A New Plastic Design Approach for the Vertical-Side-Plating Thickness of Ice-Strengthened Ships Suffering from Ice Floe Impacts
Abstract
:1. Introduction
2. Theoretical Analysis Model for the Dynamic Response of a Ship Plate under Ice Floe Impact
2.1. Introduction to the Proposed Theoretical Model
- -
- Head-on impacts of an ice floe with a vertical plate structure;
- -
- Ignoring the effects of ship motion and ice rotation during impact;
- -
- The assumed boundary condition of plate structure is fully clamped.
2.2. Theoretical Model Verification
- Rigid-mass impact
- Ice impact
2.3. Comparative Analysis of Theoretical Results of Ice Impact and Rigid-Mass Impact
3. Design Formula for Plating Thickness under Ice-Impact Load
4. Plating-Thickness Design Cases
4.1. Plating-Thickness Design Procedure
4.2. Design Curves under Different Design Parameters and Cases
4.2.1. Case 1. Different Ice Strengths (P–A Curves)
4.2.2. Case 2. Different Allowable-Permanent Set Parameters Cw
4.2.3. Case 3. Influences of Ice Shapes (θ, h) and Impact Energy
4.2.4. Case 4. Comparison of Plating-Thickness Designs for Ice Impact and Rigid-Mass Impact
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
An, A | Contact area and integral of the ship-plate-deformation area, respectively |
a1, a2, a3, b1, b2 | Constants related to plate size, ice height, and ice damage parameters |
B | Width of the ship plate |
b | Length of short side of the panel |
C | Parameter related to the ice strength and the ice belt region |
Cw | Allowable-permanent set parameter |
E | Young’s modulus of the plate material |
EARF | Energy-absorption reduction factor |
E0 | Initial total impact energy |
EI | Dissipated energy of the ice damage |
Ep | Plastic-strain energy or absorbed energy of the ship plate |
Epi, Eps | Plastic-strain energy of the ship plate under ice impact and rigid-mass impact, respectively |
Er | Rebound-kinetic energy in the rebound process |
ex | Constant |
FA | Coefficient related to the ice belt |
Fm | Maximum impact force |
H | Plating thickness |
h | Ice thickness |
L | Length of the ship plate |
Ld | Length of middle plastic hinge line |
l | Length of long side of the panel |
lm | Length of plastic hinge line |
M0, N0 | Ultimate plastic bending moment and ultimate plastic membrane force per unit length, respectively |
Mp, Np | Bending moment and membrane force of the plastic hinge line per unit length, respectively |
m0, mr | Initial ice mass and residual mass after ice impact, respectively |
P | Average pressure of ice in the process of impact extrusion |
P0 | Pressure with a contact area of 1 m2 |
PDRF | Plastic deformation reduction factor |
s | The number of plastic hinge lines |
V0 | Initial impact velocity of the ice floe and structure |
V0′ | Initial net impact velocity |
V0i′, V0s′ | Initial net impact velocities under ice impact and rigid-mass impact, respectively |
Vi | Loss velocity parameter caused by ice damage |
Vr | Rebound velocity in the rebound process |
w | Plastic deformation of the plate |
wc | Plastic deformation at the center of the plate |
wm | Maximum plastic deformation |
wpi,wps | Plastic deformation of ice impact and rigid-mass impact, respectively |
ws | Allowable-permanent set |
α | Coefficient of restitution |
αi, αs | Coefficients of restitution of ice impact and rigid-mass impact, respectively |
β | Parameter related to linear density |
δ, δm | Ice-crushing length and final ice-crushing length, respectively |
φ1, φ2 | Dimensionless coefficients related to the dimensions of the ship plate and the length of the middle hinge line |
η2, γ | Intermediate values |
μ | Unit area density of the plate |
θ | Front-end angle of the wedge-shaped ice |
Relative angular velocity between adjacent rigid plates | |
ρ | Density of the plate |
σice | Nominal ice strength |
σs | Yield strength of the plate material |
ω | Coefficient of ice damage along the height direction |
ξ | Coefficient related to P0, m0, h, θ and ex |
ζ | Value of square yield surface |
Appendix A
Appenidx A.1. Case1. Infuences of the Ice Strengths (P–A Curves)
V0 (m/s) | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | 5.5 | 6.0 | 6.5 | 7.0 | 7.5 | 8.0 |
---|---|---|---|---|---|---|---|---|---|---|---|
16.5 | 17.6 | 18.5 | 19.3 | 20.1 | 20.8 | 21.4 | 22.1 | 22.7 | 23.2 | 23.8 | |
21.1 | 22.5 | 23.8 | 24.9 | 26.0 | 27.0 | 27.9 | 28.8 | 29.7 | 30.5 | 31.2 | |
21.0 | 22.6 | 24.1 | 25.5 | 26.8 | 28.0 | 29.1 | 30.2 | 31.2 | 32.2 | 33.1 | |
28.6 | 30.4 | 31.9 | 33.4 | 34.7 | 35.9 | 37.0 | 38.1 | 39.1 | 40.1 | 41.0 | |
37.0 | 39.6 | 41.9 | 44.0 | 45.9 | 47.7 | 49.4 | 50.9 | 52.4 | 53.8 | 55.1 | |
29.0 | 30.9 | 32.7 | 34.3 | 35.7 | 37.1 | 38.4 | 39.6 | 40.7 | 41.8 | 42.9 | |
42.7 | 44.9 | 46.8 | 48.5 | 50.0 | 51.4 | 52.6 | 53.8 | 54.9 | 55.9 | 56.9 | |
29.9 | 32.1 | 34.1 | 36.0 | 37.7 | 39.4 | 40.9 | 42.3 | 43.7 | 45.0 | 46.3 |
Appenidx A.2. Case2. Influence of the Allowable-Permanent-Set Parameter Cw
V0 (m/s) | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | 5.5 | 6.0 | 6.5 | 7.0 | 7.5 | 8.0 |
---|---|---|---|---|---|---|---|---|---|---|---|
Cw = 0.1 | 24.2 | 25.6 | 26.9 | 28.0 | 29.1 | 30. 1 | 31.0 | 31.9 | 32.8 | 33.6 | 34.3 |
Cw = 0.2 | 22.6 | 24.0 | 25.3 | 26.4 | 27.5 | 28.5 | 29.4 | 30.3 | 31.2 | 32.0 | 32.8 |
Cw = 0.3 | 21.1 | 22.5 | 23.8 | 24.9 | 26.0 | 27.0 | 27.9 | 28.8 | 29.7 | 30.5 | 31.2 |
Cw = 0.4 | 19.7 | 21.1 | 22.4 | 23.5 | 24.6 | 25.6 | 26.5 | 27.4 | 28.2 | 29.0 | 29.8 |
Cw = 0.5 | 18.4 | 19.8 | 21.0 | 22.2 | 23.3 | 24.2 | 25.2 | 26.1 | 26.9 | 27.7 | 28.5 |
Appenidx A.3. Case3. Influences of Ice Shapes (θ, h) and Impact Energy
V0 (m/s) | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | 5.5 | 6.0 | 6.5 | 7.0 | 7.5 | 8.0 |
---|---|---|---|---|---|---|---|---|---|---|---|
θ = 60° | 17.0 | 18.1 | 19.2 | 20.1 | 21.0 | 21.8 | 22.6 | 23.3 | 24.0 | 24.7 | 25.3 |
θ = 90° | 18.4 | 19.8 | 21.0 | 22.2 | 23.3 | 24.2 | 25.2 | 26.1 | 26.9 | 27.7 | 28.5 |
θ = 120° | 21.1 | 22.5 | 23.8 | 24.9 | 26.0 | 27.0 | 27.9 | 28.8 | 29.7 | 30.5 | 31.2 |
θ = 150° | 24.4 | 26.0 | 27. 5 | 28.8 | 30.0 | 31.2 | 32.3 | 33.3 | 34.3 | 35.2 | 36.1 |
V0 (m/s) | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | 5.5 | 6.0 | 6.5 | 7.0 | 7.5 | 8.0 |
---|---|---|---|---|---|---|---|---|---|---|---|
h = 0.54 m | 19.2 | 20.5 | 21.7 | 22.7 | 23.7 | 24.6 | 25.5 | 26.3 | 27.1 | 27.8 | 28.6 |
h = 0.64 m | 20.2 | 21.6 | 22.8 | 23.9 | 25.0 | 25.9 | 26.8 | 27.7 | 28.5 | 29.3 | 30.0 |
h = 0.74 m | 21.1 | 22.5 | 23.8 | 24.9 | 26.0 | 27.0 | 27.9 | 28.8 | 29.7 | 30.5 | 31.2 |
h = 0.84 m | 21.8 | 23.2 | 24.5 | 25.7 | 26.8 | 27.9 | 28.8 | 29.7 | 30.6 | 31.4 | 32.2 |
h = 0.94 m | 22.3 | 23.8 | 25.1 | 26.4 | 27.5 | 28.5 | 29.5 | 30.5 | 31.4 | 32.2 | 33.0 |
h = 1.04 m | 22.7 | 24.2 | 25.6 | 26.8 | 28.0 | 29.0 | 30.1 | 31.0 | 31.9 | 32.8 | 33.6 |
V0 (m/s) | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | 5.5 | 6.0 | 6.5 | 7.0 | 7.5 | 8.0 |
---|---|---|---|---|---|---|---|---|---|---|---|
m0 = 1000 kg | 11.1 | 12.1 | 13.0 | 13.8 | 14.6 | 15.3 | 15.9 | 16.5 | 17.1 | 17.7 | 18.2 |
m0 = 5000 kg | 16.6 | 17.8 | 18.9 | 19.9 | 20.8 | 21.6 | 22.4 | 23.2 | 23.9 | 24.6 | 25.2 |
m0 = 10,000 kg | 19.4 | 20.7 | 21.9 | 23.0 | 24.0 | 24.9 | 25.8 | 26.6 | 27.4 | 28.2 | 28.9 |
m0 = 15,000 kg | 21.1 | 22.5 | 23.8 | 24.9 | 26.0 | 27.0 | 27.9 | 28.8 | 29.7 | 30.5 | 31.2 |
m0 = 20,000 kg | 22.4 | 23.8 | 25.2 | 26.4 | 27.5 | 28.6 | 29.5 | 30.5 | 31.4 | 32.2 | 33.0 |
m0 = 25,000 kg | 23.4 | 24.9 | 26.3 | 27.6 | 28.7 | 29.8 | 30.8 | 31.8 | 32.7 | 33.6 | 34.4 |
m0 = 30,000 kg | 24.3 | 25.9 | 27.3 | 28.6 | 29.8 | 30.9 | 32.0 | 32.9 | 33.9 | 34.8 | 35.6 |
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Ice Descriptions | Polar Class | |||||
---|---|---|---|---|---|---|
IACS [1] | DNV [2] | RMRS [3] | LR [4] | FSICR [5] | CCS [6] | |
Summer/autumn operation in medium first-year ice that may include old ice inclusions | PC5,6 | ICE-10 | Arc5 | 1AS | IA Super | B1* |
Summer/autumn operation in thin first-year ice that may include old ice inclusions | PC7 | ICE-05 | Arc4 | 1A | IA | B1 |
Test | Material | Plate Thickness H [mm] | Density [kg/m3] | Young’s Modulus E [GPa] | Poisson’s Ratio | Yield Strength [MPa] |
---|---|---|---|---|---|---|
Rigid-mass impact | steel | 6 | 7800 | 206 | 0.3 | 253 |
Ice impact | Steel | 2 | 7850 | 206 | 0.3 | 253 |
Test Number | Plate Specimen L × B × H [mm] | Impact Mass m0 [kg] | Initial Impact Velocity V0 [m/s] | Rebound Velocity Vr [m/s] | Vr/V0 | wcExp. [mm] | wcAna. [mm] | Relative Error (wcAna. − wcExp.)/wcExp. × 100% |
---|---|---|---|---|---|---|---|---|
1 | 400 × 400 × 2 | 93 | 1.73 | 0.89 | 0.51 | 7.00 | 8.14 | 16.26% |
2 | 800 × 400 × 4 | 90 | 3.19 | 1.56 | 0.49 | 7.06 | 7.83 | 10.90% |
Test Number | Plate Specimen L × B × H [mm] | Impact Mass m0 [kg] | Initial Impact Velocity V0 [m/s] | Vr/V0 | wcExp. [mm] | wcAna. [mm] | Relative Error (wcAna. − wcExp.)/wcExp. × 100% |
---|---|---|---|---|---|---|---|
1 | 800 × 400 × 2 | 99.21 | 3.42 | 0.20 | 11.53 | 12.70 | 10.14% |
2 | 800 × 400 × 2 | 98.99 | 3.69 | 0.15 | 12.20 | 13.35 | 9.42% |
Material | Density [kg/m3] | Young’s Modulus E [GPa] | Poisson’s Ratio | Yield Strength [MPa] |
---|---|---|---|---|
Q390D steel | 7800 | 206 | 0.3 | 390 |
Source | Equation Number | P–A Relationship | Descriptions |
---|---|---|---|
Timco and Sudom (2013) [43] | Eq. a | The data are derived from the measurement data of the Norströmsgrund lighthouse colliding with the ice floe. The lighthouse is a cylindrical concrete structure with a diameter of 7.58 m at the water line. The ice average thickness is 0.42 m. | |
DNV rules [2] | Eq. b | It is suitable for the mid-side structure of a ship with ice class of ICE-05 and ICE-10 when the value range of C is 1.4~1.95. | |
Timco and Sudom (2013) [43] | Eq. c | The majority of the data points were obtained from the Molikpaq caisson structure from interactions with first-year sea ice. Molikpaq consists of a continuous steel annulus on which sits a self-contained deck structure. The core of the annulus was filled with sand. | |
Ritch et al. (2008) [7] | Eq. d | The data were derived from the collision of the Terry Fox icebreaker with the B14 and B17 icebergs. The B 14 mass is 1900 t and the B 17 mass is 8500 t. | |
API RP 2N [45] | Eq. e | The data were derived from a combination of medium scale impact tests, dedicated ship ramming tests and measurements taken from ice load panels on the Molikpaq. Most of the ice is multi-year ice. | |
Masterson and Frederking (2007) [41] | Eq. f | First-year ice and more temperature areas. | |
ISO 19906 [46] | Eq. g | On the basis of Eq. e, the data from ship-ramming tests were deleted, and the Molikpaq data 1985 to 1986 were added. Multi-year ice was used. | |
Masterson and Frederking (2007) [41] | Eq. h | The original ice pressures were used for the design of the Molikpaq platform. Multi-year ice was used. |
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Mu, M.; Guo, K.; Cai, W.; Zhu, L.; Pi, Z.; Zhou, S. A New Plastic Design Approach for the Vertical-Side-Plating Thickness of Ice-Strengthened Ships Suffering from Ice Floe Impacts. J. Mar. Sci. Eng. 2024, 12, 233. https://doi.org/10.3390/jmse12020233
Mu M, Guo K, Cai W, Zhu L, Pi Z, Zhou S. A New Plastic Design Approach for the Vertical-Side-Plating Thickness of Ice-Strengthened Ships Suffering from Ice Floe Impacts. Journal of Marine Science and Engineering. 2024; 12(2):233. https://doi.org/10.3390/jmse12020233
Chicago/Turabian StyleMu, Mengying, Kailing Guo, Wei Cai, Ling Zhu, Zhenyu Pi, and Shuo Zhou. 2024. "A New Plastic Design Approach for the Vertical-Side-Plating Thickness of Ice-Strengthened Ships Suffering from Ice Floe Impacts" Journal of Marine Science and Engineering 12, no. 2: 233. https://doi.org/10.3390/jmse12020233
APA StyleMu, M., Guo, K., Cai, W., Zhu, L., Pi, Z., & Zhou, S. (2024). A New Plastic Design Approach for the Vertical-Side-Plating Thickness of Ice-Strengthened Ships Suffering from Ice Floe Impacts. Journal of Marine Science and Engineering, 12(2), 233. https://doi.org/10.3390/jmse12020233