- Email: [email protected]
An efficient numerical model for hydrodynamic added mass of immersed column with arbitrary cross - Section
Ocean Engineering 187 (2019) 106192
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
An efficient numerical model for hydrodynamic added mass of immersed column with arbitrary cross - Section Jiarui Zhang a, Kai Wei a, *, Shunquan Qin a, b a b
Department of Bridge Engineering, College of Civil Engineering, Southwest Jiaotong University, Chengdu, 610031, China China Railway Major Bridge Reconnaissance & Design Institute Co Ltd, Wuhan, 430050, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Immersed column Added mass Arbitrary cross Section Potential-based fluid element Hydrodynamic pressure Seismic analysis
The assessment of earthquake-induced hydrodynamic added mass is critical in the seismic design of structures supported by immersed columns. This paper developed an efficient potential-based numerical model to deter mine the hydrodynamic added mass for immersed column with arbitrary cross - section. General formulations of the hydrodynamic added mass for a column with general cross - section under horizontal earthquake were derived. The accuracy of the developed model for immersed columns was validated analytically and experi mentally. The developed model was then applied to evaluate the hydrodynamic added mass of an immersed column with dumbbell cross - section. The hydrodynamic added mass of the example dumbbell column along various earthquake directions was determined and the directional-dependent added mass was expressed as an analytical function of the added mass along the major and minor axes of the column and the earthquake di rection. The frequency- and time-domain seismic responses of the example dumbbell column were finally computed using both fluid-structure interaction (FSI) model and the developed hydrodynamic added mass model. The good agreement between seismic responses from the FSI and developed added mass model confirmed the high accuracy and efficiency of the developed model for the hydrodynamic added mass of immersed columns with arbitrary cross - section under earthquakes.
1. Introduction The dynamics of structures supported by columns submerged in water, such as sea-crossing bridge foundations (Qin and Gao, 2017), offshore platforms (Mirzadeh et al., 2016), offshore wind turbines (Wei et al., 2017), etc., require consideration of fluid-structure interaction (FSI), which does not arise for structures on land. It is a common knowledge that the vibration of immersed column subjected to earth quakes induces the motion of nearby water, producing an extra hydro dynamic force on the structure (Zhang et al., 2019; Pang et al., 2015). This extra force is usually modeled as the combination of a hypothetical mass of water and the acceleration of the structure. The hypothetical mass, typically termed as added mass, represents the effective mass of water that participates in the structural vibration through FSI. It was firstly introduced by Westergaard (1933) in the analyses of hydrody namic pressure on the dam subjected to harmonic ground motion. Based € on the concept of added mass, U�sciłowska and Kołodziej (1998), Oz (2003) and Wu and Hsu (2007) derived the analytical solutions to determine the free vibration and natural frequencies of immersed
columns. For the added mass assessment of an immersed column, the section of the column in the early literatures is usually assumed to be circular due to its accessibility to the equations of FSI interface boundary in the closed-form solutions, and many research efforts are devoted to the added mass model of columns with circular cross - section. Liaw and Chopra (1974) proposed a closed-form solution of the added mass for circular columns surrounded by water. Han and Xu (1996) presented a theoretical model of an added mass representation for a flexible immersed column vibrating in water. Different simplified equations for the added mass of immersed circular column were developed by Li and Yang (2013) and Jiang et al. (2017), respectively. Added mass models for both inner water and outer water of an immersed column with cir cular hollow cross - section have also been developed by Yang and Li (2013). The continuing development of offshore engineering contributes to the broad application of immersed columns with complex geometry other than circular cross - sections (Wang et al., 2018a). Nagaya and Hai (1985) developed a method for solving seismic response problems of a column of variable tapered cross - section. However, the added mass of immersed column with non-circular section was usually simplified as the
* Corresponding author. Room 507, Tunnel Research Building, No. 111, North 1st Section of Second Ring Road, Chengdu, Sichuan, 610031, China. E-mail address: [email protected] (K. Wei). https://doi.org/10.1016/j.oceaneng.2019.106192 Received 1 January 2019; Received in revised form 4 June 2019; Accepted 6 July 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 1. Illustration of an immersed column with arbitrary cross - section subjected to horizontal ground motion.
added mass for column with equivalent circular cross - section times a shape correction factor (Det Norske Veritas, 2010; European Committee for Standardization, 2005), which are limited to several typical cross sections, such as ellipse or rectangle, and insufficient for general engi neering practices. More recently, Yu et al. (2019) dealt with the problem of radiation from a floating cylinder with an arbitrary cross section and proposed a semi-analytical model based on linear potential flow theory. Zheng et al. (2019) further developed and expanded the model from one single to multiple floating cylinders with arbitrary cross - sections. The numerical model allows the evaluation of the earthquake induced hydrodynamics through solution of the FSI equations, which has been wildly adopted to address questions in complex fluid-structure systems. Bouaanani and Lu (2009) assessed the performance of potential-based fluid elements (PBFEs) regarding the dam-reservoir system and investigated the effect of the reservoir bottom absorption on seismic responses of dam based on this numerical method. Wei et al. (2013) conducted modal analyses of pile foundations submerged in water using PBFEs and verified the numerical model with experimental results. Wang et al. (2018b) adopted Lagrange formulation of fluid in the reservoir-dam-foundation numerical model to investigate the damage characteristic of dam. Boundary element method (BEM) (Liu, 2019; Xu et al., 2018), arbitrary Lagrangian-Eulerian (ALE) algorithm (Ozdemir et al., 2010; Zhao et al., 2014) etc., have also been applied to provide insight into the FSI issues in engineering fields. Although the seismic response of immersed column can now be computed using the FSI nu merical approaches, most of these techniques have yet been fully implemented in the common engineering practice, especially at the preliminary stage of seismic design, as they require specialized software or advanced programming and high-level expertise, and may result in extensive modeling and computational efforts. Since the added mass represents the earthquake-induced hydrodynamic force in a simplified and efficient way, a general hydrodynamic added mass model of col umns with arbitrary cross - section aided by FSI numerical method may facilitate the work of researchers and designers regarding the immersed columns. For the purpose of developing an efficient numerical model to assess the hydrodynamic added mass, this paper is organized as follows: firstly, a general formulation of hydrodynamic added mass and its corre sponding numerical framework are developed for an immersed column with arbitrary cross - section; secondly, the effectiveness of the devel oped model in the calculation of the added mass for uniform immersed columns with circular and elliptical sections and circular tapered col umns is validated with the analytical results from previous literatures; thirdly, the added mass of an immersed cylinder is calculated through
the developed model and is verified by comparison with experimental modal test results; fourthly, an immersed column with dumbbell cross section is selected as an example study, the developed model is used to calculate the hydrodynamic added mass of the column; and finally, the frequency - and time - domain seismic responses of such column are compared between the results from the FSI model and the column model with added mass and the effect of ground motion direction on the hy drodynamic added mass are investigated. 2. Model for determination of hydrodynamic added mass 2.1. Formulations of hydrodynamic added mass We consider an immersed column with arbitrary cross - section €g ðtÞ as illustrated in Fig. 1. subjected to horizontal ground acceleration u Assuming water to be inviscid and irrotational, the hydrodynamic pressure p within the water domain is governed by the wave equation in cylindrical coordinates ðr; θ; zÞ as follows:
∂2 p 1 ∂p 1 ∂2 p ∂2 p 1 ∂2 p þ þ ¼ þ ∂r2 r ∂r r2 ∂θ2 ∂z2 C2 ∂t2
(1)
where C is velocity of the compression waves in water, and t is time. Although the effect of surface gravity waves will make the hydrody namic pressure vary with the excitation frequency, but its influence on the response of engineering structures in practice is not significant (Huang and Li, 2011). The surface gravity waves are hence not included in the following derivation. The boundary conditions for the governing equation according to the aforementioned assumptions are: i) No surface gravity waves at the free surface z ¼ Hw pðr; θ; Hw ; tÞ ¼ 0
(2)
ii) No vertical motion at the bottom of the fluid domain z ¼ 0
∂p ðr; θ; 0; tÞ ¼ 0 ∂z
(3)
iii) The consistency of radial motion of structure and surrounding water at fluid-structure surface, i.e., surface function Sðr; θ; zÞ
2
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
2.2. Potential-based numerical analysis framework for hydrodynamic added mass The closed-form solution of the hydrodynamic pressure pðr; θ; zj Þ only exists for typical cross - sections, such as circle and ellipse. In order to compute the hydrodynamic pressure on the arbitrary cross - section, three-dimensional (3D) potential-based fluid elements (PBFEs), based on φ U formulation (Olson and Bathe, 1985), is implemented in the framework. In this formulation, φ and U denote the velocity potential of fluid domain and the displacement of solid domain, respectively, yielding simple FSI modeling and avoiding heavy computational burden. The velocity potential of PBFEs in the numerical model satisfies the wave equation as follows: 2
▽ φ¼
1 ∂2 φ C2 ∂t2
(7)
Fluid-structure interaction is accounted for through FSI boundary condition at the water - column interfaces, defined as
∂φ ∂U ! ¼ n ∂t ∂! n
(8)
where ! n is the unit normal vector perpendicular to the water - column interface towards the water domain. The fluid velocity potential φ at the free surface boundary and the side and bottom walls of the fluid domain hence obeys the following equation: φ¼0
Fig. 2. Flowchart of analysis framework for hydrodynamic added mass.
∂p ðr; θ; z; tÞjSðr;θ;zÞ ¼ ∂! ns
ρw €ug ðtÞ! n s ⋅! ng
(9)
For the given immersed column with arbitrary cross - section, the column and the surrounding water are discretized into solid elements and PBFEs, respectively, as shown in Fig. 2. The velocity of pressure waves in water is set to be 1440 m/s, resulting in a bulk modulus of 2.1 GPa, to include the compressibility of water. The radius of the water domain should be larger than eight times the maximum radial dimen sion of the column cross - section for the purpose of eliminating the wave reflection due to the side wall boundaries. The properties of the column are defined according to its material information. The vertical degrees of freedom for nodes at bottom of the immersed column are fully con strained, while the horizontal degrees are set to be free. Once the potential-based numerical model is developed, the hydrodynamic pressure pðr; θ; zj Þ on the column with arbitrary cross - section can be calculated using the finite element model. In order to obtain the hy drodynamic pressure at different distance z above the column base, sweeping algorithm is used to generate the hexahedral meshes for the immersed column as illustrated in Fig. 1. The division numbers of the column in the cylindrical coordinate along Sðr; θ; zÞ and z-direction are defined as n and m, respectively. The formulation of the hydrodynamic added mass can be given as
(4)
n s is the unit normal in which Hw is water depth, ρw is water density, ! vector of surface function Sðr; θ; zÞ pointing outwards, ! n g is the unit directional vector of ground motion along the direction θg , and ⋅ is the dot product between vectors. According to the work of Chopra and his colleagues (Liaw and Chopra, 1974; Goyal and Chopra, 1989a, 1989b), the amplitude of the hydrodynamic added mass per unit height ma is a function of z, the distance above the column base. Moreover, the hydrodynamic added mass ma depends on the vibration direction for columns with non-circular cross - section (Goyal and Chopra, 1989b). Therefore, a general form of hydrodynamic added mass per unit height at zj of a column vibrating along the direction θg is denoted as ma ðzj ; θg Þ, which can calculated by � �� ug m a z j ; θ g ¼ Fp z j ; θ g € (5)
n � X � Lj m a zj ; θ g ¼ n s;i ⋅! p ri ; θi ; zj ! n g ; j ¼ 1; …; m þ 1 n i¼1
where Fp ðzj ; θg Þ is the resultant hydrodynamic force along direction θg €g . per unit height at zj due to vibration of column under ground motion u When the immersed column is subjected to a unit horizontal ground €g ðtÞ ¼ 1 m/s2, along direction θg , the added motion acceleration, i.e., u mass ma ðzj ; θg Þ can also be further expressed as follows Z � �� � n s ⋅! ma zj ; θg ¼ Fp zj ; θg 1 ¼ n g dS (6) p r; θ; zj ! Sðr;θ;zj Þ
(10)
in which pðri ; θi ; zj Þ is the hydrodynamic pressure at point ðri ; θi ; zj Þ, i ¼ 1; 2; …; n; ! n s;i is the unit vector at point ðri ; θi ; zj Þ normal to the surface function Sðr; θ; zj Þ pointing outwards; Sðr; θ; zj Þ is the surface function of the column at z ¼ zj ; Lj is the circumference of Sðr; θ; zj Þ. Considering that Cartesian coordinate is more general in the finite element modeling for columns with complex cross - section, the general formulation of hydrodynamic added mass Eq. (10) can be replaced by Eq. (11) in Cartesian coordinate,
where pðr; θ; zj Þ is the hydrodynamic pressure acting on the column surface induced by its rigid motion.
n � X � � Lj n s;i ⋅ cosθg ; sinθg ; j ¼ 1; …; m þ 1 m a zj ; θ g ¼ p xi ; yi ; zj ! n i¼1
(11)
where pðxi ; yi ; zj Þ is the hydrodynamic pressure at point ðxi ;yi ;zj Þ, i ¼ 1; 2; …;n. The added mass coefficient Ca ðzÞ, defined as the added mass ma ðzÞ 3
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
divided by the column cross - section area and water density, can be calculated easily as well. The following analysis process shown in Fig. 2 can be performed to calculate the hydrodynamic added mass of the immersed column with arbitrary cross - section: i) Apply a unit acceleration to the entire column along the ground motion direction. The time-history of the acceleration with the duration of 1.2 s is given in Fig. 2. ii) Carry out transient analysis to obtain the hydrodynamic pressure pðri ; θi ; zj Þ or pðxi ; yi ; zj Þ at each node on the water - column interface. Oscillation exists in the initial zone of the time-history of hydrodynamic pressure due to the compressibility of water, but it disappears quickly. The stable value of hydrodynamic pressure at the last time-step is recommended. iii) Compute the hydrodynamic added mass at each zj , distance above the column base, through the integration of hydrodynamic pressure on the water - column interface following Eq. (10) or 11. iv) Plot the curve of hydrodynamic added mass as a function of distance above the base of immersed column. Fig. 3. Dimensions of immersed column with circular cross - section.
3. Analytical validation of the developed added mass model Immersed columns with circular and elliptical cross - section and the
Fig. 4. Comparison of added mass coefficient between results from Liaw and Chopra (Liaw and Chopra, 1974), Li and Yang (Li and Yang, 2013), Jiang et al. (Jiang et al., 2017) and developed model: (a) Hw ¼ 30m, D ¼ 4m; (b) Hw ¼ 30m, D ¼ 5m; (c) Hw ¼ 30m, D ¼ 6m; (d) Hw ¼ 30m, D ¼ 7m. 4
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
3.1. Immersed column with circular cross - section
Table 1 Comparison of integrated hydrodynamic added mass coefficient. Cases
Liaw and Chopra (Liaw and Chopra, 1974)
Developed model
Hw ¼ 30 m, D¼4m
0.924
0.920
0.39%
Hw ¼ 30 m, D¼5m
0.906
0.904
0.28%
Hw ¼ 30 m, D¼6m
0.889
0.887
0.21%
Hw ¼ 30 m, D¼7m
0.872
0.870
0.16%
An immersed column with circular cross - section, as illustrated in Fig. 3, is taken as the first example for validation. The diameter of the column is D and the water depth is Hw . The hydrodynamic added mass coefficients Ca ðzÞ of the columns with diameter D varying from 4 to 7 m and with a water depth of 30 m, are calculated using the developed model and the formulas proposed in previous literatures. The closedform analytical equation of hydrodynamic added mass for circular cyl inder proposed by Liaw and Chopra (1974) based on radiation wave theory is given in Eq. (12) and selected as the reference. Simplified formulations of the added mass for such column developed by Li and Yang (2013) and Jiang et al. (2017) shown in Eq. (13) and Eq. (14), respectively, are also used for comparison. Equation from Liaw and Chopra (1974):
Relative error
ma ðzÞ ¼ 4ρw πD
Nq X 1
� � � Iq λ’q D cos αq z Eq 2 2λ’q Hw
Equation from Li and Yang (2013): 8 2 > > �> � < 6 10 2 ρ πD D 6 ma ðzÞ ¼ w 1 exp6 1 ðz 1 5 2 4DH 3w 4 Hw > > > :
(12)
39 > > = 7> 7 Hw Þ7 5> > > ;
(13)
Equation from Jiang et al. (2017):
Table 2 Comparison of integrated hydrodynamic added mass coefficient. Cases
Fig. 5. Dimensions of immersed column with elliptical cross - section.
circular tapered column are widely used in offshore structures, such as foundations of sea-crossing bridges and offshore wind turbines. In order to validate the developed model, the hydrodynamic added mass of three immersed column examples, including two uniform columns (circular and elliptical cross - section) and one tapered column with circular cross - section is calculated by the developed model and analytical models from previous literatures. Accuracy of the hydrodynamic added mass is assessed through the comparison with the results from previous literatures.
Wang et al. (Wang et al., 2018c)
Developed model
Along majoraxis B =Hw ¼ 0.2 B =Hw ¼ 1.0
Relative error
0.847
0.845
B =Hw ¼ 0.5
0.672
0.673
0.07%
0.488
0.492
0.75%
Along minoraxis D =Hw ¼ 0.2
0.908
0.903
0.785
0.786
0.14%
D =Hw ¼ 1.0
0.625
0.630
0.85%
D =Hw ¼ 0.5
0.28%
0.52%
Fig. 6. Comparison of added mass coefficient between results from Wang et al. (Wang et al., 2018c) and developed model: (a) θg ¼ 0∘ ; (b) θg ¼ 90∘ . 5
Ocean Engineering 187 (2019) 106192
J. Zhang et al.
Table 3 Comparison of integrated hydrodynamic pressure.
Fig. 7. Dimensions of tapered column with circular cross - section.
ma ðzÞ ¼ ρw πD2
� H 1:5 w � 1 1:5 0:1D2 þ 3:8D 6:8 þ 4H w
� 2D þ 2Hw ðz exp DHw
Cases
Wang et al. (Wang et al., 2018d)
Developed model
Relative error
D1 =Hw ¼ 0.4, α ¼ 60∘
1.191
1.192
0.02%
D1 =Hw ¼ 0.4, α ¼ 70∘
1.149
1.148
0.04%
D1 =Hw ¼ 0.4, α ¼ 80∘
1.023
1.022
0.10%
D1 =Hw ¼ 1.0, α ¼ 60∘
0.561
0.561
D1 =Hw ¼ 1.0, α ¼ 70∘
0.588
0.588
0.05%
D1 =Hw ¼ 1.0, α ¼ 80∘
0.597
0.596
0.11%
0.00%
�� Hw Þ (14)
RH in which Iq ¼ 0 w cosðαq zÞdz, λ’q ¼ αq ¼ ð2q 1Þπ=2Hw , Eq ðxÞ ¼ K1 ðxÞ=ðK0 ðxÞ þ K2 ðxÞÞ, Kn ðxÞ is the modified Bessel function of order n of the second kind, and q ¼ 1; 2; ::; Nq , where Nq is the number of items, suggested to be larger than 50 for a smooth solution. The developed model is firstly used to assess the hydrodynamic added mass for the example columns. Potential-based numerical model for the immersed column with circular cross - section is built using ADINA (ADINA, 2010) according to the analysis framework depicted in Section 2. Hydrodynamic pressure on the column is calculated according to the flowchart given in Fig. 2. Because the hydrodynamic added mass ma ðzÞ of the circular column is independent of direction θg , it can be determined by Eq. (15). The analytical curves of the added mass ma ðzÞ for the example column are assessed directly using Eqs. (12)–(14), respectively. The added mass coefficient Ca ðzÞ for the column can then be calculated by Eq. (16) according to the curve of ma ðzÞ.
Fig. 9. Experimental setup: (a) positions of the cylinder and F/T transducer; (b) global view of the cylinder and the testing setup.
Fig. 8. Comparison of hydrodynamic pressure between the results from Wang et al. (Wang et al., 2018d) and the developed model: (a) tapered columns with D1 =Hw ¼ 0.4; (b) tapered columns with D1 =Hw ¼ 1.0. 6
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 10. Numerical and experimental results of wet-to-dry period ratios for the cylinder as a function of water depth. Fig. 12. Distribution of hydrodynamic added mass per unit height of the col umn along different direction θg .
Fig. 13. Relationship between hydrodynamic added mass and direction θg . n X
ma ðzÞ ¼ i¼1
Ca ðzÞ ¼
! � πD xi yi pðxi ; yi ; zÞ pffiffiffi2ffiffiffiffiffiffiffiffiffi2ffiffi; pffiffiffi2ffiffiffiffiffiffiffiffiffi2ffiffi ⋅ cosθg ; sinθg n xi þ yi xi þ yi
4ma ðzÞ ρw πD2
(15)
(16)
Fig. 4 shows the comparison of hydrodynamic added mass co efficients between the developed model and the analytical equations. It is clear that the results predicted from the developed model agree well with the closed-form results from Eq. (12) for all the circular columns. The tolerance between the developed model and results from Liaw and Chopra (1974) is significantly better than the simplified equations Eqs. (13) and (14). The differences between the Li and Yang’s results and Jiang’s results within the mediate water depth are mainly caused by the difference of the factors in the exponential function, which affects the distribution of added mass significantly. Furthermore, the coefficients Ca ðzÞ determined from developed model and results from Liaw and RH Chopra (1974) are integrated along water depth, 0 w Ca ðzÞdz =Hw , and
Fig. 11. Illustration of column with dumbbell cross - section: (a) dimensions of the example column, all dimensions are in meters; (b) FSI numerical model.
7
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 14. Frequency response of the studied column as function of the harmonic excitation frequency under ground motion along major-axis: (a) acceleration at column top; (b) displacement at column top; (c) base shear at column bottom; (d) base moment at column bottom.
the results are compared in Table 1. The relative error of the developed model is less than 0.5%, compared with the solutions from Liaw and Chopra (1974). All these results state that the developed model has high accuracy in the prediction of hydrodynamic added mass for immersed column with circular cross - section.
elliptical cross - section. For a given column with elliptical cross - section, its equivalent cir cular area of cross - section is πD2
π B2 4
for motion along major-axis, θg ¼ 0∘ ,
and is 4 for motion along minor-axis, θg ¼ 90∘ . The added mass co efficient Ca ðz; θg Þ can be defined by the following equation. 8 � > > 4ma z; θg > > > θg ¼ 0∘ ; along major axis > 2 � < ρw π B Ca z; θg ¼ (18) � > > 4m z; θ > ∘ > a 2g θ ¼ 90 ; along minor axis > g > : ρw π D
3.2. Immersed column with elliptical cross - section A column with elliptical cross - section submerged in a water depth of Hw , as shown in Fig. 5, is taken as the second example for validation. The length of major and minor axes of the elliptical cross - section is D and B, respectively. Immersed columns with B =Hw varying from 0.2 to 1.0 are selected as cases for direction of motion θg ¼ 0∘ , while columns with D =Hw varying from 0.2 to 1.0 are selected for direction of motion θg ¼ 90∘ . For each column, the aspect ratio D =B is set to 2. The analytical equation for elliptical cylinder developed by Wang et al. (2018c) is used to calculate its hydrodynamic added mass as reference. FSI model of the column and surrounding water is firstly built in ADINA (ADINA, 2010) to calculate the hydrodynamic pressure. In the application of the developed model to the column with elliptical cross section, the hydrodynamic added mass, ma ðz; θg Þ, can be determined from the pressure as follows, 1 0
The hydrodynamic added mass coefficients Ca ðz; θg Þ for all of these columns are calculated using Eqs. (17) and (18), and compared with results from Wang et al. (2018c) in Fig. 6. It can be seen that the scatters of Ca ðz; θg Þ as a function of distance to base determined by the developed model are in good agreement with the reference curves for all cases regardless of column dimension and ground motion direction. The maximum value of added mass appears at the column bottom and de creases gradually to zero at the water free surface. This tendency is similar to that of columns with circular cross - section. Because the elliptical cross - section is non-circular, the coefficients of columns with elliptical cross - section depend on the direction θg . The integrated hy drodynamic added mass coefficients are then calculated according to R Hw Ca ðz; θg Þdz =Hw for comparison in number. The results calculated 0 from developed model and Wang et al. (2018c) are listed and compared in Table 2. It is clear that the relative error between the developed model and results from Wang et al. (2018c) is less than 1%, which further proves that the developed model can evaluate the hydrodynamic added
n B C � X �L xi yi C ma z; θg ¼ pðxi ; yi ; zÞB @ qffixffi2ffiffiffiffiffiffiffiffiyffiffi2ffi; qffixffi2ffiffiffiffiffiffiffiffiyffiffi2ffiA⋅ cosθg ; sinθg n i i i i 2 2 i¼1 D D4 þ B 4 B D4 þ B 4
(17) R 2π pffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in which L ¼ 0 D2 sin θ þ B2 cos2 θ=2dθ is the circumference of the 8
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 15. Frequency response of the studied column as function of the harmonic excitation frequency under ground motion along minor-axis: (a) acceleration at column top; (b) displacement at column top; (c) base shear at column bottom; (d) base moment at column bottom.
(2018d) is selected as the reference for comparison. The hydrodynamic pressure on these circular tapered columns divided by ρw D1 =2 in the x z plane is determined through the solution of FSI models in ADINA (ADINA, 2010) and results are compared with results from Wang et al. (2018d) in Fig. 8. Results from the developed model agree well with the results from Wang et al. (2018d) regardless of taper angle α or ratio D1 =Hw . It is interesting to notice that, unlike the two types of column discussed above, the peak value of the hydrody namic pressure on the circular tapered column does not appear at the bottom of column. The influence of the taper angle on the hydrodynamic pressure is more significant for columns with relatively small value of D1 =Hw . The hydrodynamic pressure is then integrated over water depth RH following, 2 0 w pdz =ρw D1 Hw . The results from Wang et al. (2018d) and the developed model are then presented and compared in Table 3. Good consistency of the integrated hydrodynamic pressure can still be found. The maximum relative error is only 0.11%. The developed model can also be applied to the evaluation of the hydrodynamic pressure for tapered columns with circular cross - section.
Fig. 16. Accelerograms of N-S component of the Imperial Valley earthquake record at El-Centro normalized by its peak ground acceleration (PGA).
mass of immersed columns with elliptical cross - section accurately. 3.3. Tapered column with circular cross - section
4. Experimental validation of the developed added mass model
The third example for validation is a tapered column with circular cross - section, as shown in Fig. 7. The top and bottom diameter of the column are D1 , and D2 , respec tively, with a taper angle of α and submerged depth of Hw . Two cate gories of circular tapered columns including one with D1 =Hw ¼ 0:4 and the other with D1 =Hw ¼ 1:0 are adopted here. And for each category, three different taper angles varying from 60∘ to 80∘ are considered. To validate the accuracy of the developed model, the distribution of hy drodynamic pressure over water depth in the x z plane when the column moves along direction θg ¼ 0∘ investigated by Wang et al.
In this section, experimental modal tests of an immersed cylinder are conducted in a water tank to further verify the developed added mass model. Fig. 9 illustrates the configuration of the cylinder and testing devices. The F/T transducer is bolted with the lifting rod fixed on a testing trestle. The top surface of the testing cylinder is fastened with the F/T transducer, which measures the modal responses of the cylinder. The diameter and length of the cylinder are 0.15 m and 1.2 m, respec tively. The length of the gap between the bottom of water and the 9
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 17. Time-histories of the seismic responses of the column under earthquake along direction θg ¼ 0∘ : (a) acceleration at column top; (b) displacement at column top; (c) base shear at column bottom; (d) base moment at column bottom.
cylinder is 0.1 m. The cylinder is made of acrylic material, of which the elastic modulus and mass density of the cylinder are 0.65 GPa and 1470 kg/m3, respectively. Three different water depths, Hi , are consid ered: 0 m, 0.4 m, and 0.75 m. The corresponding fundamental period Ti ði ¼ 1; 2; 3Þ of the cylinder under each water depth is assessed through free vibration tests. And the wet-to-dry period ratio Ti =T1 ði ¼ 2; 3Þ can be then determined. For the purpose of comparison, finite element models of the cylinder are established. The hydrodynamic added mass for each water depth is determined using the developed model and applied to corresponding numerical models. The natural periods of the numerical models are obtained through modal analysis, and the wet-to-dry period ratios can also be calculated. The results of period ratios under different water depths are compared in Fig. 10. It can be seen that the numerical period ratios coincide with the experimental ones, which demonstrates the high accuracy of the developed model for added mass evaluation.
dimensions of the dumbbell are illustrated in Fig. 11. The x-axis and yaxis are defined as the major and minor axes of the column cross - sec tion, respectively. The material of the column is concrete, and the elastic modulus, Poisson’s ratio and mass density are set to 25 GPa, 0.2 and 2500 kg/m3, respectively. The column is fully fixed on the bottom with a tip mass of 3000 t to simulate the weight of superstructure. In the following discussion, a refined and a simplified numerical model are built for comparison. The refined model is the potential-based FSI model of the example column with dumbbell cross - section, as illustrated in Fig. 11. In the FSI model, the FSI boundary condition (Eq. (8)) is applied to the interface between solid elements and PBFEs. The free surface (Eq. (9)) and rigid wall boundary condition (Eq. (9)) are applied to free surface, side and bottom walls of water, respectively. The simplified model is a solid element made column model with added mass applied as the nodal mass to nodes on the surface of immersed part of the column. Although it is efficient to adopt added mass in a beam element model, the differences in the basic hypothesis between the beam and solid element could lead to differences in the deformation and force responses. For the purpose of eliminating additional error arising from using different element, solid elements are employed for both the refined and simplified model. The added mass of the example column with dumbbell cross - section along direction of motion θg is calculated using the developed model. The mesh generation of the column is same as the column in FSI model. The FSI model contains 67,000 solid ele ments for column and 440,000 PBFEs for water, while the column model with added mass have 67,000 solid elements only. These two types of models are adopted to perform frequency-domain and time-domain
5. Seismic analyses of immersed column with dumbbell cross section The accuracy of the developed model has been validated for some typical immersed columns in Section 3. The efficiency and accuracy of the developed added mass model in the seismic analyses of immersed column with complex cross - section still require further examination. An immersed column with dumbbell cross - section, used as deepwater bridge piers, is selected as example structure in seismic responses ana lyses. The column is 45 m high with 30 m submerged in water. The 10
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 18. Time-histories of the seismic responses of the column under earthquake along direction θg ¼ 45∘ : (a) acceleration at column top; (b) displacement at column top; (c) base shear at column bottom; (d) base moment at column bottom.
analyses to obtain seismic responses of the example column. Seismic analyses results of the FSI model are taken as the FSI results, while the results of the column model with added mass are taken as the added mass results. 5% Rayleigh damping ratio is adopted in the time-domain analyses. The deformation and force responses can be estimated by the solu tion of the following structural dynamic equation (Bathe, 2014; Chopra, 2000). € þ CU_ þ KU ¼ R MU
displacement matrix, respectively. The base shear VðωÞ and base moment MðωÞ in frequency domain are computed through the integra tion of the element stress at the bottom surface of the column as Z VðωÞ ¼ τxy ðωÞdA Z (22) MðωÞ ¼ τzz ðωÞxdA where τxy ðωÞ and τzz ðωÞ are the element shear stress and normal stress in frequency domain at the bottom surface of the column, respectively; x is the distance between the element and the neutral axis. The time domain responses of the motion and acceleration, UðtÞ and € UðtÞ, can be calculated by solving Eq. (19) through implicit time inte gration method. After that, the element stress vectors in time domain τðtÞ are determined as
(19)
where M, C, K are the mass, damping and stiffness matrices; R is the _ U € are the vectors of nodal vector of externally applied loads; and U, U, point displacement, velocity and acceleration, respectively. The frequency domain responses of the motion and acceleration can be computed through solving Eq. (19) in frequency domain based on modal orthogonality as X UðωÞ ¼ Φk Y k ðωÞ (20) X € Φk Y k ðωÞ U ðωÞ ¼ ω2
τðtÞ ¼ Ce Be UðtÞ
Similar with the calculation of frequency domain responses, the base shear VðtÞ and base moment MðtÞ in time domain are computed as Z VðtÞ ¼ τxy ðtÞdA Z (24) MðtÞ ¼ τzz ðtÞxdA
where Y k ðωÞ is the frequency dependent generalized coordinates in the kth mode of vibration; Φk is the shape of the kth mode of the column. Then the element stress vectors in frequency domain τðωÞ are deter mined as
τðωÞ ¼ Ce Be UðωÞ
(23)
where τxy ðtÞ and τzz ðtÞ are the element shear stress and normal stress in time domain at the bottom surface of the column, respectively.
(21)
where Ce and Be are the element elasticity matrix and the element strain11
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 19. Time-histories of the seismic responses of the column under earthquake along direction θg ¼ 90∘ : (a) acceleration at column top; (b) displacement at column top; (c) base shear at column bottom; (d) base moment at column bottom.
5.1. Hydrodynamic added mass of the example column
along the column major and minor axes to obtained frequency-domain responses. Fig. 14 and Fig. 15 compare the frequency responses, including the horizontal acceleration and displacement at column top, base shear and base moment, under ground motion along major and minor axes, respectively. The added mass results coincide well with the FSI results near the first modal frequency, regardless of excitation di rection. Because the hydrodynamic added mass used in this study is a frequency-independent mass, which only accounts for hydrodynamic force induced by structural rigid motion and neglects the effect of vi bration mode (Goyal and Chopra, 1989a), some differences between these curves can be observed near the second mode frequency as shown in Fig. 15. This phenomenon has also been mentioned by Goyal and Chopra (1989a) with regard to frequency responses of immersed intake towers subjected to harmonic ground motion. Seismic time-history an alyses are carried out to address its influence on time-domain structural responses. The N-S component of the 1940 Imperial Valley earthquake record at El Centro (PEER, 2013) with peak ground acceleration (PGA) scaled to 0.1 g, shown in Fig. 16, is selected as the earthquake input for time-domain seismic analyses. Three different earthquake directions θg are considered: case 1, 0∘ ; case 2, 45∘ ; case 3, 90∘ . The seismic responses of the horizontal acceleration and displacement at column top, base shear and base moment, under various earthquake input direction are obtained through time-history analyses of both FSI and column models with added mass, and are shown in Figs. 17–19. The agreement between seismic responses from the FSI model and the column model with added mass is satisfactory and confirms that the hydrodynamic added mass computed by the developed model is valid in the seismic analyses of immersed columns in time-domain. Meanwhile, the computational time
Similar with the example column with elliptical cross - section, the added mass of the column with dumbbell cross - section also differs with the direction of ground motion θg . In order to investigate the effect of direction of motion θg , the hydrodynamic added mass of the studied column along seven different directions ranging from 0∘ to 90∘ , is calculated using the refined numerical model, and presented in Fig. 12. It can be seen that the added mass of such column increases with the direction θg , because of the increasing projection length of the column cross - section along the ground motion. Furthermore, the hydrody namic added mass per unit height at z =h ¼ 0, z =h ¼ 1=3 and z =h ¼ 2= 3 against the seven different directions are scattered in Fig. 13, and the solid curves are also plotted according to Eq. (25). As shown in Fig. 13, the solid curves agree well with the scattered points, which indicates that when the added mass of the column with dumbbell cross - section along the major and minor axes is determined, the added mass along a specific ground motion direction θg can be calculated by Eq. (25). � 2 2 yy (25) ma z; θg ¼ mxx a ðzÞcos θg þ ma ðzÞsin θg yy
where mxx a ðzÞ and ma ðzÞ are the hydrodynamic added mass at z, distance above the column base, along the major and minor axes of the column cross - section, respectively. 5.2. Seismic responses of the example column €g ðtÞ ¼ eiωt with frequency ω Harmonic ground motion acceleration u ranging from 0 to 10 Hz is firstly applied to these numerical models 12
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
for the FSI model and the column model with added mass on the same computer with Intel(R) Xeon E5-2630 CPU at 2.4 GHz and 16 GB RAM is 1.3 h and 0.6 h, respectively. The assessment time of the hydrodynamic added mass using the developed model is taken into account in the computational time associated with column model with added mass. Considering that the added mass can be employed repeatedly in seismic analyses towards structures supported by the same column, the column model with added mass is obviously more efficient compared with the FSI model by avoiding solving complex FSI equations and is more useful for seismic fragility analyses.
Acknowledgments The authors would like to acknowledge the financial support from Natural Science Foundation of China (Grants No. 51708455) and Fundamental Research Fund for the Central Universities (A1920502051907-2-001). References ADINA, 2010. Theory and Modeling Guide, vols. 10–7. ADINA R&D, Watertown, MA. Tech. Rep. ARD. Bathe, K.-J., 2014. Finite Element Procedures. Klaus-Jürgen Bathe, Englewood Cliffs, N.J. Bouaanani, N., Lu, F.Y., 2009. Assessment of potential-based fluid finite elements for seismic analysis of dam–reservoir systems. Comput. Struct. 87 (3), 206–224. Chopra, A.K., 2000. Dynamics of Structures: Theory and Applications to Earthquake Engineering. Prentice Hall, Upper Saddle River, NJ. Det Norske Veritas, 2010. Environmental Conditions and Environmental Loads. DNV-RPC205, DNV. European Committee for Standardization, 2005. Eurocode 8: Design of Structures for Earthquake Resistance-Part 2. Bridges. Goyal, A., Chopra, A.K., 1989. Earthquake response spectrum analysis of intake-outlet towers. J. Eng. Mech. 115 (7), 1413–1433. Goyal, A., Chopra, A.K., 1989. Simplified evaluation of added hydrodynamic mass for intake towers. J. Eng. Mech. 115 (7), 1393–1412. Han, R.P.S., Xu, H., 1996. A simple and accurate added mass model for hydrodynamic fluid—structure interaction analysis. J. Frankl. Inst. 333 (6), 929–945. Huang, X., Li, Z.X., 2011. Influence of free surface wave and water compressibility on earthquake induced hydrodynamic pressure of bridge pier in deep water. J. Tianjin Univ. Sci. Technol. 44 (4), 319–323 (in Chinese). Jiang, H., Wang, B.X., Bai, X.Y., Zeng, C., Zhang, H.D., 2017. Simplified expression of hydrodynamic pressure on deepwater cylindrical bridge piers during earthquakes. J. Bridge Eng. 22 (6), 04017014. Li, Q., Yang, W.L., 2013. An improved method of hydrodynamic pressure calculation for circular hollow piers in deep water under earthquake. Ocean Eng. 72, 241–256. Liaw, C.-Y., Chopra, A.K., 1974. Dynamics of towers surrounded by water. Earthq. Eng. Struct. Dyn. 3 (1), 33–49. Liu, Y., 2019. HAMS: a frequency-domain preprocessor for wave-structure interactions—theory, development, and application. J. Mar. Sci. Eng. 7 (3), 81. Mirzadeh, J., Kimiaei, M., Cassidy, M.J., 2016. Performance of an example jack-up platform under directional random ocean waves. Appl. Ocean Res. 54, 87–100. Nagaya, K., Hai, Y., 1985. Seismic response of underwater members of variable cross section. J. Sound Vib. 103 (1), 119–138. Olson, L.G., Bathe, K.J., 1985. Analysis of fluid-structure interactions. a direct symmetric coupled formulation based on the fluid velocity potential. Comput. Struct. 21 (1), 21–32. € H.R., 2003. Natural frequencies of an immersed beam carrying a tip mass with Oz, rotatory inertia. J. Sound Vib. 266 (5), 1099–1108. Ozdemir, Z., Souli, M., Fahjan, Y.M., 2010. Application of nonlinear fluid–structure interaction methods to seismic analysis of anchored and unanchored tanks. Eng. Struct. 32 (2), 409–423. Pang, Y.T., Wei, K., Yuan, W.C., Shen, G.Y., 2015. Effects of dynamic fluid-structure interaction on seismic response of multi-span deep water bridges using fragility function method. Adv. Struct. Eng. 18 (4), 525–541. PEER, 2013. Pacific Earthquake Engineering Research Center Ground Motion Database. http://ngawest2.berkeley.edu/. Qin, S., Gao, Z., 2017. Developments and prospects of long-span high-speed railway bridge technologies in China. Engineering 3 (6), 787–794. U�sciłowska, A., Kołodziej, J.A., 1998. Free vibration of immersed column carrying a tip mass. J. Sound Vib. 216 (1), 147–157. Wang, P., Zhao, M., Du, X., 2018. Short-crested, cnoidal, and solitary wave forces on composite bucket foundation for an offshore wind turbine. J. Renew. Sustain. Energy 10 (2), 023305. Wang, J.-T., Zhang, M.-X., Jin, A.-Y., Zhang, C.-H., 2018. Seismic fragility of arch dams based on damage analysis. Soil Dyn. Earthq. Eng. 109, 58–68. Wang, P.G., Zhao, M., Du, X.L., 2018. Analytical solution and simplified formula for earthquake induced hydrodynamic pressure on elliptical hollow cylinders in water. Ocean Eng. 148, 149–160. Wang, P.G., Zhao, M., Du, X.L., Liu, J.B., Chen, J.Y., 2018. Simplified evaluation of earthquake-induced hydrodynamic pressure on circular tapered cylinders surrounded by water. Ocean Eng. 164, 105–113. Wei, K., Yuan, W.C., Bouaanani, N., 2013. Experimental and numerical assessment of the three-dimensional modal dynamic response of bridge pile foundations submerged in water. J. Bridge Eng. 18 (10), 1032–1041. Wei, K., Myers, A.T., Arwade, S.R., 2017. Dynamic effects in the response of offshore wind turbines supported by jackets under wave loading. Eng. Struct. 142, 36–45. Westergaard, H.M., 1933. Water pressures on dams during earthquakes. Trans. ASCE 95, 418–433. Wu, J.-S., Hsu, S.-H., 2007. The discrete methods for free vibration analyses of an immersed beam carrying an eccentric tip mass with rotary inertia. Ocean Eng. 34 (1), 54–68. Xu, H., Zou, D., Kong, X., Hu, Z., Su, X., 2018. A nonlinear analysis of dynamic interactions of CFRD–compressible reservoir system based on FEM–SBFEM. Soil Dyn. Earthq. Eng. 112, 24–34.
6. Conclusions A potential-based numerical model of the hydrodynamic added mass for immersed column with arbitrary cross - section was developed. The general formulation of the hydrodynamic added mass was derived with respect to the hydrodynamic force induced by the horizontal rigid mo tion of the column. Hydrodynamic added mass of columns with circular, elliptical cross - section and circular tapered columns was estimated through the developed model. Good agreements with results from pre vious literatures for all three examples revealed the accuracy of the developed model for these immersed columns. In order to validate the model experimentally, the modal response of an immersed cylinder as a function of water depth was obtained by both the developed added mass model and experimental modal tests. Good agreement was found be tween the developed model and experimental results. The developed model was then further applied to evaluate the hydrodynamic added mass of an immersed column with dumbbell cross - section. The effect of direction of ground motion on the added mass of the example column was investigated. Although the hydrodynamic added mass of the immersed column with dumbbell cross - section varies with the ground motion direction, the added mass along a specific direction can be calculated based on an analytical function of the added masses along the column major and minor axes and the ground motion direction. The obtained added mass was then adopted to construct the column model with added mass, while the potential-based FSI model was also estab lished as reference. The time- and frequency-domain seismic responses of the example column were obtained using both models. The column model with added mass exhibits good accuracy in the calculation of seismic responses and shows great advantage in computational effi ciency, compared with complex FSI model. These findings demonstrate that the earthquake-induced hydrody namic added mass for immersed column with arbitrary cross - section can be efficiently predicted by the developed model. And the developed model allows the implement of added mass in the seismic design of structures supported by immersed columns with complex cross - section, and waiving the substantial efforts for modeling and solving sophisti cated FSI equations. The reduction in computational time would become more significant especially when numerous nonlinear time-history an alyses for a probabilistic model are required. It will provide an efficient approach for a comprehensive understanding towards the seismic per formance and reliability of the structures supported by immersed columns. In spite of that the developed added mass model have been verified for several commonly used types of columns, additional efforts should be devoted to the throughout validation of the added mass model towards columns with more kinds of cross - section. The effects of higher order vibration modes and the surface gravity waves on the hydrodynamic added mass have not been included because the added mass developed in the framework is independent with frequency. All these limitations deserve further investigation. Moreover, BEM can be implemented instead of PBFEs to evaluate the hydrodynamic pressure in the devel oped model for the purpose of easier meshing process and faster computation with good accuracy.
13
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Yang, W.L., Li, Q., 2013. The expanded Morison equation considering inner and outer water hydrodynamic pressure of hollow piers. Ocean Eng. 69, 79–87. Yu, H., Zheng, S., Zhang, Y., Iglesias, G., 2019. Wave radiation from a truncated cylinder of arbitrary cross section. Ocean Eng. 173, 519–530. Zhang, J.R., Wei, K., Pang, Y.T., Zhang, M.J., Qin, S.Q., 2019. Numerical investigation into hydrodynamic effects on the seismic response of complex hollow bridge pier submerged in reservoir: case study. J. Bridge Eng. 24 (2), 05018016.
Zhao, C., Chen, J., Xu, Q., 2014. FSI effects and seismic performance evaluation of water storage tank of AP1000 subjected to earthquake loading. Nucl. Eng. Des. 280, 372–388. Zheng, S., Zhang, Y., Liu, Y., Iglesias, G., 2019. Wave radiation from multiple cylinders of arbitrary cross sections. Ocean Eng. 184, 11–22.
14
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
An efficient numerical model for hydrodynamic added mass of immersed column with arbitrary cross - Section Jiarui Zhang a, Kai Wei a, *, Shunquan Qin a, b a b
Department of Bridge Engineering, College of Civil Engineering, Southwest Jiaotong University, Chengdu, 610031, China China Railway Major Bridge Reconnaissance & Design Institute Co Ltd, Wuhan, 430050, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Immersed column Added mass Arbitrary cross Section Potential-based fluid element Hydrodynamic pressure Seismic analysis
The assessment of earthquake-induced hydrodynamic added mass is critical in the seismic design of structures supported by immersed columns. This paper developed an efficient potential-based numerical model to deter mine the hydrodynamic added mass for immersed column with arbitrary cross - section. General formulations of the hydrodynamic added mass for a column with general cross - section under horizontal earthquake were derived. The accuracy of the developed model for immersed columns was validated analytically and experi mentally. The developed model was then applied to evaluate the hydrodynamic added mass of an immersed column with dumbbell cross - section. The hydrodynamic added mass of the example dumbbell column along various earthquake directions was determined and the directional-dependent added mass was expressed as an analytical function of the added mass along the major and minor axes of the column and the earthquake di rection. The frequency- and time-domain seismic responses of the example dumbbell column were finally computed using both fluid-structure interaction (FSI) model and the developed hydrodynamic added mass model. The good agreement between seismic responses from the FSI and developed added mass model confirmed the high accuracy and efficiency of the developed model for the hydrodynamic added mass of immersed columns with arbitrary cross - section under earthquakes.
1. Introduction The dynamics of structures supported by columns submerged in water, such as sea-crossing bridge foundations (Qin and Gao, 2017), offshore platforms (Mirzadeh et al., 2016), offshore wind turbines (Wei et al., 2017), etc., require consideration of fluid-structure interaction (FSI), which does not arise for structures on land. It is a common knowledge that the vibration of immersed column subjected to earth quakes induces the motion of nearby water, producing an extra hydro dynamic force on the structure (Zhang et al., 2019; Pang et al., 2015). This extra force is usually modeled as the combination of a hypothetical mass of water and the acceleration of the structure. The hypothetical mass, typically termed as added mass, represents the effective mass of water that participates in the structural vibration through FSI. It was firstly introduced by Westergaard (1933) in the analyses of hydrody namic pressure on the dam subjected to harmonic ground motion. Based € on the concept of added mass, U�sciłowska and Kołodziej (1998), Oz (2003) and Wu and Hsu (2007) derived the analytical solutions to determine the free vibration and natural frequencies of immersed
columns. For the added mass assessment of an immersed column, the section of the column in the early literatures is usually assumed to be circular due to its accessibility to the equations of FSI interface boundary in the closed-form solutions, and many research efforts are devoted to the added mass model of columns with circular cross - section. Liaw and Chopra (1974) proposed a closed-form solution of the added mass for circular columns surrounded by water. Han and Xu (1996) presented a theoretical model of an added mass representation for a flexible immersed column vibrating in water. Different simplified equations for the added mass of immersed circular column were developed by Li and Yang (2013) and Jiang et al. (2017), respectively. Added mass models for both inner water and outer water of an immersed column with cir cular hollow cross - section have also been developed by Yang and Li (2013). The continuing development of offshore engineering contributes to the broad application of immersed columns with complex geometry other than circular cross - sections (Wang et al., 2018a). Nagaya and Hai (1985) developed a method for solving seismic response problems of a column of variable tapered cross - section. However, the added mass of immersed column with non-circular section was usually simplified as the
* Corresponding author. Room 507, Tunnel Research Building, No. 111, North 1st Section of Second Ring Road, Chengdu, Sichuan, 610031, China. E-mail address: [email protected] (K. Wei). https://doi.org/10.1016/j.oceaneng.2019.106192 Received 1 January 2019; Received in revised form 4 June 2019; Accepted 6 July 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 1. Illustration of an immersed column with arbitrary cross - section subjected to horizontal ground motion.
added mass for column with equivalent circular cross - section times a shape correction factor (Det Norske Veritas, 2010; European Committee for Standardization, 2005), which are limited to several typical cross sections, such as ellipse or rectangle, and insufficient for general engi neering practices. More recently, Yu et al. (2019) dealt with the problem of radiation from a floating cylinder with an arbitrary cross section and proposed a semi-analytical model based on linear potential flow theory. Zheng et al. (2019) further developed and expanded the model from one single to multiple floating cylinders with arbitrary cross - sections. The numerical model allows the evaluation of the earthquake induced hydrodynamics through solution of the FSI equations, which has been wildly adopted to address questions in complex fluid-structure systems. Bouaanani and Lu (2009) assessed the performance of potential-based fluid elements (PBFEs) regarding the dam-reservoir system and investigated the effect of the reservoir bottom absorption on seismic responses of dam based on this numerical method. Wei et al. (2013) conducted modal analyses of pile foundations submerged in water using PBFEs and verified the numerical model with experimental results. Wang et al. (2018b) adopted Lagrange formulation of fluid in the reservoir-dam-foundation numerical model to investigate the damage characteristic of dam. Boundary element method (BEM) (Liu, 2019; Xu et al., 2018), arbitrary Lagrangian-Eulerian (ALE) algorithm (Ozdemir et al., 2010; Zhao et al., 2014) etc., have also been applied to provide insight into the FSI issues in engineering fields. Although the seismic response of immersed column can now be computed using the FSI nu merical approaches, most of these techniques have yet been fully implemented in the common engineering practice, especially at the preliminary stage of seismic design, as they require specialized software or advanced programming and high-level expertise, and may result in extensive modeling and computational efforts. Since the added mass represents the earthquake-induced hydrodynamic force in a simplified and efficient way, a general hydrodynamic added mass model of col umns with arbitrary cross - section aided by FSI numerical method may facilitate the work of researchers and designers regarding the immersed columns. For the purpose of developing an efficient numerical model to assess the hydrodynamic added mass, this paper is organized as follows: firstly, a general formulation of hydrodynamic added mass and its corre sponding numerical framework are developed for an immersed column with arbitrary cross - section; secondly, the effectiveness of the devel oped model in the calculation of the added mass for uniform immersed columns with circular and elliptical sections and circular tapered col umns is validated with the analytical results from previous literatures; thirdly, the added mass of an immersed cylinder is calculated through
the developed model and is verified by comparison with experimental modal test results; fourthly, an immersed column with dumbbell cross section is selected as an example study, the developed model is used to calculate the hydrodynamic added mass of the column; and finally, the frequency - and time - domain seismic responses of such column are compared between the results from the FSI model and the column model with added mass and the effect of ground motion direction on the hy drodynamic added mass are investigated. 2. Model for determination of hydrodynamic added mass 2.1. Formulations of hydrodynamic added mass We consider an immersed column with arbitrary cross - section €g ðtÞ as illustrated in Fig. 1. subjected to horizontal ground acceleration u Assuming water to be inviscid and irrotational, the hydrodynamic pressure p within the water domain is governed by the wave equation in cylindrical coordinates ðr; θ; zÞ as follows:
∂2 p 1 ∂p 1 ∂2 p ∂2 p 1 ∂2 p þ þ ¼ þ ∂r2 r ∂r r2 ∂θ2 ∂z2 C2 ∂t2
(1)
where C is velocity of the compression waves in water, and t is time. Although the effect of surface gravity waves will make the hydrody namic pressure vary with the excitation frequency, but its influence on the response of engineering structures in practice is not significant (Huang and Li, 2011). The surface gravity waves are hence not included in the following derivation. The boundary conditions for the governing equation according to the aforementioned assumptions are: i) No surface gravity waves at the free surface z ¼ Hw pðr; θ; Hw ; tÞ ¼ 0
(2)
ii) No vertical motion at the bottom of the fluid domain z ¼ 0
∂p ðr; θ; 0; tÞ ¼ 0 ∂z
(3)
iii) The consistency of radial motion of structure and surrounding water at fluid-structure surface, i.e., surface function Sðr; θ; zÞ
2
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
2.2. Potential-based numerical analysis framework for hydrodynamic added mass The closed-form solution of the hydrodynamic pressure pðr; θ; zj Þ only exists for typical cross - sections, such as circle and ellipse. In order to compute the hydrodynamic pressure on the arbitrary cross - section, three-dimensional (3D) potential-based fluid elements (PBFEs), based on φ U formulation (Olson and Bathe, 1985), is implemented in the framework. In this formulation, φ and U denote the velocity potential of fluid domain and the displacement of solid domain, respectively, yielding simple FSI modeling and avoiding heavy computational burden. The velocity potential of PBFEs in the numerical model satisfies the wave equation as follows: 2
▽ φ¼
1 ∂2 φ C2 ∂t2
(7)
Fluid-structure interaction is accounted for through FSI boundary condition at the water - column interfaces, defined as
∂φ ∂U ! ¼ n ∂t ∂! n
(8)
where ! n is the unit normal vector perpendicular to the water - column interface towards the water domain. The fluid velocity potential φ at the free surface boundary and the side and bottom walls of the fluid domain hence obeys the following equation: φ¼0
Fig. 2. Flowchart of analysis framework for hydrodynamic added mass.
∂p ðr; θ; z; tÞjSðr;θ;zÞ ¼ ∂! ns
ρw €ug ðtÞ! n s ⋅! ng
(9)
For the given immersed column with arbitrary cross - section, the column and the surrounding water are discretized into solid elements and PBFEs, respectively, as shown in Fig. 2. The velocity of pressure waves in water is set to be 1440 m/s, resulting in a bulk modulus of 2.1 GPa, to include the compressibility of water. The radius of the water domain should be larger than eight times the maximum radial dimen sion of the column cross - section for the purpose of eliminating the wave reflection due to the side wall boundaries. The properties of the column are defined according to its material information. The vertical degrees of freedom for nodes at bottom of the immersed column are fully con strained, while the horizontal degrees are set to be free. Once the potential-based numerical model is developed, the hydrodynamic pressure pðr; θ; zj Þ on the column with arbitrary cross - section can be calculated using the finite element model. In order to obtain the hy drodynamic pressure at different distance z above the column base, sweeping algorithm is used to generate the hexahedral meshes for the immersed column as illustrated in Fig. 1. The division numbers of the column in the cylindrical coordinate along Sðr; θ; zÞ and z-direction are defined as n and m, respectively. The formulation of the hydrodynamic added mass can be given as
(4)
n s is the unit normal in which Hw is water depth, ρw is water density, ! vector of surface function Sðr; θ; zÞ pointing outwards, ! n g is the unit directional vector of ground motion along the direction θg , and ⋅ is the dot product between vectors. According to the work of Chopra and his colleagues (Liaw and Chopra, 1974; Goyal and Chopra, 1989a, 1989b), the amplitude of the hydrodynamic added mass per unit height ma is a function of z, the distance above the column base. Moreover, the hydrodynamic added mass ma depends on the vibration direction for columns with non-circular cross - section (Goyal and Chopra, 1989b). Therefore, a general form of hydrodynamic added mass per unit height at zj of a column vibrating along the direction θg is denoted as ma ðzj ; θg Þ, which can calculated by � �� ug m a z j ; θ g ¼ Fp z j ; θ g € (5)
n � X � Lj m a zj ; θ g ¼ n s;i ⋅! p ri ; θi ; zj ! n g ; j ¼ 1; …; m þ 1 n i¼1
where Fp ðzj ; θg Þ is the resultant hydrodynamic force along direction θg €g . per unit height at zj due to vibration of column under ground motion u When the immersed column is subjected to a unit horizontal ground €g ðtÞ ¼ 1 m/s2, along direction θg , the added motion acceleration, i.e., u mass ma ðzj ; θg Þ can also be further expressed as follows Z � �� � n s ⋅! ma zj ; θg ¼ Fp zj ; θg 1 ¼ n g dS (6) p r; θ; zj ! Sðr;θ;zj Þ
(10)
in which pðri ; θi ; zj Þ is the hydrodynamic pressure at point ðri ; θi ; zj Þ, i ¼ 1; 2; …; n; ! n s;i is the unit vector at point ðri ; θi ; zj Þ normal to the surface function Sðr; θ; zj Þ pointing outwards; Sðr; θ; zj Þ is the surface function of the column at z ¼ zj ; Lj is the circumference of Sðr; θ; zj Þ. Considering that Cartesian coordinate is more general in the finite element modeling for columns with complex cross - section, the general formulation of hydrodynamic added mass Eq. (10) can be replaced by Eq. (11) in Cartesian coordinate,
where pðr; θ; zj Þ is the hydrodynamic pressure acting on the column surface induced by its rigid motion.
n � X � � Lj n s;i ⋅ cosθg ; sinθg ; j ¼ 1; …; m þ 1 m a zj ; θ g ¼ p xi ; yi ; zj ! n i¼1
(11)
where pðxi ; yi ; zj Þ is the hydrodynamic pressure at point ðxi ;yi ;zj Þ, i ¼ 1; 2; …;n. The added mass coefficient Ca ðzÞ, defined as the added mass ma ðzÞ 3
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
divided by the column cross - section area and water density, can be calculated easily as well. The following analysis process shown in Fig. 2 can be performed to calculate the hydrodynamic added mass of the immersed column with arbitrary cross - section: i) Apply a unit acceleration to the entire column along the ground motion direction. The time-history of the acceleration with the duration of 1.2 s is given in Fig. 2. ii) Carry out transient analysis to obtain the hydrodynamic pressure pðri ; θi ; zj Þ or pðxi ; yi ; zj Þ at each node on the water - column interface. Oscillation exists in the initial zone of the time-history of hydrodynamic pressure due to the compressibility of water, but it disappears quickly. The stable value of hydrodynamic pressure at the last time-step is recommended. iii) Compute the hydrodynamic added mass at each zj , distance above the column base, through the integration of hydrodynamic pressure on the water - column interface following Eq. (10) or 11. iv) Plot the curve of hydrodynamic added mass as a function of distance above the base of immersed column. Fig. 3. Dimensions of immersed column with circular cross - section.
3. Analytical validation of the developed added mass model Immersed columns with circular and elliptical cross - section and the
Fig. 4. Comparison of added mass coefficient between results from Liaw and Chopra (Liaw and Chopra, 1974), Li and Yang (Li and Yang, 2013), Jiang et al. (Jiang et al., 2017) and developed model: (a) Hw ¼ 30m, D ¼ 4m; (b) Hw ¼ 30m, D ¼ 5m; (c) Hw ¼ 30m, D ¼ 6m; (d) Hw ¼ 30m, D ¼ 7m. 4
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
3.1. Immersed column with circular cross - section
Table 1 Comparison of integrated hydrodynamic added mass coefficient. Cases
Liaw and Chopra (Liaw and Chopra, 1974)
Developed model
Hw ¼ 30 m, D¼4m
0.924
0.920
0.39%
Hw ¼ 30 m, D¼5m
0.906
0.904
0.28%
Hw ¼ 30 m, D¼6m
0.889
0.887
0.21%
Hw ¼ 30 m, D¼7m
0.872
0.870
0.16%
An immersed column with circular cross - section, as illustrated in Fig. 3, is taken as the first example for validation. The diameter of the column is D and the water depth is Hw . The hydrodynamic added mass coefficients Ca ðzÞ of the columns with diameter D varying from 4 to 7 m and with a water depth of 30 m, are calculated using the developed model and the formulas proposed in previous literatures. The closedform analytical equation of hydrodynamic added mass for circular cyl inder proposed by Liaw and Chopra (1974) based on radiation wave theory is given in Eq. (12) and selected as the reference. Simplified formulations of the added mass for such column developed by Li and Yang (2013) and Jiang et al. (2017) shown in Eq. (13) and Eq. (14), respectively, are also used for comparison. Equation from Liaw and Chopra (1974):
Relative error
ma ðzÞ ¼ 4ρw πD
Nq X 1
� � � Iq λ’q D cos αq z Eq 2 2λ’q Hw
Equation from Li and Yang (2013): 8 2 > > �> � < 6 10 2 ρ πD D 6 ma ðzÞ ¼ w 1 exp6 1 ðz 1 5 2 4DH 3w 4 Hw > > > :
(12)
39 > > = 7> 7 Hw Þ7 5> > > ;
(13)
Equation from Jiang et al. (2017):
Table 2 Comparison of integrated hydrodynamic added mass coefficient. Cases
Fig. 5. Dimensions of immersed column with elliptical cross - section.
circular tapered column are widely used in offshore structures, such as foundations of sea-crossing bridges and offshore wind turbines. In order to validate the developed model, the hydrodynamic added mass of three immersed column examples, including two uniform columns (circular and elliptical cross - section) and one tapered column with circular cross - section is calculated by the developed model and analytical models from previous literatures. Accuracy of the hydrodynamic added mass is assessed through the comparison with the results from previous literatures.
Wang et al. (Wang et al., 2018c)
Developed model
Along majoraxis B =Hw ¼ 0.2 B =Hw ¼ 1.0
Relative error
0.847
0.845
B =Hw ¼ 0.5
0.672
0.673
0.07%
0.488
0.492
0.75%
Along minoraxis D =Hw ¼ 0.2
0.908
0.903
0.785
0.786
0.14%
D =Hw ¼ 1.0
0.625
0.630
0.85%
D =Hw ¼ 0.5
0.28%
0.52%
Fig. 6. Comparison of added mass coefficient between results from Wang et al. (Wang et al., 2018c) and developed model: (a) θg ¼ 0∘ ; (b) θg ¼ 90∘ . 5
Ocean Engineering 187 (2019) 106192
J. Zhang et al.
Table 3 Comparison of integrated hydrodynamic pressure.
Fig. 7. Dimensions of tapered column with circular cross - section.
ma ðzÞ ¼ ρw πD2
� H 1:5 w � 1 1:5 0:1D2 þ 3:8D 6:8 þ 4H w
� 2D þ 2Hw ðz exp DHw
Cases
Wang et al. (Wang et al., 2018d)
Developed model
Relative error
D1 =Hw ¼ 0.4, α ¼ 60∘
1.191
1.192
0.02%
D1 =Hw ¼ 0.4, α ¼ 70∘
1.149
1.148
0.04%
D1 =Hw ¼ 0.4, α ¼ 80∘
1.023
1.022
0.10%
D1 =Hw ¼ 1.0, α ¼ 60∘
0.561
0.561
D1 =Hw ¼ 1.0, α ¼ 70∘
0.588
0.588
0.05%
D1 =Hw ¼ 1.0, α ¼ 80∘
0.597
0.596
0.11%
0.00%
�� Hw Þ (14)
RH in which Iq ¼ 0 w cosðαq zÞdz, λ’q ¼ αq ¼ ð2q 1Þπ=2Hw , Eq ðxÞ ¼ K1 ðxÞ=ðK0 ðxÞ þ K2 ðxÞÞ, Kn ðxÞ is the modified Bessel function of order n of the second kind, and q ¼ 1; 2; ::; Nq , where Nq is the number of items, suggested to be larger than 50 for a smooth solution. The developed model is firstly used to assess the hydrodynamic added mass for the example columns. Potential-based numerical model for the immersed column with circular cross - section is built using ADINA (ADINA, 2010) according to the analysis framework depicted in Section 2. Hydrodynamic pressure on the column is calculated according to the flowchart given in Fig. 2. Because the hydrodynamic added mass ma ðzÞ of the circular column is independent of direction θg , it can be determined by Eq. (15). The analytical curves of the added mass ma ðzÞ for the example column are assessed directly using Eqs. (12)–(14), respectively. The added mass coefficient Ca ðzÞ for the column can then be calculated by Eq. (16) according to the curve of ma ðzÞ.
Fig. 9. Experimental setup: (a) positions of the cylinder and F/T transducer; (b) global view of the cylinder and the testing setup.
Fig. 8. Comparison of hydrodynamic pressure between the results from Wang et al. (Wang et al., 2018d) and the developed model: (a) tapered columns with D1 =Hw ¼ 0.4; (b) tapered columns with D1 =Hw ¼ 1.0. 6
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 10. Numerical and experimental results of wet-to-dry period ratios for the cylinder as a function of water depth. Fig. 12. Distribution of hydrodynamic added mass per unit height of the col umn along different direction θg .
Fig. 13. Relationship between hydrodynamic added mass and direction θg . n X
ma ðzÞ ¼ i¼1
Ca ðzÞ ¼
! � πD xi yi pðxi ; yi ; zÞ pffiffiffi2ffiffiffiffiffiffiffiffiffi2ffiffi; pffiffiffi2ffiffiffiffiffiffiffiffiffi2ffiffi ⋅ cosθg ; sinθg n xi þ yi xi þ yi
4ma ðzÞ ρw πD2
(15)
(16)
Fig. 4 shows the comparison of hydrodynamic added mass co efficients between the developed model and the analytical equations. It is clear that the results predicted from the developed model agree well with the closed-form results from Eq. (12) for all the circular columns. The tolerance between the developed model and results from Liaw and Chopra (1974) is significantly better than the simplified equations Eqs. (13) and (14). The differences between the Li and Yang’s results and Jiang’s results within the mediate water depth are mainly caused by the difference of the factors in the exponential function, which affects the distribution of added mass significantly. Furthermore, the coefficients Ca ðzÞ determined from developed model and results from Liaw and RH Chopra (1974) are integrated along water depth, 0 w Ca ðzÞdz =Hw , and
Fig. 11. Illustration of column with dumbbell cross - section: (a) dimensions of the example column, all dimensions are in meters; (b) FSI numerical model.
7
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 14. Frequency response of the studied column as function of the harmonic excitation frequency under ground motion along major-axis: (a) acceleration at column top; (b) displacement at column top; (c) base shear at column bottom; (d) base moment at column bottom.
the results are compared in Table 1. The relative error of the developed model is less than 0.5%, compared with the solutions from Liaw and Chopra (1974). All these results state that the developed model has high accuracy in the prediction of hydrodynamic added mass for immersed column with circular cross - section.
elliptical cross - section. For a given column with elliptical cross - section, its equivalent cir cular area of cross - section is πD2
π B2 4
for motion along major-axis, θg ¼ 0∘ ,
and is 4 for motion along minor-axis, θg ¼ 90∘ . The added mass co efficient Ca ðz; θg Þ can be defined by the following equation. 8 � > > 4ma z; θg > > > θg ¼ 0∘ ; along major axis > 2 � < ρw π B Ca z; θg ¼ (18) � > > 4m z; θ > ∘ > a 2g θ ¼ 90 ; along minor axis > g > : ρw π D
3.2. Immersed column with elliptical cross - section A column with elliptical cross - section submerged in a water depth of Hw , as shown in Fig. 5, is taken as the second example for validation. The length of major and minor axes of the elliptical cross - section is D and B, respectively. Immersed columns with B =Hw varying from 0.2 to 1.0 are selected as cases for direction of motion θg ¼ 0∘ , while columns with D =Hw varying from 0.2 to 1.0 are selected for direction of motion θg ¼ 90∘ . For each column, the aspect ratio D =B is set to 2. The analytical equation for elliptical cylinder developed by Wang et al. (2018c) is used to calculate its hydrodynamic added mass as reference. FSI model of the column and surrounding water is firstly built in ADINA (ADINA, 2010) to calculate the hydrodynamic pressure. In the application of the developed model to the column with elliptical cross section, the hydrodynamic added mass, ma ðz; θg Þ, can be determined from the pressure as follows, 1 0
The hydrodynamic added mass coefficients Ca ðz; θg Þ for all of these columns are calculated using Eqs. (17) and (18), and compared with results from Wang et al. (2018c) in Fig. 6. It can be seen that the scatters of Ca ðz; θg Þ as a function of distance to base determined by the developed model are in good agreement with the reference curves for all cases regardless of column dimension and ground motion direction. The maximum value of added mass appears at the column bottom and de creases gradually to zero at the water free surface. This tendency is similar to that of columns with circular cross - section. Because the elliptical cross - section is non-circular, the coefficients of columns with elliptical cross - section depend on the direction θg . The integrated hy drodynamic added mass coefficients are then calculated according to R Hw Ca ðz; θg Þdz =Hw for comparison in number. The results calculated 0 from developed model and Wang et al. (2018c) are listed and compared in Table 2. It is clear that the relative error between the developed model and results from Wang et al. (2018c) is less than 1%, which further proves that the developed model can evaluate the hydrodynamic added
n B C � X �L xi yi C ma z; θg ¼ pðxi ; yi ; zÞB @ qffixffi2ffiffiffiffiffiffiffiffiyffiffi2ffi; qffixffi2ffiffiffiffiffiffiffiffiyffiffi2ffiA⋅ cosθg ; sinθg n i i i i 2 2 i¼1 D D4 þ B 4 B D4 þ B 4
(17) R 2π pffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in which L ¼ 0 D2 sin θ þ B2 cos2 θ=2dθ is the circumference of the 8
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 15. Frequency response of the studied column as function of the harmonic excitation frequency under ground motion along minor-axis: (a) acceleration at column top; (b) displacement at column top; (c) base shear at column bottom; (d) base moment at column bottom.
(2018d) is selected as the reference for comparison. The hydrodynamic pressure on these circular tapered columns divided by ρw D1 =2 in the x z plane is determined through the solution of FSI models in ADINA (ADINA, 2010) and results are compared with results from Wang et al. (2018d) in Fig. 8. Results from the developed model agree well with the results from Wang et al. (2018d) regardless of taper angle α or ratio D1 =Hw . It is interesting to notice that, unlike the two types of column discussed above, the peak value of the hydrody namic pressure on the circular tapered column does not appear at the bottom of column. The influence of the taper angle on the hydrodynamic pressure is more significant for columns with relatively small value of D1 =Hw . The hydrodynamic pressure is then integrated over water depth RH following, 2 0 w pdz =ρw D1 Hw . The results from Wang et al. (2018d) and the developed model are then presented and compared in Table 3. Good consistency of the integrated hydrodynamic pressure can still be found. The maximum relative error is only 0.11%. The developed model can also be applied to the evaluation of the hydrodynamic pressure for tapered columns with circular cross - section.
Fig. 16. Accelerograms of N-S component of the Imperial Valley earthquake record at El-Centro normalized by its peak ground acceleration (PGA).
mass of immersed columns with elliptical cross - section accurately. 3.3. Tapered column with circular cross - section
4. Experimental validation of the developed added mass model
The third example for validation is a tapered column with circular cross - section, as shown in Fig. 7. The top and bottom diameter of the column are D1 , and D2 , respec tively, with a taper angle of α and submerged depth of Hw . Two cate gories of circular tapered columns including one with D1 =Hw ¼ 0:4 and the other with D1 =Hw ¼ 1:0 are adopted here. And for each category, three different taper angles varying from 60∘ to 80∘ are considered. To validate the accuracy of the developed model, the distribution of hy drodynamic pressure over water depth in the x z plane when the column moves along direction θg ¼ 0∘ investigated by Wang et al.
In this section, experimental modal tests of an immersed cylinder are conducted in a water tank to further verify the developed added mass model. Fig. 9 illustrates the configuration of the cylinder and testing devices. The F/T transducer is bolted with the lifting rod fixed on a testing trestle. The top surface of the testing cylinder is fastened with the F/T transducer, which measures the modal responses of the cylinder. The diameter and length of the cylinder are 0.15 m and 1.2 m, respec tively. The length of the gap between the bottom of water and the 9
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 17. Time-histories of the seismic responses of the column under earthquake along direction θg ¼ 0∘ : (a) acceleration at column top; (b) displacement at column top; (c) base shear at column bottom; (d) base moment at column bottom.
cylinder is 0.1 m. The cylinder is made of acrylic material, of which the elastic modulus and mass density of the cylinder are 0.65 GPa and 1470 kg/m3, respectively. Three different water depths, Hi , are consid ered: 0 m, 0.4 m, and 0.75 m. The corresponding fundamental period Ti ði ¼ 1; 2; 3Þ of the cylinder under each water depth is assessed through free vibration tests. And the wet-to-dry period ratio Ti =T1 ði ¼ 2; 3Þ can be then determined. For the purpose of comparison, finite element models of the cylinder are established. The hydrodynamic added mass for each water depth is determined using the developed model and applied to corresponding numerical models. The natural periods of the numerical models are obtained through modal analysis, and the wet-to-dry period ratios can also be calculated. The results of period ratios under different water depths are compared in Fig. 10. It can be seen that the numerical period ratios coincide with the experimental ones, which demonstrates the high accuracy of the developed model for added mass evaluation.
dimensions of the dumbbell are illustrated in Fig. 11. The x-axis and yaxis are defined as the major and minor axes of the column cross - sec tion, respectively. The material of the column is concrete, and the elastic modulus, Poisson’s ratio and mass density are set to 25 GPa, 0.2 and 2500 kg/m3, respectively. The column is fully fixed on the bottom with a tip mass of 3000 t to simulate the weight of superstructure. In the following discussion, a refined and a simplified numerical model are built for comparison. The refined model is the potential-based FSI model of the example column with dumbbell cross - section, as illustrated in Fig. 11. In the FSI model, the FSI boundary condition (Eq. (8)) is applied to the interface between solid elements and PBFEs. The free surface (Eq. (9)) and rigid wall boundary condition (Eq. (9)) are applied to free surface, side and bottom walls of water, respectively. The simplified model is a solid element made column model with added mass applied as the nodal mass to nodes on the surface of immersed part of the column. Although it is efficient to adopt added mass in a beam element model, the differences in the basic hypothesis between the beam and solid element could lead to differences in the deformation and force responses. For the purpose of eliminating additional error arising from using different element, solid elements are employed for both the refined and simplified model. The added mass of the example column with dumbbell cross - section along direction of motion θg is calculated using the developed model. The mesh generation of the column is same as the column in FSI model. The FSI model contains 67,000 solid ele ments for column and 440,000 PBFEs for water, while the column model with added mass have 67,000 solid elements only. These two types of models are adopted to perform frequency-domain and time-domain
5. Seismic analyses of immersed column with dumbbell cross section The accuracy of the developed model has been validated for some typical immersed columns in Section 3. The efficiency and accuracy of the developed added mass model in the seismic analyses of immersed column with complex cross - section still require further examination. An immersed column with dumbbell cross - section, used as deepwater bridge piers, is selected as example structure in seismic responses ana lyses. The column is 45 m high with 30 m submerged in water. The 10
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 18. Time-histories of the seismic responses of the column under earthquake along direction θg ¼ 45∘ : (a) acceleration at column top; (b) displacement at column top; (c) base shear at column bottom; (d) base moment at column bottom.
analyses to obtain seismic responses of the example column. Seismic analyses results of the FSI model are taken as the FSI results, while the results of the column model with added mass are taken as the added mass results. 5% Rayleigh damping ratio is adopted in the time-domain analyses. The deformation and force responses can be estimated by the solu tion of the following structural dynamic equation (Bathe, 2014; Chopra, 2000). € þ CU_ þ KU ¼ R MU
displacement matrix, respectively. The base shear VðωÞ and base moment MðωÞ in frequency domain are computed through the integra tion of the element stress at the bottom surface of the column as Z VðωÞ ¼ τxy ðωÞdA Z (22) MðωÞ ¼ τzz ðωÞxdA where τxy ðωÞ and τzz ðωÞ are the element shear stress and normal stress in frequency domain at the bottom surface of the column, respectively; x is the distance between the element and the neutral axis. The time domain responses of the motion and acceleration, UðtÞ and € UðtÞ, can be calculated by solving Eq. (19) through implicit time inte gration method. After that, the element stress vectors in time domain τðtÞ are determined as
(19)
where M, C, K are the mass, damping and stiffness matrices; R is the _ U € are the vectors of nodal vector of externally applied loads; and U, U, point displacement, velocity and acceleration, respectively. The frequency domain responses of the motion and acceleration can be computed through solving Eq. (19) in frequency domain based on modal orthogonality as X UðωÞ ¼ Φk Y k ðωÞ (20) X € Φk Y k ðωÞ U ðωÞ ¼ ω2
τðtÞ ¼ Ce Be UðtÞ
Similar with the calculation of frequency domain responses, the base shear VðtÞ and base moment MðtÞ in time domain are computed as Z VðtÞ ¼ τxy ðtÞdA Z (24) MðtÞ ¼ τzz ðtÞxdA
where Y k ðωÞ is the frequency dependent generalized coordinates in the kth mode of vibration; Φk is the shape of the kth mode of the column. Then the element stress vectors in frequency domain τðωÞ are deter mined as
τðωÞ ¼ Ce Be UðωÞ
(23)
where τxy ðtÞ and τzz ðtÞ are the element shear stress and normal stress in time domain at the bottom surface of the column, respectively.
(21)
where Ce and Be are the element elasticity matrix and the element strain11
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Fig. 19. Time-histories of the seismic responses of the column under earthquake along direction θg ¼ 90∘ : (a) acceleration at column top; (b) displacement at column top; (c) base shear at column bottom; (d) base moment at column bottom.
5.1. Hydrodynamic added mass of the example column
along the column major and minor axes to obtained frequency-domain responses. Fig. 14 and Fig. 15 compare the frequency responses, including the horizontal acceleration and displacement at column top, base shear and base moment, under ground motion along major and minor axes, respectively. The added mass results coincide well with the FSI results near the first modal frequency, regardless of excitation di rection. Because the hydrodynamic added mass used in this study is a frequency-independent mass, which only accounts for hydrodynamic force induced by structural rigid motion and neglects the effect of vi bration mode (Goyal and Chopra, 1989a), some differences between these curves can be observed near the second mode frequency as shown in Fig. 15. This phenomenon has also been mentioned by Goyal and Chopra (1989a) with regard to frequency responses of immersed intake towers subjected to harmonic ground motion. Seismic time-history an alyses are carried out to address its influence on time-domain structural responses. The N-S component of the 1940 Imperial Valley earthquake record at El Centro (PEER, 2013) with peak ground acceleration (PGA) scaled to 0.1 g, shown in Fig. 16, is selected as the earthquake input for time-domain seismic analyses. Three different earthquake directions θg are considered: case 1, 0∘ ; case 2, 45∘ ; case 3, 90∘ . The seismic responses of the horizontal acceleration and displacement at column top, base shear and base moment, under various earthquake input direction are obtained through time-history analyses of both FSI and column models with added mass, and are shown in Figs. 17–19. The agreement between seismic responses from the FSI model and the column model with added mass is satisfactory and confirms that the hydrodynamic added mass computed by the developed model is valid in the seismic analyses of immersed columns in time-domain. Meanwhile, the computational time
Similar with the example column with elliptical cross - section, the added mass of the column with dumbbell cross - section also differs with the direction of ground motion θg . In order to investigate the effect of direction of motion θg , the hydrodynamic added mass of the studied column along seven different directions ranging from 0∘ to 90∘ , is calculated using the refined numerical model, and presented in Fig. 12. It can be seen that the added mass of such column increases with the direction θg , because of the increasing projection length of the column cross - section along the ground motion. Furthermore, the hydrody namic added mass per unit height at z =h ¼ 0, z =h ¼ 1=3 and z =h ¼ 2= 3 against the seven different directions are scattered in Fig. 13, and the solid curves are also plotted according to Eq. (25). As shown in Fig. 13, the solid curves agree well with the scattered points, which indicates that when the added mass of the column with dumbbell cross - section along the major and minor axes is determined, the added mass along a specific ground motion direction θg can be calculated by Eq. (25). � 2 2 yy (25) ma z; θg ¼ mxx a ðzÞcos θg þ ma ðzÞsin θg yy
where mxx a ðzÞ and ma ðzÞ are the hydrodynamic added mass at z, distance above the column base, along the major and minor axes of the column cross - section, respectively. 5.2. Seismic responses of the example column €g ðtÞ ¼ eiωt with frequency ω Harmonic ground motion acceleration u ranging from 0 to 10 Hz is firstly applied to these numerical models 12
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
for the FSI model and the column model with added mass on the same computer with Intel(R) Xeon E5-2630 CPU at 2.4 GHz and 16 GB RAM is 1.3 h and 0.6 h, respectively. The assessment time of the hydrodynamic added mass using the developed model is taken into account in the computational time associated with column model with added mass. Considering that the added mass can be employed repeatedly in seismic analyses towards structures supported by the same column, the column model with added mass is obviously more efficient compared with the FSI model by avoiding solving complex FSI equations and is more useful for seismic fragility analyses.
Acknowledgments The authors would like to acknowledge the financial support from Natural Science Foundation of China (Grants No. 51708455) and Fundamental Research Fund for the Central Universities (A1920502051907-2-001). References ADINA, 2010. Theory and Modeling Guide, vols. 10–7. ADINA R&D, Watertown, MA. Tech. Rep. ARD. Bathe, K.-J., 2014. Finite Element Procedures. Klaus-Jürgen Bathe, Englewood Cliffs, N.J. Bouaanani, N., Lu, F.Y., 2009. Assessment of potential-based fluid finite elements for seismic analysis of dam–reservoir systems. Comput. Struct. 87 (3), 206–224. Chopra, A.K., 2000. Dynamics of Structures: Theory and Applications to Earthquake Engineering. Prentice Hall, Upper Saddle River, NJ. Det Norske Veritas, 2010. Environmental Conditions and Environmental Loads. DNV-RPC205, DNV. European Committee for Standardization, 2005. Eurocode 8: Design of Structures for Earthquake Resistance-Part 2. Bridges. Goyal, A., Chopra, A.K., 1989. Earthquake response spectrum analysis of intake-outlet towers. J. Eng. Mech. 115 (7), 1413–1433. Goyal, A., Chopra, A.K., 1989. Simplified evaluation of added hydrodynamic mass for intake towers. J. Eng. Mech. 115 (7), 1393–1412. Han, R.P.S., Xu, H., 1996. A simple and accurate added mass model for hydrodynamic fluid—structure interaction analysis. J. Frankl. Inst. 333 (6), 929–945. Huang, X., Li, Z.X., 2011. Influence of free surface wave and water compressibility on earthquake induced hydrodynamic pressure of bridge pier in deep water. J. Tianjin Univ. Sci. Technol. 44 (4), 319–323 (in Chinese). Jiang, H., Wang, B.X., Bai, X.Y., Zeng, C., Zhang, H.D., 2017. Simplified expression of hydrodynamic pressure on deepwater cylindrical bridge piers during earthquakes. J. Bridge Eng. 22 (6), 04017014. Li, Q., Yang, W.L., 2013. An improved method of hydrodynamic pressure calculation for circular hollow piers in deep water under earthquake. Ocean Eng. 72, 241–256. Liaw, C.-Y., Chopra, A.K., 1974. Dynamics of towers surrounded by water. Earthq. Eng. Struct. Dyn. 3 (1), 33–49. Liu, Y., 2019. HAMS: a frequency-domain preprocessor for wave-structure interactions—theory, development, and application. J. Mar. Sci. Eng. 7 (3), 81. Mirzadeh, J., Kimiaei, M., Cassidy, M.J., 2016. Performance of an example jack-up platform under directional random ocean waves. Appl. Ocean Res. 54, 87–100. Nagaya, K., Hai, Y., 1985. Seismic response of underwater members of variable cross section. J. Sound Vib. 103 (1), 119–138. Olson, L.G., Bathe, K.J., 1985. Analysis of fluid-structure interactions. a direct symmetric coupled formulation based on the fluid velocity potential. Comput. Struct. 21 (1), 21–32. € H.R., 2003. Natural frequencies of an immersed beam carrying a tip mass with Oz, rotatory inertia. J. Sound Vib. 266 (5), 1099–1108. Ozdemir, Z., Souli, M., Fahjan, Y.M., 2010. Application of nonlinear fluid–structure interaction methods to seismic analysis of anchored and unanchored tanks. Eng. Struct. 32 (2), 409–423. Pang, Y.T., Wei, K., Yuan, W.C., Shen, G.Y., 2015. Effects of dynamic fluid-structure interaction on seismic response of multi-span deep water bridges using fragility function method. Adv. Struct. Eng. 18 (4), 525–541. PEER, 2013. Pacific Earthquake Engineering Research Center Ground Motion Database. http://ngawest2.berkeley.edu/. Qin, S., Gao, Z., 2017. Developments and prospects of long-span high-speed railway bridge technologies in China. Engineering 3 (6), 787–794. U�sciłowska, A., Kołodziej, J.A., 1998. Free vibration of immersed column carrying a tip mass. J. Sound Vib. 216 (1), 147–157. Wang, P., Zhao, M., Du, X., 2018. Short-crested, cnoidal, and solitary wave forces on composite bucket foundation for an offshore wind turbine. J. Renew. Sustain. Energy 10 (2), 023305. Wang, J.-T., Zhang, M.-X., Jin, A.-Y., Zhang, C.-H., 2018. Seismic fragility of arch dams based on damage analysis. Soil Dyn. Earthq. Eng. 109, 58–68. Wang, P.G., Zhao, M., Du, X.L., 2018. Analytical solution and simplified formula for earthquake induced hydrodynamic pressure on elliptical hollow cylinders in water. Ocean Eng. 148, 149–160. Wang, P.G., Zhao, M., Du, X.L., Liu, J.B., Chen, J.Y., 2018. Simplified evaluation of earthquake-induced hydrodynamic pressure on circular tapered cylinders surrounded by water. Ocean Eng. 164, 105–113. Wei, K., Yuan, W.C., Bouaanani, N., 2013. Experimental and numerical assessment of the three-dimensional modal dynamic response of bridge pile foundations submerged in water. J. Bridge Eng. 18 (10), 1032–1041. Wei, K., Myers, A.T., Arwade, S.R., 2017. Dynamic effects in the response of offshore wind turbines supported by jackets under wave loading. Eng. Struct. 142, 36–45. Westergaard, H.M., 1933. Water pressures on dams during earthquakes. Trans. ASCE 95, 418–433. Wu, J.-S., Hsu, S.-H., 2007. The discrete methods for free vibration analyses of an immersed beam carrying an eccentric tip mass with rotary inertia. Ocean Eng. 34 (1), 54–68. Xu, H., Zou, D., Kong, X., Hu, Z., Su, X., 2018. A nonlinear analysis of dynamic interactions of CFRD–compressible reservoir system based on FEM–SBFEM. Soil Dyn. Earthq. Eng. 112, 24–34.
6. Conclusions A potential-based numerical model of the hydrodynamic added mass for immersed column with arbitrary cross - section was developed. The general formulation of the hydrodynamic added mass was derived with respect to the hydrodynamic force induced by the horizontal rigid mo tion of the column. Hydrodynamic added mass of columns with circular, elliptical cross - section and circular tapered columns was estimated through the developed model. Good agreements with results from pre vious literatures for all three examples revealed the accuracy of the developed model for these immersed columns. In order to validate the model experimentally, the modal response of an immersed cylinder as a function of water depth was obtained by both the developed added mass model and experimental modal tests. Good agreement was found be tween the developed model and experimental results. The developed model was then further applied to evaluate the hydrodynamic added mass of an immersed column with dumbbell cross - section. The effect of direction of ground motion on the added mass of the example column was investigated. Although the hydrodynamic added mass of the immersed column with dumbbell cross - section varies with the ground motion direction, the added mass along a specific direction can be calculated based on an analytical function of the added masses along the column major and minor axes and the ground motion direction. The obtained added mass was then adopted to construct the column model with added mass, while the potential-based FSI model was also estab lished as reference. The time- and frequency-domain seismic responses of the example column were obtained using both models. The column model with added mass exhibits good accuracy in the calculation of seismic responses and shows great advantage in computational effi ciency, compared with complex FSI model. These findings demonstrate that the earthquake-induced hydrody namic added mass for immersed column with arbitrary cross - section can be efficiently predicted by the developed model. And the developed model allows the implement of added mass in the seismic design of structures supported by immersed columns with complex cross - section, and waiving the substantial efforts for modeling and solving sophisti cated FSI equations. The reduction in computational time would become more significant especially when numerous nonlinear time-history an alyses for a probabilistic model are required. It will provide an efficient approach for a comprehensive understanding towards the seismic per formance and reliability of the structures supported by immersed columns. In spite of that the developed added mass model have been verified for several commonly used types of columns, additional efforts should be devoted to the throughout validation of the added mass model towards columns with more kinds of cross - section. The effects of higher order vibration modes and the surface gravity waves on the hydrodynamic added mass have not been included because the added mass developed in the framework is independent with frequency. All these limitations deserve further investigation. Moreover, BEM can be implemented instead of PBFEs to evaluate the hydrodynamic pressure in the devel oped model for the purpose of easier meshing process and faster computation with good accuracy.
13
J. Zhang et al.
Ocean Engineering 187 (2019) 106192
Yang, W.L., Li, Q., 2013. The expanded Morison equation considering inner and outer water hydrodynamic pressure of hollow piers. Ocean Eng. 69, 79–87. Yu, H., Zheng, S., Zhang, Y., Iglesias, G., 2019. Wave radiation from a truncated cylinder of arbitrary cross section. Ocean Eng. 173, 519–530. Zhang, J.R., Wei, K., Pang, Y.T., Zhang, M.J., Qin, S.Q., 2019. Numerical investigation into hydrodynamic effects on the seismic response of complex hollow bridge pier submerged in reservoir: case study. J. Bridge Eng. 24 (2), 05018016.
Zhao, C., Chen, J., Xu, Q., 2014. FSI effects and seismic performance evaluation of water storage tank of AP1000 subjected to earthquake loading. Nucl. Eng. Des. 280, 372–388. Zheng, S., Zhang, Y., Liu, Y., Iglesias, G., 2019. Wave radiation from multiple cylinders of arbitrary cross sections. Ocean Eng. 184, 11–22.
14