# On the Resistance to Buckling Loads of Idealized Hull Structures: FE Analysis on Designed-Stiffened Plates

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Characteristics of Ship Hull Structure

## 3. Theoretical Concept of Plate Buckling

#### 3.1. Kirchhoff Plate Theory

- the deflection in the plate is small, i.e., less than the thickness of the plate;
- during bending, the middle plane of the plate extant in the neutral surface and does not stretch;
- the plane sections of the plate can rotate during bending to extant normal to the neutral surface, and distortion does not occur. This state makes the stresses and strains proportional to the distance from neutral surface;
- the effect of shearing forces is neglected and the loads are completely resisted by the bending moments that are induced in the elements plate;
- the thickness of the plate is not higher compared to the other dimensions.

#### 3.2. Plate Buckling

## 4. Fundamental FE Algorithm

- Hooke’s law is applied to the plate and stiffener material;
- Mindlin’s theory is applied to the bending deformation, which is a state before the bending in the plate happens, and the linear elements remain straight when they are perpendicular to the middle plane of the plate;
- The x- and y-directions become a function of the deflection that occurs in the z-direction;
- The deflections of the plate are small—less than the thickness of the plate;
- After bending occurs, it is common and normal for the plate and stiffener to stand straight before bending.

## 5. Benchmarking Analysis

#### 5.1. Analysis Setup and Configuration

#### 5.2. Validation of FE Methodology

^{3}, a Young’s modulus of 73.4 GPa, and a Poisson’s ratio of 0.33 [43,44]. Table 3 details the material properties of Aluminum Alloy 2024-T3. The total mass of the specimens was not significantly different compared with the experimental data, shown in Table 4. Specimen A has a total mass of 1.974 kg and Specimen B has 1.993 kg, compared with the experimental data at 1.959 kg and 1.968 kg, respectively. Hence, Specimen B’s design was marginally heavier than Specimen A’s. The mass percentage difference of Specimen A and Specimen B were less than 1.27% compared with the experimental data.

#### 5.3. Mesh Convergence Study

## 6. Integrated FE Study

#### 6.1. Geometrical Design

#### 6.2. Material Model

^{3}, a Young’s Modulus of 210 GPa, and a Poisson’s ratio of 0.3.

#### 6.3. Boundary Condition and Scenario

## 7. Results: Linear Buckling

#### 7.1. Effect of Thickness Change

#### 7.2. Effect of Geometrical Change

## 8. Results: Nonlinear Buckling

#### 8.1. Effect of Thickness Change

#### 8.2. Effect of Geometrical Change

#### 8.3. Effect of Material Change

## 9. Overall Discussion: Linear vs. Nonlinear Behaviors

## 10. Conclusions

- Aspect ratio of rectangular plate a/b = 3 has a higher critical buckling load value compared to a/b = 2 and 4. This particular phenomenon is due to the dimension of the width of the plate b (where the applied load works) as a/b = 3 is smaller compared with a/b = 2 and 4 even though the buckling coefficients ${k}_{c}$ are the same;
- The method of the mesh size selection (i.e., the element length to thickness (ELT) ratio) can provide a fairly good critical buckling load value compared with the analytical data. In the buckling analysis of rectangular plates, the ELT ratio 7 gave an error value of 0.65% compared with the analytical result;
- A change in the thickness of the plate significantly increased the strength and the generated energy during the buckling of the stiffened panels for both Specimen A and Specimen B;
- For Specimen A, increase in the thickness of the longitudinal stringers moderately increased the strength of the stiffened panels;
- For Specimen B, an increase in the thickness of the sub-stiffeners did not significantly increase the strength of the stiffened panels;
- Material S355JR-EN10210 produced a higher ultimate panel collapse load compared to S235JR-EN10025 (A) and S235JR-EN10025 (B).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Yao, T.; Fujikubo, M. Buckling and Ultimate Strength of Ship and Ship-Like Floating Structures; Butterworth-Heinemann; Elsevier: Oxford, UK, 2016. [Google Scholar]
- Durban, D.; Zuckerman, Z. Elastoplastic buckling of rectangular plates in biaxial compression/tension. Int. J. Mech. Sci.
**1999**, 41, 751–765. [Google Scholar] [CrossRef] - Elgaaly, M. Post-buckling behavior of thin steel plates using computational models. Adv. Eng. Softw.
**2000**, 31, 511–517. [Google Scholar] [CrossRef] - Petry, D.; Fahlbusch, G. Dynamic buckling of thin isotropic plates subjected to in-plane impact. Thin Walled Struct.
**2000**, 38, 267–283. [Google Scholar] [CrossRef] - Cheung, Y.K.; Au, F.T.K.; Zheng, D.Y. Finite strip method for the free vibration and buckling analysis of plates with abrupt changes in thickness and complex support conditions. Thin Walled Struct.
**2000**, 36, 89–110. [Google Scholar] [CrossRef] - Shimizu, S. Tension buckling of plate having a hole. Thin Walled Struct.
**2007**, 45, 827–833. [Google Scholar] [CrossRef][Green Version] - Amani, M.; Edlund, B.L.O.; Alinia, M.M. Buckling and post buckling behavior of unstiffened slender curved plates under uniform shear. Thin Walled Struct.
**2011**, 49, 1017–1031. [Google Scholar] [CrossRef] - Xing, Z.; Kucukler, M.; Gardner, L. Local buckling of stainless steel plates in fire. Thin Walled Struct.
**2020**, 148, 106570. [Google Scholar] [CrossRef] - Danielson, D.A.; Wilmer, A. Buckling of stiffened plates with bulb flat flanges. Int. J. Solids. Struct.
**2004**, 41, 6407–6427. [Google Scholar] [CrossRef] - Kwon, Y.B.; Park, H.S. Compression tests of longitudinally stiffened plates undergoing distortional buckling. J. Constr. Steel. Res.
**2011**, 67, 1212–1224. [Google Scholar] [CrossRef] - Layachi, H.; Xu, Y.M. Performance and Stability Analysis of Sub-stiffening for Mechanical Buckling and Post Buckling: A Selective Study. MATEC Web. Conf.
**2017**, 95, 01004. [Google Scholar] [CrossRef][Green Version] - Kong, X.; Yang, Y.; Gan, J.; Yuan, T.; Ao, L.; Wu, W. Experimental and numerical investigation on the detailed buckling process of similar stiffened panels subjected to in-plane compressive load. Thin Walled Struct.
**2020**, 148, 106620. [Google Scholar] [CrossRef] - Srivastava, A.K.L.; Datta, P.K.; Sheikh, A.H. Buckling and vibration of sti ened plates subjected to partial edge loading. Int. J. Mech. Sci.
**2003**, 45, 73–93. [Google Scholar] [CrossRef] - Kumar, M.S.; Alagusundaramoorthy, P.; Sundaravadivelu, R. Ultimate Strength of Ship Plating under Axial Compression. Ocean Eng.
**2006**, 33, 1249–1259. [Google Scholar] [CrossRef] - Seifi, R.; Khoda-yari, N. Experimental and numerical studies on buckling of cracked thin-plates under full and partial compression edge loading. Thin Walled Struct.
**2011**, 49, 1504–1516. [Google Scholar] [CrossRef] - Sujiatanti, S.H.; Tanaka, S.; Shinkawa, S.; Setoyama, Y. Experimental and numerical studies for buckling and collapse behaviors of a cracked thin steel panel subjected to sequential tensile and compressive loading. Thin Walled Struct.
**2020**, 157, 107059. [Google Scholar] [CrossRef] - Xu, M.C.; Song, Z.J.; Zhang, B.W.; Pan, J. Empirical formula for predicting ultimate strength of stiffened panel of ship structure under combined longitudinal compression and lateral loads. Ocean Eng.
**2018**, 162, 161–175. [Google Scholar] [CrossRef] - Jung, G.; Huang, T.D.; Dong, P.; Dull, R.M.; Conrardy, C.C.; Porter, N.C. Numerical prediction of buckling in ship panel structures. J. Ship Prod. Des.
**2007**, 23, 171–179. [Google Scholar] [CrossRef] - Zhang, S.; Khan, I. Buckling and ultimate capability of plates and stiffened panels in axial compression. Mar. Struct.
**2009**, 22, 791–808. [Google Scholar] [CrossRef] - Stamatelos, D.G.; Labeas, G.N.; Tserpes, K.I. Analytical calculation of local buckling and post-buckling behavior of isotropic and orthotropic stiffened panels. Thin Walled Struct.
**2011**, 49, 422–430. [Google Scholar] [CrossRef] - Rahbar-Ranji, A. Elastic buckling analysis of longitudinally stiffened plates with flat-bar stiffeners. Ocean Eng.
**2012**, 58, 48–59. [Google Scholar] [CrossRef] - Wang, J.; Ma, N.; Murakawa, H. An efficient FE computation for predicting welding induced buckling in production of ship panel structure. Mar. Struct.
**2015**, 41, 20–52. [Google Scholar] [CrossRef] - Reddy, J.N. Theory and Analysis of Elastic Plates and Shells, 2nd ed.; Taylor & Francis Group, LLC: Boca Raton, FL, USA, 2007. [Google Scholar]
- Wang, C.M.; Zhang, Y.P.; Pedroso, D.M. Hencky bar-net model for plate buckling. Eng. Struct.
**2017**, 150, 947–954. [Google Scholar] [CrossRef] - Love, A.E.H. The small free vibrations and deformation of a thin elastic shell. Proc. R. Soc. Lond.
**1888**, 43, 352–353. [Google Scholar] - Ridwan Putranto, T.; Laksono, F.B.; Prabowo, A.R. Fracture and Damage to the Material accounting for Transportation Crash and Accident. Procedia Struct. Integr.
**2020**, 27, 38–45. [Google Scholar] [CrossRef] - Ridwan. Failure Assessment of Ship Hull Materials Under Tension Loading as Part of Impact Phenomena Using Finite Element Approach. Master Thesis, Universitas Sebelas Maret, Surakarta, Indonesia, 2021. [Google Scholar]
- Ridwan, R.; Prabowo, A.R.; Muhayat, N.; Putranto, T.; Sohn, J.M. Tensile analysis and assessment of carbon and alloy steels using fe approach as an idealization of material fractures under collision and grounding. Curved Layer Struct.
**2020**, 7, 188–198. [Google Scholar] [CrossRef] - Prabowo, A.R.; Muttaqie, T.; Sohn, J.M.; Bae, D.M.; Setiyawan, A. On the failure behaviour to striking bow Penetration of impacted marine-steel structures. Curved Layer Struct.
**2018**, 5, 68–79. [Google Scholar] [CrossRef][Green Version] - Prabowo, A.R.; Sohn, J.M.; Bae, D.M.; Cho, J.H. Performance assessment on a variety of double side structure during collision interaction with other ship. Curved Layer Struct.
**2017**, 4, 255–271. [Google Scholar] [CrossRef] - Prabowo, A.R.; Sohn, J.M. Nonlinear dynamic behaviors of outer shell and upper deck structures subjected to impact loading in maritime environment. Curved Layer Struct.
**2019**, 6, 146–160. [Google Scholar] [CrossRef][Green Version] - Prabowo, A.R.; Tuswan, T.; Ridwan, R. Advanced development of sensors’ roles in maritime-based industry and research: From field monitoring to high-risk phenomenon measurement. Appl. Sci.
**2021**, 11, 3954. [Google Scholar] [CrossRef] - Rahai, A.R.; Alinia, M.M.; Kazemi, S. Buckling analysis of stepped plates using modified buckling mode shapes. Thin Walled Struct.
**2008**, 46, 484–493. [Google Scholar] [CrossRef] - Moen, C.D.; Schafer, B.W. Elastic buckling of thin plates with holes in compression or bending. Thin Walled Struct.
**2009**, 47, 1597–1607. [Google Scholar] [CrossRef] - Rad, A.A.; Panahandeh-Shahraki, D. Buckling of cracked functionally graded plates under tension. Thin Walled Struct.
**2014**, 84, 26–33. [Google Scholar] - Ndubuaku, O.; Liu, X.; Martens, M.; Cheng, J.J.R.; Adeeb, S. The effect of material stress-strain characteristics on the ultimate stress and critical buckling strain of flat plates subjected to uniform axial compression. Constr. Build Mater.
**2018**, 182, 346–359. [Google Scholar] [CrossRef] - Tenenbaum, J.; Eisenberger, M. Analytic solution for buckling of rectangular isotropic plates with internal point supports. Thin Walled Struct.
**2021**, 163, 107640. [Google Scholar] [CrossRef] - Turner, M.J.; Clough, R.W.; Martin, H.C.; Topp, L.J. Stiffness and deflection analysis of complex structures. J. Aeronaut. Sci.
**1956**, 23, 805–823. [Google Scholar] [CrossRef] - Wegmuller, A.W. Finite Element Analyses of Elastic-Plastic Plates and Eccentrically Stiffened Plates. Ph.D. Thesis, Lehigh University, Bethlehem, PA, USA, 1971. [Google Scholar]
- Mukhopadhyay, M.; Mukherjee, A. Finite element buckling analysis of stiffened plates. Comput. Struct.
**1990**, 34, 795–803. [Google Scholar] [CrossRef] - Quinn, D.; Murphy, A.; McEwan, W.; Lemaitre, F. Stiffened panel stability behaviour and performance gains with plate prismatic sub-stiffening. Thin Walled Struct.
**2009**, 47, 1457–1468. [Google Scholar] [CrossRef][Green Version] - ANSYS. Academic Research Mechanical, Release 21.1, Help System, Coupled Field Analysis Guide; ANSYS Inc.: Canonsburg, PA, USA, 2021. [Google Scholar]
- Goldarag, F.E.; Babaei, A.; Jafarzadeh, H. An experimental and numerical investigation of clamping force variation in simple bolted and hybrid (bolted-bonded) double lap joints due to applied longitudinal loads. Eng. Fail. Anal.
**2018**, 91, 327–340. [Google Scholar] [CrossRef] - Goldarag, F.E.; Barzegar, S.; Babaei, A. An experimental method for measuring the clamping force in double lap simple bolted and hybrid (bolted-bonded) joints. Trans. Famena.
**2015**, 39, 87–94. [Google Scholar] - Abubakar, A.; Dow, R.S. Simulation of ship grounding damage using the finite element method. Int. J. Solids Struct.
**2013**, 50, 623–636. [Google Scholar] [CrossRef][Green Version] - Prabowo, A.R.; Bae, D.M.; Sohn, J.M.; Zakki, A.F.; Cao, B.; Cho, J.H. Effects of the rebounding of a striking ship on structural crashworthiness during ship-ship collision. Thin Walled Struct.
**2017**, 115, 225–239. [Google Scholar] [CrossRef] - Alsos, H.S.; Amdahl, J. On the resistance to penetration of stiffened plates, Part I—Experiments. Int. J. Impact Eng.
**2009**, 36, 799–807. [Google Scholar] [CrossRef] - Alsos, H.S.; Amdahl, J.; Hopperstad, O.S. On the resistance to penetration of stiffened plates, Part II: Numerical analysis. Int. J. Impact Eng.
**2009**, 36, 875–887. [Google Scholar] [CrossRef] - Lloyd, G. Rules for Classification and Construction; DNV GL SE: Hamburg, Germany, 2009. [Google Scholar]
- Prabowo, A.R.; Cao, B.; Bae, D.M.; Bae, S.Y.; Zakki, A.F.; Sohn, J.M. Structural analysis of the double bottom structure during ship grounding by finite element approach. Lat. Am. J. Solids Struct.
**2017**, 14, 1106–1123. [Google Scholar] [CrossRef][Green Version] - AbuBakar, A.; Dow, R.S. The impact analysis characteristics of a ship’s bow during collisions. Eng. Fail. Anal.
**2019**, 100, 492–511. [Google Scholar] [CrossRef] - Zucco, G.; Weaver, P.M. The role of symmetry in the post-buckling behaviour of structures. Proc. R. Soc. A
**2020**, 476, 20190609. [Google Scholar] [CrossRef] - Prabowo, A.R.; Bae, D.M.; Sohn, J.M. Comparing structural casualties of the Ro-Ro vessel using straight and oblique collision incidents on the car deck. J. Mar. Sci. Eng.
**2019**, 7, 183. [Google Scholar] [CrossRef][Green Version] - Bae, D.M.; Prabowo, A.R.; Cao, B.; Sohn, J.M.; Zakki, A.F.; Wang, Q. Numerical simulation for the collision between side structure and level ice in event of side impact scenario. Lat. Am. J. Solids Struct.
**2016**, 13, 2991–3004. [Google Scholar] [CrossRef][Green Version] - Prabowo, A.R.; Ridwan, R.; Tuswan, T.; Sohn, J.M.; Surojo, E.; Imaduddin, F. Effect of the selected parameters in idealizing material failures under tensile loads: Benchmarks for damage analysis on thin-walled structures. Curved Layer Struct.
**2022**, 9, 258–285. [Google Scholar] [CrossRef] - Prasetya, L.W.; Prabowo, A.R.; Ubaidillah, U.; Istanto, I.; Nordin, N.A.B. Design of crashworthy attenuator structures as a part of vehicle safety against impact: Application of waste aluminum can-based material. Theor. Appl. Mech. Lett.
**2021**, 11, 100235. [Google Scholar] [CrossRef] - Prabowo, A.R.; Prabowoputra, D.M. Investigation on Savonius turbine technology as harvesting instrument of non-fossil energy: Technical development and potential implementation. Theor. Appl. Mech. Lett.
**2020**, 10, 262–269. [Google Scholar] [CrossRef] - Sakuri, S.; Surojo, E.; Ariawan, D.; Prabowo, A.R. Investigation of Agave cantala-based composite fibers as prosthetic socket materials accounting for a variety of alkali and microcrystalline cellulose treatments. Theor. Appl. Mech. Lett.
**2020**, 10, 405–411. [Google Scholar] [CrossRef]

**Figure 1.**Simply supported thin rectangular plate subjected to an in-plane uniaxial load [1].

**Figure 3.**Buckling coefficient of rectangular plates with various boundary conditions [1].

**Figure 5.**Typical finite element mesh on Specimen A. (

**a**) FE model; (

**b**) FE result with the deformation contour.

**Figure 6.**Comparison between experimental [41] and FE on Specimen A.

**Figure 7.**Typical finite element mesh on Specimen B. (

**a**) FE model; (

**b**) FE result with the deformation contour.

**Figure 8.**Result comparison between experimental data [41] and FE on Specimen B.

**Figure 9.**Load versus displacement curves based on comparison with laboratory experiment by Quinn et al. [41]. (

**a**) Specimen A and (

**b**) Specimen B.

**Figure 10.**Geometrical design of specimens with three different aspect ratio a/b and three different thickness h.

**Figure 16.**First four mode shapes in the 2, 3, and 4 aspect ratio a/b and 3 mm thickness of the plate.

**Figure 17.**Effect of the a/b ratio obtained from the 3 mm thickness of the plate on the critical buckling load: (

**a**) 3 mm plate thickness; (

**b**) 4 mm plate thickness; (

**c**) 5 mm plate thickness.

**Figure 18.**Effect of thickness change to the load versus displacement curve. (

**a**) Specimen A; (

**b**) Specimen B.

**Figure 19.**Effect of thickness change to energy versus displacement curve. (

**a**) Specimen A; (

**b**) Specimen B.

**Figure 20.**Equivalent strain contour in Specimen A: (

**a**) 3 mm plate thickness; (

**b**) 4 mm plate thickness; (

**c**) 5 mm plate thickness.

**Figure 21.**Equivalent strain contour in Specimen B: (

**a**) 3 mm plate thickness; (

**b**) 4 mm plate thickness; (

**c**) 5 mm plate thickness.

**Figure 22.**Effect of stiffeners thickness to the load versus displacement curve. (

**a**) Specimen A; (

**b**) Specimen B.

**Figure 23.**Effect of stiffeners thickness to the energy versus displacement curve. (

**a**) Specimen A; (

**b**) Specimen B.

**Figure 24.**Equivalent strain contour in Specimen A: (

**a**) 4 mm longitudinal stringers thickness; (

**b**) 5 mm longitudinal stringers thickness; (

**c**) 6 mm longitudinal stringers thickness.

**Figure 25.**Equivalent strain contour in Specimen B. (

**a**) 2 mm sub-stiffeners thickness; (

**b**) 2.5 mm sub-stiffeners thickness; (

**c**) 3 mm sub-stiffeners thickness.

**Figure 26.**Effect of materials change to the load versus displacement curve. (

**a**) Specimen A; (

**b**) Specimen B.

**Figure 27.**Effect of materials change to the load versus displacement curve. (

**a**) Specimen A; (

**b**) Specimen B.

Milestone | Author(s) | Subject | Methodology | Important Remarks |
---|---|---|---|---|

2003 | Srivastava et al. [13] | Buckling and vibration of stiffened plates subjected to partial edge loading | Numerical analysis | They found that with the higher aspect ratio of the stiffened plates, the buckling loads are not significantly affected by the position of the load. Furthermore, increasing the compressive load causes a decrease in natural frequencies. |

2006 | Kumar et al. [14] | Ultimate strength of ship plating under axial compression | Experimental test and numerical analysis | The strength of stiffened plates is decreased by 24 to 37% when a square cutout is present in the center of the plate. Compared to panels without a cutout, the strength of the panel is decreased by 23% for stiffener-initiated failure, and 31 to 46% for plate-initiated failures when a rectangular cutout is present in the middle of the plate, respectively. |

2007 | Jung et al. [18] | Buckling analysis in ship panel structures | Numerical analysis using Q-WELD™ ABAQUS software | Buckling resistance of the stiffened panels can be effectively increased by transient thermal tensioning (TTT). Center-to-edge welding has decreased the buckling of panels compared with the edge-to-edge welding. |

2009 | Zhang and Khan [19] | Buckling and ultimate capability of plates and stiffened panels in axial compression | Analytical and numerical analysis using ABAQUS software | The ultimate strength of the simply supported plates in axial compression can be determined using Faulkner’s formula with an immensely satisfactory prediction. In the case $\beta <2.5$, the dissimilarity results in Faulkner’s formula being not more than 1.5% compared with the finite element results. |

2011 | Seifi and Khoda-yari [15] | Buckling analysis of cracked, thin plates under full and partial compression edge loading | Experimental test and numerical analysis using ABAQUS software | Cracks in plates influence the critical buckling load. Increasing the crack length in the plate leads to a reduction in the critical buckling load. Furthermore, the critical buckling load is extremely decreased when the cracks are perpendicular to the loading compared with the other crack directions. |

2011 | Stamatelos et al. [20] | Buckling analysis of composite bladed stiffened plates with different aspect ratios and different numbers of stiffeners | Analytical and numerical analysis using ANSYS | The proposed analytical method for buckling and post-buckling analysis was based on the classical lamination plate theory and two-dimensional Ritz displacement functions for arbitrary edge supports. The results show that the prediction of the critical buckling load obtains satisfactory agreement with an error of not more than 8% compared with the finite element results. |

2012 | Rahbar-Ranji [21] | Buckling analysis of longitudinally stiffened plates with flat-bar stiffeners | Analytical and numerical analysis using ANSYS | It is noted that the Euler stress for tripping is not influenced by web buckling when the attached plate is neglected. |

2015 | Wang et al. [22] | Buckling analysis when producing the ship panel structure | Experimental test and numerical analysis using JWRIAN (Joining and Welding Research Institute Analysis) | An elastic finite element analysis to predict welding-induced buckling based on the inherent deformation method is proposed. The result of the welding angular distortion computed by the proposed FE method has a satisfactory agreement with the experimental result for the sequential welding of the fillet welded joint. |

2018 | Xu et al. [17] | Ultimate strength analysis of the stiffened panel of the ship structure | Experiment, analytical, and numerical analysis using ANSYS | The load carrying capacity of the stiffened panel is immensely influenced by the plate slenderness and column slenderness. For the tripping buckling, flexural or torsional rigidity of stiffener is strongly considered. |

2020 | Sujiatanti et al. [16] | Buckling and collapse analysis of a cracked thin steel panel | Experimental test and numerical analysis using LS-DYNA | The presence of the crack in plate appears to decrease the ultimate plate collapse load. The result shows that the cracked plate has a small value for the ultimate buckling load (e.g., 35 kN and 30 kN for 50 mm and 75 mm cracks, respectively, compared with an intact plate which is 45 kN). |

Milestone | Author(s) | Subject | Methodology | Important Remarks |
---|---|---|---|---|

2008 | Rahai et al. [33] | Buckling analysis of stepped plates | Analytical and numerical analysis using ANSYS | A novel, approximate procedure using an energy method based on modified buckling mode shapes for buckling analysis using simply supported rectangular steps has been introduced in this study. As noted, this novel, approximate method can be applied to all plates for which the shape mode is identified. |

2009 | Moen and Schafer [34] | Elastic buckling of thin plates with holes | Numerical analysis using ABAQUS | The presence of holes in the plate can create unique buckling modes. Depending on the size and geometry of the hole, the critical elastic buckling stress on the plate can either decrease or increase. |

2014 | Rad and Panahandeh-Shahraki [35] | Buckling analysis of cracked functionally graded plates | Numerical analysis (Finite element) | This study suggests that as the Poisson ratio increases in both uni-axial and bi-axial loads, the critical load decreases. |

2018 | Ndubuaku et al. [36] | The material stress-strain characteristics on the ultimate stress and critical buckling of flat plates | Numerical analysis using ABAQUS | This study indicates that the ultimate compressive strength and strain on the plate are affected by the strain hardening. The phenomenon of this effect is seen more clearly when the plate thickness is thicker. |

2021 | Tenenbaum and Eisenberger [37] | Analytic solution for the buckling of rectangular isotropic plates with internal point supports | Analytical | The boundary conditions and the loading type (uni-axial or bi-axial) are seen to considerably affect the buckling load factor λ. |

Property | Value |
---|---|

Density | 2780 kg/m^{3} |

Young’s modulus | 73.4 GPa |

Yield stress | 315 MPa |

Tensile strength | 550 MPa |

Poisson’s ratio | 0.33 |

Exponent | 0.406 |

Failure strain | 0.18 |

Mass [kg] | Mass Difference [kg] | Mass Percentage Difference [%] | ||
---|---|---|---|---|

Quinn et al., [41] | Specimen A | 1.959 | - | - |

Specimen B | 1.968 | - | - | |

Current study | Specimen A | 1.974 | +0.015 | +0.77 |

Specimen B | 1.993 | +0.025 | +1.27 |

Mesh Size [mm] | Ultimate Panel Collapse Load [kN] | Load Difference Percentage [%] | |||
---|---|---|---|---|---|

Specimen A | Specimen B | Specimen A | Specimen B | ||

Experimental data [41] | - | 216.6 | 255.0 | - | - |

Current study | 5 | 223.8 | 257.5 | +3.33 | +0.99 |

7.5 | 219.8 | 259.9 | +1.48 | +1.93 | |

10 | 217.4 | 257.1 | +0.38 | +0.84 | |

12.5 | 219.2 | 260.2 | +1.20 | +2.03 | |

15 | 218.7 | 262.3 | +0.96 | +2.87 | |

17.5 | 220.5 | 262.1 | +1.80 | +2.76 | |

20 | 222.7 | 238.7 | +2.83 | −6.38 | |

22.5 | 224.2 | 239.2 | +3.50 | −6.20 | |

25 | 225.5 | 237.3 | +4.09 | −6.96 | |

27.5 | 225.3 | 237.8 | +4.02 | −6.74 |

Material Type | Material Grade | K (MPa) | Exponent, n (-) | ε_{f}(-) | Yield Stress (MPa) | Ultimate Stress (MPa) |
---|---|---|---|---|---|---|

A | S235JR-EN10025 | 740 | 0.24 | 0.35 | 285 | 416 |

B | S235JR-EN10025 | 760 | 0.225 | 0.35 | 340 | 442 |

C | S355JR-EN10210 | 830 | 0.18 | 0.28 | 390 | 495 |

Plate Thickness | Critical Buckling Load [N/m] | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Analytical (a/b ratio) | Finite element result (a/b ratio) | |||||||||||

2 | 3 | 4 | 2 | 3 | 4 | |||||||

- | 5 | 6 | 7 | 5 | 6 | 7 | 5 | 6 | 7 | |||

3 | 20,498 | 81,994 | 20,498 | 20,494 | 20,496 | 20,500 | 82,072 | 82,111 | 82,164 | 20,494 | 20,496 | 20,500 |

4 | 48,589 | 194,355 | 48,589 | 48,601 | 48,612 | 48,626 | 194,700 | 194,880 | 195,100 | 48,601 | 48,612 | 48,626 |

5 | 94,900 | 379,600 | 94,900 | 94,958 | 94,997 | 95,031 | 380,690 | 381,180 | 382,070 | 94,958 | 94,998 | 95,031 |

ELT Ratio | Ultimate Panel Collapse Load (kN) | Energy (kJ) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Plate Thickness (Specimen A) | Plate Thickness (Specimen B) | Plate Thickness (Specimen A) | Plate Thickness (Specimen B) | |||||||||

3 mm | 4 mm | 5 mm | 3 mm | 4 mm | 5 mm | 3 mm | 4 mm | 5 mm | 3 mm | 4 mm | 5 mm | |

5 | 477.2 | 655.6 | 790.7 | 633.0 | 825.8 | 995.2 | 1.1 | 1.3 | 1.5 | 1.3 | 1.7 | 2.1 |

6 | 461.4 | 653.2 | 795.2 | 629.9 | 824.3 | 999.6 | 1.1 | 1.3 | 1.5 | 1.3 | 1.8 | 2.1 |

7 | 463.3 | 652.3 | 793.6 | 633.1 | 825.3 | 1003.0 | 1.1 | 1.3 | 1.6 | 1.4 | 1.8 | 2.1 |

ELT Ratio | Ultimate Panel Collapse Load [kN] | Energy [kJ] | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Thickness of the Longitudinal Stringers (Specimen A) | Thickness of the Sub-Stiffeners (Specimen B) | Thickness of the Longitudinal Stringers (Specimen A) | Thickness of the Sub-Stiffeners (Specimen B) | |||||||||

4 mm | 5 mm | 6 mm | 2 mm | 2.5 mm | 3 mm | 4 mm | 5 mm | 6 mm | 2 mm | 2.5 mm | 3 mm | |

5 | 790.7 | 839.6 | 884.0 | 995.2 | 1010.0 | 1025.1 | 1.5 | 1.7 | 1.9 | 2.1 | 2.1 | 2.1 |

6 | 795.2 | 844.8 | 889.0 | 999.6 | 1015.3 | 1030.7 | 1.5 | 1.8 | 2.0 | 2.1 | 2.1 | 2.1 |

7 | 793.6 | 843.0 | 887.4 | 1003.0 | 1018.3 | 1034.3 | 1.6 | 1.8 | 2.0 | 2.1 | 2.1 | 2.2 |

Material | ELT Ratio | Ultimate Panel Collapse Load [kN] | Energy [kJ] | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Plate Thickness (Specimen A) | Plate Thickness (Specimen B) | Plate Thickness (Specimen A) | Plate Thickness (Specimen B) | ||||||||||

3 mm | 4 mm | 5 mm | 3 mm | 4 mm | 5 mm | 3 mm | 4 mm | 5 mm | 3 mm | 4 mm | 5 mm | ||

S235JR-EN10025 (A) | 5 | 477.2 | 655.6 | 790.7 | 633.0 | 825.8 | 995.2 | 1.1 | 1.3 | 1.5 | 1.3 | 1.7 | 2.1 |

6 | 461.4 | 653.2 | 795.2 | 629.9 | 824.3 | 999.6 | 1.1 | 1.3 | 1.5 | 1.3 | 1.8 | 2.1 | |

7 | 463.3 | 652.3 | 793.6 | 633.1 | 825.3 | 1003.0 | 1.1 | 1.3 | 1.6 | 1.4 | 1.8 | 2.1 | |

S235JR-EN10025 (B) | 5 | 516.9 | 743.3 | 899.3 | 700.9 | 913.0 | 1105.8 | 1.2 | 1.4 | 1.6 | 1.4 | 1.9 | 2.3 |

6 | 512.1 | 737.1 | 900.7 | 698.3 | 912.5 | 1113.8 | 1.2 | 1.4 | 1.7 | 1.4 | 1.9 | 2.3 | |

7 | 514.2 | 736.9 | 898.9 | 701.4 | 914.4 | 1117.7 | 1.2 | 1.4 | 1.7 | 1.5 | 1.9 | 2.3 | |

S355JR-EN10210 | 5 | 553.1 | 816.6 | 990.7 | 756.3 | 965.2 | 1165.4 | 1.2 | 1.4 | 1.6 | 1.5 | 2.0 | 2.2 |

6 | 549.9 | 797.6 | 976.6 | 754.5 | 965.9 | 1179.6 | 1.3 | 1.4 | 1.7 | 1.5 | 2.0 | 2.4 | |

7 | 554.9 | 796.9 | 976.7 | 753.3 | 968.6 | 1194.6 | 1.3 | 1.5 | 1.7 | 1.5 | 1.9 | 2.3 |

Material | ELT Ratio | Ultimate Panel Collapse Load [kN] | Energy [kJ] | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Thickness of the Longitudinal Stringers (Specimen A) | Thickness of the Sub-Stiffeners (Specimen B) | Thickness of the Longitudinal Stringers (Specimen A) | Thickness of the Sub-Stiffeners (Specimen B) | ||||||||||

4 mm | 5 mm | 6 mm | 2 mm | 2.5 mm | 3 mm | 4 mm | 5 mm | 6 mm | 2 mm | 2.5 mm | 3 mm | ||

S235JR-EN10025 (A) | 5 | 790.7 | 839.6 | 884.0 | 995.2 | 1010.0 | 1025.1 | 1.5 | 1.7 | 1.9 | 2.1 | 2.1 | 2.1 |

6 | 795.2 | 844.8 | 889.0 | 999.6 | 1015.3 | 1030.7 | 1.5 | 1.8 | 2.0 | 2.1 | 2.1 | 2.1 | |

7 | 793.6 | 843.0 | 887.4 | 1003.0 | 1018.3 | 1034.3 | 1.6 | 1.8 | 2.0 | 2.1 | 2.1 | 2.2 | |

S235JR-EN10025 (B) | 5 | 899.3 | 950.2 | 997.5 | 1105.8 | 1123.0 | 1025.1 | 1.6 | 1.9 | 2.1 | 2.3 | 2.3 | 2.1 |

6 | 900.7 | 952.8 | 1000.5 | 1113.8 | 1131.0 | 1147.9 | 1.7 | 1.9 | 2.1 | 2.3 | 2.3 | 2.3 | |

7 | 898.9 | 950.8 | 998.7 | 1117.7 | 1135.8 | 1153.5 | 1.7 | 1.9 | 2.1 | 2.3 | 2.3 | 2.3 | |

S355JR-EN10210 | 5 | 990.7 | 1041.0 | 1086.7 | 1165.4 | 1185.1 | 1204.3 | 1.6 | 1.9 | 2.1 | 2.2 | 2.2 | 2.2 |

6 | 976.6 | 1036.9 | 1084.8 | 1179.6 | 1197.9 | 1217.2 | 1.7 | 1.9 | 2.2 | 2.4 | 2.3 | 2.4 | |

7 | 976.7 | 1032.8 | 1081.8 | 1194.6 | 1213.0 | 1231.1 | 1.7 | 1.9 | 2.2 | 2.3 | 2.4 | 2.5 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Prabowo, A.R.; Ridwan, R.; Muttaqie, T. On the Resistance to Buckling Loads of Idealized Hull Structures: FE Analysis on Designed-Stiffened Plates. *Designs* **2022**, *6*, 46.
https://doi.org/10.3390/designs6030046

**AMA Style**

Prabowo AR, Ridwan R, Muttaqie T. On the Resistance to Buckling Loads of Idealized Hull Structures: FE Analysis on Designed-Stiffened Plates. *Designs*. 2022; 6(3):46.
https://doi.org/10.3390/designs6030046

**Chicago/Turabian Style**

Prabowo, Aditya Rio, Ridwan Ridwan, and Teguh Muttaqie. 2022. "On the Resistance to Buckling Loads of Idealized Hull Structures: FE Analysis on Designed-Stiffened Plates" *Designs* 6, no. 3: 46.
https://doi.org/10.3390/designs6030046