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ISSN : 1598-6721(Print)
ISSN : 2288-0771(Online)
The Korean Society of Manufacturing Process Engineers Vol.20 No.1 pp.80-87
DOI : https://doi.org/10.14775/ksmpe.2021.20.01.080

# A Study on the Buckling Strength of Stern Skeg Shell Plate

Kyung-Shin Choi*, Sang-Seok Seol*, Jin-Woo Kim*, Seok-Hwan Kong*, Won-Jee Chung*#
*School of Mechanical Design Engineering, Changwon National University
#Corresponding Author : wjchung@chanwon.ac.kr Tel: +82-55-213-3624, Fax: +82-55-263-5221
04/11/2020 26/11/2020 27/11/2020

## Abstract

Most container ships are currently being constructed as Ultra-Large Container Ships. Hence, the equipment of the ships is also becoming relatively large. In particular, propellers, rudders, and rudder stocks are large in the stern structure, and in relation, efficient design of the hull structures to safely secure these parts is important. The bottom shell plate surface of a stern skeg is a perforated plate from which the rudder stock penetrates, so it is an important component for the stern structure. In this paper, to determine the critical buckling of the shell plate, an interaction curve equation for the two-axis compression of the shell plate was derived using the maximum value of the static structural stress multiplier in a load multiplier mode. This equation predicts the timing of the buckling occurrence. By analyzing this interaction curve equation, the buckling behavior of the plates subjected to a combination load was determined and the usefulness of applying it to ship building was investigated.

# 선미 스케그 외판의 좌굴강도에 관한 연구

최 경신*, 설 상석*, 김 진우*, 공 석환*, 정 원지*#
*창원대학교 일반대학원 기계설계공학과

## 초록

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

## 2. Skeg Bottom plate Model for Actual Vessels

### 2.1 Target Model

Buckling, which lacks structural rigidity, typically represents a failure mode that loses stability. Because Euler's formula, which generates a narrow curve with respect to critical stress, does not provide an accurate expression for a specific shape, the critical buckling strength value in most of the classification rules is calculated by substituting and plastically modifying the elastic buckling strength using the approach by Johnson-Ostenfeld. However, because the critical buckling strength value obtained by this plastic modification tends to be higher than the actual value in the perforated plate, the calculated values are different from the property values of the perforated plates used in actual vessels. In particular, the form factor of the bottom section in the case of the stern skeg bottom plate is nonlinear, which is not found in the non-perforated and perforated plates of conventional vessels. Because the stern skeg bottom plate is a part that receives large force in the stern of the vessel, and the perforation ratio varies depending on the size of the rudder stock, the stern skeg bottom plate is crucial in the stern of the vessel. The shape of the stern skeg bottom plate is determined by the form factor of the section to calculate the area of the rudder and the rudder force. In this study, the form factor of the rudder section in the actual vessels used in shipyards was surveyed, as shown in Table 1. The vessels were sorted by vessel type, and the one with the highest rudder force from the data was selected as the target model. In the survey of the rudder force by vessel type, and the cross-sectional form factor of the rudder, which are most commonly used in actual vessels, the rudder force of the large container vessel was the largest, and the cross-sectional form factor in this case was confirmed to be of the hollow type. Fig. 1 shows the shape of the nonlinear perforated skeg bottom plate in container vessels.

### 2.2 Finite Element Analysis (FEA)

To perform the FEA, the size of the perforation (d), aspect ratio (a/b), and thickness (t) were set as the main variables for the stern skeg, as shown in Fig. 2. The main dimensions and properties used in the analysis are as follows: length of plate (a): 10400mm; maximum width (b): 2718mm; thickness: 100mm in the center of the perforation and 49mm in the bow/stern direction; Poisson's ratio (ν): 0.3; elastic modulus (E): 205.8 GPa; and yield stress (σy ): 235 MPa. The skeg has a nonlinearity different from the shape of the conventional square-shaped perforated plate, which was obtained by calculating the ratio of the length and width of the plate as area. The thickness of the plate is 100mm from the center of the perforated circle by 2m in the forward and rear directions, and the thickness otherwise is 49mm. Furthermore, as the length of the plate exceeds 10m, the buckling factor was introduced, and the critical stress value was calculated as the boundary condition of the combined load subjected to simultaneous biaxial compression due to the characteristics of the structure applied in actual vessels. The analysis was performed on the assumption of elastic-perfectly plastic behavior of property materials.

### 2.3 Elastic Buckling by Biaxial Compression

The boundary condition that best represents the behavior related to the buckling strength in the actual plate member is the four-sided simple support condition presented by each class. Thus, the plate member constituting the structure is a continuous structure that is connected to other plate members in the structure. The coupling condition constraining the displacement in the in-plane direction is applied, and the analysis is performed to maintain all four sides straight with respect to the in-plane load. However, because the buckling calculation formula in each class is composed of only a function of the size of the perforated plate and the plate’s aspect ratio tends to be evaluated higher than the actual final strength, the structures in actual vessels are often subjected to complex compression of two or more axes and buckling of simple support plates. As in Fig. 2, the curve (a) is the side of the length parallel to σax and (b) is parallel to σay . Thus, the work (W) applied to biaxial compression can be expressed as Equation (1)[10]as follows.

$W = t π 2 8 ∑ m = 1 ∞ ∑ n = 1 ∞ C m n 2 ( b σ a x a m 2 + a σ a y b n 2 )$
(1)

Here, the length of the plate a is in the x direction, and its width b is in the y direction. m, n refers to the number of sine half-waves in a x, y axis direction, and the aspect ratio (a/b) is defined as δ and shown in Equation (2). The strain energy of the plate subjected to biaxial compression is also expressed as the aspect ratio δ, as shown in Equation (3).

$W = a t π 2 8 b ∑ m = 1 ∞ ∑ n = 1 ∞ C m n 2 [ ( m δ ) 2 σ a x + n 2 σ a y ]$
(2)

$U = a b π 2 8 D ∑ m = 1 ∞ ∑ n = 1 ∞ C m n 2 [ ( m a ) 2 + ( n b ) 2 ] 2 = a π 4 8 b 3 D ∑ m = 1 ∞ ∑ n = 1 ∞ C m n 2 [ ( m δ ) 2 + n 2 ] 2$
(3)

If work (W) and strain energy (U) are equal, it can be expressed as in Equation (4) according to the principle of virtual work.

$[ ( m δ ) 2 σ a x + n 2 σ a y ] c r = D π 2 b 2 t [ ( m δ ) 2 + n 2 ] 2$
(4)

In addition, to derive the interaction curve for biaxial compression, the critical stress equation of the long plate and the wide plate was substituted, as shown in Equation (5), where (σax)cr refers to the critical stress in the σax direction and (σay )cr refers to the critical stress applied only in the (σay ) direction. D refers to the bending stiffness of the plate, which is a function of t as a variable where the plate thickness is determined at the design stage.

$( σ a x ) c r , 1 = 4 D π 2 b 2 t , ( σ a y ) c r , 1 = [ 1 + ( 1 δ ) 2 ] 2 D π 2 b 2 t$
(5)

Based on Equation (5), Equation (4) was made dimensionless, and the interaction curve for biaxial compression, which is a complex stress state, was expressed as in Equation (6).

$m δ 2 σ a x σ a x c r , 1 + n 2 σ a y σ a y c r , 1 1 4 δ 2 + 1 δ 2 2 c r = 1 4 m δ 2 + n 2 2$
(6)

To obtain the curve for the critical buckling stress of the biaxial load, the graph shown in Fig. 3 was obtained by substituting the interaction curve in Equation (6) when the aspect ratio (a/b) of the skeg deck is 2, 3, and 4, respectively. The x and y axes correspond to σax/(σax )cr,1 and σay/(σay )cr,1, and various straight lines are illustrated according to the value of m, which is the number of sine half-waves in each axis direction. Moreover, the critical buckling stress curve for the biaxial load can be obtained by connecting the tangent lines to the innermost lines. Based on this calculation, because the stress ratio of the critical stress value of the wide surface vulnerable to buckling around the perforated circle in the nonlinear skeg bottom is σax = 0.39 σay, δ = 3.9, a substitution by Equation (4) results in simplification, as in Equation (7). Fig. 4 represents the critical buckling stress for each axis direction as a result of the number of sine half-waves for the short and long axes, where n is always 1 as the number of sine half-waves for the short axis. In addition, when m is 1, 2, 3, and 4,

$m 3.9 2 + 0.2 n 2 σ a x c r = π 2 718 2 100 * 205 , 800 100 3 100 1 − 0.3 2 * m 3.9 2 + n 2 2 m 3.9 2 + 0.2 n 2 2$
(7)

which is the number of sine half-waves for the long axis, it shows the results of the critical stress values. As shown in the results, for the maximum strain m, which represents the strain the skeg bottom can withstand without being destroyed, σax is 1,386 MPa and σay becomes 355 MPa at the stress ratio of 3.9.

## 3. Evaluation of Strength of Skeg

### 3.1 Strength Analysis of Skeg under Biaxial Compressive Load

To examine the buckling safety of perforated plates subjected to combined load, this study set design parameters for the perforation diameter of the skeg bottom based on actual vessels and further considered the size of the perforation, slenderness ratio, and properties of plate thickness. ANSYS linear buckling V13, a commercial finite element structural analysis program, was used to perform the buckling strength analysis for the target model. A 100-mm (wide/long) grid was formed to closely discriminate the behavior of the perforated column, which has 68,870 nodes and 12,950 elements, and a shell element with six degrees of freedom was applied at each node of the finite element. Because the skeg bottom is a typical cantilevered structure, the fixed support point was selected as the end of the hull 3m above the skeg bottom. The size of the perforation occupies 20% of the skeg deck, which is 2,000mm long in the bow and stern directions of the hull from the center of the aforementioned perforation and 100mm thick. The thickness otherwise is set as 49mm. The analysis was conducted with the strongest rudder force of 9,672kN among the possible resistances at the upper structure of the vessel’s stern. Fig. 5 shows the result of static structural stress, indicating that the load is concentrated due to the influence of the width direction where the section modulus is small around the perforation and that the allowable stress is satisfied because the maximum stress is 78 MPa.

Fig. 6 shows the results for the stress equivalent to elastic strain, and as shown in the results, the strain due to compression from the vertical strain is determined as 0.0003mm/mm compared to the initial length in the longitudinal direction, which indicates that there is a sufficient margin to reach the ultimate strength. Fig. 7 shows that the load in the width direction with a smaller section modulus than the longitudinal direction with a large section modulus around the perforation occurred as the maximum strain.

## 4. Conclusion

The hull’s skeg bottom is located at the stern of the vessel, and despite its 100% exposure to the rudder force in the form of a cantilever, there is no calculation formula for calculating the thickness, because each classification rule is missing. For this reason, the thickness has been generally applied based on experience. In particular, the skeg bottm has a typical shape of a hull with a perforated plate through which the rudder stock passes while functioning as a watertight bulkhead. The perforated plate of the hull is the most important design factor for a vessel, and the phenomenon of buckling behavior must be understood. In addition, when a notch occurs at the thick plate of the hull of 65mm or more, the stress typically propagates straight through the cracks regardless of the stress proportionality. Thus, although a thick plate that guarantees brittle crack arrestability should also be considered, a general thick steel plate is used. To solve this problem, first, the critical stress was theoretically calculated using the interaction curve equation according to the combined load for the buckling behavior of the stern skeg bottom in which the biaxial compression acts, and the following conclusions were drawn through the linear buckling analysis.

• 1. The excessive thickness of the hull plate around the perforation was applied in the design although the stress was 78 MPa, which is far below the allowable yield stress, even when applying the rudder force of a large container ship, which can occur at the stern of the vessel, as the maximum force.

• 2. According to the result obtained by the theoretical curve equation for the critical buckling stress regarding elastic buckling by biaxial compression, σax is 1386 MPa and σay is 355 MPa. Considering 1404 MPa at the stress ratio 3.9 in the analysis, σay is 360 MPa, which is similar to the theoretical value. As a result, the maximum strain was confirmed at the aspect ratio of 3, and this result shows that the section modulus in the width direction from the perforated skeg bottm is more important than the thickness. An optimized design is necessary with its focus on section geometry to secure the section modulus.

• 3. In the case of the skeg bottom, the thickness has been determined based on experience without the calculation formula of the classification due to the long-lasting patents of foreign companies. However, due to the expiration of the patent period, revision of the calculation formula is required for each class. This study first identified the elastic buckling phenomenon due to the biaxial compressive load of the skeg bottom for the buckling behavior and further investigated the suitability of the skeg bottom applied to actual vessels. The results of this study will provide basic data to develop a calculation formula that determines the thickness of the stern skeg bottom in the future by evaluating the final strength with buckling.

## Figure

Nonlinear perforated SKEG bottom plate
Critical stress of plates with different aspect ratio under 2 axis compression
Buckling critical stress value for sine half waves length
Von-Mises stress for SKEG bottom plate
Elastic strain for SKEG bottom plate
Deformation for SKEG bottom plate
Linear buckling for SKEG bottom plate

## Table

Investigated data of rudder profiles

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