# A Design Method to Assess the Primary Strength of the Delta-Type VLFS

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Verification

#### 3.1.1. Comparison with Geometrical CAD Calculations

#### 3.1.2. Comparing the Quasi-Static Primary Strength Loads with Hydrodynamic Splitting Forces

**A.****External hydrodynamic**

**B.****Gravitational:**

**C.****Inertial:**

#### 3.2. Total Hydrodynamic Load Comparison: Validation of the Procedure

#### 3.3. Implementation of the Design Procedure: Initial Results for Prismatic Side-Hulls

- Wave height: $10\mathrm{m}$;
- Wavelength: $100,150,200,250,300,350,400\mathrm{m}$;
- Wave phase: $0\xb0$ to $330\xb0$, in $30\xb0$ intervals. Phase $0\xb0$ is where the wave crest is at LCG;
- Wave direction: $45\xb0,30\xb0,15\xb0,0\xb0,-15\xb0,-30\xb0,-45\xb0$;
- The required bending moduli of the critical cross section are in the order of 1000 m
^{3}.

#### 3.4. Implementation of the Design Procedure: More Efficient Narrow Stern Design

#### 3.4.1. Load Results

- Wave height: $10\mathrm{m}$;
- Wavelength: $200,250,300,350,400,450\mathrm{m}$;
- Wave phase: $0\xb0$ to $330\xb0$, in $30\xb0$ intervals. Phase $0\xb0$ is where the wave crest is at LCG;
- Wave direction: $45\xb0,30\xb0,15\xb0,0\xb0,-15\xb0,-30\xb0,-45\xb0$.

#### 3.4.2. Initial Scantlings

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 2.**The Delta VLFS initial configuration (top view). The assumed critical cross section, marked with a red line, is at the connection of the side-hull to the front delta. The origin of the global coordinate system $\left(\mathit{x},\mathit{y}\right)$ is at the center of gravity of the structure, and the vertical axis is positive upwards from the calm free surface.

**Figure 3.**Global and local coordinate systems. In this analysis, the origin of the global coordinate system is at C.G. and the origin of the local system is at the connection point of the side-hull, as presented in Figure 2.

**Figure 4.**A demonstration of load calculation by CAD: (

**a**) positioning the structure at the equilibrium position as calculated by GHS (draft, list, and trim); (

**b**) setting the incoming wave surface; (

**c**) trimming the solid hull by the wave surface, keeping the underwater part; (

**d**) cutting the side-hull at the critical cross section (marked in orange in (

**c**)) and finding the center of floatation, the moment arm (blue arrow in (

**d**)), and the volume for the load calculations.

**Figure 5.**A comparison of bending moments between the CAD and analytical models, for three wave directions: $45\xb0$ (

**top**), $30\xb0$ (

**middle**), $15\xb0$ (

**bottom**).

**Figure 6.**A comparison of vertical bending moments (${M}_{y}$). The incident wave splitting forces calculation by AQWA is in blue, while the hydrostatic wave load by analytic calculation is in magenta. From top to bottom, left to right: bending moment plots for wavelengths of 200 m, 250 m, 300 m, 350 m, 400 m, 450 m.

**Figure 7.**A comparison of vertical bending moments (${M}_{y}$). The incident wave splitting forces calculation by AQWA is in blue, while the hydrostatic wave load by analytic calculation is in magenta. From top to bottom, left to right: bending moment plots for wavelengths of 200 m, 250 m, 300 m, 350 m, 400 m, 450 m.

**Figure 8.**A comparison of the maximal bending moments ${M}_{y}$ (upper sub figure) and ${M}_{z}$ (lower sub figure). The wave splitting forces calculation by AQWA (complete hydrodynamic solution) is in blue. The hydrostatic wave load by analytic calculation is in magenta. Each subplot presents a wave direction. For each direction, the wavelength increased from 200 to 450 m.

**Figure 10.**Illustration of the interactive plot (procedure output)- Shear and axial stresses using the simplified analytical model for hull girder primary strength. From top to bottom: shear stress (results), axial stress (results), wavelength (input), wave direction (input), and wave phase (input). The x axis presents the serial number of the load case (wave combinations). As can be observed, the potentially critical load cases are readily identified as the two peak axial stress value of the 200 m’ $-45\xb0$ wave (on the left side of the axial stress plot). Note that the interactive property of this plot could not be presented here.

**Figure 11.**Delta VLFS with stern radius ${\mathit{R}}_{\mathbf{3}}=\mathbf{10}m$ (20 m stern diameter).

**Figure 12.**Delta VLFS with stern radius ${\mathit{R}}_{\mathbf{3}}=\mathbf{15}m$ (30 m stern diameter).

**Figure 13.**Delta VLFS with stern radius ${\mathit{R}}_{\mathbf{3}}=\mathbf{20}m$ (40 m stern diameter).

**Figure 14.**A comparison of the maximal axial stress in the X direction at the critical cross section. The wave splitting forces calculation by AQWA (complete hydrodynamic moments) is in blue. The hydrostatic analytic calculation is in magenta. Each subplot presents a wave direction. For each direction, the wavelength increased from 200 to 450 m.

**Figure 15.**Initial scantling presented at the critical cross section. The plate thicknesses and stiffeners’ dimensions and spacing are typical for the entire structure. All dimensions are in millimeters.

**Figure 16.**Compartment division on a single floor. Each compartment is about $30\mathrm{m}$ in length.

$\mathbf{Vertical}\mathbf{Bending}\mathbf{Moment}\mathbf{(}\mathit{M}\mathit{y}\mathbf{)}$ | $\mathbf{Horizontal}\mathbf{Bending}\mathbf{Moment}\mathbf{(}\mathit{M}\mathit{z}\mathbf{)}$ | |
---|---|---|

Prismatic (initial) | $1.85\times {10}^{11}\mathrm{Nm}$ | $4.7\times {10}^{10}\mathrm{Nm}$ |

$\mathrm{Tapered}({R}_{3}=10$) | $1.17\times {10}^{11}\mathrm{Nm}$ | $6.4\times {10}^{9}\mathrm{Nm}$ |

Reduction (%) | $\mathbf{37}\%$ | $\mathbf{86}\%$ |

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**MDPI and ACS Style**

Gafter, R.; Drimer, N. A Design Method to Assess the Primary Strength of the Delta-Type VLFS. *J. Mar. Sci. Eng.* **2021**, *9*, 1026.
https://doi.org/10.3390/jmse9091026

**AMA Style**

Gafter R, Drimer N. A Design Method to Assess the Primary Strength of the Delta-Type VLFS. *Journal of Marine Science and Engineering*. 2021; 9(9):1026.
https://doi.org/10.3390/jmse9091026

**Chicago/Turabian Style**

Gafter, Roy, and Nitai Drimer. 2021. "A Design Method to Assess the Primary Strength of the Delta-Type VLFS" *Journal of Marine Science and Engineering* 9, no. 9: 1026.
https://doi.org/10.3390/jmse9091026