The aim of this paper is to apply the elastic wave motion theory and the classical one-dimensional cavitation theory to analyze the response of a typical double-bottom structure subjected to underwater blast. The section-varying bar theory and the general acoustic impedance are introduced to get the simplified analytical models. The double-bottom structure is idealized by the basic unit of three substructures which include the simple panel, the panel with stiffener (T-shaped), and the panel associated with girder (I-shaped). According to the simplified models, the analytical models for the corresponding substructures are set up. By taking the cavitation effect into account, the process of fluid-structure interaction can be thoroughly understood, as well as the stress wave propagation. Good agreement between the analytical solution and the finite element prediction is achieved. On the other hand, the Taylor predictions for the panel associated with girder (I-shaped) including the effects of cavitation are invalid, indicating a potential field for the analytical method. The validated analytical models are used to determine the sensitivity of structure response to dimensionless geometric parameters
A considerable amount of literature exists on the feature of underwater explosion load and the dynamic response of ship structure subjected to underwater explosion [
First theoretical studies on FSI date back to World War II. Taylor [
The loads due to cavitation need to be taken into consideration. By treating water as a bilinear elastic medium, Kennard [
A significant part of the recent literature on blast loading has concentrated on the underwater blast loading of sandwich structure [
Although considerable efforts have been devoted to understanding the effects of FSI on the 1D response of monolithic plates and sandwich panels, it still remains unclear how the wave propagation and FSI affect the response of a complex structure such as double-bottom structures. Analytical models for study of the double-bottom structures response are rare. When analyzing a tanker grounding accident, the total response of the assembly is obtained through the summation of the responses of all structural members [
In this study, we examined the dynamic response of the free-standing double-bottom structure plates in contact with a liquid on one side to underwater explosion. We proposed analytical models taking into consideration the stress wave propagation in the structure, the cavitation effects in the fluid, and the corresponding effects on the plates. The general acoustic impedance [
The outline of the paper is as follows: in Section
A typical double-bottom structure of a naval ship subjected to underwater explosion is shown in Figure
Sketches of problem geometry for double-bottom structure: (a) typical double-bottom structure to underwater blast; (b) three substructures for the double-bottom structure.
The profiles for the panel, the panel with stiffener (T-shaped), and the panel with girder (I-shaped) are shown in Figure
Parameter for the substructures (unit: m).
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Simple panel | 0.01 | — | — | — | — | — |
Panel with stiffener (T-shaped) | 0.01 | 0.01 | — | 0.07 | 0.07 | — |
Panel with girder (I-shaped) | 0.01 | 0.01 | 0.01 | 0.07 | 0.8 | 0.07 |
The geometries of the three substructures used in this analysis.
Panel
T-shaped plate
I-shaped plate
Due to the varying cross section of the structure, the stress wave will be redistributed since a portion will be reflected and the other will be transmitted at the interface (see Figure
The wave reflection and transmission at the interface.
According to the force equilibrium and compatibility at the interface, we can obtain
In this paper, the general acoustic impedance [
Underwater explosions give rise to spherical shockwaves, traveling in water at approximately sonic speed and impinging on structures. At sufficient distance from the point of denotation, spherical shockwaves can be taken as one-dimensional planar wave. Assume that the double-bottom structure is loaded by a planar, exponentially decaying pressure wave with the peak pressure A steel panel plate with one medium (mediums A panel with stiffener (T-shaped) is treated as a structure with two mediums (mediums An panel with girder (I-shaped) is treated as a structure with three mediums (mediums
Sketches of geometry, reference system, and loading case for (a) the simple panel; (b) the panel with stiffener (T-shaped); (c) the panel with girder (I-shaped).
According to the reference system shown in Figure
In this paper, we assume that the water cavitation pressure is zero, in line with the assumptions of previous studies in this field [
We would like to reemphasize one important difference between our studying and that of Jin et al.’s work [
The wave propagation process in the panel: (a) the wave not transmitted into the steel plate (b) the wave transmitted into steel plate once or more.
Therefore, the reflection pressure on the wet surface can be expressed as
The value of
The wave propagation process in the panel with stiffener (T-shaped): (a)
Consider the wave propagation process sketched in Figure
We discuss this problem considering four scenarios (see Figure
The different cases of the input wave propagates in the panel with girder (I-shaped).
If
If
If
If
To sum up, the reflected pressure on the wet surface of the panel with girder (I-shaped) at the specific time
The pressure of the wave that transmits
As listed above, the plate velocity and pressure for three substructures prior to the onset of first cavitation can be summarized as
In this section, the analysis is performed with the assumption that the water response can be described by a linear-elastic, reversible pressure versus volumetric strain constitutive relation for
The pressure in the water between the reflected wave front and the wet face is
For
The cavitation time
If
If
The conditions for propagation of a breaking front in the water are given by Kennard [
According to Figure
In the first step,
In the second step,
In the third step,
Based on (
Schematic illustration of the phenomena of initial cavitation, emergence and propagation of breaking fronts, and development of a closing front.
As shown in Figure
A sketch of the boundary conditions in the analysis of free-standing panel with girder (I-shaped) impinged by a water blast.
Three-dimensional FE simulations were performed to provide more insight into fluid and structural response. For the corresponding substructures, we consider a homogeneous material with
The mesh of fluid and structure: (a) the water column; (b) the panel with stiffener (T-shaped); (c) the panel with girder (I-shaped).
In ABAQUS, by using a Mie-Gruneisen equation of state with a linear Hugoniot relation, we can model the water with the linear-elastic relation described in (
The water cavitation effects are realized by total wave formulation. We set the cavitation limit to be zero and the hydrostatic pressure field in the water column to be sufficiently small. A uniform pressure is imposed on the left surface of the water column. The impulses used all have peak pressure
In this section, we compare the analytical and FE predictions for fluid-structure interface pressure and velocity histories, as well as the motion of breaking and closing fronts, to assess the accuracy of the analytical models.
For simplicity, we define nondimensional parameters:
Figure
Analytical predictions and FE prediction of the spatial variation of (a) the time to cavitation and (b) the velocity
For the simple panel, cavitation initiates at a very small distance from the wet surface and gives rise to a breaking front traveling away from the interface, while the breaking front approaching the interface cannot be arrested before reaching the wet surface. For the panel with stiffener (T-shaped) and the panel with girder (I-shaped), the first cavitation is located within the fluid and the breaking front traveling towards the structure would reverse its motion and become the closing front. Figure
As shown in Figure
The expressions (
Figure
Nondimensional time histories of (a) nondimensional wet surface pressure and (b) nondimensional velocity for the simple panel.
To reveal the fluid-structure interaction of the double-bottom structure, the nondimensional time histories of wet surface pressure and velocity for the panel with stiffener (T-shaped) and the panel with girder (I-shaped) are shown in Figures
Nondimensional time histories of (a) nondimensional wet surface pressure and (b) nondimensional velocity for the panel with stiffener (T-shaped).
Nondimensional time histories of (a) nondimensional wet surface pressure and (b) nondimensional velocity for the panel with girder (I-shaped).
Note that the obvious drops in the analytical solutions indicate the superposition of the reflected wave from the rear surface of the structure and the incident wave. For the panel with stiffener (T-shaped), this reflected wave is the rarefaction wave which causes the great drop in the pressure and the decrement of decay constant (see Figure
In Figure
The intensity of loads impacted on the structure consequent to underwater explosion, as quantified by the specific impulse (per unit area) applied on the wet surface, is deduced by solving the following equation:
Define the maximum incident impulse
The analytical solutions for the nondimensional impulse
Analytical and FE predictions of normalized impulse.
Case | Analytical results | FE predictions | Error |
---|---|---|---|
Simple panel | 0.0681 | 0.0673 | 1.25% |
The panel with stiffener (T-shaped) | 0.1217 | 0.1184 | 2.71% |
The panel with girder (I-shaped) | 0.3987 | 0.3977 | 0.25% |
The analytical solutions and FE predictions of the nondimensional impulse for (a) simple panel; (b) the panel with stiffener (T-shaped); (c) the panel with girder (I-shaped).
Having established the accuracy of the analytical predictions, the analytical models are now used to determine the sensitivity of the plate response to the nondimensional parameters of
The nondimensional time histories of the nondimensional pressure
In Figure
Sensitivity of the nondimensional impulse
To investigate the effect of the single parameter
(a) The normalized time histories of the nondimensional pressure
Figure
The normalized time histories of the nondimensional pressure
By integrating the pressure histories, we can obtain the analytical maximum nondimensional impulse (see Figure
Sensitivity of the nondimensional impulse
As shown in Figure
(a) The normalized time histories of the nondimensional pressure
To investigate the effect of
(a) The normalized time histories of the nondimensional pressure
Taylor’s predictions for monolithic plates of equivalent mass are also shown in Figures
In Section
It is noted that the parameter
The sketch of the double-bottom structure cell subjected to underwater explosion.
To obtain the solution of (
Figure
Comparison of analytical solution and FE predictions of (a) the nondimensional maximum impulse and (b) the nondimensional impulse time histories.
By setting
In summary, the steps of the analytical method for an arbitrary double-bottom structure in this paper are as follows: the first step is to divide a double-bottom structure into several substructures according to the nondimensional parameters
In this paper, the response of double-bottom structure to underwater explosion has been investigated by analytical approaches. We have constructed and validated theoretical models for the substructures (the simple panel, the panel with stiffener (T-shaped), and the panel with girder (I-shaped)) loaded by planar, exponentially decaying underwater shock wave. Based on the dynamic response of the substructures, we established the approximate analytical models which are able to predict the response of the double-bottom structure to an underwater explosion. The shock propagation mechanism in the substructures, FSI effects, and cavitation phenomenon were considered in detail. According to the analysis, the main conclusions are as follows: The analytical models for substructures loaded by exponentially decaying water blast were validated by comparing their predictions to those obtained from fully coupled 3D FE simulations and good agreements were found. However, Taylor’s predictions for rigid plates of equivalent areal mass including the effects of cavitation fail to capture the reflected wave from the varying cross section of the structure. It shows a potential application field for the analytical method proposed in this paper. Writing the expression of dynamic response on the wet surface in nondimensional form shows that the response of substructures to underwater blast depends on three parameters, namely,
The major limitation of the analytical method in the present paper is the assumption that the structure is completely elastic, neglecting plasticity, buckling, and fracture. When a double-bottom structure subjected to underwater blast, the structure would suffer severe deformation. However, the effects of plasticity and buckling cannot be solved by this method. The spherical fashion of the reflected wave at the abrupt change in cross section and the propagation of flexural wave due to bending cannot be captured. Therefore, more detailed calculations of the plastic wave propagation in 2D and 3D need to be done to reveal the fluid-structure effect on a double-bottom subjected to underwater explosion. This will from part of our future research.
Despite these limitations, the current double-bottom model and analyses presented in this paper are useful in understanding the fluid-structure interaction, and the elastic stress wave propagation process in complex structures before the impulse cause the bending or the deformation of the whole structure.
Based on the wave motion theory, we propose analytical models for the simple panel, the panel with stiffener (T-shaped), and the panel associated with girder (I-shaped) to study underwater explosion problem. Based on the analysis of substructures, we also propose the approximate analytical models for a typical double-bottom structure to investigate underwater explosion problem. The cavitation effect is taken into consideration to reveal the fluid-structure interaction. The geometric parameter
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank Dr. Christiaan Adika Adenya for checking the grammar in this paper. The authors also gratefully acknowledge the financial support provided by the National Natural Science Foundation of China (U1430236, 51479041, 51279038, and 51509228).