Study on Modal Dynamic Response and Hydrodynamic Added Mass of Water-Surrounded Hollow Bridge Pier with Pile Foundation

Associate Professor, Department of Road and Bridge Engineering, Wuhan University of Technology, Wuhan 430063, China M.S. Candidate, Department of Road and Bridge Engineering, Wuhan University of Technology, Wuhan 430063, China Ph.D. Candidate, Department of Road and Bridge Engineering, Wuhan University of Technology, Wuhan 430063, China Associate Professor, Department of Civil Engineering, Chongqing Jiaotong University, Chongqing 400074, China


Introduction
Long-span bridges with high piers, such as the Miaoziping Bridge [1,2], the Sutong Bridge [3,4], and the China East Sea Bridge [5], are sound solutions for crossing deep-water rivers, reservoirs, straits, and so on.Generally, the high piers are built with voided sections for the sake of structural efficiency and low cost.e bridge substructures are usually composed of a group of piles connected by a large concrete pile cap and a hollow pier body which is usually topped by a cap beam.During the service life of deep-water bridges, the substructures are constantly submerged in deep water, which induces additional force on the structures when exposed to various dynamic loads, like seismic excitations.
e cause of the additional force is the fluid-structure interaction which has long been deemed as an important factor to be considered particularly in seismic design of dams, offshore platforms, and wind farms.It is worthwhile to mention that during the 2008 Wenchuan Earthquake, the Miaoziping Bridge, a typical large-span deep-water bridge with high and hollow piers, suffered severe damage and the fluid-structure interaction between the oscillating piers and surrounding water has been regarded as a crucial trigger [1,2].
Until recently, analytical approaches to solve the fluidstructure interaction problem fall approximately into two categories: one is the semianalytical and seminumerical approach and the other is the numerical approach.In the former one, theoretical derivation is performed to obtain analytical formulations for the hydrodynamic pressure which are then applied to the finite element models of structures to account for the fluid-structure interaction, hence the analytical formulations are of utmost importance to the prediction accuracy.Two methods are widely used to formulate the hydrodynamic pressure of cylindrical structures surrounded by water: (1) a method based on the Morison equation or modified Morison equation [6][7][8][9][10][11]; (2) a method based on the radiation wave theory [12][13][14][15].e semianalytical and seminumerical approach has advantages of both analytical formulation and numerical simulation and hence is widely used to predict the effects of fluid-structure interaction.Currently available formulations for the hydrodynamic pressure, however, still remain to be improved.For example, the method based on the Morison equation is semitheoretical and semiempirical, and the inertia coefficient and drag coefficient defined in the equation are difficult to determine precisely.Moreover, the method cannot be used to take into consideration the effect of inner water inside a hollow structure.Another example, the method based on radiation wave theory is an analytical solution with rigorous theoretical derivation and is applicable when the influence of water compressibility and surface waves should be taken into account, but its formulations are very complicated and inconvenient for practical application.Besides, the method applies only to cylindrical structures with uniform cross section, and therefore it is inappropriate for high and hollow piers whose cross sections usually vary along the height in practical engineering.With technological innovations in engineering computation, numerical methods, such as finite element method and boundary element method, are referred more and more frequently as a solution to the dynamic response of fluid-structure interaction [16][17][18][19][20][21][22].But the numerical models are usually complicated with a huge number of elements, and the solutions are computationally inefficient.Furthermore, the numerical methods cannot account accurately for the nonlinearities such as structural damage, concrete cracking, and so on.
To validate the theoretical and numerical methods, some experiments have been conducted to study fluidstructure interaction and supportive test data have been obtained [20,[23][24][25][26][27].It can be found that most of these experiments were conducted on cantilever columns, towers, and other cylindrical structures with solid cross sections.But in this study, an experimental program was designed to investigate the dynamic responses of a high and hollow bridge pier submerged in deep water.During the study, the first two lateral vibration modes along the x-axis and y-axis of the specimen were tested in various combined conditions of water levels and tip masses.e tests were conducted for three cases where the specimen contacts with only outer water, only inner water, and both outer and inner water.Using potential-based fluid elements, 3D finite element models were established for each case and each condition.e numerical models were verified since the numerical results were successfully validated against the obtained test data.Based on the verified models, the hydrodynamic added mass (HAM) and the modal hydrodynamic added mass (MHAM) along the x-axis and y-axis of the specimen were studied from the three cases, respectively.According to the distribution of hydrodynamic added mass (HAM) along the specimen, the equivalent analytical model was finally proposed for calculating the dynamic characteristics of hollow bridge piers surrounded by water.

Selection of Similarity Ratio.
A specimen was designed based on a reduced scale.During dynamic similarity ratio design, the inertial force, the gravity, and the restoring force of the specimen need to satisfy the equation written as where S E is the similarity constant of the elastic modulus; S g is the similarity constant of the gravitational acceleration; S ρ is the similarity constant of the density; and S l is the similarity constant of the geometrical dimensions.It is important to note that the equation above can hardly be achieved especially for the specimen composed of two materials (reinforced concrete), so the similarity ratio is approximate.
In order to strictly satisfy the similarity ratio between the self inertia force of the structure and the additional inertia force of the water, the similarity constants of the structure density and the water density need to be kept the same.e water used in the tests is the same as the prototype water, so the similarity constant of the structural density S ρ is 1. e similarity constant of the gravitational acceleration is also 1.According to the dimensions of the prototype bridge pier and other conditions, S l was selected as 1/40 and the actual dimensions of the specimen were adjusted, for example, the pier height of the specimen was adjusted to 1.6 m because the bridge pier is partly submerged in water.
e similarity constants of other physical properties needed are listed in Table 1.

Design of Test Specimen.
e specimen was inspired by a prototype deep-water bridge with large spans and hollow piers.e prototype is the Miaoziping Bridge which is located near the epicenter of the Wenchuan Earthquake, 29 km from Yingxiu town in southwestern China, as schematically shown in Figure 1. e main bridge is a threespan continuous framed bridge with two intermediate hollow piers about 100 m high.e hollow piers, designated as No. 3 pier and No. 4 pier, are submerged in the Zipingpu reservoir to a depth of 50 m.At the time of the Wenchuan Earthquake, the construction of the bridge was almost completed except for the installation of expansion joints.During the earthquake, the intermediate hollow piers suffered severe damage with several penetrating cracks propagating at the bottom of the piers.Figure 2  concrete [28], and Young's modulus of C30 concrete is 30 GPa.As shown in Figure 3, a reduced scale hollow pier specimen was built.e specimen consists of six steel tube   e hollow pier body is 1.6 m in height, with a single-cell cross section of 0.30 × 0.20 × 0.025 m.It is made of C15 concrete according to the Chinese code , with a density of 2,300 kg/m 3 and Young's modulus of 22 GPa.Young's modulus of the material selected (C15) is smaller than the original material (C50), mainly to guarantee the measurement requirements.
e pier body is reinforced by ten Chinese Grade HRB400 longitudinal steel bars and HPB300 [29] stirrups.e bars have a diameter of 8 mm, and the stirrups have a diameter of 6 mm which are placed at a vertical spacing of 0.1 m.It is essential to note that the change rules of the specimen-water interaction as a function of water level were studied in this study and the reduction scale effects will not affect the change rules, and some numerical and experimental analyses can be referred from previous study [30].

Instrumentation and Test
Procedure.In this experimental research, a steel water tank was designed and built in advance to create a deepwater environment for the test specimen.e detailed introduction of the water tank (including the size effect analysis), experimental instruments, and experimental method were presented in a previous study conducted by Deng et al. [30].ree different cases were considered in the research and designated as case I (where the specimen was in contact with the water only surrounding the outside surfaces of the specimen, or precisely, contact only with outer water), case II (where the specimen interacted with the water only existing inside the hollow pier body, or precisely, interacted only with inner water), and case III (where the specimen was in contact with both outer and inner water).Forced vibration tests were carried out repeatedly to investigate the modal dynamic response of the specimen with different tip masses: (1) surrounded by 16 levels of outer water, varying from 0 to 1.8 m for an empty to full water tank, respectively; (2) interacting with 11 levels of inner water, varying from 0.8 m to 1.8 m (corresponding to a water height of 0.05 m to 1.05 m inside the hollow pier body); (3) contact with both outer and inner water for the same 16 levels of water heights as defined previously.
e designed water levels can be seen in Figure 4. e tests for the above three cases were all repeated for a tip mass of 0, 11.4, 23.4, and 31.7 kg, respectively.Various tip masses were used to examine the effect of bridge superstructure mass, and they were imposed at the top of the pier through steel bricks.
A small hole was drilled at the bottom of the pier body in advance; during the tests for case I and case II, the hole was plugged through viscose rayon so that these two cases could be conducted separately.After the tests for case II were successfully performed, the hole was released and the water inside the hollow pier body was poured out.en, the tests for case III could be conducted, and due to the hole, the water levels inside and outside the hollow pier body were purposely kept the same.Advances in Civil Engineering the procedure of fluid-structure interaction, and the corresponding fluid element is the potential-based fluid element.In the ϕ − U formulation, U, representative of displacement, is used as a static variable in the solid domain and ϕ, representative of velocity potential, is used in the fluid domain.e potential-based fluid element is assumed to be in viscid, irrotational, compressible, or almost incompressible (slightly compressible), with no heat transfer and relatively small displacements on the fluid-structure boundaries.e potential-based fluid element has been proved to be reliable and computationally efficient and has been widely applied in modeling the water-surrounded dams, cylinders, towers, and so on [16,18,19,31].So, it is adopted in this work.According to the potential-based fluid theory, the wave equation is written as

Numerical Simulations
where C w is velocity of compressive waves within water and t is time variable.In order to solve equation ( 2) with respect to a hollow bridge pier with pile foundation, the following boundary conditions are specified for the velocity potential ϕ: (i) An essential boundary condition at the free surface and the four lateral and bottom rigid walls of the water tank: (ii) A natural boundary condition at fluid-structure interfaces, including the water-piles, water-cap, and water-pier body interfaces: where _ u n is positive normal velocity corresponding to unit surface normal vector and n pointing into the water tank out of the structure.Equation (2) can be derived through standard techniques and discretized to obtain the following nonstandard eigenvalue problem: where F (j) � −iϕ (j) in which i � �� � −1 √ and ϕ (j) is fluid potential eigenvector; U (j) is modal displacement eigenvector in which M is structural mass matrix; M FF is potential energy matrix of water domain; C FU is matrix coupling the velocity potential to displacements on the fluid-structure interfaces; K is structural stiffness matrix; K FF is the kinetic energy matrix for water; (K UU ) S is the matrix relating to the structural stiffness to water motion; and ω j is coupled frequency of water-structure system along mode j.
Equation ( 5) is a nonstandard eigenvalue problem in which all of the eigenvalues are real and nonnegative.It can be solved using either the Determinant search method or the Lanczos iteration method.

Finite Element Models.
Generally, the potential-based finite elements have been used in numerical simulations.e purpose is to compare the results obtained experimentally and numerically, respectively, and to validate the numerical procedures used widely to explore the dynamic characteristics of fluid-structure interaction.
ree-dimensional (3D) finite-element models are established using the finite element code ADINA [31] to model the tested hollow pier specimen and water.As illustrated in Figure 5, the 3D 8-node shell finite elements available in ADINA are used to model the steel tube piles, and the 3D 20-node solid finite elements are used to model the rest of the specimen including the pier body and pile cap.
e tip mass is represented by the concentrated mass elements, and the water domains, including outer water and inner water, are meshed using the 3D 20-node potentialbased fluid elements.e ADINA code provides a convenient function to generate automatically fluid-structure interface elements along the boundaries between the fluid and structure.e bounding surfaces of the water domain are modeled using various types of potential interfaces, like free surface, fluid-structure interface, and rigid wall.Fluidstructure interaction is accounted for through these potential interfaces defined at the water-structure interfaces.
e physical properties of the tested hollow pier specimen and water are listed in Table 2.
For each value of the tip mass, finite-element models were established for the dry structure, i.e., without water, and the wet structures submerged by 15 water levels in case I and III, and by 11 inner water levels in case II, as shown in Figure 4. is resulted in a total of 168 finite-element models with 42 for each of the four tip masses.

Definition of the Vibration Period Increase
Rate.In order to discuss the effect of fluid-structure interaction on the modal dynamic response of the specimen, vibration period increase rate is defined herein to show the tendency of the specimen vibration period in different test cases.e vibration periods T 0 xj and T 0 yj , j � 1, 2, are the results of the specimen without water, corresponding to the first two modes along the x-axis and y-axis, respectively.e vibration periods T w xj , j � 1, 2, and T w yj , j � 1, 2, correspond to the first two modes along the x-axis and y-axis, respectively, which were determined for each water level considered in this paper.
During this study, higher orders of mode are not easy to be excited in a steady condition based on traditional forced vibration tests, and they were not the major concern of this experimental research.e torsional mode was not taken into consideration during the tests, because in most cases the torsional mode contributes little to the seismic response of bridges, especially to that of straight bridges.

Effect of Outer Water-Structure Interaction on Vibration
Periods.In case I, only the water surrounding the outer surfaces of the specimen was considered during the tests with a purpose of investigating the effect of outer waterstructure interaction on the hydrodynamic modal response of the specimen.Figure 6 presents the trends of the periodical percentage increases of the specimen with respect to the outer water level changing from 0 to 1.8 m, for an empty to full water tank, for the four tip masses, respectively.Along with the experimental (EXP) results, the numerical results (FEM) obtained using the finite-element models described previously are also given in these figures.
As shown in these figures, a satisfactory agreement is displayed between the numerical results and the experimental data for all combinations of water level and tip mass.It can be revealed that (1) when the water level is below the pile cap, i.e., H w < 55 cm, the period percentage increases of the four lateral modes remain practically constant under the four tip masses, implying that the hydrodynamic effect of water-piles interaction is negligibly weak when bridges are submerged by water up to the bottom of pile caps; (2) when the water level is just over the entire height of the pile cap, i.e., 55 cm < H w < 75 cm, the curves of the four lateral modes under the four tip masses ascend obviously but limitedly, indicating that the fluid-structure interaction between water and pile cap plays an more important role in the hydrodynamic response of the specimen than that of between water and piles; (3) and when the pier body is submerged in water, i.e., H w > 75 cm, a continuous rise of the curves can be

Material Elasticity modulus Mass density Poisson ratio Velocity of compression waves
Advances in Civil Engineering observed in the four lateral modes under the four tip masses, especially in those along the x-axis due to the fact that the area of fluid-structure potential interfaces perpendicular to the x-axis is larger than that perpendicular to the y-axis.
Besides, the figures also reveal a different trend between the results of first lateral modes and second lateral modes for the four tip mass cases.With the increase of water levels, the period percentage increases for the second modes grow quickly in the beginning and then in a much slower pattern, whereas the period percentage increases for the first modes grow contrarily.
It is important to note that the maximum period percentage increases are approximately 23% for the first lateral modes and about 15% for the second lateral modes along the x-axis , both occurring when the water tank is full.erefore, it might be neoconservative to overlook the hydrodynamic loads in design and construction of bridges with hollow piers submerged in deep water.

Effect of Inner Water-Structure Interaction on Vibration
Periods.Since small holes, intentionally reserved for construction purposes, are commonly found in reinforced concrete piers, it is not difficult to accept that the hollow chambers of pier body could be filled with water when it is submerged in deep water.It is necessary to point out that case II where the bridge interacts only with the water inside its hollow piers is not common in practical engineering, but it is still of a certain significance in understanding the hydrodynamic behavior of hollow structures.Figure 7 illustrates the trends of the period percentage increases of the specimen with respect to the inner water level changing from 0.8 to 1.8 m for the four tip masses, respectively.For the sake of comparison, the abscissas of the figures still start from 0 to 180 cm, from which the ranges of 0-80 cm correspond to the results obtained using dry specimen (the specimen without water).
Figure 7 manifests a sound agreement between the numerical results and test data for all considered cases.It can be seen that the vibration periods of the specimen become larger as the water level increases, indicating that the effect of fluid-structure interaction is more remarkable with increasing water level.It is not surprising that similar observations can be made in case II just as in case I.But a discrepancy can still be noticed that in case II: the results along the x-axis are closer to those along the y-axis.Similar to the trends in case I, the period percentage increases for the first modes for both directions in case II decrease when the tip mass becomes larger, whereas the period percentage increases of the second modes show an opposite tendency.By comparison between Figures 6 and 7, it can be found that the effect of inner water-structure interaction is not that notable when compared with outer water-structure interaction, and in case II, the maximum period percentage increases are approximately 10% for the first lateral modes and about 4% for the second lateral modes.

Effect of Water-Structure Interaction on Vibration Periods Taking into Consideration Both Outer and Inner Water.
e period percentage increases of the specimen with respect to the outer water level are shown in Figure 8 for the four tip masses, respectively.A satisfactory agreement can also be found between the numerical results and experimental data for all considered combinations of water level and tip mass.For the same lateral mode, the variation trends of the period percentage increases with the water level and tip mass are closer to those observed in case I; this provides further evidence that the effect of outer water plays a dominant part.It is worthwhile to point out that the period percentage increases for the four lateral modes in case III are the largest among the three considered cases.For example, the period percentage increases reach a maximum of about 30% for the first lateral mode and nearly 20% for the second lateral mode.9.
It is interesting to notice that, under both cases, the period percentage increases of the first modes for both directions decrease when the tip mass becomes larger, whereas the period percentage increases of the second modes show an opposite trend.It is implied that the effect of fluid-structure interaction is attenuated in the first modes but strengthened in higher order modes as the tip mass increases.e same trends can be found in case II which is not presented here.It is also indicated from each plot of Figure 9 that the effect of fluid-structure interaction in case III is always greater than that in case I for each water level and tip mass, because of the existence of inner water in case III.

Modal Participation Mass of Fluid-Structure System.
Both experimental and numerical results revealed that the vibration period increase is noticeable when the specimen is submerged in water (or interacts with inner water).Generally, it is accepted that, in water-structure interaction system, water has little impact on the stiffness of structure and the additional damping of the system induced by water, can be neglected, and therefore the main factor leading to the increase of vibration periods is attributed to the hydrodynamic added mass [11,12,24].Several formulations for the hydrodynamic added mass have been proposed by means of theoretical derivation during past decades [12,14,32], but the formulations have been derived based on column structures with a uniform circular or rectangular cross section.As for the hollow pier investigated herein, it might be difficult to derive a theoretical formulation for the hydrodynamic added mass.With technological competence of numerical methods in computing the problem of fluidstructure interaction, the hydrodynamic added mass for complex structures can be obtained through numerical simulations [16,31,33].
In a water-structure system, the modal participation mass for each vibration mode consists of two sources: one originating from the structural participation mass and the other from hydrodynamic added mass [33].In a specific direction, the hydrodynamic added mass for each water level can be obtained through subtraction of the accumulated modal participation masses between the water-structure system and dry-structure system, as presented in detail in Section 5.2.e principle and procedure for determining the modal participation mass of water-structure system can be found in ADINA [31]; and they are briefly introduced below.e motion equation of a fluid-structure interaction system under ground motion loading is given as follows: where M is the mass matrix for the system; K is the stiffness matrix for the system; M FF is the potential energy matrix for water; K FF is the kinetic energy matrix for water; (K UU ) S is the matrix relating the structural stiffness to water motion; C FU is the matrix coupling the velocity potential to displacements on the water-structure interfaces; U r is the vector  Advances in Civil Engineering for nodal displacements with respect to the ground motion; ϕ is the velocity potential corresponding to the absolute (not relative) velocities; _ u gk is the ground displacements in direction k; and d k is the vector of nodal point values in which d k (i) � 1 if equation i corresponds to a translation in direction k and d k (i) � 0 otherwise.Equation ( 6) can be solved by mode superposition if the structural damping is neglected, the assumption of infinitesimal velocity is warranted, and no infinite boundaries are defined.
e modal expansions are given in the following: where x j is generalized coordinate of the jth mode; ω j is the jth modal frequency; and U (j) and F (j) are, respectively, the displacement eigenvector and fluid potential eigenvector of the jth mode, which can be determined by eigenvalue analysis.Generally, the eigenvectors are scaled according to the following orthogonality condition: where δ ij is the Kronecker delta, u (i) and u (j) are the structural displacement vectors for mode i and j respectively; F (i) and F (j) are the fluid plural velocity potentials for mode i and j respectively; and ω i and ω j are the ith and jth modal frequency respectively.Notice that when no fluid exists, equation ( 8) reduces to the usual orthogonality condition u (i) M UU u (j) � δ ij .
With modal damping ratio ζ j for the jth mode, the modal equation of motion can be written as where the load vectors are Note that the term (K UU ) S is numerically very small compared with the rest of the structural stiffness matrix, when there is a structure adjacent to the fluid.erefore, the term (U (j) ) T (K UU ) S d k u gk can be neglected.en, equation ( 11) can be rewritten as where the jth modal participation factor is If the eigenvectors meet the orthogonality condition ( 7), the jth modal participation mass m j and the accumulated modal participation mass of the first n modes are

Definition of Hydrodynamic Added Mass. For convenience, two symbols of m 0
x and m 0 y are introduced to denote the accumulated modal participation masses along the x-axis and y-axis for the hollow pier specimen without water, and accordingly the symbols of m w x and m w y are used for the specimen submerged by different levels of water.So the hydrodynamic added masses can be defined as (HAM) ΔM x � m w x − m 0 x and ΔM y � m w y − m 0 y .Besides, the symbols of m 0 xj and m 0 yj , j � 1 and 2, are also introduced to represent the modal participation masses of the first two modes along the x-axis and y-axis for the specimen without water.Similarly, the symbols of m w xj and m w yj , j � 1 and 2, are defined for the specimen surrounded in different levels of water.So the modal hydrodynamic added masses (MHAMs) can be obtained through Δm xj � m w xj − m 0 xj and Δm yj � m w yj − m 0 yj .In the finite element code ADINA, the modal participation mass for each vibration mode of a water-structure system can be obtained using the modal participation factor (modal added mass and accumulated added mass) which is provided by ADNIA.It is important to point out that the modal analysis should include sufficient numbers of vibration modes to guarantee the accuracy of the accumulated modal participation mass.

Effect of Outer Water-Structure Interaction on Hydrodynamic Added Mass.
e HAMs along the x-axis and yaxis, respectively, are shown in Table 3 for the specimen with a tip mass equal to 0, 11.4, 23.4, and 31.7 kg, respectively, submerged by outer water to various levels.e HAMs along a specific direction are determined using the first 30 vibration modes along the same direction, since higher modes contribute little to the results.e MHAMs defined previously are depicted in Figure 10 with respect to the water levels for the specimen with a tip mass equal to 0, 11.4, 23.4, and 31.7 kg, respectively.
As indicated in Table 3, both the HAMs along the x-axis and y-axis, respectively, increase with water level, and the HAMs along the x-axis are evidently larger than those along the y-axis for the same water level.It can be seen in Figure 10 that the MHAMs for the four vibration modes present different trends with different water levels.e MHAMs of  the first vibration modes along both directions grow significantly as water level rises, with a maximum value of around 119 kg along the x-axis and around 62 kg along the yaxis, and each maximum value is pretty close to the HAM of corresponding direction.e MHAMs of the second vibration modes along both directions display different trends from those of first vibration modes where monotonic increases can no longer be observed, and the values of the MHAM are much smaller for the second vibration modes along both directions.

Advances in Civil Engineering
It is interesting to note that for a specific water level, the tip mass has little effect on the HAM but has a remarkable influence on the MHAM of different vibration modes.e MHAM of the first modes along both directions decrease with the increase of tip mass, whereas the MHAM of the second modes show an opposite trend.erefore, a conclusion can be deduced that the HAM of the watersurrounded specimen for both directions are closely related to pier dimensions and water level but are of little relevance to the tip mass.It is worthwhile to point out that the curves of the MHAM show a similar tendency to the curves of the vibration period percentage increases as indicated in Figure 6, which further validates the numerical simulations against the obtained experimental data.4 lists the HAMs along the xaxis and y-axis, respectively, for the specimen with a tip mass equal to 0, 11.4, 23.4, and 31.7 kg, respectively, interacting with various levels of inner water.

Effect of Inner Water-Structure Interaction on Hydrodynamic Added Mass. Table
e MHAMs of the specimen corresponding to the first two vibration modes along the x-axis and y-axis, respectively, are depicted with respect to inner water levels for the four tip masses in Figure 11.For the sake of comparison with case I, the abscissas of the figures still range from 0-180 cm, among which the portion of 0-80 cm corresponds to the results obtained without inner water.
It can be seen in Table 4 that the HAMs for both directions increase with inner water level, and their values are close to each other for a specific level of inner water.As shown in Figure 11, the curves of the MHAM corresponding to the four vibration modes display different trends with different water levels.e curves corresponding to the first vibration mode along both directions ascend monotonically as the inner water height increases, with slightly larger values found along the xaxis than those along the y-axis for the same level of inner water.e curves for the second vibration mode along both directions exhibit a bit complicated trend with inner water level, although the variation amplitudes of the curves remain negligibly small.As in case I, the tip mass imposed at the top of the hollow pier plays a similar role in affecting the MHAM induced by inner water in case II.Hence, a similar conclusion can be drawn in case II that the HAMs for both directions of the specimen interacting with inner water have little relevance to the tip mass which is closely related to the MHAM of each vibration mode.
e observations made from Figure 11 are consistent with those from Figure 7, which also provides convincing validation of the numerical simulations against the obtained experimental data.

Effect of Water-Structure Interaction on Vibration Periods Taking into Consideration Both Outer and Inner Water.
Table 5 presents the HAMs of the specimen along the x-axis and y-axis, respectively, for the tip masses of 0, 11.4, 23.4, and 31.7 kg, respectively, in case III where the specimen interacts with both outer and inner water.As shown in Figure 12, the MHAMs of the specimen corresponding to the first two vibration modes along the x-axis and y-axis, respectively, are plotted with respect to the water levels considered herein for the four tip masses.
As indicated in Table 5 and Figure 12, the variations of the HAM and the MHAM of the specimen in case III show similar trends to those in case I for both directions, with an obvious difference, it was found that noticeably larger values are obtained in case III than in case I.And again, the observations indicated in Figure 12 are consistent with those in Figure 8, and the numerical results are successfully validated against the experimental data.

Equivalent Model Based on Hydrodynamic
Added Mass

Distribution of Hydrodynamic Added Mass.
is section is intended to discuss the distribution of HAM along the height of the specimen.As indicated in Tables 3 and 5, the HAMs induced by pile-water interaction in various cases are negligibly small, and no HAMs are considered to the piles in the equivalent models to be discussed.erefore, when H w � 75 cm, the HAM can be deemed to be induced by cap-water interaction; and when H w > 75 cm, the HAM consists of two parts: one is induced by cap-water interaction and the other is induced by pier-water interaction.Supposing the HAM induced by cap-water interaction remain constant with the increase of water level (H w > 75 cm), the HAM to the pile cap and pier body, as shown in Tables 6 and 7, can be readily calculated for various cases using the data presented in Tables 3-5.It can be seen from Tables 6 and 7 that the HAM obtained in case III (taking into consideration both the effects of outer and inner water) approximate the sum of those in case I (taking into consideration only the effect of outer water) and case II (taking into consideration only the effect of inner water) under same analytical or experimental conditions.
e HAMs to the pile cap are assumed to be evenly distributed along the height since the pile caps in the practical engineering are commonly small in height when compared with piers, and the HAM induced by pier-water interaction is distributed along the pier submerged portion according to its actual distribution pattern.Previous researchers pointed out that the HAM at a particular point is proportional to the hydrodynamic pressure on the surface and the acceleration of the structure at that point, which are irrelevant to the vibration shape of the structure [11,14,24].Figures 10-12 reveal that the HAM of the specimen along the x-axis (or y-axis) is mainly dependent on the Advances in Civil Engineering contributions of the fundamental vibration mode along the same direction.Hence, the hydrodynamic pressure and the modal acceleration in the fundamental mode are utilized to obtain the distribution of the HAM along the specimen.respectively, corresponding to a water level of H w � 1.8 m in different cases.It is shown that the tip mass has little effect on the magnitude or distribution of the HAM along the specimen.is observation can also be made from the results corresponding to other water levels (for brevity, they are not detailed herein).

Equivalent Models.
e assumption that the HAMs are uniformly distributed along the structure submerged by water [8,9,11] is applied to established Equivalent model I of the specimen.Meanwhile, the distributions of the HAM along the specimen, discussed in Section 6.1, are also applied and incorporated into a finite element model of the specimen which is denoted as Equivalent model II.In order to provide reasonable comparisons, the modeling details of piles, pile cap, and hollow pier body remain the same as elaborated in Section 3.2.e HAMs are modeled through modified material density in Equivalent model I and through distributed added masses directly to the nodes in Equivalent model II, as illustrated in Figure 17.

Comparison of Vibration Periods.
e symbols  T e vibration period ratios of the first two modes along the x-axis and y-axis, respectively, are plotted against water level as shown in Figures 18-20, corresponding to case I, case II, and case III.In Figures 18-20, the results of the specimen with four tip masses of 0, 11.4, 23.4, and 31.7 kg, are included corresponding to water levels not lower than H w � 0.8 m, since the pier body cannot contact water if lower water levels are considered in analysis (Figure 4).As illustrated in Figure 18, convincing agreement can be reached between the results obtained by Equivalent model II and the FEM mentioned in Section 3.2, with a maximum error smaller than 1% provided the results of the FEM are exact, comparing with a maximum error of

18
Advances in Civil Engineering than 0.6%, while Equivalent model II displays better agreement with the FEM.It can be drawn from above discussions that it is not always appropriate to assume a uniform distribution of hydrodynamic added masses in performing simplified analysis of fluid-structure interaction.For the hollow pier specimen, the HAM induced by outer water should be distributed along the submerged portion in its practical distribution pattern, but the HAM induced by inner water can still be taken to be uniformly distributed along the submerged portion.e distribution of HAM considering both the effects of outer and inner water is approximately the superposition of those considering only the effect of outer and inner water, respectively.It is also important to note that the conclusions herein might only be applicable to the hollow pier specimen considered in this paper and should not be extrapolated directly to seismic or other dynamic load response analyses of structures surrounded by water.

Summary and Conclusions
is paper presents an experiment program conducted to study the modal dynamic response of hollow bridge pier with pile group foundation submerged in water.A reduced scale hollow pier was designed and tested in three cases where the specimen contacts with only outer water, only inner water, and both outer and inner water, respectively.Four tip masses, that is, 0, 11.4, 23.4,31.7 kg, were considered in the tests to present the effect of bridge superstructure mass.Various water levels were defined as a function in the tests.ree-dimensional finite element models, corresponding to the specimen tested under combination conditions of the three cases, the four tip masses, and various water levels mentioned above, were established.
e vibration periods of the first four lateral modes, including the first two modes along the x-axis and the first two modes along the y-axis, were measured and calculated.e experimental results and the numerical results were then validated against each other.Based on the validated models, the hydrodynamic added masses were calculated and analyzed.An equivalent model (Equivalent model II) was proposed based on the hydrodynamic added mass distribution which was obtained by using the hydrodynamic pressure and the modal acceleration.Equivalent model I was also established through uniform distribution of the hydrodynamic added mass.Vibration periods of Equivalent model I and Equivalent model II were calculated and compared.e following conclusions can be made based on this research: (1) In the three cases that structure contacts with only outer water, only inner water and both outer, and inner water, the vibration periods of the hollow pier specimen increases as water level rises, indicating the effect of fluid-structure interaction on the modal response of the specimen increases with water level.Among the three cases, the effect of fluid-structure

Figure 1 : 3 Figure 2 :
Figure 1: General layout of the Miaoziping Bridge; all dimensions are in meters.(a) Picture of the Miaoziping Bridge.(b) Elevation view.

3. 1 .
Fluid-Structure Interaction Formulation.In a fluidstructure system, the vibration of structure induces water flow normal to the structural boundaries which produces additional hydrodynamic pressure acting on the structure.ewell-known ϕ − U formulation can be used to represent

Figure 3 :
Figure 3: Dimensions and details of the hollow pier specimen; all dimensions are in meters unless otherwise specified.

Figure 4 :
Figure 4: Water levels considered during the tests; all dimensions are in meters.

Figure 5 :
Figure 5: Finite-element model of the hollow pier specimen and water domain: (a) overview; (b) elevation view through a central cutting plane.

Figure 6 :
Figure 6: Measured and computed percentage increases of vibration period for the hollow pier specimen in case I as a function of water level: (a) without tip mass; (b) with an 11.4 kg tip mass; (c) with a 23.4 kg tip mass; and (d) with a 31.7 kg tip mass.
level (cm) x-axis 1st mode by FEM y-axis 1st mode by FEM x-axis 2nd mode by FEM y-axis 2nd mode by FEM x-axis 1st mode by EXP y-axis 1st mode by EXP x-axis 2nd mode by EXP y-axis 2nd mode by EXP level (cm) x-axis 1st mode by FEM y-axis 1st mode by FEM x-axis 2nd mode by FEM y-axis 2nd mode by FEM x-axis 1st mode by EXP y-axis 1st mode by EXP x-axis 2nd mode by EXP y-axis 2nd mode by EXP 1st mode by FEM y-axis 1st mode by FEM x-axis 2nd mode by FEM y-axis 2nd mode by FEM x-axis 1st mode by EXP y-axis 1st mode by EXP x-axis 2nd mode by EXP y-axis 2nd mode by EXP 1st mode by FEM y-axis 1st mode by FEM x-axis 2nd mode by FEM y-axis 2nd mode by FEM x-axis 1st mode by EXP y-axis 1st mode by EXP x-axis 2nd mode by EXP y-axis 2nd mode by EXP (d)

Figure 7 :
Figure 7: Measured and computed percentage increases of vibration period for the hollow pier specimen in case II as a function of water level: (a) without tip mass; (b) with an 11.4 kg tip mass; (c) with a 23.4 kg tip mass; and (d) with a 31.7 kg tip mass.

Figure 8 :
Figure 8: Measured and computed percentage increases of vibration period for the hollow pier specimen in case III as a function of water level: (a) without tip mass; (b) with an 11.4 kg tip mass; (c) with a 23.4 kg tip mass; and (d) with a 31.7 kg tip mass.

Figure 9 :
Figure 9: Computed period percentage increases for the pier specimen as a function of water level: (a) first mode along the x-axis; (b) first mode along the y-axis; (c) second mode along the x-axis; and (d) second mode along the y-axis.

4Figure 10 :
Figure 10: MHAM of the specimen for the four tip masses in case I as a function of water level, corresponding to (a) first mode along the xaxis; (b) first mode along the y-axis; (c) second mode along the x-axis; and (d) second mode along the y-axis.

Figures 13 and 14 Figure 11 :
Figures 13 and 14 present the relative accelerations distribution, the relative hydrodynamic pressures distribution, and the relative HAM distribution for the first vibration modes along the x-axis and y-axis of the specimen without

4Figure 12 :
Figure 12: MHAM of the specimen for the four tip masses in case III as a function of water level, corresponding to (a) first mode along the x-axis; (b) first mode along the y-axis; (c) second mode along the x-axis; and (d) second mode along the y-axis.
were obtained taking the point in each graph whose ordinate value is 1 as reference point, separately).Two sets of results are given for each figure, and they are corresponding to the middle vertical line and the side vertical line (to refer them hereinafter as mid-line and side-line for short), respectively, on the surface of the pier perpendicular to the vibration direction.It is evident that the distributions of the HAM obtained using the results along mid-line are consistent with those obtained using the results along sideline in both case I and case II.For convenience, the HAM corresponding to the mid-line is computed for other tip masses and water levels, and typical results are discussed below.Figures15 and 16illustrate the HAM distribution of the specimen with tip masses of 0, 11.4, 23.4, and 31.7 kg,

Figure 13 :Figure 14 :Figure 15 :
Figure 13: Distribution of relative modal acceleration, relative hydrodynamic pressure, and relative HAM along the specimen without tip mass corresponding to a water level of H w � 1.8 m in case I: (a) relative modal acceleration; (b) relative hydrodynamic pressure; (c) relative HAM.
w xj and  T w yj , j � 1, 2, are introduced to represent the vibration periods of the two equivalent models along the x-axis and y-axis, respectively, with the subscript j � 1 representing Equivalent model I and the subscript j � 2 representing Equivalent model II.e vibration periods T w xj , j � 1, 2 and T w yj , j � 1, 2 are obtained using the FEM established in Section 3.2, corresponding to the first two vibration modes along the xaxis and y-axis, respectively.So, the vibration period ratios R xj �  T w xj /T w xj and R yj �  T w yj /T w yj , j � 1, 2, are defined to discuss the application of the two equivalent models.

Figure 16 :
Figure 16: Comparison of the relative added mass distributions along the specimen corresponding to a water level of H w � 1.8 m between: (a) case III and case II along the x-axis; (b) case III and case II along the y-axis.

Figure 19 :Figure 20 :
Figure 19: Vibration period ratios of the specimen in case II as a function of water level correspond to: (a) first mode along the x-axis; (b) first mode along the y-axis; (c) second mode along the x-axis; and (d) second mode along the y-axis.

Table 1 :
Physical properties of the designed specimen.

Table 3 :
HAM along the x-axis and y-axis for the four tip masses in case I.

Table 4 :
HAM along the x-axis and y-axis for the four tip masses in case II.

Table 5 :
HAM along the x-axis and y-axis for the four tip masses in case III.

Table 6 :
HAM of the pile cap along the x-axis and y-axis in the three cases.

Table 7 :
HAM of the pier along the x-axis and y-axis in the three cases.