# Application of Equivalent Single Layer Approach for Ultimate Strength Analyses of Ship Hull Girder

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Overview of the ESL Approach

**ABCD**stiffness matrices are obtained from the first derivative of membrane forces and bending moments of the unit cell (UC) simulations under six loading conditions, as can be seen in Figure 2. Compared to full 3D FEM, the generation of stiffness matrices requires an additional modeling and computational effort, which however can be made fairly automatic using programming.

#### 2.2. Implementation of ESL in Abaqus

## 3. Case Study of One Compartment Aluminium Box Girder

## 4. Case Study of Full-Scale Steel Ship Structure

#### 4.1. Ship Particulars

#### 4.2. Loading and Boundary Conditions

#### 4.3. Damage Scenario

#### 4.4. Full 3D Finite Element (3D FE) Model

#### 4.5. Incremental-Iterative Method

## 5. Ultimate Strength Analysis

#### 5.1. Vertical Bending Moment

#### 5.2. Horizontal Bending Moment

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Undeformed and deformed geometries of an edge of a plate under the assumption of the first-order shear deformation theory (FSDT).

**Figure 2.**Six different unit cell (UC) configurations with boundary conditions needed for ABCD stiffness matrix definition. Forces and moments shown with arrows are associated with respective stiffness components. Boundary conditions for edges with the same color are identical.

**Figure 5.**Bending moment–curvature curves and deformation shapes obtained from Benson et al. [14], 3D FEM and ESL models. U is in mm.

**Figure 7.**Design of hull structure for case study: midship section (

**left**) and side view and top view (

**right**).

**Figure 8.**(

**a**) Boundary condition applied in the tanker under vertical and horizontal bending moments. (

**b**) Distributed pressure applied on the surface.

**Figure 11.**Deflection of ship’s hull along its length at the ultimate stage under vertical and horizontal bending moments.

**Figure 12.**Response of intact ship under vertical bending moment. (

**a**) Bending moment–deflection curves. (

**b**) Longitudinal stress distribution to vertical coordinate at the midship under hogging condition.

**Figure 13.**Response of ship grounding under vertical bending moment. (

**a**) Bending moment–deflection curves. The maximum bending moments are used in calculating the histogram of moment reduction ratio. (

**b**) Longitudinal stress distribution to vertical coordinate at the midship under hogging condition.

**Figure 14.**Response of ship collision under vertical bending moment. (

**a**) The bending moment–deflection curves. The maximum bending moments are used in calculating the histogram of moment reduction ratio. (

**b**) The longitudinal stress distribution to vertical coordinate at the midship under hogging condition.

**Figure 15.**Collapse mode in double bottom and main deck structures under hogging and sagging conditions in post-ultimate stage, respectively. Deformation scaling factor is 5×.

**Figure 16.**The moment reduction ratio under grounding and collision damage in the hogging and sagging conditions using the 3D FEM, ESL, and CSR methods.

**Figure 17.**Horizontal bending moment-deflection relationships of tanker in the condition of (

**a**) intact, (

**b**) grounding, (

**c**) collision with damage in compression, and (

**d**) collision with damage in tension. The maximum bending moments are used in calculating the histogram in Figure 18.

**Figure 18.**The moment reduction ratio of horizontal bending moment in grounding and collision cases for the 3D FEM, ESL, and CSR methods.

Parameter | Symbol | Unit | Value |
---|---|---|---|

Overall length | ${L}_{OA}$ | m | 182.2 |

Length between perpendiculars | ${L}_{PP}$ | m | 175.3 |

Moulded breadth | B | m | 32.2 |

Depth | H | m | 15.0 |

Design draught | B | m | 11.1 |

Displacement | ∇ | t | 52,298 |

Double bottom height | ${H}_{DB}$ | m | 2.21 |

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**MDPI and ACS Style**

Putranto, T.; Kõrgesaar, M.; Tabri, K.
Application of Equivalent Single Layer Approach for Ultimate Strength Analyses of Ship Hull Girder. *J. Mar. Sci. Eng.* **2022**, *10*, 1530.
https://doi.org/10.3390/jmse10101530

**AMA Style**

Putranto T, Kõrgesaar M, Tabri K.
Application of Equivalent Single Layer Approach for Ultimate Strength Analyses of Ship Hull Girder. *Journal of Marine Science and Engineering*. 2022; 10(10):1530.
https://doi.org/10.3390/jmse10101530

**Chicago/Turabian Style**

Putranto, Teguh, Mihkel Kõrgesaar, and Kristjan Tabri.
2022. "Application of Equivalent Single Layer Approach for Ultimate Strength Analyses of Ship Hull Girder" *Journal of Marine Science and Engineering* 10, no. 10: 1530.
https://doi.org/10.3390/jmse10101530