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Fluid–structure interaction on concentric composite cylinders containing fluids in the annulus

  • D. Alaei
  • Y. W. KwonEmail author
  • A. Ramezani
Original Paper
  • 46 Downloads

Abstract

Dynamic response of two concentric horizontal composite cylinders containing water in the annulus was investigated under impact loading so as to examine the load transfer from the outer cylinder to the inner through a fluid medium. Different water filling levels in the annulus were considered along with different magnitudes of impact loading. Both of the composite cylinders were 254 mm long, had a diameter of 76.2 mm and 88.9 mm, respectively, and were assembled concentrically. Both experimental and numerical studies were conducted to supplement each other. The experimental set-up was designed and constructed. Both cylinders were constrained at both ends, and the water level was varied in the annulus of the two cylinders. The experimental set-up used strain gages at certain locations. For each experiment, the strain data were collected and examined. Then, the fast Fourier transform was applied to the strain data to identify major vibrational frequencies and to examine the effect of the added mass. The numerical study provided additional results which were not measured by the experiment, such as the fluid pressure in the annulus and the dynamic motion of the cylinders. The fluid–structure interaction resulted in significant coupling of the outer and inner composite cylinders.

Keywords

Fluid-structure interaction Structural coupling by fluid Composite structure Impact loading 

1 Introduction

With composite materials advanced in the twentieth century, these materials have been used with growing interest in vehicles, planes, aircrafts and ships. The first vehicle to have a composite part was the Corvette C1 in 1950, and the airplane manufacturer Airbus S.A.S used composites for the rudder of their first plane family A300/A310 or the glass fiber hull in the Eurofighter Typhoon. The aerospace industry has consistently increased the usage percentage of composites because high specific strength and stiffness are important for aircrafts. On the other hand, the marine industry has also increased the usage of composite materials.

One of the unique environmental factors for marine applications is the structural interaction with water, mostly seawater. Water has approximately 800 times higher density than air. Thus, composites used for marine applications encounter greater drag forces than their flying counterparts from the fluid–structure interaction (FSI). This phenomenon gained great attention in 1940 when the Tacoma Narrows Bridge collapsed due to FSI. Airflow around the bridge induced vibrations on the bridge structure that hit resonance and led to failure.

There has been an extensive research on FSI. There are two groups of FSI problems depending on which medium initiates the FSI. In one group, the fluid medium starts the interaction with the structure. A good example of this group is the flow-induced vibration of pipes and pipe bundles (Blevins 1990; Paidoussis 1983; Weaver and Fitzpatrick 1988; Weaver et al. 2000). Another set of examples are vortex-induced vibration (Sarpkaya 2004; Williamson and Govardhan 2004; Bearman 2011; Zhang et al. 2018). The other group is the FSI problems excited by the solid medium, i.e. structures. Sloshing problems in storage tanks subjected to seismic loading belong to this group (Vathi et al. 2017; Matsui 2006). Other FSI studies considered mechanical dynamic loading to structures (Kwon et al. 2010, 2012, 2013, 2016a, b; Kwon 2011; Kwon and Plessas 2014).

While the vast majority of the FSI studies examined steel structures, a much smaller number of papers investigated FSI of polymer composite structures. Some of the studies on composite structures were published in the Refs. (Kwon et al. 2010, 2012, 2013, 2016a, b; Kwon 2011; Perotti et al. 2013; You and Inaba 2013; Kwon and Plessas 2014; Kwon and Bowling 2018; Bowling and Kwon 2018).

Very recently, coupling effects of composite structures via a fluid medium were studied (Kwon and Bowling 2018; Bowling and Kwon 2018]. This research considered two independent structures with water between them. As dynamic loading was applied to one structure, the motion of the structure resulted in pressure wave in water, which finally excited the other structure. Both experimental and numerical studies were conducted. The water level between the structures was also varied from no water to full water incrementally. The study showed the coupling effect depended on many parameters such as the stiffness of the structures, water level, the distance between the structures, and loading type.

Previous research was concentrated on flat composite plates while this research examined the coupling of curved composite structures. In other words, two same-length cylinders with different diameters were arranged concentrically with different water volumes in the annulus of the cylinders. Both experimental and numerical studies were conducted to supplement each other.

The next section describes the experimental set-up that was designed and fabricated for this study. Numerical modeling is then presented followed by discussion of results. Finally, conclusions are provided.

2 Experimental set-up

The experimental set-up used in this study is described here, and the final assembly is shown in Fig. 1. The set-up consists of two concentric horizontal cylinders with equal length. The outer cylinder has the inner diameter 88.9 mm (5.5 in.) while the inner cylinder has the inner diameter 76.2 mm (3.0 in.). Both cylinders were fabricated using the filament winding process with ± 45° layer orientations. Both cylinders are 1 mm thick. The composite cylinders were made of T700S fibers and ProSetM1002 resin. The material properties of the composite are given in Table 1.
Fig. 1

Assembled experimental set-up

Table 1

Material properties of composites made of T700S and ProSetM1002

EL (GPa)

ET (GPa)

GLT (GPa)

GTT (GPa)

ν LT

ν TT

ρ (kg/m3)

139

7.35

2.18

0.432

0.236

0.216

1485

Strain gages were attached to both cylinders. Since water was filled in the annulus between the two cylinders, strain gages were attached to the outer surface of the outer cylinder and to the inner surface of the inner cylinder. The span length of the cylinder was divided into four equal lengths as shown in Fig. 2, and strain gages were attached to the “middle” and “left” locations in the figure assuming the strains at the “right” location would be closer to those at the “left” location from symmetry. For the outer cylinder, four strain gages were used as shown in Fig. 2; front (i.e. the impact side), back, top, and bottom. However, the attachment of strain gages to the inner surface of the inner cylinder was difficult because of the small inner diameter. As a result, only three gages were attached to the middle section of the inner cylinder. Strains were measured along the hoop direction.
Fig. 2

Strain gage locations of outer and inner cylinders

Two cylinders were inserted into the grooves of the supporting structures as shown in Fig. 3. The supporting structures were fabricated using the 3-D printing technique with polycarbonate. Proper watertight sealing was done to prevent water leaking from the annulus of the two concentric cylinders. Then, the supporting structures were secured to the base as shown in Fig. 1.
Fig. 3

Support structures for composite cylinders

Impact loading was applied using a pendulum mechanism. The spherical shape of impactor made of steel had a load cell to measure the impact force, and its magnitude was controlled from the drop angle of the pendulum. The drop angle was measured from the vertical downward reference line. In other words, when the pendulum of 1.46 kg is freely hanging, the angle is zero. The drop angle varied such as 20°, 30° and 45°. The impactor was set to apply the force to the front side of the middle section as sketched in Fig. 2.

Throughout the paper, the following notations are used consistently. The mid-section of a cylinder is denoted by “M” while the left section was designated by “L” as far as the location along the longitudinal direction of the cylinder is concerned. Along the circumferential direction, “F” indicates the front (i.e. impact side), “T’ is for the top, “B” is for the bottom, and “P” is for the posterior side. Furthermore, OC is used for the outer cylinder and IC is used for the inner cylinder. As a result, any location of a cylinder is denoted using three symbols such as OC–M–F. The first symbol indicates the cylinder, the second one is for the axial direction and the third one is for the hoop direction. Thus, OC–M–F indicates the front side of the mid-section of the outer cylinder. One exception is IC–M–T–P. Because of the limited accessibility, the strain gage could not be attached to the top and posterior sides, respectively. As a result, one gage was attached between the top and posterior sides. Therefore, T–P suggests the middle location of the top and posterior sides. The water level in the annulus was measured by the percentage ratio of the water to the volume of the annulus. It was designated as “50 W” for the 50% water level. Finally, different drop angles are denoted by “20”, “30” and “45”, respectively.

3 Numerical model

A numerical analysis was conducted to supplement the experimental study qualitatively. Some parameters were not measured from the experiments, and those values could be obtained from the numerical study. Those parameters are transient fluid pressure, displacement, velocity, acceleration of the structures.

The two-way FSI was conducted using the ANSYS program (Kohnke 1999). The analysis consists of both structures and fluid coupled at their interface boundary surfaces where tractions and deformations should maintain their continuity. The ‘Transient Structural’ module was used for the structural part. Two composite cylinders were modelled using four-noded shell elements. The formulation of shell elements is very lengthy so that it is omitted here, but can be found in Ref. (Kwon and Bang 2000). Each composite cylinder has 5408 nodes and 5304 shell elements. Figure 4 shows two concentric cylinders with the nodal locations for mechanical transient loading. A triangular shape of force time-history was applied to the nodal points as plotted in Fig. 5. That is, the force increased linearly until 0.05 s. and decreased linearly until 0.1 s. This force was applied to the 16 nodal points of 3 × 3 shell elements located at the front center of the outer cylinder as shown in Fig. 4.
Fig. 4

Isometric view of the cylinders with the nodal impact location

Fig. 5

Profile of applied force to the numerical study

ANSYS CFX was utilized for modeling the water between the two cylinders. The incompressible Navier–Stokes equation is solved using CFX, whose equation is shown below:
$$\frac{{\partial \vec{u}}}{\partial t} + \left( {\vec{u} \cdot \nabla } \right)\vec{u} = - \frac{1}{\rho }\nabla p + \nu \nabla^{2} \vec{u},$$
(1)
where \(\vec{u}\) is the fluid velocity vector, \(\rho\) is the fluid density, \(p\) is the pressure, \(\nu\) is the kinematic viscosity, and \(t\) denotes time. The nominal properties of water at room temperature were used in the study.

Three different water volumes were considered between the cylinders: 0% (no water), 50% and 100% (full) water. When water was filled partially in the annulus, the remaining space was modelled as air. Conforming meshes were generated for both structures and fluid such that nodal points of the fluid and structures could match at their interface boundary surfaces. At the fluid–structure interfaces, equilibrium of loads and deformation compatibility were applied between the two media. Two different analyses for structures and fluid, respectively, were solved in the staggered manner and repeated until the interface conditions were satisfied for each time increment.

4 Results and discussion

4.1 Experimental results

The experiments were repeated ten times for each test case to check the repeatability of the test data. Figure 6 shows the impact force time-history with the 45° drop angle when the concentric cylinders had 50% water full in their annulus. The figure confirmed the repeatability such that every test resulted in the same impact force time-history. The same observation was made for strain time-responses. Because all the test data were very close one another, one representative data set was selected for comparison instead of averaging all ten data set.
Fig. 6

Impact force plot of ten test runs with filling level 50% and drop angle 45°

The annulus space between the two concentric cylinders was filled with water incrementally such as 25, 50, 75 and 100% of the space volume. Impact testing was conducted at each water level. The impact was controlled to have one contact for each impact test, i.e. no secondary contact. Time-histories of the impact forces are plotted in Fig. 7 with the 45° drop angle at different water amounts. The contact period between the impactor and the composite cylinder was approximately 16 ms regardless of the water level.
Fig. 7

Comparison of impact force with different water levels with drop angle 45°

The peak impact force is plotted in Fig. 8 as a function of the water level for three different drop angles. The figure shows that the impact force varies as a function of the water level even if the impact mass and its drop angle remained constant. This is because of the change in the overall mass and stiffness of the entire structure consisting of the two composite cylinders and the water in their annulus as the water level changes. The peak impact force decreased until the water level was 75%. This was more severe with the 45° drop angle. As the water became full in the annulus, the peak impact force increased. While the peak impact force was smaller with partially filled water compared to the no water case, the resultant strains in the outer composite cylinder were greater with the partially filled water. This is considered to be resulted from the added mass effect. When there is water in contact with a structure, the effect of the added mass decreased the vibrational frequency of the structure. The change of the vibrational frequency as a function of the water level is plotted in Fig. 9. The vibrational frequency was obtained from the fast Fourier transform (FFT) of the strain–time histories at each water level. The plot was normalized in terms of the natural frequency of the dry outer cylinder. As seen in the figure, the natural frequency decreased as the water level increased.
Fig. 8

Plot of peak impact force as a function of water level

Fig. 9

Plot of vibrational frequency as a function of water level

Considering a single degree of freedom mass–spring system subjected to a harmonic force, the magnitude of the vibrational displacement,\(d\) is expressed as
$$d = \frac{{d_{\text{st}} }}{{\left| {1 - \frac{\omega }{{\omega_{\text{n}} }}} \right|}},$$
(2)
where \(d_{st}\) is the static displacement of the spring for the given magnitude of the harmonic force, \(\omega\) is the excitation frequency of the applied force, and \(\omega_{n}\) is the natural frequency of the spring–mass system. For the current system, the natural frequency of the cylinder was much larger than the excitation frequency. As the natural frequency decreases with the added mass effect, the denominator of Eq. (2) becomes smaller, and subsequently this results in the larger displacement \(d\), or larger strain (Kwon et al. 2016b). In addition, 75% and 100% water cases had the peak forces reached quicker than other cases. In other words, the peak force is reached in a shorter period of time. This can also contribute to a larger displacement and strain as explained in Ref. (Kwon and Bowling 2018).
As expected, the smaller drop angle produced smaller peak impact forces. Neglecting friction, the ratios of the impact velocities are \(v_{20} /v_{45} = 0.206\) and \(v_{30} /v_{45} = 0.457\), respectively, in which subscript indicates the drop angle. Figure 10 compares the impact force–time histories of three different drop angles while the concentric cylinders are full of water. The ratios of the peak impact forces are \(P_{20} /P_{45} = 0.404\) and \(P_{30} /P_{45} = 0.532\). Therefore, there was no clear relationship between the impact velocity and the impact force.
Fig. 10

Comparison of forces at different drop angles with 100% water

When the drop angle was 20°, the impact force had a gradual change as a function of time without a clear peak force. However, as the drop angle was increased to 45°, the impact force had a very clear peak force. The case of the 30° drop angle was between the two other cases. Comparison of the contact time periods showed that the 20° drop angle had a shorter contact duration while the 45° drop angle had a longer contact duration. However, the difference among the contact durations was not significant. The FFT of the force time-histories showed that the frequency distribution of the impact forces resulting from three different drop angles was more or less similar.

Strain responses were compared at the front side of the outer cylinder (OC) at its mid-section with the 45° drop angle as seen in Fig. 11. The shape of the strain curve resembles that of the impact force as shown in Fig. 7. This suggests that the front-side strain is directly related to the impact force as expected. However, the maximum magnitude of strain is the largest with 50% water and the smallest with 100% water. This behavior is different from the peak impact force which was the greatest with 100% water. Certainly, the strain gage location at the front side is not the same as the impact location. This may cause such a discrepancy between the maximum strain case and the maximum force case. As soon as the contact between the impactor and the OC was over, the strain at the front side of OC diminished quickly. As the drop angle varied, there was no noticeable difference in the strain measured at the front side of OC other than the increase in magnitude as a function of the drop angle.
Fig. 11

Strains at the front side of OC at three different water levels and 45° drop angle

The posterior side of OC at the mid-section showed much more interesting strain responses. Figure 12 compares the strain time history for three different water levels when the drop angle was 45°. Strain responses are markedly different depending on the water level. While the strain variation was more gradual for the no water case, the strain showed more oscillatory behaviors with water. Even though the strain response was complex, it was very repeatable for the same test condition.
Fig. 12

Strains at the posterior side of OC at three different water levels and 45° drop angle

The 50% water case had the largest positive strain while the 100% water case had the greatest negative strain. The frequency spectrum of the strain responses in Fig. 12 is plotted in Fig. 13 using FFT. There is a change in the major vibrational frequencies depending on the water level.
Fig. 13

Frequency spectrum of strains at the posterior side of OC at three different water levels and 45° drop angle

Figure 14 compares the strains at OC–M–P with 100% water for three different drop angles 40°, 45° and 50°. The major frequency of the strain responses was the same for different drop angles. The strain magnitude increased generally along with the drop angle as expected. However, the 50° drop angle did not necessarily produce the largest strain at every peak points of the strain time-history plots.
Fig. 14

Strains at the posterior side of OC with 100% water level and different drop angles

Figure 15 compares the strains at the OC–M–F and IC–M–F. Strain gages were attached to the outer surface of the OC while they were attached to the inner surface of the IC. Therefore, if both OC and IC deformed in the same direction, either inward or outward, the strain gages would have the opposite signs. That is, one strain is positive and the other strain is negative. To avoid the confusion, the sign of the IC strains was changed such that the strain would indicate the outer surface of the IC. This is considered in Fig. 15 as well as all other plots including strains at IC.
Fig. 15

Comparison of strain responses at the front of the mid-section between IC and OC with 100% water and 45° drop angle

As impactor stroke the OC at the front side of the mid-section, the strain at OC went to compression. On the other hand, the strain at IC showed an early compression followed by a major tension. For the most of time, both cylinders move out of phase at the front side of the mid-section. The OC had a larger peak strain than the IC.

A similar plot is seen in Fig. 16 at the bottom side of the mid-section with 100% water and the 45° drop angle. The bottom sides had more oscillatory motions than the front side. Except for the time between 0.005 and 0.012 s approximately, both cylinders vibrated in phase motion for the most of time. Furthermore, the magnitude of the peak strain was larger for the IC than OC at the bottom side. At the posterior location of the left section, IC also had the larger absolute peak strain than OC as shown in Fig. 17.
Fig. 16

Comparison of strain responses at the bottom of the mid-section between IC and OC with 100% water and 45° drop angle

Fig. 17

Comparison of strain responses at the posterior of the left-section between IC and OC with 100% water and 45° drop angle

The absolute peak strains were plotted as a function of the water filling level for the drop angle of 45°. Figure 18 shows those at the OC while Fig. 19 is for the IC. Except for the M–F location, all other locations of the OC had the maximum strain with the 50% water fill. On the other hand, the IC had the maximum strains with the 50, 75 or 100% water level depending on the location.
Fig. 18

Maximum magnitude of strains at OC as a function of water fill level with 45° drop angle

Fig. 19

Maximum magnitude of strains at IC as a function of water fill level with 45° drop angle

4.2 Numerical results

First, the contours of water pressure were plotted at different times to examine how the pressure wave propagated through the water medium when there was 100% water. Figure 20 exhibits the pressure distribution of the middle cross-section along the longitudinal axis at two specific times. Figure 21 shows the deformation of the inner cylinder with 800 times magnification. The deformation plot indicates that the inner cylinder deforms in response to the water pressure propagation. That is, the inner cylinder had a greater deformation where there was high pressure in water. The pressure time histories at the four locations (front, posterior, top and bottom) are plotted in Fig. 22. The maximum pressure occurred at the front side followed by the posterior side.
Fig. 20

Water pressure contour plots at a time 0.0039 s and b time 0.0104 s

Fig. 21

800 times magnified deformed shape of inner cylinder at a time 0.0039 s and b time 0.0104 s

Fig. 22

Pressure time history with 100% water

The FFT plots of the numerical results are shown in Fig. 23 for two different water levels. The numerical results showed that the first major frequency decreased from the 50% water level to the 100% water level by approximately 50%. The experimental results gave an approximately 60% reduction. Considering the difference between the numerical and experimental models, the difference can be acceptable. For example, the experimental boundary condition is not fully clamped and the wall thickness is not uniform as assumed in the numerical analysis.
Fig. 23

Plots of strain magnitude vs. frequency of the numerical results of the inner cylinder with a 50% water and b 100% water

The displacements of representative positions of IC and OC are displayed. Figure 24 shows the displacements of the location at IC–M–P. It is noticeable that IC had a higher magnitude and frequency in the displacement at the filling level 50% when compared to the displacement history of the filling level 100%. Both 50% and 100% filling levels show harmonic responses.
Fig. 24

Displacement at IC–M–P at two different filling levels

The displacement at the position OC–M–P is seen in Fig. 25. Comparing the plots of various filling levels, it is recognizable that the displacement history of the filling level 50% illustrates a higher oscillation, the same as in Fig. 24 at IC–M–P. The time to reach a plateau at the filling level 50% took much longer compared to those at the filling levels 0% and 100%. Figure 26 plots the displacement time-history of IC–M–F. Here the trend continued with more oscillations at the filling level of 50% as noticeable in the previous figures. However, the displacement was greater with the 100% water.
Fig. 25

Displacement at OC–M–P at three different filling levels

Fig. 26

Displacement at IC–M–F at two different filling levels

Displacement at the position OC–M–F shows higher values compared to the previous displacement history as shown in Fig. 27. However, the biggest displacement occurred at filling level 50%. Unlike the plot of filling level 50%, the other two plots of filling levels 100% and 0% stabilized quicker.
Fig. 27

Displacement at OC–M–F at three different filling levels

Acceleration and velocity went hand in hand with deformations caused by impact loading. Therefore, accelerations are discussed here. In general, the acceleration of IC reached its highest value at the filling level 100%, while the acceleration is comparatively small at the filling level 50%, which can be seen in Fig. 28. The acceleration plots of OC, however, did not follow this trend, with greater oscillation at a filling level 50% throughout the OC–M–F and OC–M–P locations. As to the acceleration plots at filling levels 100% and 0%, they stabilized themselves sooner without a long-lasting oscillation, as seen in Fig. 29.
Fig. 28

Acceleration plots at a IC–M–F and b IC–M–P

Fig. 29

Acceleration plots at a OC–M–F and b OC–M–P

5 Conclusions

Structural coupling of two independent, concentric cylindrical shells via an internal fluid was studied numerically and experimentally. The composite cylinders were constructed using the filament winding technique, and the test set-up was designed and fabricated for this study. External impact loading was applied to the center of the outer cylinder. The dynamic response of both cylinders as well as the fluid pressure were either measured or calculated.

The results showed that the coupling effect was significant and dependent on the water amount between the two cylinders. The water level influenced the vibrational frequency of both outer and inner cylinders. As expected, the frequency decreased as the water level increased. However, the degree of reduction was different between the outer and inner cylinder.

While the same impact condition was used for the testing, the water level also affected the impact force time-histories. The peak impact force did not necessarily increase with the increase of the water level. The peak impact force was smallest with the 75% water level and largest with the 100% water level. Furthermore, the hoop strain at OC–M–F was largest for the 50% water level even though the peak impact force was not necessarily the largest at the 50% water level.

The strains at IC–M–F and OC–M–F with 100% water showed the opposite signs in their major responses. In other words, the outside surface of OC had the maximum compressive strain while the outside of IC had the maximum tensile strain. The outer cylinder had an approximately 20% larger strain than the inner cylinder. On the other hand, at the L–P location, the IC showed a larger strain than the OC.

The numerical study showed that the 50% water level resulted in larger displacements with higher oscillatory motions than 0% and 100% water levels at the locations of IC–M–P and OC–M–P. However, the displacement at IC–M–F was greater for the 100% water level, and the displacement at OC–M–F was the largest for the no water case. The inner cylinder had a small acceleration for the 50% water level and a much larger acceleration for the 100% water level, which was also comparable to that of the outer cylinder.

Both experimental and numerical studies demonstrated significant coupling effects of two concentric cylinders. As shown in a previous study (Bowling and Kwon 2018), the coupling effect would vary depending on the stiffness of each cylinder and their radial spacing. Additional studies will be conducted and reported later.

Notes

Acknowledgements

The technical assistance from Chanman Park and Jarema Didoszak is greatly appreciated. In addition, one of the authors (YW Kwon) acknowledges the financial support from the Solid Mechanics Program of the Office of Naval Research. Dr. Yapa Rajapakse is the program manager.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringNaval Postgraduate SchoolMontereyUSA
  2. 2.Helmut Schmidt UniversityHamburgGermany

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