate the seismic responses and natural frequencies of the circular, elliptical and rectangle cylinders. The results indicate that water-cylinder interaction can reduces the natural frequency of the cylinder, and this influence decreases as width-depth ratio and aspect ratio increase. It is also obtained that water compressibility has little influence on the natural frequency of cylinder.

Nomenclature

P(x, y, z, ω) hydrodynamic pressure expressed in frequency domain

ω circle frequency

c velocity of the wave propagate in water

Pj(x, y, ω) the modal hydrodynamic pressure in xy-plane

N mapping function

J Jacobi matrix

Availability of data and materials

All data, models, and code used during the study appear in the published article.

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Acknowledgements

The support is gratefully acknowledged. The results and conclusions presented are of the authors and do not necessarily reflect the view of the sponsors.

Funding

This work is supported by the National Natural Science Foundation of China (52078010) and Ministry of Education Innovation Team of China (IRT_17R03).

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MZ carried out the manuscript writing and numerical analysis, PW provided guidance in methodology development, PW and XW provided guidance in technical writing, PW, XW and CZ carried out the analytical model, and XD participated in the design of the study. The authors read and approved the final manuscript.

Corresponding author

Correspondence to Piguang Wang.

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Zhao, M., Wang, X., Wang, P. et al. An accurate frequency-domain model of water interaction with cylinders of arbitrary shape during earthquakes. ABEN 2, 5 (2021). https://doi.org/10.1186/s43251-020-00026-3

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Keywords

  • Water-cylinder interaction
  • Surface wave
  • Water compressibility
  • Seabed flexibility
  • Added mass
  • Wave and earthquake action