An Updated Design Formula for Predicting the Compressive Strength of Plate: Elastic Buckling and Ultimate Compressive Strength

Abstract

In the present study, a simplified and useful design formula is proposed to predict the ultimate strength of a plate under longitudinal compression. The shape of the elastic buckling strength (${\sigma }_{xE}$) equation is utilised and adjusted to predict the ultimate compressive strength of the plate. In total, 600 cases of reasonable plate scenarios are selected to update the design formula by broadly considering the plate geometry (i.e., plate length, breadth, and thickness), material property (i.e., elastic modulus and yield strength), and initial deflection. The proposed formula, including the factor or coefficient for correction ($C_f$) may help ocean and shore (including onshore, offshore and nearshore) structural engineers improve safety and design the unstiffened plate element used in shipbuilding and oil and gas.

1. Introduction

Generally, plated structures utilised in ships and offshore structures are assembled by welding with various stiffeners to satisfy the required structural performance. As highlighted by Paik [1] and IACS [2], in recent years, the Ultimate Limit State Design (ULSD) approach has received increasing attention in structural design from Working Stress Design (WSD). Due to the predominant vertical bending moments on ocean mobilities, such as ships and ship-shaped offshore structures, the tertiary members (i.e., plates and stiffeners) that compose the mid-ship section of the ocean mobility are subjected to compressive and tensile loads. In particular, stiffened panels at the deck and bottom are severely exposed by axial loadings that are proportionate to their distance from the neutral axis. In the case of tension, as shown in Figure 1 (green-coloured line), it is often assumed that the material model maintains its yield strength after yield, which is typically Elastic Perfectly Plastic (EPP) or perfectly bi-linear behaviour, while in the case of compression, the prediction of ultimate strength performance, including buckling collapse characteristics, is important.

Regarding the ultimate compressive strength of the local elements, several studies have been conducted to investigate the collapse strength due to the compression being significantly lower than its expected strength (or yield strength), as shown in Figure 1 (red-coloured line). Historically, numerous studies have been carried out, and von Karman’s elastic buckling strength of the plate equation, which adopts the effective width (b_e) concept, can be considered a significant improvement in technology development. This is a result of the two-dimensional (2D) element (plate or shell). There was also a significant development in the one-dimensional element (column), which was established much earlier than the 2D.

For 1D, the simplest but well-known approach is the elastic buckling force ($P_E$) or elastic buckling strength (${\sigma }_E$) proposed by Euler. The well-known equations and detailed information can be referred to in Case I (Column buckling) and Case II (Plate buckling).

The well-known elastic buckling strength equation is summarised in Equation (1). In general, it is most commonly used for simple support conditions where the member is subjected to compression. In addition, it is characterised by the function of column slenderness ratio ($\lambda$) shown in Equation (1) and plate slenderness ratio ($\beta$) presented in Equation (2). An additional characteristic is that both of them (=slenderness ratios) are divided into geometric and material properties.

• Elastic buckling strength of the column (1D)

$${\sigma }_{xE(column)}={\displaystyle \frac{P_E}{A}}={\displaystyle \frac{{\pi }^{2}EI}{A{L}^2}}={\displaystyle \frac{{\pi }^{2}E}{\left(A/I\right)L^2}}={\displaystyle \frac{{\pi }^{2}E}{{\left(L/r\right)}^2}}={\displaystyle \frac{{\sigma }_Y}{{\lambda }^2}}$$

(1)

$$where\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\lambda =\sqrt{{\displaystyle \frac{{\sigma }_Y}{{\sigma }_E}}}=\underset{Geometric}{\underbrace{{\displaystyle \frac{L}{\pi r}}}}\times \underset{Material}{\underbrace{\sqrt{{\displaystyle \frac{{\sigma }_Y}{E}}}}}$$

• Elastic buckling strength of the plate (2D)

$${\sigma }_{xE(plate)}=k_{x\hspace{0.17em}\hspace{0.17em}(plate)}{\displaystyle \frac{{\pi }^{2}E}{12\left(1-{\nu }^2\right)}}{\left({\displaystyle \frac{t}{b}}\right)}^2=k_{x\hspace{0.17em}\hspace{0.17em}(plate)}{\displaystyle \frac{{\pi }^{2}D}{b^{2}t}}=k_{x\hspace{0.17em}\hspace{0.17em}(plate)}{\displaystyle \frac{{\pi }^2}{12\left(1-{\nu }^2\right)}}\left({\displaystyle \frac{1}{{\beta }^2}}\right)$$

(2)

$$where\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{k}_x={\left({\displaystyle \frac{mb}{a}}+{\displaystyle \frac{a}{mb}}\right)}^2\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\beta =\underset{Geometric}{\underbrace{{\displaystyle \frac{b}{t}}}}\times \underset{Material}{\underbrace{\sqrt{{\displaystyle \frac{{\sigma }_Y}{E}}}}}\hspace{0.17em}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}D={\displaystyle \frac{E{t}^3}{12\left(1-{\nu }^2\right)}}$$

Based on the elastic buckling strength concept, various formulae have been progressively and continuously developed over time to take into account the phenomena that may occur in reality (i.e., imperfections, yield effects, residual stress effects, slenderness effects, and many others). These are often referred to as plasticity-corrected formulae, and the most commonly used formula in the International Association of the Classification Societies (IACS) is the Johnson–Ostenfeld (J-O) formula, as presented in Equation (3). This is one of the concepts that compensates for the limitations of the Euler or Elastic buckling strength formula, which overestimates the buckling capacity of structures in relatively small slenderness regions. The intention behind the formula is to ensure that the elastic buckling strength of a structure is always less than the yield strength if it exceeds 50% of the material yield strength which was derived from the Johnson parabola equation.

• Johnson–Ostenfeld (J-O) formula

$${\sigma }_{cr(J-O)}=\left\{\begin{array}{c}{\sigma }_E\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}{\sigma }_E\le 0.5\hspace{0.17em}{\sigma }_Y\\ {\sigma }_Y\left(1-{\displaystyle \frac{{\sigma }_Y}{4{\sigma }_E}}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}{\sigma }_E>0.5\hspace{0.17em}{\sigma }_Y\hspace{0.17em}\end{array}\right.$$

(3)

Beyond the critical buckling strength (${\sigma }_{cr}$), recent attempts have been made to develop limit a state-based (${\sigma }_u$) structural safety assessment of structures. This basically adopts the effective width concept, as illustrated in Equation (4). For more information, the reader is recommended to refer to the ultimate limit state section of Paik’s book [1].

$${\sigma }_{xu}={\sigma }_Y\left({\displaystyle \frac{b_e}{b}}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{{\sigma }_{xu}}{{\sigma }_Y}}={\displaystyle \frac{b_e}{b}}$$

(4)

In this regard, a number of studies have been carried out to predict the ultimate compressive strength of structures. In particular, many studies have been carried out for plate elements or unstiffened panels [3,4,5,6,7,8,9,10] and stiffened plates or panels [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25] under compression. In some cases, analytical solutions have been derived based on governing equations and relevant boundary conditions, but recent studies have also proposed empirical expressions in the form of curve fitting based on various parametric studies [8,26,27,28,29,30] as shown in the Figure 2. In particular, many empirical expressions include the slenderness ratio of plates and columns (Plate slenderness ratio, $\beta$ and Column slenderness ratio, $\lambda$). As a typical example, the following equations have been developed for the empirical ultimate strength of plates (Equations (5a) to (5d)). A number of equations have also been developed for the compressive ultimate strength of stiffened panels (Equations (6a) to (6d)).

• <Case I>: ULS of un-stiffened panel (= plate) in longitudinal compression: Typical example of the existing formulae.

• Faulkner [31]

$${\displaystyle \frac{{\sigma }_{xu}}{{\sigma }_Y}}=\left\{C_1/\beta -{C_2/\beta }^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\beta \ge 1.0\right.$$

(5a)

where C₁ = 2.0 and C₂ = 1.0 for the four-edge simply-supported condition, C₁ = 2.25 and C₂ = 1.25 for the four-edge clamped condition.

• Cui and Mansour [32]

$${\displaystyle \frac{{\sigma }_{xu}}{{\sigma }_Y}}=\left\{\begin{array}{c}1.0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\beta \le 1.9\\ 0.08+1.09/\beta +{1.26/\beta }^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\beta >1.9\end{array}\right.$$

(5b)

• Paik et al. [33]

$${\displaystyle \frac{{\sigma }_{xu}}{{\sigma }_Y}}=\left\{\begin{array}{l}-0.032{\beta }^4+0.002{\beta }^2+1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\beta \le 1.5\\ 1.274/\beta \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.5<\beta \le 3.0\\ 1.248/{\beta }^2+0.283\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\beta >3.0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\end{array}\right.$$

(5c)

• Kim et al. [8]

$${\displaystyle \frac{{\sigma }_{xu}}{{\sigma }_Y}}=1-e\left({\displaystyle \frac{c_1}{\beta }}+{\displaystyle \frac{c_2}{{\beta }^2}}+{\displaystyle \frac{c_3}{{\beta }^3}}+c_4\right)$$

(5d)

where sub-coefficients (for $c_1$ to $c_4$) represent the amount of the initial deflection and can be referred to in Kim et al. [8].

• < Case II > ULS of the stiffened panel in longitudinal compression: Typical example of the existing formulae.

• Paik and Thayamballi [34]

$${\displaystyle \frac{{\sigma }_{xu}}{{\sigma }_{Yeq.}}}={\displaystyle \frac{1}{\sqrt{0.095+0.936{\lambda }^2+0.17{\beta }^2+0.188{\lambda }^2{\beta }^2-0.067{\lambda }^4}}}\le {\displaystyle \frac{1}{\sqrt{{\lambda }^2}}}$$

(6a)

• Zhang and Khan [10]

$${\displaystyle \frac{{\sigma }_{xu}}{{\sigma }_{Yeq.}}}={\displaystyle \frac{1}{{\beta }^{0.28}}}{\displaystyle \frac{1}{\sqrt{1+{\lambda }^{3.2}}}}$$

(6b)

• Kim et al. [30]

$${\displaystyle \frac{{\sigma }_{xu}}{{\sigma }_{Yeq.}}}={\displaystyle \frac{1}{0.884+e^{{\lambda }^2}}}+{\displaystyle \frac{1}{0.4121+e^{{\lambda }^2}}}$$

(6c)

• Kim et al. [28]

$${\displaystyle \frac{{\sigma }_{xu}}{{\sigma }_{Yeq.}}}=\left[\begin{array}{l}{c}_0+\left(c_1+c_2\sqrt{\lambda }+{\displaystyle \frac{c_3}{\beta }}+c_4{\displaystyle \frac{h_w}{t_w}}+c_5\sqrt{{\displaystyle \frac{I_{pz}}{I_{sz}}}}\right)\sqrt{\lambda }\\ +\left(c_6+{\displaystyle \frac{c_7}{\beta }}+c_8{\displaystyle \frac{h_w}{t_w}}+c_9\sqrt{{\displaystyle \frac{I_{pz}}{I_{sz}}}}\right){\displaystyle \frac{1}{\beta }}\\ +\left(c_{10}+c_{11}{\displaystyle \frac{h_w}{t_w}}+c_{12}\sqrt{{\displaystyle \frac{I_{pz}}{I_{sz}}}}\right){\displaystyle \frac{h_w}{t_w}}+\left(c_{13}+c_{14}\sqrt{{\displaystyle \frac{I_{pz}}{I_{sz}}}}\right)\sqrt{{\displaystyle \frac{I_{pz}}{I_{sz}}}}\end{array}\right]\le 1.0$$

(6d)

where sub-coefficients for T-bar can be referred to in Kim et al. [28].

A detailed review of empirical equations for stiffeners can be found in Kim et al. [35]. However, the existing empirical formulae are still complex in terms of their shape, and it would be more effective if simple equations could be used to predict the ultimate strength performance. It is worth remembering that even very complex formulae can be analysed using deep learning techniques (ANN, DNN, many others) [36,37,38,39,40,41,42,43,44], but classification rules and design guidelines still favour simple formulae. In this regard, this study aims to propose a simplified empirical formula for predicting the ultimate strength performance of flat-plate elements used in ocean mobilities subjected to longitudinal compression. For simplification, the well-known elastic buckling strength expression for plates (Equation (2)) is utilised, and a new correction factor ($C_f$) is derived to estimate the compressive ultimate strength behaviour of plates by utilising $k_x$, which usually implies boundary condition effects. The results of this study are considered to be valuable data for evaluating the ultimate compressive strength performance of plate elements used in ocean mobilities.

Figure 2. Comparison of the existing design formula for un-stiffened panel (plate element) [8,31,32,33].

Figure 2. Comparison of the existing design formula for un-stiffened panel (plate element) [8,31,32,33].

2. Methodology

2.1. General

As mentioned earlier, the elastic buckling strength of the plate (${\sigma }_{xE}$) under longitudinal compression can be predicted by Equation (3), and it can also be expressed by adding coefficient ($C_f$), as summarised in Equation (7).

$$ULS={\displaystyle \frac{{\sigma }_{xu}}{{\sigma }_Y}}=\boxed{C_f}\times \underset{{\sigma }_{xE}}{\underbrace{k_x{\displaystyle \frac{{\pi }^{2}E}{12(1-v^2)}}{\left({\displaystyle \frac{t}{b}}\right)}^2}}\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}{\sigma }_{xu}={\sigma }_Y\boxed{C_f}\times {\sigma }_{xE}$$

(7)

where $k_x={\left({\displaystyle \frac{mb}{a}}+{\displaystyle \frac{a}{mb}}\right)}^2$, $\upsilon$ = Poisson’s ratio, $t$ = plate thickness, $b$ = plate width, $E$ = material elastic modulus, $C_f$ = coefficient to convert elastic buckling to ultimate compressive strength behaviours.

In particular, $C_f$, the main idea of this study, can consider the boundary conditions of the plate, the initial deflection amount, and have excellent scalability by considering the wide range of the plate slenderness ratios (=$\beta$, function of plate geometry and material property presented in Equation (2)). In addition, the relationship between the elastic buckling strength and the ultimate strength of the plate is compared and verified. Finally, various types of deflection shapes exist for the initial deflection, as stated by researchers [7,45,46]. However, the buckling mode shape-based initial deflection, which is most commonly used, was assumed. In reality, the initial deflection shape of the plate is complicated, i.e., hungry-horse mode, mountain mode, spoon mode, sinusoidal mode, buckling mode and others [45,47]. It means that the mode shape is not clear due to the welding [33]. However, when comparing the effects on ULS of the hungry-horse (HH) mode, Admiralty Research Establishment (ARE) mode, and critical buckling (CM) mode (represented by the sine wave) as recently investigated by shown in Shen et al. [48], it can be seen that the buckling mode provides the most conservative ULS estimates for the four specific grillage models. The situation reverses, however, when the initial deflection of the plate is smaller than the average level (0.1β²t). Since the average level is generally assumed, it is considered logical to apply it in the present study. In this regard, we have included additional explanations in the manuscript.

The objectives of the present study are summarised as follows.

• To propose an empirical expression to predict the ultimate compressive strength of a plate and derive a correction factor ($C_f$) that can take into account initial deflection and different plate sizes.
• To verify the applicability of the developed empirical formula.
• To investigate the correlation between elastic buckling strength and the ultimate strength of a plate.

The results of this study (=updated formulae) are expected to help structural engineers predict the ultimate strength performance of plate elements used in ocean mobilities.

2.2. Development of the Correction Factor (C_f)

The overall figure for predicting the ultimate limit state (ULS) of the plate under longitudinal compression, including the correction factor ($C_f$) development procedure, is summarised in Figure 3.

2.2.1. Collection of Plate Data and Definition of Plate Structure Characteristics

In the case of plates, 600 cases of plates were selected. In brief, the selection process was based on the results of Kim et al. [8], presented in Figure 4, which extracted the plate properties used in a total of 12 ships of various sizes, especially in the mid-ship section, and analysed their probabilistic characteristics. The geometries such as plate slenderness ratio, length ($a$, length), width ($b$, breadth), and thickness (t, thickness) are defined. In addition, material properties should also be defined. Poisson’s ratio is typically assumed to be 0.3, and for elastic modulus ($E$), 205.8 GPa is applied in shipping industries. The material yield strength is one of the important factors where mild steel (${\sigma }_Y$ = 235 MPa) and high tensile steel (AH32 = 315 MPa) are generally used in shipyards. In this study, additional values of 355 MPa and 390 MPa were considered to investigate the material yield strength effect.

2.2.2. Selection of the Scenarios

As mentioned at the end of Section 2.2.1, an improved empirical formula is proposed in this study that can take into account both the effect of material yield strength and the initial deflection of the plate. In the case of the material model, we employed the Elastic Perfectly Plastic (EPP) model. The details of the selected scenarios can be referred to in

Table 1. Summary of the analysis scenarios adopted.

Properties & Conditions		Number of Scenarios	Selected Scenarios
Geometric	Plate length (a)	1	4150mm
	Plate breadth (b)	1	830mm
	Plate thickness (t)	50	See Appendix A
Material	Yield strength (σY)	4	235,315,355 and 390 MPa
	Elastic modulus	1	205.8 GPa
Initial imperfections	Initial deflection (CID = initial deflection coefficient)	3	0.025 (slight), 0.10 (average) and 0.30 (severe)
	Residual stress	N/A	N/A
Boundary condition		1	Simply-supported
Loading condition		1	Longitudinal compression
$Total scenarios : 1\times 1\times 50\times 4\times 1\times 3\times 1\times 1=600 cases$

.

2.2.3. Calculation of Ultimate Limit Strength for Selected Scenarios

In this study, the ultimate strength characteristics of plate elements were obtained using a numerical method (= ANSYS nonlinear finite element analysis, NLFEA). It is recognised that there are various ways to predict structural behaviour, such as experimental, numerical, and analytical methods. Regarding the ultimate compressive strength behaviour of the plate and stiffened panels, several studies [49,50,51,52,53,54,55,56] have been conducted on the experimental method, and the numerical methods are widely adopted from the validation results nowadays. One typical result can also be referred to in Kim et al. [5,57]. In their study, the ultimate compressive strength test of the curved plates was conducted and validated with the design formula.

The compressive load related to buckling collapse was applied. In the case of boundary condition (BC), the simply-supported BC was considered, as illustrated in Figure 5 (lefthand side), which enables the confirmation of the lowest structural capacity. Mesh size also plays an important role in FE analysis. A previous study by Kim et al. [58] recommended a number of elements (NoE = 10) in the plate width direction for flat plates and NoE = 20 or more for curved plates. Recently, Wang et al. [3] also provided relevant guidance to utilise the 3D solid element for ULS analysis. Since this study deals with flat plates, the NoE = 10 elements in plate width direction using 2D shell elements are reasonably applied, as shown in Figure 5. In the case of the initial deflection mode, the first buckling mode from the Eigen buckling analysis was utilised. Regarding the FE analysis technique, displacement control was used. It is also available to achieve the ULS from the load control option.

The ULS data were collected from the selected scenario-based NLFEM, and the effect of the initial deflection amount was analysed by yield strength, as shown in Figure 6. It can be seen that the larger (=severe) the initial deflection amount, the lower the behaviour of ULS, regardless of the yield strength. It means that the initial deflection amount more significantly affects the ULS than material yield strength. In Equation (7), where $C_f$ is the FEM data of the ULS on the left side, the right side can be calculated from the plate dimensions. Since simply-supported BC is only considered in this study, the value of $k_x$ is considered to be 4.0. Figure 7 is a $C_f$ versus plate slenderness ratio ($\beta$) diagram, as initial deflection amount and yield strength vary.

2.2.4. Development of Correction Factor (or Sub-Equation) ($C_f$)

To calculate the coefficient $C_f$ in Equation (7), the sub-equations were developed by regression analysis based on polynomial and logarithmic forms. From Figure 8, it can be seen that the behaviour of $C_f$ changes around beta at 1.9. Several scholars, i.e., von Karman and Cui and Mansour, derived that a plate slenderness ratio of 1.9 is often considered to distinguish between thin and thick plates – for example, a thin plate when it is larger than 1.9 and a thick plate when it is smaller. Similarly, when analysing the present results, we also found that the tendency varies around the 1.9 threshold, which we used to develop the empirical formula.

It seems that the influence of the plate’s ULS behaviour also affects the $C_f$ trend. To ensure accuracy, correction factors (or sub-equations) (Equation (8)) were developed by dividing the ranges based on beta at 1.9.

Table 2. (a) Obtained sub-coefficients from curve fitting (β < 1.9); (b) Obtained sub-coefficients from curve fitting (β ≥1.9).

(a)
β < 1.9	Initial deflection amount (CID)
Sub-coefficients	0.025	0.05	0.10	0.15	0.20	0.25	0.30
f1	−0.212	0.057	0.204	0.185	0.131	0.085	0.049
f2	0.804	−0.518	−1.149	−0.967	−0.647	−0.385	−0.188
f3	−0.846	1.447	2.341	1.844	1.165	0.637	0.252
f4	0.684	−1.003	−1.515	−1.025	−0.452	−0.028	0.266
f5	−0.154	0.293	0.394	0.236	0.070	−0.047	−0.124
R2	0.999	0.999	0.999	0.999	0.999	0.999	0.999
(b)
β ≥ 1.9	Initial deflection amount (CID)
Sub-coefficients	0.025	0.05	0.10	0.15	0.20	0.25	0.30
g1	0.125	0.127	0.118	0.117	0.118	0.123	0.124
g2	2.042	2.036	2.095	2.112	2.119	2.099	2.100
g3	0.389	0.342	0.303	0.261	0.229	0.191	0.170
R2	0.999	0.999	0.999	0.999	0.999	0.999	0.999

a,b summarise the coefficients obtained from the regression analysis according to the amount of initial deflection. To verify the accuracy, we calculated the statistical value of $R^2$ and confirmed its applicability (above 0.99 on average).

$$\hspace{0.17em}{C}_f=\left\{\begin{array}{c}{f}_1{\beta }^4+f_2{\beta }^3+f_3{\beta }^2+f_4\beta +f_5\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\hspace{0.17em}\beta <1.9\\ g_1{\beta }^{\left(g_2\right)}+g_{3\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}\beta \ge 1.9\end{array}\right.$$

(8)

where f₁ to f₅ = sub-coefficients for β < 1.9 (Table 2a) and g₁ to g₃ = sub-coefficients for β $\ge$ 1.9 (Table 2b).

2.2.5. Validation of the Correction Factor ($C_f$) Developed

Figure 9 presents the validation results of the ultimate strength predicted by the proposed formula in this study (Equation (9)) with the NLFEM analysis results. It can be seen from the mean values and coefficient of variation (COV) that the developed design formula predicts the compressive ultimate strength of the plate considering three initial deflection levels (0.025, 0.1 and 0.3${\beta }^{2}t$) with good agreement. Figure 10 compares the ULS behaviour according to beta with different amounts of initial deflection.

• Proposed design formula

$$\hspace{0.17em}{\displaystyle \frac{{\sigma }_{xu}}{{\sigma }_Y}}=C_F\times {\sigma }_{xE}$$

(9)

where ${\sigma }_{xE}=k_x{\displaystyle \frac{{\pi }^{2}E}{12(1-v^2)}}{\left({\displaystyle \frac{t}{b}}\right)}^2$ = elastic buckling strength of the plate.

3. Discussions on the Proposed Design Formula (PROS and CONS)

The ultimate strength of the plate predicted by the updated design formula was compared with the elastic buckling strength calculated from Equation (3). Figure 11 shows the deviation of the ultimate compressive strength from the elastic buckling strength, and Figure 12 presents the comparison results by initial deflection.

Using the elastic buckling strength formula instead of the plastic correction Johnson–Ostenfeld equation, it was found that for a plate slenderness ratio ($\beta$) at 2.2 and below, the elastic buckling strength is on average 51.7% higher than the ultimate strength, while for $\beta$ at 2.2 and above, the ultimate strength is on average 29.7% higher than the elastic buckling strength. Although the existing design criteria are based on the elastic zone, the safety margin should be accurately predicted because it directly relates to cost efficiency and safety. It is believed that the proposed formula enables the calculation of a more accurate structural capacity by utilising the elastic buckling strength and ultimate limit state, which can be calculated simultaneously with a simplified correction factor (or sub-equation $C_f$).

The obtained formula was compared with the NLFEM results and existing formulae, as shown in Figure 13. In the case of the Faulkner equation (Eq. 5a), ULS is set as 1.0 when the plate slenderness ratio ($\beta$) is less than 1.0, which means that the obtained ultimate compressive strength of the plate cannot exceed the material yield strength. One more thing here is that the proposed formula is based on the NLFEM results from the present study. This means that the other formulae are derived from the other datasets. For example, Kim et al. [8] proposed an advanced design formula for predicting the ULS of the plate under compression. They only considered two types of materials, i.e., mild steel and HT32 (yield strength = 235 and 315 MPa). In this study, we added two more materials, HT 36 and HT40, and mean and COV were slightly updated. It is important to clearly acknowledge the limitations of claiming superiority of empirical results obtained from one dataset over other formulae.

In addition, we can also consider the effect of the plate aspect ratio (=a/b or plate length/plate width) on the ultimate strength of the plate in longitudinal compression. In this study, we adopted a single value of the plate aspect ratio (a/b = 5.0), because the aspect ratio has less effect on the ULS of the plate as shown in Figure 14. If the aspect ratio (a/b) is not an integer, it is available to calculate the ULS by utilising the result that satisfies the nearest integer to that value. This may enable predicting the lowest ULS.

4. Conclusions and Limitations

In this study, the elastic buckling strength formula for plates was updated to include the initial deflection, boundary conditions and material yield strength of plates. In particular, Equation (7) represents the core of the present study, and the developed $C_f$ is used to implement the various effects mentioned above. The results obtained in this study are summarised as follows.

• A design equation for predicting the ultimate compressive strength of plates was developed, which took the form of updating the elastic buckling strength equation. In particular, through the development of the $C_f$, an equation was developed that can take into account various conditions such as initial deflection and yield strength.
• The developed empirical formula showed an accuracy of $R^2$ = 0.99 compared to the results of nonlinear finite element analysis (NLFEA), which proves its applicability.
• The correlation between the elastic buckling strength and the ultimate strength of the plate was investigated. Compared to the elastic buckling strength, the compressive ultimate strength decreases as beta increases, with an average decrease of 29.7%. The details can be found in the developed $C_f$ and ULS relationship. However, for thin plates with a large slenderness ratio, the ULS being greater than the elastic buckling strength is the background for introducing the ultimate strength criteria. Therefore, the percentage numbers many not be importantly considered.
• The applicable range of the plate slenderness ratio shall be clearly presented. As shown in Figure 15, it is recommended to utilise the proposed formula in the $0.067\le \beta$ range.

Figure 15. Applicable range of the proposed formula.

Figure 15. Applicable range of the proposed formula.

This study conducted an in-depth investigation of the elastic buckling and ultimate compressive strength of a plate subjected to longitudinal compression. This is because it is the dominant load direction resulting from vertical bending motions on ocean mobilities (i.e., ships, ship-shaped offshore structures, and others). However, these structures are exposed to a variety of combined loads, and further research should be carried out on combined loading, along with lateral pressure, transverse compression, and shear. In addition, other boundary conditions should also be considered, which will be the subject of future research.