Do Kyun Kim , , , ,Imjun Bn , ,,B Y Poh ,Sung-Chul Shin ,

a Department of Naval Architecture and Ocean Engineering,College of Engineering,Seoul National University,Seoul,South Korea

b Research Institute of Marine Systems Engineering,Seoul National University,Seoul,Korea

c Group of Marine,Offshore and Subsea Technology (MOST),School of Engineering,Newcastle University,Newcastle upon Tyne,UK

d Department of Naval Architecture and Ocean Engineering,Pusan National University,Busan,South Korea

e Ocean and Ship Technology Research Group,Department of Civil and Environmental Engineering,PETRONAS University of Technology,Seri Iskandar,Perak,Malaysia

Keywords: Ultimate strength Unstiffened panel Longitudinal compression Mesh convergence Finite-element method

ABSTRACT The present study aims to determine the appropriate size of mesh or the number of the element (NoE)for flat-and curved plates,which is suggested to assess its safety subjected to axial compression based on the ultimate limit state (ULS) design and analysis concept.The unstiffened panel (=plate) and stiffened panel,considered primary members of ships and ship-shaped offshore structures,are subjected to repeated axial compression and tension caused by continued vertical bending moments applied to the hull girder.Plates are attached with stiffeners by welding,and 6,8 or 10 elements are generally recommended to allocate in flat-plate’s breadth direction in between stiffeners for finite-element (FE) modelling,which enables the presentation of the shape of initial deflection applied to the plate.In the case of the load-shorting curve for curved plate,it is reported that the nonlinear behaviour characteristics,i.e.,snap-through,snap-back,secondary buckling and others,appear in typical flank angle.To take this into account,we investigated the preferred number of elements (6,8 or 10) generally applied to the flat plate whether it is an appropriate or more fine-sized element (or mesh) that should be considered.A useful guide is documented based on obtained outcomes which may help structural engineers select optimised mesh-size to predict ultimate strength and understand its characteristic of the flat and curved plates.?Corresponding authors.

1.Introduction

The global strength assessment (=safety assessment of hull girder) shall be done before conducting local strength assessment by checking buckling or collapse of primary structural elements such as plates and stiffened panels [1] .There are various types of safety assessment methods that exist.The numerical simulation based structural design,which ensures reasonable price enclosing actual cost and computational cost (or time) and relatively high accuracy,is gradually preferred to be applied for designing the infrastructures among other methods,i.e.,experimental and analytical [2–4] .Nowadays,the finite-element method (FEM),known as one of the powerful techniques,is preferred for performing structural analysis and design in general [5],while each classification society or shipyard also has its own in-house numerical simulation code that meets its own needs.In the present study,the ANSYS nonlinear finite element method (NLFEM),widely used in the various engineering fields,is adopted to understand the effect of mesh-size on the ultimate strength performance of flat and curved plates subjected to longitudinal compression.

Several studies on the ultimate strength behaviour,including the critical-and elastic buckling strength of plate elements,have been conducted in various ways over the past century.In particular,various types of empirical formulations have been proposed based on the effective width concept proposed by von Karman in the early 19thcentury,and verification of this concept has also been made through experimental and numerical studies [6–18] .Recent advances in the concepts of plate-effectiveness evaluation are also summarised by Paik [19] .In addition,there exist many empirical formulations in predicting the ultimate strength of local structural components,i.e.,the unstiffened panel as cited above and stiffened panel [20–26],developed to date.Recently,Kim et al[27] .proposed an empirical formulation in predicting ULS of initially deflected plate elements subjected to longitudinal compression.Detailed technical reviews may be referred to by Cui and Mansour [7],Zhang [28],and Kim et al [4] ..Besides,ISSC [5,29-32] also continuously put their effort s to verify the finite-element(FE) modelling techniques by conducting a wide range of benchmark studies in predicting ULS of ships and offshore structures.

The curved plates are also widely used in the SHORE structures such as onshore (civil),offshore (floating structures including subsea systems),nearshore (coastal) structures.In particular,several researchers have conducted several studies to understand the ultimate compressive strength characteristics of the curved plate [ 33–36,49 ].Recently,Martins et al [37] .technically reviewed the thinwalled curved plates subjected to in-plane stresses.The empirical formulas in predicting the ultimate strength of the cylindrically curved panels are also provided by Tran [38,39],DNV [40],Martins[41] and Park [42] .

When performing numerical analysis through FEM,the essential things that dealt with can be summarised as follows.

?Pre-processing technique: Modelling

?Element modelling (type,size,others);Material properties(type,characteristics);Boundary condition;Applied loading;Initial imperfection;and many others

?Processing technique: Analysis

?Post-processing technique: Data processing

It is recognised that modelling technique,including input data,should be considered as the most important part when we conduct structural analysis by FEM [43,44] .Most of the designs or research reports present the mesh convergence study results as an initial outcome.The selected mesh-size,rationally decided by the engineer’s judgment,is applied to all scenarios to produce an outcome.In other words,it can be understood that the finite-element analysis (FEA),through the selection of irrelevant mesh-sizes,has a significant impact on its accuracy.We are aware that an even number of elements is recommended to implement the initial deflection shape accurately and maximum deflection in the plate’s length and breadth directions,as shown in Fig.1 (a).In other words,the ULS can be overrated if odd numbers of elements are allocated,as illustrated in Fig.1 (b).This is because the numerical simulation is performed by assuming a maximum deflection that is less than the actual deflection,resulting in accuracy matters.

In the case of simply supported rectangular plate (=flat plate),the buckling half wave number in the longitudinal direction (m)can be decided based on given loading condition by Eq.(1.1),which can be applied when the shape of initial deflection is buckling mode.This can be simplified as presented in Eq.(1.2) by omitting the loading ratio term (σy/σx) when no compression in the transverse direction (σy) is applied [45] .It means that the applied loading component is solely longitudinal compression (σx).Based on Eq.(1.2),mcan be determined.

For biaxial compression

For longitudinal compression

where,a/b=aspect ratio of the plate,m=buckling half wave number in longitudinal (x) direction.a=plate thickness,b=plate breadth,σx=longitudinal compression andσy=transverse compression.

There are two recommended methods for mapping the initial deflection shape into the modelled plate in FEM software.First,we can calculate buckling half wave number (m) by Eq.(1.2),and initial deflection (wo) can mathematically be expressed by the sinusoidal shape that is preferred to be adopted by meeting the plate edge’s boundary condition with simply supported shown in Eq.(2.1) .The obtainedmby Eq.(1.2) can now substitute into Eq.(2.1) .

Furthermore,we can produce more complex shapes of initial deflection,i.e.,hungry horse mode,mountain mode,spoon mode,sinusoidal mode,by taking superposition of sine functions in Eq.(2.2) .The amplitude of the buckling mode initial deflection for longitudinal compressive loading is presented asAomwhich decides the amplitude of initial deflection in the plate shown in Eq.(2.1) .In addition,Bomcan be expressed as a ratio betweenAomand the maximum initial deflection (wopl) shown in Eq.(2.3) .For the practical design purpose,Bomis assumed to 1.0,in general [45] .In the case of maximum initial deflection (wopl) values,it may be referred to Eq.(2.4) proposed by Smith et al [13] .

where,wo=initial deflection,Aom=amplitude of the buckling mode initial deflection for longitudinal compressive loading,a=plate length;b=plate breadth.

where,Bom=welding-induced initial deflection amplitude normalised by maximum initial deflection (=wopl).

The other way is eigenvalue analysis which provides feasible mode shapes.We then normally select the first mode,known as the recommended shape requiring minimum potential energy to be deflected,and update the geometry by applying scale factor based on the assumed maximum initial deflection level shown in Eq.(2.4) .It is also possible to adopt the different amounts of maximum initial deflection (wopl=b/ 200) proposed in Common Structural Rules (CSR) by IACS [46,47] .

Therefore,both methods are available for flat plates,while curved plates or plates with higher nonlinearity are recommended to select eigenvalue analysis.The shapes of initial deflection for flat-and curved plates are discussed and shown in Table 1 .Furthermore,it is encouraged to allocate 6 or 8 or 10 elements for meshing in the flat plate’s breadth direction when we model the plate surrounded by longitudinal stiffeners by FEM.It depends on the engineer’s judgement made by considering computational cost as well as the accuracy of the results.

This study’s problem statement is the applicability of a similar number of elements (=6,8 or 10) used in flat-plate to the curved plate.Regarding this,the effect of mesh-size (or element numbers) on the ultimate compressive strength behaviour of flatand curved plates is investigated and documented.Furthermore,reliable scenarios have also been selected by considering the effect of the plate’s aspect ratio,flank angle,and plate slenderness ratio.Detailed scenarios adopted in this study can be found in Eq.(3) .

where,a/b=aspect ratio (=1,3 and 5) by adopting fixed value of plate breadth (b=830mm),t=plate thickness (t=8,14 and 28 mm which will generate 3.506,2.003 and 1.002 of beta values closed to 3.5,2.0 and 1.0,respectively),θ=flank angle (=0o,5o,10o,20oand 30o),σY=material yield strength (=235 MPa),E=elastic modulus (=205.8 GPa),NoE=Number of element in plate’s breadth (=6,8,10,12,14,20 and 30).

Fig. 1.How to allocate the number of elements in the plate by meshing?

2.Research procedure

This section introduces the detailed procedure of pre-processing of numerical analysis.Fig.2 shows the definition of flat-and curved plates,which can be simply distinguished by flank angle.From here,the necessary geometric information and applied loading are provided.In the following section,processing (FE analysis)and post-processing are going to be delivered.

2.1.Consideration of mesh-size

2.1.1.Shapeoftheinitialdeflectedplates:flatandcurved

As mentioned earlier,simply supported flat-plate can mathematically be expressed by a sine function,and the same outcome can be obtained by eigenvalue analysis Table 1 .shows typical examples of the initial deflection shape of flat-and curved plates by eigenvalue analysis.As would be expected,the shape of initial deflection for flat-plate is buckling mode,meaning that it can be rendered by Eq.(2.1) or made visible by eigenvalue analysis.In contrast,the shape of the initial deflection of the curve-plate does not fit by buckling mode.It seems that the hungry horse mode shape in Fig.3 (a) might suit the curved plate case shown in Table 1 .In this regard,we conclude that eigenvalue analysis is recommended for the curved plate case,besides other types of initial deflection shape are presented in Fig.3 .

2.2.Generation of input data for ultimate strength calculation of numerical simulation

2.2.1.Materialproperties(Yieldstrength)

The material of MS24,known as mild-steel with 24kgf (or 235MPa) of yield strength (σY) and 205.8GPa of Elastic modulus(E),is only considered.This material is widely used in shipyards as a primary material,with HT32 (=315 MPa) and HT 36 (=355 MPa).The effect of material yield strength on the ultimate strength performance of flat-and curved plates under compression is not discussed further here.

Table 1 Typical examples of initial deflection shape of flat-and curved plates (Note: Scale varies).

Fig. 2.Schematic view of to define flat-and curved plates.

2.2.2.Plategeometry

Details of geometric properties are illustrated in Fig.2 for flatand curved plates,respectively.The plate-element characteristics are presented as the plate slenderness ratio shown in Eq.(4) .Theβis a function of the plate’s geometry and material properties.In general,the ultimate limit state (ULS) behaves asβincreased.

where,b=plate breadth,t=plate thickness,σY=material yield strength,andE=elastic modulus.

Kim et al [27] .provided probability density distributions (PDFs)of plate length,breadth and thickness information by collecting 3077 cases of the actual plate from commercial ships.They have provided 4150mm and 815mm of plate length (=a) and breadth(=b) as mean values,respectively,from the obtained probability density functions (PDFs).These scantlings of plate length and breadth are adopted as the fixed values in this study.Three plate aspect ratio cases (a/b=1,3 and 5) are considered.This means that three plate lengths (=1 × b,3 × b and 5 × b) are selected.In addition,three representative plate slenderness ratios (=1.0,2.0,and 3.0) are targeted by taking a plate thickness of 10,14,and 28.5mm.Based on Eq.(4),three plate slenderness ratios can be calculated as 0.9841,2.003,and 2.8047,which are not precisely the same value but close to 1.0,2.0,and 3.0,respectively.

2.2.3.Flankangle

The flank angle is defined in Fig.2 and can be presented as a function of the curved plate’s breadth and radius.

In the present work,0 to 30 degrees of flank angles,i.e.,0(flat-plate),10,20,and 30 (curved plates),were considered for the study purpose.Recently,Park et al [42] .highlighted a secondary buckling phenomenon observed at a 5 ° curved plate’s slenderness ratio at 3.45.In this regard,5 degrees is also considered in this study.

2.3.FE modelling by ANSYS nonlinear finite-element method (NLFEM)

It is recognised that the numerical simulation results are significantly affected by the modelling technique,and reliable and verified techniques,i.e.,material,element types,loading,boundary condition and many others,are necessary.Especially,the element type and formulation selections are important to produce the actual structural behaviour of the structure,i.e.,buckling,postbuckling,large deflection,plasticity effects,and many others.As informed,the ANSYS nonlinear finite element simulation code has been adopted to calculate the ultimate strength of the flat-and curved plates in this study.The shell element in FEM is widely adopted to reveal the abovementioned complex behaviours [49–51] .Several ANSYS users adopted the shell 181 element,which is a 3D isoparametric quadrilateral element with four nodes and 6 degrees of freedom (DOF),to analyse the thin-walled structures [4,5,21,27] .As confirmed by many studies,shell 181 element,which provides reasonable outcomes,is adopted in this study.

In the case of the material properties adopted in FE simulation,a simplified bilinear stress-strain curve with no tangential angle (=no strain hardening effect considered) was used.This can be found in the ANSYS classical metal plasticity model,and it can be understood that stress levels do not allow to exceed yield strength as plastic strain increases.Therefore,the material curve (=perfectly plastic) can be defined by elastic modulus,yield strength information.

Finally,the arc-length method was applied in predicting the nonlinear buckling behaviour of the flat-and curved plates.Previous research [8] reported that the conventional Newton-Raphson method is challenging to accurately predict the complex structural collapse behaviour due to the stiffness matrix’s singularity issue and diverge the solution.However,the arc-length method suits to obtain the nonlinear quasi-static or static solution of the unstable condition by tracking the complex buckling behaviours of the thinwalled plate,i.e.,snap-back,snap-through,secondary buckling,and others.The detailed explanations of arc-length method may be referred to [52,53] .

2.3.1.Mesh-size(orelementnumberinplate’sbreadthdirection)

Fig. 3.Various plate’s initial deflection shapes [ 45,48] .

It is now clear why the even number of finite-elements should be considered in the plate’s breath and length directions by Fig.1 .More specifically,6,8 or 10 elements in Fig.4 (a) to (c) are recommended to be allocated in the plate’s breadth direction to present a realistic shape of the flat plate’s initial deflection,in general.However,those suggested number of elements (6,8,and 10)enable to present nonlinear behaviours in a load-shortening curve,i.e.,secondary buckling,snap-through,snap-back and others,may occur when curved plates in compression should be more thoroughly investigated.

In this regard,additional cases (12,14,20 and 30 elements),which enable an in-depth analysis of curved plates with elastoplastic large deformation,are generated as shown in Fig.4 (d) to(g).The effect of the selected mesh-size or element numbers on the ultimate strength performance of flat-and curved plates in longitudinal compression will be discussed later.

2.3.2.Boundarycondition

It is well-known that the plate edge condition welded by supporting stiffeners is neither simply supported nor clamped.Because the torsional rigidity of the stiffeners (=supporting members) attached at the plate edges is neither zero (0) nor infinite(∞) [2] .It may be recommended to accurately take into account the boundary condition for analysing the behaviour of plate under compression,including buckling and post-buckling behaviour [3] .

However,it is challenging to consider the plate’s actual torsional rigidity surrounded by support members.Therefore,simply supported boundary condition,which allows securing the additional safety margin,is widely used for FE analysis.Therefore,the simply supported boundary condition is adopted,and details may be referred to as Fig.5 .

2.3.3.Appliedloading

The longitudinal compression caused by vertical bending moments is solely considered for the ultimate strength analysis of flat-and curved plates by ANSYS NLFEM.

2.4.Expected outcome: ultimate strength performance of the plate

The typical samples of the expected outcome for flat-and curved plates are presented in Fig.6 (a) and (b) by pointing out the ultimate limit state (ULS) as well as other states such as initial condition,proportional limit,and post-buckling state in the loadshortening curves.These ULS data for all scenarios were extracted from the obtained FE results by ANSYS NLFEM commercial simulation code.

2.5.Verification of the FE technique

Before conducting FE simulations for the selected scenarios,we selected sample scenarios in Table 2 to validate the FE technique.They adopted ANSYS NLFEM,which is also employed in this study.The details of plate geometric and material properties are summarised in Table 2 .For the reader’s information,the original data can be referred to Figure 12 in the literature by Park et al [42] ..

Fig. 4.Examples of the considered number of elements (or mesh-size) in plate’s breadth direction(Typical scenarios with a/b=3).

Fig. 6.Typical results of ultimate strength calculation by ANSYS nonlinear finite-element method (NLFEM).

Table 2 Selected sample scenario from Park et al [42] .for the validation of the FE modelling technique.

Fig.7.Comparison of load-shortening curves of the curved plate between Park et al [42] and the present study.

The comparison results between Park et al [42] .and the present study are shown in Fig.7 .They provided load-shortening curves varying plate aspect ratio (a/b=1.0 to 6.0) with flank angles of 5oand 45ounder longitudinal compression.As informed,we selected a typical case illustrated in Table 2 from Park et al [42] .to verify the FE modelling technique in this study.

The obtained load-shortening curve was well-matched with the previous outcome,as shown in Fig.7 .In particular,we confirmed that nonlinear load-shortening behaviours,including the secondary buckling phenomenon,were captured by the present FE technique with good agreements.Details may also be referred to Table A1.Therefore,the remaining scenarios presented in Eq.(3) are simulated by ANSYS NLFEM,and the results are discussed in the following section.

We have also reviewed the analytical method in addition to the FE method validation.In order to calculate the stress-strain(or load-shortening) behaviour,it can be derived from the governing equation and application of the appropriate boundary conditions.Those techniques are nearly similar for the flat-and curved plates.They can be started from the two nonlinear governing differential equations,i.e.,equilibrium &compatibility equations,of large-deflection plate theory as shown in Eq 6.1 and 6.2 .

Equilibriumequation

CompatibilityequationforthincircularcylinderbyDonnell(1934)

where,w,wo=added and initial deflections;F=Airy’s stress function.

In the case of the cylindrically curved plate with all edges simply-supported condition,the initial (wo) and total deflection (w)functions can be expressed as shown in Eq 6.3 and 6.4 .

where,b=Rχ,y=Rθ.

WhenAiry’sstressfunctionandtheaddeddeflectionaredefined,thein-planestressescanbecalculatedasshowninEq 7.1 .

Assumingtheplane-stressconditionofthedeflectedplateelement,thenonlinearstraindisplacementrelationshipscanbesummarisedas follows(Eq 7.2) [54].

From here,the end-shortening displacement (u) can be obtained,and it can be solved by adopting the principle of virtual work or many other techniques.This can be referred to Park and Seo [55] .For deriving the average stress and strain relationship,they adopted virtual work with an incremental form,and the arclength method was adopted.They concluded that the proposed analytical method could be applicable in predicting the nonlinear buckling strength of a curved plate.The details may be referred to Park and Seo [55] .

3.FE analysis results and discussions

3.1.Stress distribution,strain energy and ultimate limit state (ULS)

Table 3a shows the von-Mises stress distributions of the typical plates (a/b=1,3 and 5) having 30oof flank angle with 1.002 of plate slenderness ratio (=β) at ultimate limit state (=ULS)by considering the effect of element-sizes.The number of elements (=NoE) in length direction can be calculated by multiplying the plate’s aspect ratio.The mesh aspect ratio (=mesh length/ mesh breadth) maintains 1.0,which is mostly preferred condition in FEA.The von-Mises stress distributions show similarities across the board.As it would be expected,it can,however,be concluded that the fine mesh (or higher number of elements) tends to provide precise stress distribution,as shown in Table 3a .Nowadays,meshless method,NURBS,Isogeometric Analysis (IGA) technique and adaptive mesh technique and many others are widely used and highlighted by many researchers [56–61] .Nevertheless,it is common to solve various engineering problems using the traditional FEM technique,and for that,mesh convergence study is essential.

In general,the von-Mises stress distribution in plates with various aspect ratios and element-sizes shows a similar shape,but stress and strain values obtained by the load-shortening curve should be compared.In the ships and offshore construction industry,structural and fatigue analyses are essential for the welded zone with complex structural shapes.In particular,the stress level on the welded zone due to repeated environmental loads within the design life shall be carefully checked during the design phase.Fine mesh is generally recommended in the welded zone,where the stress is expected to be concentrated,while the coarse mesh is used for general structural members.Sometimes,the mesh-size is revised by taking upscaling based on the engineer’s judgement when the obtained stress level is too high (or the element’s colour by von-Mises stress distribution is red).The stress level can be adjusted by meeting the safety criteria.With regards to this,the agreement between structural engineer or designer (=constructer) and customers (=owner),including surveyors (=classification society),should be required for obtained results whether it is acceptable or not.

In this typical case in Table 3a,it is observed that the effect of mesh-size does not significantly impact the ultimate strength performance of curved plate (β=1.002 &θ=30o) as shown in Fig.8 (a) to (c).Moreover,stress concentration at the plate edges region is clearly observed proposed by von-Karman named as “effective width concept” [17,19,62,63] .Detailed results of ULS may be referred to in sections 3.2-3.4.

While the strain energy,which represents the area under the obtained load-shortening curve,tends to behave differently as plate aspect ratio (a/b) increases observed in Fig.8 (a) to (c).When a/b is 1.0,there were no significant changes in load-shortening curves meaning that the effect of mesh-size can be neglected in this case.As a/b increased,such as 3.0 and 5.0,we observed that coarse mesh (or plate with a smaller number of elements) could overestimate strain energy capacity.When we consider optimised mesh or element-size for finite-element analysis (FEA),it is reminded.Until now,we targeted 1.002 of plate slenderness ratio,which presents a thick plate (t=28mm).

The next research question should head to the thinner plates,generating a higher valueβthan the 1.002 covered in Fig.8 and Table 3a .Therefore,we compared von-Mises stress distributions for 2.003 and 3.506 of plate slenderness ratios in Table 3b and 3c,respectively.Besides,each load-shortening curve is compared in Figs.9 and 10 .In general,a similar trend was observed with 1.002 ofβ.

It is observed from load-shortening curves that ULS tends to vary as the plate gets thinner (i.e.,β: 1.002 → 2.003 → 3.506).This means that the difference of ULS between each mesh-size in-creases when the plate slenderness ratio gets higher.In addition,strain energy also depends on mesh-size.On the other hand,the mesh-size or element-size effect should be carefully taken into account for the numerical simulation by FEM.In this comparison,the plate thickness is solely changed for three plate aspect ratios(a/b=1.0,3.0 and 5.0 cases) by fixing other parameters consisting of plate slenderness ratio (β).

Table 3b von-Mises stress views at ULS for typical plate (flank angle=30,beta=2.003).

Table 3c von-Mises stress views at ULS for typical plate (flank angle=30,beta=3.506).

Fig. 8.Typical examples of the considered number of elements (NoE or mesh-size) in the plate’s breadth direction (Flank angle=30 o,β=1.002).

3.2.Effect of flank angle,plate aspect ratio,and plate slenderness ratio

Other research groups have widely studied the effect of flank angle on the ULS of the curved plate [34,39,41,42,64,65],so details may not be discussed further here.The effect of mesh-size or the number of elements on the flat-and curved plates defined based on the flank angle in Fig.5 is investigated in this section.Flat-plate can be expressed when the flank angle equals zero,while the flank angle in a curved plate exceeds zero.

3.2.1.Reminderoftheassumedconditionandselectedscenariosfor outcomeanalysis

As would be expected,the ULS tends to change as the flank angle increases Table 4 .shows the detailed investigation results by representing the effect of mesh-size (or the number of elements in plate breadth’s direction=6,8,10,12,14,20 and 30 shown in Fig.4) on the ultimate strength of flat-plate (θ=0) and curved plate (θ=5,10,20 and 30) under longitudinal compression.Three levels of plate slenderness ratio (β=1.0 02,2.0 03 and 3.506) are considered by changing plate thickness (t=8,14 and 28 mm) and plate length (a=b,3band 5b) while other parameters such as plate breadth (b=830 mm).Material properties (E=205800 MPa andσY=235 MPa) are considered as a constant value.

Prior to investigating the details,it is required to remind that the term of the number of elements (NoE) in this study represents how many elements are allocated in the plate’s breadth direction,affecting the mesh-size.Visualised NoE information may be referred to as Fig.4 and Table 3 .We assumed that the obtained ULS results by 30 of the number of elements in the plate’s breadth direction,namely NoE (30),can be considered as a refined outcome by fine mesh.The histograms presented in Table 4 show the relationship among NoE (horizontal-axis),plate aspect ratio (verticalaxis) and difference (%).In the case of difference (%),the ratio can be calculated between ULS by each NoE and ULS by NoE (30).Therefore,the histogram height gives a useful guide for structural engineers in determining the relevant number of elements for their FE analysis.In this study,we set 1.0% of the ULS difference with NoE (30) as a criterion.For example,the difference exceeding 1.0% is highlighted as red,shown in Tables 4(b),(f) to (o).As an example of not following the general trend by showing lower ULS than NoE 30 results,it is highlighted in blue presented in Table 4 (l).All calculated differences are shown in Table 4 so that engineers could decide mesh-size based on their judgement.

Table 4 Comparison of ULS resu lt s considering mesh -siz e effect.

In addition,based on plate slenderness ratios selected,we categorised three plate types basedβsuch as think (=3.506),middle (=2.003) and thin plates (1.002).Therefore,we investigate the effect of flank angle in this section and the effect of plate slenderness ratio (β),and plate aspect ratio (a/b) will be discussed further in the next section.

In the following sections,i.e.,3.2.2 to 3.2.4,the ultimate limit state (ULS) difference between each result and 30 of the number of the element (NoE) in the plate’s breadth direction will be mainly discussed.Therefore,the difference will represent the following Eq.(8) .For example,when theβis 2.003,the ULS difference between NoE (30) and NoE (6) of flat-plate with a/b=5 is 1.33,as shown in Table 4 (f).

ThedifferencewithNoE(30)

Note: NoE (30) represents 30 elements in the plate’s breadth direction.

3.2.2.Effectofslendernessratio(β)

In the case of plate slenderness ratio effects on ULS of flatand curved plate,it can be seen by each column representing the changes of thickness.For example,theβeffects on ULS of flatplate can be investigated by comparing column 1 (C1) shown in Table 4 (a),(f) and (k).In the same manner,theβeffects on ULS of curved plates may be referred to as C2 to C5.

We found out that mesh-size or NoE significantly affects ULS when the plate gets thinner from the observation.This should be carefully taken into account when engineers decide the plate’s mesh-size for FE analysis.In the flat-plate case,the ULS difference tends to increase as the plate thickness becomes thin or asβincreases.In particular,this should be carefully reminded when we model the thin-plates,as shown in Table 4 (k).

A similar trend is captured in curved plates shown in C2 to C5.The larger the flank angle,the greater the ULS difference.Here,two things should be highlighted.If the plate is thick (β=1.002),referred to as R1 in Table 4,mesh-size might not affect the ULS of the flat and curved plates significantly.As shown in R1,differences were less than 1.0% except for one case at 5oof flank angle case.Even we consider this case,NoE (8) or NoE (10) in the plate’s breadth direction could be recommended for ULS assessment of flat-and curved plates under longitudinal compression.However,middle and thin plates should carefully take into account for element-number effect,as shown in R2 and R3.Highlighted boxes by red may help readers understand the ULS difference in a positive value,while blue represents a negative value.A similar trend captured for those middle (β=2.003) and thin plate(β=3.506) cases,

Fig.10.Typical examples of the considered number of elements (NoE or mesh-size) in the plate’s breadth direction (Flank angle=30 o,β=3.506).

In conclusion,NoE (10),which is generally used,is applicable to flat and curved plate meshing.When the curved plate is getting thinner,NoE (20) might be recommended to minimise the difference with NoE (30).

3.2.3.Effectofplateaspectratio(a/b)

In general,the effect of plate aspect ratio (a/b) was not significant from the calculated difference (Δ) shown in thick plate (=R1).In the case of middle (=R2) and thin (=R3) plates,similar trends have been observed,but the case of 5oof flank angle should be carefully taken into account.A secondary buckling phenomenon was captured by FE simulation in a curved plate with a flank angle of 5oin Park et al.[ 42 ]shown in Fig.7 .which can be considered as one of the good examples.They mentioned that the fluctuation of ULS is captured in a curved plate with a flank angle of 5o.This may be caused by the deflection shape,which changes continuously shown in Fig.7 .The initial deflection shape before we apply the compression (①) is changing slightly to the down when it reaches proportional (②) and ultimate limit states(③) by maintaining the half-wave number.A significant change is captured when it follows the secondary buckling path by changing deflection shape (④),and progressive collapse behaviours can be observed in the post-buckling region (⑤and ⑥).The fluctuation behaviour in Table 4 (l) when a/b=3.0 can be considered the effect of lower flank angle and initial deflection shape.

3.2.4.Effectofflankangle(θ)

In 3.2.2 and 3.2.3,we have discussed the typical flank angle (=5o),which generates the fluctuation behaviour of the ULS by taking the 2ndbuckling path.

Apart from typical flank angle (=5o) cases,it is generally observed that the greater the flank angle,the more significant the difference (Δ) defined in Eq.(5) .As mentioned earlier in Section 3.2.2,the effect of mesh-size or NoE is not significantly affected when the plate is getting thicker.The initial deflection shape can explain this.

Fig. 11.The effect of NoE on the load-shortening curves of the curved plate at specific conditions(i.e.,lower flank angle with curved plate which generates snap-back,snap-through,secondary buckling etc.)

The initial deflection shape can be obtained by eigenvalue analysis with a geometry update method based on the assumed amount of initial deflection of the plate (wopl=0 .1β2t) in ANSYS NLFEM.Also,buckling mode shape and hungry horse mode shape of initial deflections are obtained for flat-and curved plates,respectively.In this regard,curved plate (θ>0) tends to have relatively significant differences compared to flat-plate (θ=0).

As would be expected,the thinner plate could be initially deflected easier than the thicker plate.Regardless of the plate’s thickness,flank angle and ULS difference tend to increase together,except for a few cases.

3.3.Additional investigation on the effect of mesh-size at specific conditions

Apart from the above results shown in Sections 3.1 and 3.2,the effect of element (or mesh) size is investigated further at the specific conditions such as lower flank angle,which generates snapback,snap-through or secondary buckling.Three specific scenarios are selected based on the manual scenario selection procedure because the specific cases,shown in Fig.11 (a) to (c),do not always occur.

As would be expected,the following conclusions have been obtained.

?As the number of the element (NoE) increases,strain energy(=area under the load-shortening curve) decreases.This means that the structural designer should take into account the effect of number of elements or mesh-sizes when we conduct the FE model for curved plate.

?Different NoE may express the similar shape of load-shortening curves of the curved plate at the specific conditions,i.e.,lower flank angle with curved plate,which generates snap-back,snapthrough or secondary buckling phenomenon.

?In order to select the reliable number of the element,at least 10 NOE or more might be needed.

In this simplified parametric study,a limited number of cases are discussed and investigated to consider complex behaviour such as snap-back,snap-through or secondary buckling.Therefore,additional studies are recommended to be conducted further in the future.

4.Concluding remarks

A piece of helpful information in selecting the appropriate size of mesh or number of the element (NoE) for both flat-and curved plates are documented by conducting the parametric study.In total,315 reliable scenarios in Eq.(3) were adopted to consider the effects of geometric (=plate aspect ratio,plate slenderness ratio,and flank angle),material (=material yield strength and Young’s modulus) properties together with finite-element size (=NoE in plate’s breadth direction).An essential tip is how to allocate the number of finite-elements in the plate,which should be able to accurately and smoothly implement the initial deflection shape and maximum deflection in the plate’s length and breadth direction highlighted in Fig.1 .

Based on assumed NoE cases in Fig.4,a comparison of studies in terms of the ultimate limit state (ULS) was conducted in detail.The obtained outcomes and insights are summarised as follows.

Outcomesofthisstudy

?[Effectofplateaspectratio]It is found that mesh-size or NoE significantly affects ULS when the plate gets thinner.NoE (10),which is generally used,is applicable to flat and curved plate meshing.When the curved plate is getting thinner,NoE (20)might be recommended to minimise the difference with NoE(30).

?[Effectofplateslendernessratio:Thickvs.Middlevs.Thin plate]In general,the effect of plate aspect ratio (a/b) was not significant from the calculated difference (Δ) shown in thick plate (=R1).In the case of middle (=R2) and thin (=R3)plates,similar trends have been observed,but the case of 5oof flank angle should be carefully taken into account.

?[Effectofflankangle]Apart from the typical flank angle(θ=5),it is generally observed that the greater the flank angle,the more significant the difference (Δ) defined in Eq.(5) .This means that additional consideration may be required as the flank angle increases.The effect of mesh-size or NoE is not significantly affected when the plate is getting thicker.Details may be referred to as Table 4 .

Limitationofthisstudy&Furtherstudyrequired

?In this study,only three (3) plate thicknesses,such as 8,14,and 28mm,were selected in Table 4 .Addition scenarios of plate slenderness ratio could be considered by selecting the different thicknesses of the plate.

?The effect of secondary buckling behaviour should be further investigated how and when it occurs.The condition that causes secondary buckling should also be studied by investigating the effect of initial deflection shape,amount,plate thickness,material properties,applied loading,boundary condition,and others.?In this study,the 1:1 mesh aspect ratio,mostly preferred by FE engineers,is only considered.Different mesh-aspect ratios and element types,and formulations can also be investigated further.

DeclarationofCompetingInterest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported under the framework of international cooperation program managed by the National Research Foundation of Korea (NRF-2022K2A9A2A230 0 0266).This study is also supported by the start-up funding provided by Seoul National University .Some part of this study has been presented in MSc final report by Mr Im-jun Ban,supervised by Dr Do Kyun Kim,when he studied at Newcastle University.

Appendix

Table A.1 Deformed shape at each load-shortening curve stage