Appl Compos Mater DOI 10.1007/s10443-017-9646-0
Numerical Evaluation of Dynamic Response for Flexible Composite Structures under Slamming Impact for Naval Applications O. H. Hassoon 1,2 & M. Tarfaoui 1 & A. El Moumen 1 & H. Benyahia 1 & M. Nachtane 1
Received: 6 September 2017 / Accepted: 11 September 2017 # Springer Science+Business Media B.V. 2017
Abstract The deformable composite structures subjected to water-entry impact can be caused a phenomenon called hydroelastic effect, which can modified the fluid flow and estimated hydrodynamic loads comparing with rigid body. This is considered very important for ship design engineers to predict the global and the local hydrodynamic loads. This paper presents a numerical model to simulate the slamming water impact of flexible composite panels using an explicit finite element method. In order to better describe the hydroelastic influence and mechanical properties, composite materials panels with different stiffness and under different impact velocities with deadrise angle of 100 have been studied. In the other hand, the inertia effect was observed in the early stage of the impact that relative to the loading rate. Simulation results have been indicated that the lower stiffness panel has a higher hydroelastic effect and becomes more important when decreasing of the deadrise angle and increasing the impact velocity. Finally, the simulation results were compared with the experimental data and the analytical approaches of the rigid body to describe the behavior of the hydroelastic influence. Keywords Composite panels . Flexible composites . Fluid-structure interaction . Hull slamming
1 Introduction Water entry problem is the impact between the free water surface and structures that considered one of critical design area in ship structures, which concentrated to determine the global and local hydrodynamic forces and pressures. The main early efforts in this area are intensified analytically in the case of rigid structure to predicate the hydrodynamic forces [1–3]. The fast * A. El Moumen
[email protected]
1
ENSTA Bretagne, FRE CNRS 3744, IRDL, F-29200 Brest, France
2
University of Technology, Baghdad, Iraq
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development of composite materials in the last decade motivates to exploit these materials in naval vessels structure, due to their lightweight, high strength and stiffness to density ratios. For these reasons, various researchers have been study their behavior to ensure the performance and reliability over the life time of these structures. The main difference between the rigid body and deformable structures is the presence of the hydroelastic influence along the fluid-structure interface, which can be generated a high hydrodynamic peak pressure during the impact duration. However, a deadrise angle between water and structure is considered as important factor to presence this phenomenon, especially with the small deadrise angle [4]. The hydroelastic effects consist of both dynamic and kinematic influence. The dynamic effects are happen due to the interaction between the water and structure, while the kinematic effect produced due to the inertia effect and the change of the local deadrise angle along fluid-structure interface [5]. Finite element method (FEM) and experiment investigations have been considered to determine the slamming impact problem for deformable structures and surround the damage types in the impacted structures. Various efforts are attempted to simulate the slamming impact using FEM. Peseux et al. [6] have been studied numerically and experimentally the slamming impact for both rigid and deformable structures. The authors have considered a cone shape samples and perform experimentally a series of tests with different deadrise angles and thickness. The variational formulations with FEM are implemented to solve Wagner problem. Huera and Gharib [7] have presented the experimental study of the slamming impact for flat plates with the free surface during the first phase of the slamming. Their study concentrated about the effect of the cushioning, using different velocities and a small deadrise angles. It was also noted there, that the air trapped phenomena increases with the angles less than 5°. Siyauan and Mahfuz [8] introduce the numerical simulation technique to study the fluid structure interaction (FSI) for the sandwich structure. They performed it, by coupling the finite element analysis (FEA) with the computational fluid dynamic (CFD) model, and used the global model to construct the composite and the foam of the sandwich structure. Panciroli et al. [9, 10] have investigated on the water impact problem for the deformable wedge using experimental and numerical approaches. Their results show that, under different boundary conditions, it appears that the hydroelastic influence depends highly on the ratio (R) between the natural frequency of the structure and the wetting time. Hydroelastic influence is neglected for a value of R lower than 50, since the same hydrodynamic pressure has been occurred when it’s compared with rigid bodies. Kevin et al. [11] have studied the impact of the wedge onto the clam water, coupling CFD and FEM, in order to predicate the hydroelastic response of the structure. This is done by determining the hydrodynamic pressure for the rigid body and then transforms these loads to the finite element structure model. A good agreement between theoretical methods and simulation models is useful when the objective for estimating the maximum deflections. Kaushik and Batra [12] have studied the water slamming of deformable sandwich panels using the Arbitrary Lagrangian Eulerian (ALE) formulation in LSDYNA software. All geometric nonlinearities are considered when determining panel deformations, accounted for inertia effects in the fluid and the structure, and examined delamination between the core and the face sheets. Stenius et al. [13] investigate on the hydroelastic influence of water entry problem for high-speed craft using FEM. The study consists of a systematic series of simulations of the structure behavior using different velocities impact, boundary conditions and deadrise angles. The simulation results of deformable structures are comparing with the reference solution as rigid/quasi-static simulations. De Rosis et al. [14] investigate on hydroelastic analysis of slamming problem coupling lattice Boltzmann and finite element methods. Numerical results are confronted to analytical and experimental findings for rigid and compliant wedges. A good
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agreement is observed between different approaches; namely analytical, experimental and numerical, demonstrating the feasibility to use numerical methods for slamming impact analyses. Stenius et al. [15] have used the numerical simulations for estimating the hydroelastic problem related to the panel-water impacts for high-speed craft. Experimentally, deflections and strains are compared with that corresponding to non-hydroelastic reference solutions. The finite element simulation determined for the quasi-static reference of rigid body, indicates that the hydroelastic effects are located close to the panel supports for the very flexible panels and sandwich constructions. Hassoon et al. [16, 17] presented experimental and numerical study of the dynamic behaviour of laminate composite and sandwich panels under constant velocity. They investigated the hydroelastic effects due to the flexibility of these structures on the hydrodynamic loads and the resulting deformations. Moreover, different damage modes were incorporated into numerical model using user-materials routine VUMAT to predict and understood the damage mechanism. In the present work, the numerical simulation model of the slamming impact for deformable laminate composite structures was developed. Due to the lightweight and high strength properties of composites compared to rigid structures. The numerical model has been performed based on the Eulerian- Lagrangian model built in Abaqus software, which is capable to simulate the fluid-structure interaction that causes large deformation in the fluid. The composite panels consist of vinylester resin reinforced with glass fibers. The pressure distribution, deflection of the panel, local deadrise angle and the deformation of panels are investigated as indicators to describe the hydroelastic effect during water-structure interaction.
2 Numerical Slamming Model 2.1 Characterization of Fluid Domain In slamming impact model, a fluid-structure interaction phenomenon requires to be solved simultaneously. Coupled Eulerian Lagrange (CEL) built-in Abaqus 14.6, which can be used more than one materials model that couples the Lagrangian and Eulerian model. Volume of Fluid (VOF) method is capable to solve a wide range of non-linear problems in fluid and solid mechanics [18], because it allows arbitrary large deformations and enables free surfaces to evolve, as shown in Fig. 1. The Eulerian was modeled as multi-materials (water and air) based on the equation of state (EOS) means that fluid is considered to be thermodynamically equilibrium at all time intervals. The EOS can be describing by linear Us - Up formulation of the Mie-Gruneisen equation of state, which can be exploited for the water entry problem. The pressure is determined as a function of the density (ρ) and the internal energy Em per unit mass as: P ¼ f ðρ; E m Þ
ð1Þ
U s ¼ C 0 −sU p
ð2Þ
where Us, Up, C0 and s = dUs/dUp, are the shock velocity, particle velocity, the velocity of the sound in the fluid and linear Hugoniot slope coefficient, respectively [19]. ρ0 C 0 2 μ Г 0μ P¼ 1− ð3Þ þ Г 0 ρ0 Em 2 ð1−sμÞ2
Appl Compos Mater
(a) Analytically slamming description
(b) CEL model based on Volume of Fluid method (VOF) Fig. 1 Slamming phenomenon
where μ = 1-(ρ0/ρ) is the nominal volumetric compressive strain, ρ0 is the initial density and Г0 is parameter of the approximation. Defined the water parameters in Table 1 with density of 1000 kg/m3, the pressure of the fluid can be calculated as: P ¼ ρ0 C 0 2 μ
ð4Þ
The fluid is considered as the Newtonian fluid flow, which means that the viscosity depends on the change of the temperature, the temperature is constant then viscosity is also constant in this model.
2.2 Fluid-Structure Interaction Algorithm The CEL formulation both the Eulerian and the Lagrangian equations of the model are depended on the same classical formulations of the conservation equations mass, energy and
Table 1 Mie-Gruneisen EOS Parameters C0 (m/s)
μ (−)
s (−)
Г0 (−)
ρ(kg/m3)
1420
0.001
0
0
1000
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momentum. To couple the response of Eulerian and Lagrangian as the Lagrangian mesh moves in interface of the Eulerian mesh, both the Eulerian and Lagrangian occupied initial boundary conditions. Consequently, the coupling calculated by apply pressure boundary conditions on the Lagrangian mesh from the Eulerian mesh, while the Lagrangian mesh boundaries supplied velocity boundary conditions on the Eulerian mesh. Based on the volume of the fluid (VOF), the Eulerian cell has assumes a constant pressure, since when it interacts with the Lagrangian domain the cell area will be change. The area centroid calculated by the Eulerian area that not intersects with Lagrangian and the reconstruction of pressure conditions. Thus, velocity conditions of the Lagrangian mesh are updated after calculation of the force from the Eulerian mesh. According to [20], the element volume is calculated by subtracting the Lagrangian volume from the volume of the Eulerian element. Once these quantities have been established, the stress and pressure are updated. The velocities of the uncovered parts of the element are updated in the normal manner. For more information concerning Abaqus CEL and a detailed description of the mathematical expressions, see [21, 22]. To prevent the Lagrangian body from overlap with the underlying Eulerian elements, the coupling force is calculated relative with the penetration rate based on the penalty coupling method, which working such as spring system attached in each of both node ends of the Lagrangian and Eulerian. 0
F ¼ k :d
ð5Þ
Where k′ and d representing the spring system stiffness and penetration respectively. The spring stiffness is depended on the scaling factor, bulk modulus of the fluid and the mesh size of the fluid as descripted in Eq. (6). 0
k ¼ pf
KA v
ð6Þ
Where pf, K , A and v are respectively the penalty scale factor, the bulk modulus, the average area of the structure element and the volume of the fluid element, which are in the coupling state.
2.3 Model, Meshing and Composite Specimens Generating Due to the symmetry of the wedge slamming problem, a 3D half model was created; the water domain is limited with dimensions (1.5 m × 1 m × 0.86 m. Due to a high consuming calculation time, water domain is divided in many regions. The 3D model is meshed with Fig. 2 Mesh convergence for the fluid domain, half model
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(a) Pressure distribution
(b) Pressure coefficient
Fig. 3 Mesh convergence for fluid domain, β =30° and V = 13 m/s
EC3D8R linear element Eulerian brick. The main effective parameter in the finite element method is the element mesh density. Therefore, mesh convergence was applied for the fluid domain which was close to the impact region that subjected to slamming impact of rigid body, using different mesh size of 0.5, 1, 1.5 and 2.5 mm, as mentioned in Fig. 2. The impact between composite panels and water domain was conducted for different mesh resolutions. It can be observed according to the amplitude and the oscillations of pressure as shown in Fig. 3a, that the best element mesh size was 0.5 mm with less noise and good agreement with analytical methods as well as the non-dimensional coefficient pressure Fig. 3b. In the first phase of the impact, some of the high frequency oscillations occurred due to some numerical penetration of the fluid in the body, which can be overcome by refining the mesh in the initial impact position. The total mesh of slamming model was composed of 1,078,000 cubic elements. It should be noted that the mesh have to be refined uniformly in all directions close to impact location between composite panels and fluid surface and it became coarse toward the extremes. The composite panel was meshed with 8-nodes linear reduced integration solid elements (C3D8R); each lamina has a solid element in the through-thickness direction. Figure 4 illustrates a schematic representation and mesh of 3D model for present simulations. Composite panels (495 × 250 mm) have clamped boundaries condition on two edges and free boundary conditions on the other, while the Eulerian fluid domain of exterior boundaries is defined as non-reflecting boundaries to avoid reflection pressure waves, as illustrated in Fig. 5. The glass-fiber vinylester composite panels with different thickness of 13 mm for semiflexible (SF) and 8 mm for flexible have been studied, analyzed and compared with rigid wedge. Steel panel with thickness of 13 mm and high stiffness (Young’s modulus of E = 210 GPa and Poisson ratio of ν =0.3) is considered as a rigid wedge. The hydroelasticity effects highly appears in impacted wedges with deadrise angle from 5 to 20°, while the 30° behaves like as rigid case [10]. Therefore, in this study, all of the composite panels are examined with deadrise angle of 10°. The strain gauges (SG) and the pressure sensors are located on the inside of wedge panels at three different positions along the span panel to cover mode shapes, Fig. 5b. The details of the panel mechanical properties are illustrated in the Table 2.
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(a) Tank and fixture system
(b) Composite panelin fixture system
(c) Mesh of model: composite panel (white color), fluid domain (green color) and fixture system (red color). Fig. 4 Schematic representation of full model and their FE mesh
3 Numerical Results 3.1 Performance of Composite Panels Subjected to Water Impact Firstly, one will be interested on effect of the structure flexibility on performance of panels. Therefore, two different thicknesses of composite panels are tested: semi-flexible and flexible. However, the hydroelastic structure response is affected by flexibility of panels, as the results, the water-structure boundary conditions have been changing due to variation in the local deadrise angle and the fluid flow as shown in Fig. 6. In the case of flexible panels, Fig. 7 shows more flexibility of the panel leading to an increase these effects. This can explain the reason behind high pressure near the chine due to the change in kinematic influence and loading of the fluid particles close to the interface. The numerical results of longitudinal strain indicate that the panel sustained to different vibrations modes, consequently a high local pressure can be happening. In the
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(a) Boundary conditions
(b) Panel dimensions and strain gauge location
Fig. 5 Model configuration and boundary conditions
Table 2 Considered composite panel properties with two thicknesses (8 mm and 13 mm) [17, 23] Elastic moduli (GPa)
Position ratios (−)
E11 E22 E33 ν12 ν13 48.16 11.21 11.21 0.274 0.274 Ultimate tensile and compression strengths (MPa) Xt Xc Yt Yc Zt 1021 978 29.5 171.8 15
10
Local deadrise angle (Degree)
Deflection (mm)
T=10 ms T=12 ms T=14 ms
6 4 2 0 -2
0
0.1
ν23 0.096 Zc 171.8
G12 G13 G23 4.42 4.42 9 Ultimate Shear strengths (MPa) S12 S13 S23 70 70 30
10.06
T=2 ms T=4 ms T=6 ms T=8 ms
8
Shear moduli (GPa)
0.2
0.3
0.4
Span of the plate (m)
(a) Panel deflexion
0.5
10.04 10.02 10
T=2 ms T=4 ms T=6 ms T=8 ms T=10 ms T=12 ms T=14 ms
9.98 9.96 9.94 9.92
0
0.1
0.2
0.3
0.4
Span of plate (m)
(b) Local deadrise angle
Fig. 6 Histories of kinematic effect on semi-flexible panels, for deadrise angle 10° and V = 6 m/s
0.5
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Local deadrise angle (Degree)
15
Deflection (mm)
10.15
T=2 ms T=4 ms T=6 ms T=8 ms T=10 ms T=12 ms T=14 ms
10
5
0
-5 0
0.1
0.2
0.3
0.4
10.1 10.05 10 9.95 9.9 9.85
0.5
T=2 ms T=4 ms T=6 ms T=8 ms T=10 ms T=12 ms T=14 ms 0
0.1
0.2
Span of the plate (m)
0.3
0.4
0.5
Span of plate (m)
(b) Local deadrise angle
(a) Panel deflexion
Fig. 7 Histories of kinematic effect on flexible panels for deadrise angle 10° and V = 6 m/s
initial time impact, the inertia effect of the panel dominates the structural response that being decelerated during the impact. For this reason, negative values of the strain have 2500
SG-low
2000
Deflection (mm)
SG-middle
Strain (µ )
SG-high
1500 1000 500 0 -500 -1000 0
5
10
9 8 7 6 5 4 3 2 1 0
15
0
0.1
0.2
0.3
0.4
0.5
Span of panel (m)
Time (ms)
(a) V= 4 m/s 5000
Strain (µ )
3000
Deflection (mm)
SG-low SG-middle SG-high
4000 2000 1000 0 -1000 -2000 -3000 0
5
10
15
20 18 16 14 12 10 8 6 4 2 0 0
0.1
0.2
0.3
0.4
Span of panel (m)
Time (ms)
(b) V= 6 m/s Fig. 8 Strain along span panel and max deflection for flexible composite panel
0.5
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1500
4
1000
3.5
Deflection (mm)
Strain (µ )
been seen. Increase of wetted area, the deflection dominates the hydroelastic influence leading to positive values of strain at the centre and the edges of the panel, as shown in Figs. 8 and 9. In order to understand the effect of vibration mode on performance of composite panels used in naval application, we consider the time history of maximum pressure for semi-flexible and flexible panels. Pressure-time histories along the panel-water interaction of semi-flexible panel at different velocities were compared to those of the rigid body. The results of semi-flexible composite panels were reported in Figs. 10 and 11 for velocity of 4 m/s and 6 m/s, respectively. From these figures, it appears that the inertia effect obvious in all pressure time histories corresponds to that of the rigid body. Pressure amplitude close to the keel has a lower value than the pressure for the rigid body, and continues to decrease especially in maximum panel deflection. This reduction was attributed to the change of kinematic conditions along the fluid-structure interface, increase of the local deadrise angle and decrease of the local impact velocity. In the position close to the chine, a high peak pressure is observed which occurred due to great variation in the kinematic effects. In the other hand, hydroelastic inertia effects of the panel weight are more appearance in high flexible structure response and increasing relatively with decrease the panel
500 0 -500 SG-low
-1000
SG-middle
-1500
SG-high
3 2.5
2 1.5 1 0.5
-2000
0
-2500 0
5
10
15
0
20
0.1
0.2
0.3
0.4
0.5
Span of panel (m)
Time (ms)
(a) V= 4 m/s 4000
Strain (µ )
3000
Deflection (mm)
SG-low SG-middle
2000
SG-high
1000 0 -1000 -2000 -3000 0
5
10
15
10 9 8 7 6 5 4 3 2 1 0 0
0.1
0.2
0.3
0.4
Span of panel (m)
Time (ms)
(b) V=6 m/s Fig. 9 Strain along span panel and max deflection for semi-flexible composite panel
0.5
900 800 700 600 500 400 300 200 100 0
pressure-low pressure-middle pressure-high
Pressure (kPa)
Pressure (kPa)
Appl Compos Mater 900 800 700 600 500 400 300 200 100 0
pressure-low pressure-middle pressure-high
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5
Time (ms)
Time (ms)
(b)
rigid body
Pressure (kPa)
(a)
900 800 700 600 500 400 300 200 100 0
semi-flexible
pressure-low pressure-middle pressure-high
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5
Time (ms)
(c)
flexible
Fig. 10 Pressure time histories, V = 4 m/s
stiffness. For the high flexible panel with lower stiffness bending, Figs. 10 and 11 show that a great hydroelastic effects are occurred. However, the pressure close to the chine edge has very high amplitude and exceeds the pressure in the rigid body which implies that a high hydroelastic influence has been appeared. Figure 12 illustrates snapshots of pressure distribution for flexible composite panels and rigid body after slamming impact. No deformation of panels was observed for the case of rigid body. Pressures close to the chine in both flexible and semi-flexible composite panels are compared with the theoretical approaches of Van Kármàn [1] and Wagner [2] and experimental results obtained by Battley and Allen [24] to identify the flexibility effects relative with impact velocity. Mathematical expressions of analytical approaches are given by Eqs. 7 and 8. The comparison of different approaches is presented in the Fig. 13. This figure shows a good correlation between experiment and numerical model. It appears also that the analytical model of Van Kármàn diverges compared with experimental and numerical results, especially for high velocity impact, while Wagner model provide a reasonable estimation. We have Analytical formula of maximum pressure proposed by:
&
Von Karman [1]: Pmax ¼ 0:5ρ V 2 π cotβ
ð7Þ
2000 1800 1600 1400 1200 1000 800 600 400 200 0
pressure- low pressure-Middle pressure- high
0
2.5
5
7.5
Pressure (kPa)
Pressure (kPa)
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10
12.5
2000 1800 1600 1400 1200 1000 800 600 400 200 0
15
pressure-low pressure-middle pressure-high
0
2.5
5
Time (ms)
10
12.5
15
(b) semi-flexible
rigid body
Pressure (kPa)
(a)
7.5
Time (ms)
2000 1800 1600 1400 1200 1000 800 600 400 200 0
pressure- low pressure- middle pressure- high
0
2.5
5
7.5
10
12.5
15
Time (ms)
(c) flexible Fig. 11 Pressure time histories, V = 6 m/s
&
Wagner [2]: Pmax ¼ 0:5ρ V 2
π2 tanβ 2
ð8Þ
3.2 Mechanical Response from Hydroelastic Effects The structural response has been analyzed with regard to stresses and hydrodynamic force acting on the panel. A high pressure in the end of the panel edge is regarded as significant source of local damage in composite panels due to normal and transverse shear stress. Hence, it is generally considered a critical region in the marine structure design. Stress concentration in the local positions can be caused different failure modes in the composite materials [25–27]. Figure 14 illustrates stress concentration induced by slamming impact on our composite panels. Repeating of the slamming impact leads to reduction of the material stiffness properties [28, 29]. As for the total hydrodynamic forces that obtained during the impact duration, more flexible panels have a great peak value of the force comparing with high stiffness panels. Allen and Battley [30] have recently investigated this situation experimentally. It can be seen that in the early impact stage, the time-force histories in different panels have the
Appl Compos Mater
(a) Flexible composite t= 12 ms
(b) Rigid body t= 10.2 ms Fig. 12 Pressure distribution after slamming impact
same profile, they consequently behave like a rigid body. In contrast, more divergence in their forms and amplitudes relative to the development of panel deflection toward the chine can be observed, as shown in Fig. 15. In brief, this situation of the force peak can
2000
Von Karman (1929) Wagner (1932) Experiment rigid-M. Battley (2012) Flexible panel Semi-flexible panel
Pressure (KPa)
1800
1600 1400 1200 1000 800 600 400 200 0 0
2
4
Velocity impact (m/s) Fig. 13 Maximum pressure close to the chine comparing with theoretical approaches
6
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(a) Flexible composite panel
(b) Semi-flexible composite panel Fig. 14 Concentration of the stress close to the chine edge, t = 12 ms
be interpreted as occurrence of greater presence of hydroelastic influence in low stiffness panels. Therefore, the hydroelastic effects were worked to increase the panel deflection. 30
70
Flexible panel Semi-flexible panel Rigid body
Flexible panel
60
Semi-flexible panel
Force (kN)
Force (kN)
25
Rigid body
20 15
10 5
50 40 30 20 10
0
0 0
5
10
15
20
Time (ms)
(a) impact velocity 4 m/s Fig. 15 Slamming force of the wedge
25
0
2.5
5
7.5
10
12.5 15
Time (ms)
(b) impact velocity 6 m/s
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4 Hydroelastic Coefficient and Non-Dimensional Slamming Force In order to show how the hydroelastic influence depends on the impact velocity, bending stiffness, the nondimensional parameter used to frame the impact velocities caused hydroelastic effects. Faltinsen [28] was introduced the nondimensional parameter as the ratio between the wetting time and the lowest natural frequency of the structure. He considered that the hydroelasticity effect become important for a wedge shaped cross section when: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . tanB ð9Þ V ρL3 EI ≤0:25 Where EI is bending stiffness of the structure, L is the length of the structure, ρ is the density of the fluid and β is the deadrise angle. This nondimensional parameter is dependent upon the change of the velocity, the relative deadrise angle and structural stiffness. In the other hand, Det Norske Veritas (DNV) classification societies [31] for structural requirements of naval vessel have been considered. The DNV classification requires that the allowable deflection of the panel not exceed 2% of the shortest panel span and for the strain: εallowable = 0.3 εultimate. Figure 16 shows the variation of the deflection vs a variety of impact velocities of simulation results in both flexible and semi-flexible composite 7000
35
6000
Hydroelastic influence
Strain (µ )
Deflection (mm)
DNV Deflection design limit
30 25 20 15 10 5
5000 4000 3000
1000
0
0 0
2
4
6
8
10
Velocity impact (m/s)
(a)
0
8000
DNV Deflection design limit Hydroelastic influence
30
2
4
6
8
10
Velocity impact (m/s) Flexible composite panel
35
DNV Max tensiel strain limit
7000
25
Strain (µ )
Deflection (mm)
DNV Max tensiel strain limit Hydroelastic influence
2000
20 15 10
Hydroelastic influence
6000 5000 4000 3000 2000
5
1000
0
0 0
2
4
6
8
Velocity impact (m/s)
10
0
2
4
6
8
Velocity impact (m/s)
(b) Semi-Flexible composite panel Fig. 16 Deflection design limit and strain respect with impact velocity for composite panel
10
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panels for 10° deadrise angle, respectively. In this figure, we can conclude on the impact velocity according to hydroelastic criteria and DNV classification. The hydroelasticity becomes important, when the impact velocity excesses of 4 m/s for flexible panels and 6 m/s for semi-flexible panels, as shown in Fig. 16. These results were also shown in [32, 33] for other composites. Generalize the slamming force of composite wedge related to the deadrise angle, a nondimensional slamming force (Cs) for the wedge shape with constant velocity was calculated as follows: Cs ¼
F 0:5 ρ V 2 A
ð10Þ
(A ¼ 2W * Vt=tanβÞ is the projection of the panel area on the water surface. The comparison of the non-slamming forces is mentioned in Fig. 17. The flexibility of the panel is an important factor in slamming loads as there is change in the local deadrise angle and the projection area of the panel on the water surface. Therefore, it can be observed that when the flexibility of the panel increases, the force coefficient value begins to deviate from that of more stiff panels. This helps the ship designer to take this coefficient into consideration throughout the design phase. Actually, this coefficient differs in deformable structures from that in rigid structures, according to the von Kármàn models (f(β) = π) and Wagner (f(β) = 3π/2) [1, 2].
5 Conclusions This study presents a numerical investigation of the water entry problem for deformable composite panels to analysis the behavior of hydroelastic effects, thus, assisting the ship designer predicts the global and local hydrodynamic loads. Multi-material Eulerian-Lagrangian model builds in finite element method (ABAQUS 6.14), which was exploited to represent the water domain that has more feasibility of modeling and allows high deformation in the fluid. Two different stiffness panels subjected to different impact velocities were studied. For further understanding of the hydroelastic influence, deflection and local strain in different Fig. 17 Comparison of the nondimensional slamming force (β = 10°) with different thicknesses and velocities
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locations along the span panel are analyzed. Firstly, the hydroelastic influence decomposes in two categories effects: inertia and the kinematic. Inertia effect has been detected in the early stage of the impact relative with loading rate that appears in lower impact velocity, and has longer wetting time than the first natural frequency of the structure. Kinematic effects along water-panel interface exhibit a larger hydroelastic influence due to structure flexibility and the change in local deadrise angle. Consequently, these effects can be reducing and increasing the panel response, which significantly appear in higher impact velocity, especially in the center and the end edges of the panel. Numerical critical pressure results (at chine) have been compared with experimental findings and analytical approaches for rigid wedge. Finally, structural response has been analyzed according to stresses and the total acting force on panels. Flexible panels have a great amplitude peak force and hydrodynamic pressure close to the chine relative with high stiffness panels. Observing that stresses concentration at the upper edge of the panel has a clear increasing in magnitudes, which represent the initial local failure in composite structures. The failure modes, based on the Hashin’s damage criteria for fibers and the Puck’s damage criteria for the matrix, will be addressed in the future works.
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