A Simple Empirical Formula for Predicting the Ultimate Strength of Ship Plates with Elastically Restrained Edges in Axial Compression

An investigation is conducted on the static ultimate limit state assessment of ship hull plates with elastically restrained edges subjected to axial compression. Both material and geometric non-linearities were considered in finite element (FE) analysis. The initial geometric imperfection of the plate was considered, while the residual stress introduced by welding was not considered. The ultimate strength of simply supported ship hull plates compared well with the existing empirical formula to validate the correctness of the applied boundary conditions, initial imperfection and mesh size. The extensive FE calculations on the ultimate strength of ship hull plates with elastically restrained edges are presented. Then a new simple empirical formula for plate ultimate strength is developed, which includes the effect of the rotational restraint stiffness, rotational restraint stiffness, and aspect ratios. By applying the new formula and FE method to ship hull plates in real ships, a good coincidence of the results between these two methods is obtained, which indicates that the new formula can accurately predict the ultimate strength of ship hull plates with elastically restrained edges.


Introduction
The ultimate strength of ship hull plates is perhaps one of the most important issues to be considered in ship design. The collapse of the ship hull plates can lead to serious consequences for property and life security. Therefore, it is necessary to calculate the ultimate strength of structural components for ship structure design and strength assessment (IACS, 2006a).
The main research methods on the ultimate strength of ship hull plates are analytical methods, experimental tests, empirical approaches, and non-linear finite element (FE) simulation.
Up to now, a lot of researchers have devoted to the investigation on the ultimate strength of ship structures under axial compression. Paik and Pedersen (1996) and Cui et al. (2002) adopted analytical methods to study the ultimate strength ship structures, and some simplified analytical formulas were proposed to calculate the ultimate strength of ship plates. Since analytical methods can only deal with simple cases, more works are done by using other methods.
Experimental study on the ultimate strength of ship structures provides first-hand information to understand the collapse behaviors of ship structures and valuable data for verification of theoretical and numerical methods. A large number of experimental studies on the ultimate strength have been conducted, e.g. Tanaka and Endo (1988), Ghavami (1994), Chen et al. (2004), Ghavami and Khedmati (2006), Guedes Soares (2008a, 2008b). Based on a large number of test data, some simple available empirical formulas were developed to predict the ultimate strength of ship structures e.g. Guedes Soares and Gordo (1997), Frieze and Lin (1991). λ With the rapid development of computing technology, FE method has become a powerful tool for the non-linear analysis of large-scale finite element model of ship structures. Many investigators studied the ultimate strength of ship structures by FE method and derived some simple accurate empirical formulas. Paik and Duran (2004) derived an empirical formulation to calculate the ultimate strength of aluminum stiffened panels using the finite element method (FEM) with in the range of 0.23-2.24. Fujikubo et al. (2005aFujikubo et al. ( , 2005b proposed a formula to calculate the ultimate strength of steel stiffened plate under combined transverse compression and lateral pressure, which can account for the interactions influence of adjacent plate and stiffener. Paik (2007) carried out a series of axial compression tests and numerical simulations to study the ultimate strength of aluminum stiffened plates in marine applications and derived an empirical formula for the calculation of the ultimate compressive strength of aluminum stiffened plates. Khedmati et al. (2010) have also proposed an empirical formula for predicting the ultimate strength of aluminum stiffened plates subjected to combined axial compression and lateral pressure. Xu et al. (2018) studied the effect of lateral pressure and stiffener type on the ultimate strength of steel stiffened plates in in-plane compression using the FE method and derived an empirical equation for the estimation of the ultimate strength of stiffened plates subjected to combined axial compression and lateral pressure. Kumar et al. (2009), Cho et al. (2013 and Choung et al. (2014) have derived some useful empirical formulas to estimate the ultimate strength of stiffened plates, which consider different configurations of loads combinations including axial compression, transverse compression, shear force, and lateral pressure. Zhang (2016) presented a comprehensive review and study on ultimate strength analysis methods for steel plates and stiffened plate subjected to axial compressive loading and proposed a new simple formula to estimate ultimate strength, and the validation of the formula was verified by the experimental and numerical data of 180 specimens. Khedmati et al. (2016) developed empirical expressions to estimate the ultimate compressive strength of welded aluminum stiffened plates subjected to combined transverse in-plane compression and lateral pressure. Kim et al. (2017) developed an advanced empirical formulation to estimate the ultimate strength of the plate under longitudinal compression. Besides, Kim et al. (2018) presented a comprehensive review of existing empirical formulas for predicting the ultimate strength of stiffened plates subjected to longitudinal compressive loading and suggested that further studies be conducted on advanced empirical formulations shortly. Yang et al. (2018aYang et al. ( , 2019 proposed an empirical formula for predicting the dynamic ultimate compressive strength of steel ship plates subjected to in-plane impact loading. It is well known that the ultimate strength of ship structures has been well studied. However, the ultimate strength of ship hull plates with elastically restrained edges subjected to axial compressive loading is seldom reported. Some publications have pointed out that the boundary condition is a very important factor in the dynamic behavior of engineering structures (Paik and Thayamballi, 2000;Wang, 2016, 2017). In practice, a stiffened plate in a ship is supported by a variety of components along the edges which offer a certain amount of rotational restraint stiffness. Therefore, the assumption of simply supported or clamped boundary conditions may introduce errors. Here, we examine the effect of the rotational restraint stiffness on the dynamic buckling behavior of stiffened plate.
This paper aims to study the ultimate strength of ship hull plates with elastically restrained edges subjected to axial compression and develop a new empirical formula by using a large number of numerical results. The effects of rotational restraint stiffness, aspect ratio, initial imperfection, and plate slenderness ratio are included in the new empirical formula.
2 Non-linear FE analysis procedure calibrations

Empirical formula verification
In this section, the existing empirical formula is used to verify the FE modelling (boundary conditions, loading conditions, mesh sizes, and initial imperfection, etc.) established in this paper. The ship hull plates in the study of Paik et al. (2008) were used for the ultimate strength calculation. The geometrical dimensions of the plate are plate length a=4300 mm and plate width b=815 mm, which are chosen from a double hull oil tanker with 230 m in length. The plate thickness is in the range of 11.8 mm to 32 mm and the corresponding plate slenderness ratio is in the range of 1.0 to 2.7, where, t is the plate thickness, is the material yield stress, E is Yong's modulus. All of these plates are under simply supported boundary conditions. The initial geometric imperfection w 0 of the plate is usually represented by a Navier's double Fourier series function.
where the values of M and N depend on the complexity of the initial geometric imperfection shape. i and j are the numbers of half-waves in the plate length direction and plate width direction for initial deflection, respectively. w 0max is the maximum value of initial deflection function.
In general, only one component in Eq.
(1) is enough to express the initial geometric imperfection (Zhang and Khan, 2009).
The initial geometric imperfection shape adopted in this ship hull plate has five half-waves in the longitudinal direction and one half-wave in the transverse direction with the amplitude of 4.1 mm (b/200). This level of initial geometric imperfections satisfies the requirement of IACS Shipbuilding and Repair Quality Standard, (IACS, 2006b) which defines an average value of 4 mm and a limit of 8 mm for deck plate at parallel part.
The ultimate compressive strength of these ship hull plates is summarized in Table 1. It can be seen that the FE analysis results accord with available Faulkner formula results with a maximum error of 2.8%, which confirms that the present FE model analysis has good accuracy in predicting the ultimate compressive strength of ship hull plates. It is noteworthy that the average initial im-σ u σ y perfections and simply supported boundary conditions are considered in Faulkner's formula. The non-dimensional ultimate compressive strength ( / ) under clamped boundary conditions is also shown in Table 1.
The dimensionless ultimate strength ( / ) of the ship hull plate under the simply-supported and clamped boundary conditions against plate slenderness ratio curves are shown in Fig. 1. As shown, very significant differences can be found and the difference value between the two curves increases with plate slenderness ratio . The ultimate strength of ship hull plates under clamped boundary conditions is nearly the same as the one under simply supported boundary conditions when . This is because the plate with is so thick that the collapse mode is gross yielding, so the influence of boundary constraint is very limited. Fig. 2 shows the stress distribution and geometric deformation of the rectangular plate with at the ultimate limit state subjected to axial compressive loading under simply supported and clamped boundary conditions, respectively. It can be seen that gross yielding occurs at the ultimate limit state for both simply supported clamped constraints. A slight difference between the collapse modes for these two cases leads to very small differences of ultimate strength.

Experiment verification
The plate ultimate strength tests from Cui et al. (2002) are compared with the present FE results. Table 2 shows the comparison of the FE results with Cui et al.'s and tests res-ults of plates in Max (1971). Three different plates with simply-supported boundary conditions (a×b×t = 533 mm× 889 mm×12.44 mm, 533 mm×1371.6 mm×11.43 mm, 1526.36 mm×875.7 mm×14.17 mm) were chosen for validation of the present method. The specific boundary constraint can be found in Fig. 3, U and UR represent linear displacement and angular displacement, respectively. The present FE results agree well with the test and Cui et al.'s results. Cui et al. (2002) explained the reason for the large discrepancy for plate No. 02 that the slenderness ratio for this plate is relatively large and in this case the collapse modes and the critical load are hard to identify.
It can be concluded that the present FE model can ef- fectively predict the ultimate strength of a plate subjected to axial compressive loading. Then the effects of rotational restraint stiffness (K), aspect ratio (a/b), slenderness ratio ( ), ratio of b to t ( ) and material yield stress ( ) are studied by FE method, and the ultimate strength formula of ship hull plates with elastically restrained edges is derived.

FE models for analysis
3.1 Geometric data and material modelling ν ρ σ y β The geometry properties and coordinate system of the ship hull plate are shown in Fig. 4. Longitudinal, transverse and vertical directions are denoted as x, y, and z. Parameters a, b and t represent the length, width, and thickness, respectively. The material properties are Young's modulus E=2.06 MPa, Poisson's ratio =0.3, and density =7.8×10 −6 kg/mm 3 . The plate length used in this work is a=500 mm, and the aspect ratios are in the range of 1 to 4. The material yield stress of the typical ship steel plate is 315 MPa. The plate slenderness ratio of the deck or bottom structures in merchant ranges from 1.5 to 3.0 (Paik et al., 2008).
In the present study, the influence of aspect ratio, plate slenderness ratio, material yield stress and non-dimensional rotational restraint stiffness (Ka/D) on the ultimate strength are studied. Rotational restraint stiffness at boundary edges depends on the stiffener, which can be represented as K=GJ/a 2 (G=shear modulus, J=torsion constant of support member at edges, GJ=stiffeners torsional stiffness). The values for these parameters are summarized in Table 3. D is the panel stiffness which is expressed as follows:

Boundary and loading conditions
For the ultimate strength analysis, simplified boundary conditions (simply supported and clamped) are used to simulate the boundary constraints of support members at four edges. Pham and Hong (2017) and Xu et al. (2013) have pointed out that the boundary condition is a very important factor on the dynamic behaviour of engineering structures. In practice, the ship hull plate is supported by a variety of components along the edges which offer a certain amount of rotational restraint stiffness. Therefore, the assumption of simply supported or clamped boundary condition may introduce errors. Here, we examine the effect of the rotational restraint stiffness on the static ultimate strength of ship hull plate. Yang et al. (2018b) expressed that the effect of rotational restraint is considered by using the FE model extended along the longitudinal and transverse direction. As Tanaka et al. (2014) and Xu et al. (2018) indicated that the research models can be divided into four types according to the modelling size in each direction along the longitudinal girder and transverse frame, i.e. the "1/2+1/2 span & 1/2+1/2 bay", "1/2+1/2 span & 1/2+1+1/2bay", "1/2+1+1/2 span & 1/2+1/2 bay" and "1/2+1+1/2 span & 1/2 + 1+1/2 bay" models ("span" denotes the scope in longitudinal direction while "bay" denotes the transverse direction). In this paper, a model simply supported at transverse edges and elastic-  ally restrained at longitudinal edges is adopted to study the effect of rotational restraint stiffness and derive the new empirical for predicting the ultimate strength of ship hull plates.

Initial geometric imperfections
Initial geometric imperfections and welding residual stresses are produced due to processing and manufacturing, and they can result in lower stiffness and carrying capacity of ship structures. In the present study, only the influences of the initial geometric imperfections are discussed. It is thought that initial deflection which has the same shape with the plate buckling mode may give the lowest resistance to the external loading. Steen et al. (2004) studied the ultimate strength of the stiffened plate by considering the influence of the initial deflection of which the amplitude is and the shape is buckling mode. In the present paper, the plate buckling modes of rectangular plates with different aspect ratios are regarded as the initial deflection of the ship plating which follows Eq. (2) with an average level (w 0max =b/200), as shown in Fig. 5 (Paik et al., 2008).

Nonlinear FE mesh modelling
The S4R element type is a four-node doubly curved general-purpose shell that offers reduced integration to control hourglass instability, which is used to model the ship plating. There are six degrees of freedom for each element node. The influences of three different mesh densities on ultimate strength are studied. The ultimate strength of the ship hull plate for the three mesh sizes (b/l=100, b/l=50, and b/l=25) are 2.351×10 6 , 2.353×10 6 and 2.362×10 6 , respectively. l is the element length. To keep the balance between the result accuracy and calculation time, the model with a mesh size of b/l=50 is chosen.

Formulation development of ultimate strength of ship hull plates
To present the influence degree of rotational restraint stiffness on the ultimate strength of ship hull plates, a plate σ y with a=b=500 mm, b/t=51.6, =315 MPa, and E=2.058× 10 5 MPa is modelled. Fig. 6 shows the non-dimensional ultimate strength against axial displacement curves of ship hull plates under different boundary constraints. It can be seen that the ultimate strength increases with the rotational restraint stiffness. The ultimate strength of the ship hull plate with Ka/D≤0.5 is in a good agreement with that of the simply supported edges, while the ultimate strength is almost the same as the results at clamped boundary conditions when Ka/D≥250. The ultimate strength of the ship hull plates subjected to axial compressive load is calculated by FE method. The rotational restraint stiffness was applied to the FE model by creating spring at the four edges in the "Interaction module" of ABAQUS software. Table 4 shows the non-dimensional ultimate strength ( / ) of the ship hull plates with a=b=500 mm, w 0 =2.5 mm, and different plate slenderness ratios. It can be seen that the ultimate strength of ship plates decreases with the increase of the plate slenderness ratio while increases with the rotational restraint stiffness (Ka/D). The maximum value at the case Ka/D=500, is nearly three times greater than the minimum value at the case Ka/D=0.5, . The curves of the non-dimensional ultimate strength against plate slenderness ratio are shown in Fig. 7. To present the variation laws of each curve clearly, only five cases are chosen to show in the figure. As shown, the curve for Ka/D=0.5, 250 almost coincides with the curves for the cases at simply supported and clamped boundary conditions, respectively. It also can be seen that the various laws of the non-dimensional ultimate strength on plate slenderness ratio for all curves are nearly the same. Therefore, it is helpful to derive a new empirical formula for predicting the non-dimensional ultimate strength of ship hull plates with elastically restrained edges. stiffness. It can be seen that the influence degree of rotational restraint stiffness on the non-dimensional ultimate strength ( / ) changes with the plate slenderness ratio . Therefore, the interaction between the rotational restraint stiffness and plate slenderness ratio needs to be considered in the new empirical formula. According to the results in Table 4, a binary quadratic equation is derived from the data fitting method as shown in Eq. (4). Fig. 9 shows the non-dimensional ultimate strength ( / ) against plate slenderness ratio and rotational restraint stiffness three-dimensional surface map.
denotes the ultimate strength obtained by proposed formula when , Ka/D=500. The surface diagram is from the empirical formula and the red scatter diagram is from FE results. As shown, a good agreement can be found. The effects of aspect ratios and initial imperfection amplitudes will be added in Eq. (4) in the form of correction coefficients.
where, 'e' refers to the base of natural logarithms, approximately 2.71.

Effect of aspect ratio
In this section, the influence of aspect ratio (a/b) on the ultimate strength of ship hull plates is studied by using the FE method. Table 5 shows the dimensionless ultimate strength ( / ) of ship hull plates under different aspect ratios. As shown, the non-dimensional ultimate strength of ship hull plates is insensitive to aspect ratio. This is because the different aspect ratios correspond to different initial imperfection shapes, and the half wave numbers of initial imperfection are defined by the integer value of a/b. Then, due to the interaction between aspect ratios and half wave numbers, the effect of aspect ratio is very limited to the ultimate strength of ship hull plates. Some initial imperfection shapes adopted in the present study are depicted as in Fig. 10. The maximum standard deviation of ultimate strength in Table 5 is only 4.436, which further confirms the independence of aspect ratios on the dimensionless ultimate strength of the rectangular plate. The distributions of dimensionless ulti-   Fig. 11. It is seen that the dimensionless ultimate strength of rectangular plates with different aspect ratios is very close to the mean values. This conclusion is also given in Faulkner's formula.
Thus, there is no need to define a correction factor for the term of an aspect ratio in Eq. (4).

Effect of initial imperfections
It is well realized that the magnitude of initial distortion is an important parameter of influence on the ultimate limit state design of welded steel-plated structures Vhanmane and Bhattacharya, 2008;Khedmati et al., 2014). Therefore, the term of initial imperfection amplitudes should be included in the empirical formula. The non-dimensional ultimate strength of ship hull plates with different initial imperfection amplitudes are summarized in Table 6. We can see that the ultimate strength decreases nearly linearly with the increase of initial imperfection amplitudes. The maximum difference value in Table 6 is 0.238 which implies that the influence of initial imperfection amplitudes needs to be considered in the new empirical formula. Fig. 12 shows the dimensionless ultimate strength against initial imperfection shape curves. It can be seen that the relationship between the dimensionless ultimate strength and initial imperfection amplitudes is almost linearly decreased. The mean value of slopes for these curves is −0.0837. A linear equation was used to express the change law of the ultimate strength-imperfection amplitude curve. Therefore, the new equation considering the effects of initial imperfection amplitudes is written as: 5 Applications to existing ship hull plates σ y υ ρ To validate the new empirical formula, it is necessary to apply the formula to the existing ship hull plates. The outer bottom plate of 3100TEU and 10000TEU container ships are adopted as the calculation model, which originates from Yang et al. (2019). The material properties used for the ship plates is that =315 N/mm 2 , Young's modulus E=2.06×10 5 MPa, Poisson's ratio =0.3, density =7.8×10 −6 kg/mm 3 . The geometrical dimensions of these ship plates are summarized in Table 7.
The dimensionless ultimate strength of the real ship hull  plates with simply-supported edges in Table 7 calculated by three different methods are summarized in Table 8. The maximum error between the proposed method and Faulkner's method is −4.41%, and in comparison with the FE results, with a maximal error of −4.59%. This implies that the new empirical formula proposed in the present study can predict the ultimate strength of ship hull plates with simplysupported edges accurately. Fig. 13 presents the mean value and error bar of the ultimate strength of ship hull plates between FE results and empirical formula results. The maximum error occurs when Ka/D=150 and corresponding values are 6.07%, 5.10%, 7.30%, 7.74%, and 5.67%, respectively for the cases C1, C2, C3, C4, and C5, which means that the comparison between FE results and empirical formula results are in a good agreement. It can be concluded that the new empirical formulation proposed in the present paper can be used to predict the ultimate strength of ship hull plates with rotational restraint stiffness subjected to axial compressive loads.

Conclusions
In this paper, the effects of rotational restraint stiffness on the ultimate strength of ship hull plates under axial compressive loading are studied by using FE method. The parametric sensitivity analysis using a variable-controlling approach was conducted. A new empirical formula that can predict the ultimate strength of ship hull plates with elastically restrained edges subjected to axial compression is developed using a large number of FE results. Some important conclusions are summarized as follows.
The ultimate strength of ship plates subjected to axial compression decreases with an increase of plate slenderness ratio while increases with rotational restraint stiffness (Ka/D). The boundary condition can be dealt with as simply supported boundary condition when K/D≤0.5; while can be regarded as clamped boundary condition when K/D≥250. σ u σ y σ y   The aspect ratio (a/b) has a little impact on the non-dimensional ultimate strength of ship hull plates. The reason is that due to the interaction between aspect ratios and half wave numbers, the effect of aspect ratio is very limited to the non-dimensional ultimate strength of ship hull plates.
The ultimate strength of ship hull plates decreases almost linearly with the increase of initial imperfection amplitudes. Therefore, a linear function which depicts the relationship between initial imperfection amplitudes and non-dimensional ultimate strength of ship hull plates are introduced into the empirical formula.
The new empirical formula derived from a large number of FE results is validated by the existing ship plates. The maximum error occurs when Ka/D=150 and corresponding values are 6.07%, 5.10%, 7.30%, 7.74% and 5.67%, respectively for Cases C1, C2, C3, C4 and C5, which implies that the empirical formula can be used to predict the ultimate strength of ship hull plates with elastically restrained edges subjected to axial compressive loading.