Evaluation of Vibration Characteristics of an Existing Harbor Caisson Structure Using Tugboat Impact Tests and Modal Analysis

Abstract

Ambient and forced vibration tests are proposed to evaluate dynamic characteristics of a caisson-type breakwater, including natural frequencies and modal damping ratios. The feasibility of a numerical analysis model with fluid-structure-soil interaction effects, which play an important role in evaluating structural performance and safety, was investigated by comparing the numerical results with experimental results. The Oryukdo breakwater in Busan, Republic of Korea, was employed as the target structure. This breakwater was once heightened by installing additional parapet structures of about 4 m tall to improve the harbor tranquility in 2005. Vibration tests were carried out in 2000 (before heightening) and in 2011 (after heightening). Most caissons were tested in the first test, while only one caisson was tested in 2011. It was found that natural frequencies were reduced by 1.7%–4.3% after heightening, and similar results were observed from the numerical analysis. It was also found that forced vibration tests can yield more reasonable results than ambient vibration tests. Even though there is some discrepancy between experimental and numerical results, numerical analysis can be carried out to analyze dynamic characteristics and evaluate structural performance and safety.

1. Introduction

In the recent years, strong typhoons (e.g., Typhoon Maemi in 2003 and Typhoon Bolaven in 2012) have slammed into the Korean Peninsula. Tremendous economic losses were incurred, including the severe damage to harbor structures. To protect harbor infrastructures and facilities from future typhoons and storms, design specifications for breakwaters should be revised to take into account gradually increasing extreme wind speed and wave height during typhoons due to climate change in the northeast Asian region [1]. Accordingly, existing structures designed to old design specifications need to be upgraded to meet the recently revised design codes based on performance evaluations.

It is possible to evaluate existing structures by carrying out numerical analysis with simulation models based on design drawings. However, it would be very difficult or infeasible to quantitatively evaluate structural performance when design drawings are unclear or when the structure is modified without documentation. In such cases, the structural performance can be evaluated using nondestructive techniques or loading tests, which can also be applied to assess the reusability of deteriorated facilities (e.g., armor blocks, tetra pods, and caissons) in harbor remodeling projects. Among available methods, visual inspection is a practical and widely applied means of inspecting caisson-type breakwaters and can be supplemented with nondestructive evaluation when necessary. Loading tests such as static and dynamic tests allow for loading conditions similar to service loading states and hence can more accurately evaluate structural performance. However, such tests require a lot of time and expense to be carried out and are very difficult or impossible to apply to large infrastructures like caisson-type breakwaters. Alternatively, ambient and forced vibration tests (AVTs and FVTs), which are actively applied to large inland infrastructures such as bridges and buildings, can be also applied for performance evaluations of caisson-type breakwaters.

Vibration tests on caisson-type breakwaters were previously reported by many researchers [2–5]. Boroschek et al. [6] evaluated a pile-supported breakwater using AVTs and FVTs to identify the dynamic characteristics of a pile-supported breakwater. Gao et al. [2] analyzed dynamic characteristics such as natural frequencies in the 0–30 Hz range for a newly constructed caisson-type breakwater in Shandong Province, China, using six tons of exciter, and identified two rigid body modes at about 3.8 Hz and 8.0 Hz. Lamberti and Martinelli [3] carried out FVTs using impact loading on two breakwaters in Genoa Voltri Port and Punta Riso Port in Italy using a sand bag of 2 tons and tugboats of 100 and 500 tons as a part of PROVERBS, a research project to develop reliability-based design guidelines. It was found that the estimated natural frequencies, that is, 1.4–1.8 Hz for the first mode and 2.3–3.6 Hz for the second mode, were very close to the numerically calculated values. Furthermore, it was observed that the horizontal and rotational modes are highly correlated. Boroschek et al. carried out similar vibration tests on a pile-supported breakwater in Ventanas Port, Chile, through pull-back tests (a type of FVTs) and AVTs. It was reported that clearer and more distinct peaks were observed from FVTs than AVTs, and the first and second natural frequencies were estimated to be around 1.6 Hz and 2.7 Hz, respectively. They also investigated the effects of the external loading level (i.e., pull-back force) on natural frequencies, damping ratios, and initial deflections.

In this study, vibration tests were performed on the caisson-type Oryukdo breakwater in Busan, Republic of Korea, which was heightened about 4 m by installing additional parapet structures to increase harbor tranquility. Two rounds of tests were carried out: in 2000 before heightening and in 2011 after heightening. The first tests were FVTs via tugboat and were conducted to investigate the structural safety of most of the caissons. The stochastic subspace identification method was applied to estimate natural frequencies and modal damping ratios of the caissons. The more recent vibration tests were carried out to investigate the effect of the additionally installed parapets. The practicality of the numerical analysis model was then investigated by comparing the analysis results with experimental results.

2. Theoretical Background

2.1. Experimental Modal Analysis Method

The stochastic subspace identification method utilizes the singular value decomposition of a block Hankel matrix with a cross-correlation matrix of responses. The fundamental basis is the stochastic state-space equation, which expresses the system dynamics under the stochastic random excitation as

$$\begin{array}{cc}& \mathbf{z}\left(k+\mathrm{1}\right)=\mathbf{A}\mathbf{z}\left(k\right)+\mathbf{w}\left(k\right),\\ \mathbf{y}\left(k\right)=\mathbf{C}\mathbf{z}\left(k\right)+\mathbf{v}\left(k\right),\end{array}$$

(1)

where $\mathbf{z}(k)$ and $\mathbf{y}(k)$ are the state and observation vectors at kth time step, respectively, and $\mathbf{w}(k)$ and $\mathbf{v}(k)$ are statistically uncorrelated Gaussian random vector sequences with zero means representing the process and measurement noises, respectively. A and C are the system and observation matrix, respectively. Then, the cross-correlation function $\mathbf{R}(k)$ can be calculated as

$$\begin{array}{cc}& \mathbf{R}\left(k\right)=E\left[\mathbf{y}\left(k+m\right)\mathbf{y}{\left(m\right)}^T\right]\\ =\mathbf{C}{\mathbf{A}}^{k-\mathrm{1}}E\left[\mathbf{z}\left(m+\mathrm{1}\right)\mathbf{y}{\left(m\right)}^T\right]\\ =\mathbf{C}{\mathbf{A}}^{k-\mathrm{1}}\mathbf{G},\end{array}$$

(2)

where $\mathbf{G}\triangleq E[\mathbf{z}(m+\mathrm{1})\mathbf{y}(m{)}^T]$ . Constructing the block Hankel matrix with the cross-correlation matrix $\mathbf{R}(k)$ , this block Hankel matrix ( ${\mathbf{H}}_{n_{\mathrm{1}},n_{\mathrm{2}}}$ ) can be decomposed into an observability matrix ( ${𝒪}_{n_{\mathrm{1}}}$ ) and an extended controllability matrix ( ${𝒞}_{n_{\mathrm{2}}}^{\text{ext}})$ as follows:

$$\begin{array}{cc}& {\mathbf{H}}_{n_{\mathrm{1}},n_{\mathrm{2}}}=\left[\begin{array}{ccc}{\mathbf{R}}_{\mathrm{1}}& \cdots & {\mathbf{R}}_{n_{\mathrm{2}}}\\ ⋮& \ddots & ⋮\\ {\mathbf{R}}_{n_{\mathrm{1}}}& \cdots & {\mathbf{R}}_{n_{\mathrm{1}}+n_{\mathrm{2}}-\mathrm{1}}\end{array}\right]\\ =\left[\begin{array}{ccc}\mathbf{C}\mathbf{G}& \cdots & \mathbf{C}{\mathbf{A}}^{n_{\mathrm{2}}-\mathrm{1}}\mathbf{G}\\ ⋮& \ddots & ⋮\\ \mathbf{C}{\mathbf{A}}^{n_{\mathrm{1}}-\mathrm{1}}\mathbf{G}& \cdots & \mathbf{C}{\mathbf{A}}^{n_{\mathrm{1}}+n_{\mathrm{2}}-\mathrm{2}}\mathbf{G}\end{array}\right]\\ =\left[\begin{array}{c}\mathbf{C}\\ ⋮\\ \mathbf{C}{\mathbf{A}}^{n_{\mathrm{1}}-\mathrm{1}}\end{array}\right]\left[\begin{array}{ccc}\mathbf{G}& \cdots & {\mathbf{A}}^{n_{\mathrm{2}}-\mathrm{1}}\mathbf{G}\end{array}\right]={𝒪}_{n_{\mathrm{1}}}{𝒞}_{n_{\mathrm{2}}}^{\text{ext}},\end{array}$$

(3)

where ${𝒪}_{n_{\mathrm{1}}}\triangleq {\left[\begin{array}{ccc}{\mathbf{C}}^T& \cdots & (\mathbf{C}{\mathbf{A}}^{n_{\mathrm{1}}-\mathrm{1}}\end{array}{)}^T\right]}^T$ , ${𝒞}_{n_{\mathrm{2}}}^{\text{ext}}\mathrm{\hspace{0.17em}\hspace{0.17em}}\triangleq \left[\begin{array}{ccc}\mathbf{G}& \cdots & {\mathbf{A}}^{n_{\mathrm{2}}-\mathrm{1}}\mathbf{G}\end{array}\right]$ , and $n_{\mathrm{1}}$ and $n_{\mathrm{2}}$ are the numbers of the cross-correlation matrix in rows and columns in the block Hankel matrix. Then, the system matrix A can be obtained using the upper ( $n_{\mathrm{1}}-\mathrm{1}$ ) block matrix deleting the last block row of ${𝒪}_{n_{\mathrm{1}}}$ and the lower ( $n_{\mathrm{1}}-\mathrm{1}$ ) block matrix of the upper shifted matrix by one block row as

$$\begin{array}{cc}& {𝒪}_{n_{\mathrm{1}}-\mathrm{1}}^{\uparrow }={𝒪}_{n_{\mathrm{1}}-\mathrm{1}}\mathbf{A},\end{array}$$

(4)

where ${𝒪}_{n_{\mathrm{1}}-\mathrm{1}}^{\uparrow }\triangleq {\left[\begin{array}{ccc}{(\mathbf{C}\mathbf{A})}^T& \cdots & {(\mathbf{C}{\mathbf{A}}^{n_{\mathrm{1}}-\mathrm{1}})}^T\end{array}\right]}^T$ , and ${𝒪}_{n_{\mathrm{1}}-\mathrm{1}}\triangleq {\left[\begin{array}{ccc}{\mathbf{C}}^T& \cdots & {(\mathbf{C}{\mathbf{A}}^{n_{\mathrm{1}}-\mathrm{2}})}^T\end{array}\right]}^T$ . The eigenvalues and vectors of the discrete system can be calculated from the eigenvalue decomposition of the system matrix A as follows:

$$\begin{array}{l}\mathbf{A}\mathrm{\Psi }=\mathrm{\Psi }\text{M}(\text{M}=\mathrm{diag}\left({\mu }_{\mathrm{1}},\dots ,{\mu }_N\right)\in {\mathbf{R}}^{N\times N},\\ \hspace{0.5em}\mathrm{\Psi }=\left[\begin{array}{lll}{\psi }_{\mathrm{1}}& \cdots & {\psi }_N\end{array}\right]\in {\mathbf{R}}^{N\times N}).\end{array}$$

(5)

Finally, the eigenvalue, modal damping ratio, natural frequency, and modal vector for the physical system can be obtained as follows:

$$\begin{array}{cc}& {\lambda }_k=\frac{\mathrm{1}}{\mathrm{\Delta }t}\ln {\mu }_k,\\ {\xi }_k=-\frac{\mathrm{Re}\left({\lambda }_k\right)}{\left|{\lambda }_k\right|},\\ {\omega }_k=-\frac{\mathrm{Im}\left({\lambda }_k\right)}{\sqrt{\mathrm{1}-{\xi }_k^{\mathrm{2}}}},\\ {\varphi }_k=\mathbf{C}{\psi }_k.\end{array}$$

(6)

2.2. Numerical Modal Analysis Method

Dynamic analysis with fluid-structure-soil interactions was carried out to analyze the dynamic characteristics of a caisson-type breakwater. In this method, the region of the structure and near-field soil is modeled using finite elements, and the far-field soil medium is modeled using infinite elements as shown in Figure 1. The fluid region is modeled with inviscid and incompressible fluid elements for the near-field region and viscous dampers at the interface between the near-field and far-field regions.

Figure 1

Modeling of fluid-structure-soil interaction system.

The dynamic stiffness matrix for structure and near-field soil medium ( ${\mathbf{S}}^{\text{SN}}(\omega )$ ) can be obtained using the stiffness matrix ( ${\mathbf{K}}^{\text{FE}}$ ) and mass matrix ( ${\mathbf{M}}^{\text{FE}}$ ) for 2D plane strain finite elements as follows:

$$\begin{array}{cc}& {\mathbf{S}}^{\text{SN}}\left(\omega \right)=\left(\mathrm{1}+i\mathrm{2}h\right){\mathbf{K}}^{\text{FE}}-{\omega }^{\mathrm{2}}{\mathbf{M}}^{\text{FE}},\end{array}$$

(7)

where $i=\sqrt{-\mathrm{1}}$ , h is the hysteretic damping ratio, and ω is the circular frequency. The modeling procedure for structure and near-field soil using finite elements is almost the same as the procedure used in finite element analysis except in the case of saturated soil. For saturated soil with a Poisson ratio of 0.5, the selective reduced integration technique was applied in this study to avoid the locking phenomenon on volumetric deflection and to simulate saturated soil behavior accurately. The dynamic stiffness matrix for far-field soil ( ${\tilde{\mathbf{S}}}^F(\omega )$ ) can be obtained using stiffness and mass matrices for dynamic infinite elements as follows [7–10]:

$$\begin{array}{cc}& {\tilde{\mathbf{S}}}^F\left(\omega \right)=\left(\mathrm{1}+i\mathrm{2}h\right){\tilde{\mathbf{K}}}^{\text{IE}}\left(\omega \right)-{\omega }^{\mathrm{2}}{\tilde{\mathbf{M}}}^{\text{IE}}\left(\omega \right),\end{array}$$

(8)

where the stiffness and mass matrices for dynamic infinite elements are obtained as follows [11]:

$$\begin{array}{cc}& {\tilde{\mathbf{K}}}^{\text{IE}}\left(\omega \right)={\iint }_{{\Omega }^{\left(e\right)}}^{}{\tilde{\mathbf{B}}}^T\left(\omega \right)\mathbf{D}\tilde{\mathbf{B}}\left(\omega \right)dx\hspace{0.17em}dy,\\ {\tilde{\mathbf{M}}}^{\text{IE}}\left(\omega \right)={\iint }_{{\Omega }^{\left(e\right)}}^{}{\tilde{\mathbf{N}}}^T\left(\omega \right)\rho \tilde{\mathbf{N}}\left(\omega \right)dx\hspace{0.17em}dy.\end{array}$$

(9)

In (9), $\tilde{\mathbf{B}}(\omega )$ and D are the strain-displacement and stress-strain matrices, respectively, and ρ and $\tilde{\mathbf{N}}(\omega )$ are the mass density and shape function vectors, respectively, for an infinite element with complex wave functions. In the far-field region, horizontally layered soil and infinite half-space soil were modeled using horizontal infinite elements (HIEs) and vertical infinite elements (VIEs), respectively. The soil medium at the corner was modeled using corner infinite elements (CIEs). Transitional horizontal infinite elements (THIEs) were also employed to satisfy the continuity of the displacement by linearly varying the wave numbers associated with body waves in the vertical direction. For modeling the fluid, the fluid was assumed to be shear-free, inviscid, and incompressible. Hence, it was modeled based on the following Navier equation [12]:

$$\begin{array}{cc}& K_f\nabla {\nabla }^T{\mathbf{u}}_f-{\rho }_f{\ddot{\mathbf{u}}}_f=\mathrm{0}\hspace{1em}\text{in}\hspace{0.17em}\hspace{0.17em}\Omega ,\end{array}$$

(10)

where ${\mathbf{u}}_f$ , ${\rho }_f$ , and $K_f$ are, respectively, the displacement vector, mass density, and bulk modulus of the fluid. Equation (10) can be transformed into the weak form using the natural and essential boundary conditions as follows:

$$\begin{array}{l}{\int }_{\Omega }^{}\left\{{\rho }_f{\mathbf{w}}^T{\ddot{\mathbf{u}}}_f+K_f{\left({\nabla }^T\mathbf{w}\right)}^T\left({\nabla }^T{\mathbf{u}}_f\right)\right\}d\Omega \\ \hspace{1em}+{\rho }_{f}g{\int }_{{\mathrm{\Gamma }}_f}^{}\left({\mathbf{w}}^T\mathbf{n}\right)\left({{\mathbf{n}}_g}^T{\mathbf{u}}_f\right)d\mathrm{\Gamma }\\ =-{\int }_{{\mathrm{\Gamma }}_p}^{}\left({\mathbf{w}}^T\mathbf{n}\right)\stackrel{-}{p}\hspace{0.17em}d\mathrm{\Gamma },\end{array}$$

(11)

where w and $\stackrel{-}{p}$ are the virtual displacement field and prescribed pressure at the interface ( ${\mathrm{\Gamma }}_p$ ), respectively; n and ${\mathbf{n}}_g$ are the direction vector normal to the boundary and the gravitational direction vector at the interface, respectively; and g denotes the gravitational acceleration. The matrix equation can be expressed by imposing variational formulation as follows:

$$\begin{array}{cc}& \mathbf{M}{\ddot{\mathbf{d}}}_f+\left(\mathbf{K}+\mathbf{S}\right){\mathbf{d}}_f=\mathbf{f},\end{array}$$

(12)

where M, K, and S are the mass matrix of fluid, stiffness matrix associated with volumetric deflection, and stiffness matrix against sloshing motion, respectively, and f denotes the external loading vector. Modeling the fluid using four-node quadrilateral displacement-type fluid elements, the stiffness and mass matrices of the fluid medium can be obtained by combining the constant-strain mode ( ${\mathbf{K}}_{\mathrm{0}}^e$ and ${\mathbf{M}}_{\mathrm{0}}^e$ ) and bending mode ( ${\mathbf{K}}_B^e$ and ${\mathbf{M}}_B^e$ ) as follows:

$$\begin{array}{cc}& {\mathbf{K}}^e={\mathbf{K}}_{\mathrm{0}}^e+{\mathbf{K}}_B^e,\hspace{1em}\hspace{1em}{\mathbf{M}}^e={\mathbf{M}}_{\mathrm{0}}^e+{\mathbf{M}}_B^e.\end{array}$$

(13)

Herein, because the bending mode can reduce the accuracy of the fluid element associated with volumetric deflection, (1 × 1)-reduced integration and mass projection techniques were applied for constructing stiffness and mass matrices, respectively, to resolve the accuracy problem. The spurious zero-energy modes in rotational displacement were solved by forcing a rotational penalty method with irrotational flow conditions. Finally, the stiffness and mass matrices for the four-node quadrilateral displacement-type fluid element can be obtained as follows [12]:

$$\begin{array}{cc}& {\mathbf{K}}^e={\mathbf{K}}_{\mathrm{0}}^e+{\mathbf{K}}_R^e,\hspace{1em}\hspace{1em}{\mathbf{M}}^e={\mathbf{M}}_{\mathrm{0}}^e+{\mathbf{M}}_R^e.\end{array}$$

(14)

By using the mass and stiffness matrices obtained through the previously mentioned formulation procedure, the equation of motion for the fluid-structure-soil interaction system can be expressed by the following:

$$\begin{array}{l}\left[\begin{array}{lllll}{\mathbf{S}}_{ff}\left(\omega \right)& {\mathbf{S}}_{fn}\left(\omega \right)& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ {\mathbf{S}}_{nf}\left(\omega \right)& {\mathbf{S}}_{nn}\left(\omega \right)& {\mathbf{S}}_{ns}\left(\omega \right)& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& {\mathbf{S}}_{ss}\left(\omega \right)& {\mathbf{S}}_{ff}\left(\omega \right)& {\mathbf{S}}_{si}\left(\omega \right)& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathbf{S}}_{is}\left(\omega \right)& {\mathbf{S}}_{ii}\left(\omega \right)+{\tilde{\mathbf{S}}}_{ii}\left(\omega \right)& {\tilde{\mathbf{S}}}_{ie}\left(\omega \right)\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& {\tilde{\mathbf{S}}}_{ei}\left(\omega \right)& {\tilde{\mathbf{S}}}_{ee}\left(\omega \right)\end{array}\right]\\ \hspace{1em}\times \left\{\begin{array}{l}{\mathbf{d}}_f\left(\omega \right)\\ {\mathbf{d}}_n\left(\omega \right)\\ {\mathbf{d}}_s\left(\omega \right)\\ {\mathbf{d}}_i\left(\omega \right)\\ {\mathbf{d}}_e\left(\omega \right)\end{array}\right\}=\left\{\begin{array}{l}\mathbf{0}\\ {\mathbf{f}}_n\left(\omega \right)\\ \mathbf{0}\\ \mathbf{0}\\ \mathbf{0}\end{array}\right\},\end{array}$$

(15)

where subscripts f, n, s, i, and e denote the fluid region, interface between the fluid and structure, region of the structure and near-field soil, interface between the near-field and far-field regions, and far-field region, respectively. $\mathbf{S}(\omega )$ and $\tilde{\mathbf{S}}(\omega )$ denote the dynamic stiffness matrices for finite elements and infinite elements, respectively, and ${\mathbf{f}}_n(\omega )$ represents the external loading vector. For simulating the radiational energy dissipation effect in the fluid region, the viscous damper was added at the interface between the near-field and far-field regions, and the damping parameter ( $C_p$ ) was calculated using the p-wave propagation velocity ( $V_p$ ), mass density (ρ), and projection area (A) as follows [13]:

$$\begin{array}{cc}& C_p=\rho V_{p}A.\end{array}$$

(16)

3. Dynamic Characteristics by Forced Vibration Tests

3.1. Layout of Target Structure and Experimental Setups

Vibration-based structural tests were carried out on the Oryukdo breakwater in Busan, Republic of Korea, in 2000, as a part of a structural safety assessment project on that breakwater. At that time, FVTs using impact loading applied by a tugboat were carried out on most of the caissons. Three successive caissons were combined into one test case, and 14 test cases were conducted as shown in Table 1. The center caisson in each group was excited with a tugboat, and impulse responses were measured using eight accelerometers, as shown in Figure 3(a). The caisson IDs and case numbers are indicated in Figure 2(a) and Table 1. The accelerometers were installed on the top surface of the cap concrete, as shown in Figure 3(a). Meanwhile, in 2005, the height of this breakwater was increased by constructing additional parapet structures of 4 m to protect naval facilities in the harbor. To investigate the effect of the additional parapet structures on the dynamic characteristics, AVTs and FVTs were performed in 2011 using the experimental setup shown in Figure 3(b). Details of the setup are described in Yoon et al. [14].

Table 1

Test cases and caisson IDs.

Test cases	Caisson IDs
Case  1	1, 2, 3
Case  2	7, 8, 9
Case  3	15, 16, 17
Case  4	19, 20, 21
Case  5	25, 26, 27
Case  6	31, 32, 33
Case  7	32, 33, 34
Case  8	33, 34, 35
Case  9	35, 36, 37
Case  10	36, 37, 38
Case  11	39, 40, 41
Case  12	41, 42, 43
Case  13	42, 43, 44
Case  14	44, 45, 46

Figure 2

Location of target breakwater and impact vibration test.

Figure 3

Experimental setups.

3.2. Vibration Tests in 2000 before Parapet Installation

Figures 4(a)–4(c) show impulse responses for Cases  3 and 4, and Figure 4(d) shows the stabilization chart for selecting stable modes and an appropriate model order [15] by stochastic subspace identification during the experimental modal analysis. From this stabilization chart, it can be seen that there are two clear and distinct modes around 1.5 Hz and 2.7 Hz. The estimated natural frequencies are listed in Table 2 along with the modal damping ratios.

Table 2

Experimental natural frequencies from vibration tests in 2000.

	Experimental natural frequency (Hz)		Caisson size and depth of sand-fill
	$f_1$	$f_2$	Size	Depth (m)
Case  1	1.982	3.720	A	11.5
Case  2	1.748	3.351	B	13.8
Case  3	1.499	2.835	B	14.4
Case  4	1.488	2.651	B	14.5
Case  5	1.477	2.672	B	13.7
Case  6	1.608	2.886	B	10.3
Case  7	1.586	2.811	B	10.3
Case  8	1.582	2.758	B	10.3
Case  9	1.590	2.781	B	10.3
Case  10	1.613	2.765	B	10.0
Case  11	1.695	2.972	B	7.5
Case  12	1.750	3.031	B	7.4
Case  13	1.752	3.510	B	7.4
Case  14	1.957	4.122	A	5.0

Figure 4

Measured impulse responses and stabilization chart.

From Table 2, it can be observed that the estimated natural frequencies are obviously reduced as the sand-fill layer is thickened as shown in Figure 5. It means that the structural system becomes more flexible as the thickness of the sand-fill layer increases. This is because of the larger flexibility of the sand-fill layer (shear wave velocity of about 250 m/s) compared to that of the bedrock (shear wave velocity of about 400 m/s). Figure 6 shows the estimated modal damping ratios. Although there is no clear relationship between the damping ratios and depth of the sand-fill layer, it can be observed that the modal damping ratios for the second mode are generally higher than those for the first mode. Furthermore, the estimated damping ratios are in the range of 4%–15%, which is relatively higher than the range for general concrete structures. This is attributed to the additional hydrodynamic damping effects owing to the adjacent sea water in this structural system.

Figure 5

Estimated natural frequencies and thickness of sand-fill.

Figure 6

Estimated modal damping ratios and thickness of sand-fill.

3.3. Vibration Tests in 2011 after Parapet Installation

Figures 7 and 8 show the measured acceleration responses and their corresponding stabilization charts for the 18th caisson structure, with the results of AVTs shown in Figure 7 and the results of FVTs shown in Figure 8. For the field works, wireless system is good to reduce installation time and cost [17]. Therefore, instead of the wired system used in the earlier test in 2000, wireless sensors were used in the tests. Seven high-sensitivity ICP-type accelerometers with a measurement range of ±0.5 g were used, and the acceleration responses were sampled at 100 Hz. Ambient vibration responses were also measured for three minutes after impact-based FVTs. The estimated dynamic characteristics are listed in Table 3.

Table 3

Experimental natural frequencies from vibration tests in 2011.

Modes	AVTs		FVTs
	Frequency (Hz)	Damping ratio (%)	Frequency (Hz)	Damping ratio (%)
1st	1.471	4.536	1.430	2.760
2nd	2.753	6.183	2.695	0.888

Figure 7

Measured responses and stabilization chart for AVTs.

Figure 8

Measured responses and stabilization chart for FVTs.

The natural frequencies for the first and second modes are estimated to be 1.471 Hz and 2.753 Hz, respectively, from AVTs; they are estimated to be 1.430 Hz and 2.695 Hz, respectively, from FVTs. From the results, it is observed that the estimated natural frequencies from FVTs are lower than those from AVTs by 2.1%–2.8%. This can be explained by considering the level of external forces: in the case of AVTs, the external force is relatively small, and, hence, the responses are also very small and the soil behaves relatively stiffer owing to the initial stiffness effects. Similar trends were found in the research by Boroschek et al. [6], who reported that the estimated natural frequencies from a pull-back test were 1.63 Hz and 2.74 Hz for the first and second modes, respectively, and that these values were lower than those from AVTs (1.68 Hz and 2.82 Hz) by 2.8%-2.9%. Considering these observations, it can be preliminarily concluded that the natural frequencies estimated from FVTs are slightly lower than those obtained from AVTs by about 2%-3%. Considering that strong typhoons generate high levels of external loads, FVTs can exert more realistic loading conditions. In other words, the structural stiffness can be overestimated by AVTs. Compared with the previous results obtained in 2000, the first and second natural frequencies decreased by about 1.7%–4.3% after heightening owing to the additional mass effects as shown in Table 4.

Table 4

Changes in natural frequencies after parapet installation.

	1st mode	2nd mode
Test in 2000 without parapet (Hz)	1.493	2.743
Test in 2011 with parapet (Hz)	1.430	2.695
Rate of change (%)	−4.30	−1.70

4. Numerical Modal Analysis Results

4.1. Layout of Numerical Analysis

Figure 9(a) shows the sectional dimension of the Oryukdo breakwater after it was heightened, and Figure 9(b) represents the corresponding numerical analysis model using finite, infinite, and viscous damper elements. This numerical analysis model built for the heightened structure was constructed by simply adding the installed parapet (circled in Figure 9(b)) to the model of the unheightened breakwater. In 2011, only one caisson (the 18th) was tested among the 46 total caissons for the purpose of this comparison study. It is noted that the sand-fill depth is 14.8 m, the region of the structure and near-field soil was modeled using 2D plane finite elements, the fluid region was modeled using quadrilateral four-node displacement-type fluid elements, and the horizontal infinite far-field soil region was modeled using 2D dynamic infinite elements. A viscous damper was also added for simulating the radiational energy dissipation.

Figure 9

Sectional dimension and numerical model of breakwater after parapet installation.

The material properties used in the numerical analysis are listed in Table 5, with general concrete properties, assigned to the cap concrete, caisson, and concrete block, and ordinary sea water properties used for the fluid region. The soil properties listed in Table 5 were assigned for the armor blocks, sand-fill, and infinite homogenous half-space region.

Table 5

Material properties of caisson and soil for numerical analysis.

	Elastic modulus (kN/m2)	Poisson's ratio	Density (×103 kg/m3)	Shear wave velocity (m/s)
Cap concrete	2.800 × 107	0.20	2.50	—
Caisson concrete	2.800 × 107	0.20	2.50	—
Sea water	1.967 × 106	0.49	1.02	—
Armor stone	2.800 × 106	0.20	2.00	—
Rubble stone	7.450 × 104	0.40	2.20	—
Sand-fill soil	—	0.43	2.00	250.0
Bedrock soil	—	0.45	2.40	400.0

4.2. Analysis Results

In this study, the KIESSI-2D software (a computer program for soil-structure interactions using finite and infinite element techniques) was used for forced vibration analysis on the breakwater structure considering fluid-structure-soil interactions [18]. The dynamic characteristics were analyzed using numerical models of the structure before and after parapet installation, and the results were compared with experimentally evaluated natural frequencies.

To excite this structural system, impulse loading was applied, where the loading spectrum has unit amplitude at all frequencies ( $\mathrm{1}(\omega )$ ) and the frequency response function (FRF) can be obtained by dividing the dynamic response ( $U(\omega )$ ) with the static response ( $U(\mathrm{0})$ ) as follows:

$$\begin{array}{cc}& H\left(\omega \right)=\frac{U\left(\omega \right)}{U\left(\mathrm{0}\right)}.\end{array}$$

(17)

The impulse loading was horizontally applied at the upper-left corner of the cap concrete from the inner harbor side (i.e., point A in Figure 9(b)), and the vertical and horizontal responses were obtained at the upper-right corner of the cap concrete in the outer harbor side (i.e., point B in Figure 9(b)). The displacement ( $u(t)$ ), velocity ( $\dot{u}(t)$ ), and acceleration ( $\ddot{u}(t)$ ) were obtained using the inverse Fourier transform of responses in the frequency domain.

Figures 10 and 11 show the calculated horizontal and vertical responses and the corresponding FRFs, respectively. The resonant frequencies obtained for the first and second modes are 1.514 Hz and 2.441 Hz, respectively, for the cases of the unheightened breakwater. The frequencies are 1.416 Hz and 2.344 Hz for the heightened case. Compared with the measured values shown in Table 6, the natural frequencies of the first mode are closely matched, while the second mode frequencies deviate by about 10%. Moreover, the frequency changes after structural heightening were 1.7%–4.3% in the experimental tests and about 4.6%–6.5% in the numerical tests. Hence, further studies are recommended for reducing discrepancies between experimental and numerical results to some extent.

Table 6

Experimental and numerical natural frequencies before and after parapet installation.

Modes	Vibration tests		Numerical analysis
	1st	2nd	1st	2nd
No parapet (Hz)	1.494	2.743	1.514	2.441
With parapet (Hz)	1.430	2.695	1.416	2.344
Change rate (%)	−4.3	−1.7	−6.5	−4.0

Figure 10

Impulse responses at upper-right corner of cap concrete by FSSI analysis (Dotted line: before installation, solid line: after installation).

Figure 11

FRFs of responses at upper-right corner of cap concrete by FSSI analysis (dotted line: before installation; solid line: after installation).

5. Conclusions

Dynamic characteristics were analyzed using experimental and numerical tests for the caisson-type Oryukdo breakwater, in Busan, Republic of Korea, and the practical applicability of a numerical analysis model was investigated by a comparison study. From the tests, it was found that the natural frequencies decrease as the sand-fill layer is thickened as a result of the lower shear wave velocity of sand-fill ( $V_s=250$  m/s) compared to that of bedrock ( $V_s=400$  m/s). Consequently, the entire structural system becomes more flexible as the sand-fill layer becomes thicker. Recently, AVTs and FVTs were also conducted to analyze dynamic characteristics after heightening by parapet installation. It was found that the natural frequencies estimated from FVTs are lower owing to the high level of external loads. Therefore, it is recommended that FVTs be applied when possible for more accurate evaluation of practical cases.

It was also found that the estimated natural frequencies after heightening are lesser by 1.7%–3.7% compared to those before heightening owing to the additional mass effect to the parapet structure, which is not affected by stiffness change. A similar trend was found from numerical analysis as well. However, there is some degree of discrepancy (about 2%-3%) between the experimental and numerical results. Hence, further studies are recommended to investigate the source of this discrepancy and identify effective remedies. It is noteworthy that the methods presented in this study can be applied in the global monitoring of damage such as scouring at the foundation of caisson-type structures. Moreover, vibration tests can be a good alternative to the visual inspections popularly performed by divers, which are generally time-consuming and dangerous.