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Contents lists available at ScienceDirectThin-Walled Structuresjournal homepage: /locate/twsFull length articleUltimate limit state based design versus allowable working stress baseddesign for box girder crane structuresDong Hun Leea, Sang Jin Kimb, Man Seung Leec, Jeom Kee Paika,b,d,aDepartment of Naval Architecture and Ocean Engineering, Pusan National University, Busan 46241, Republic of KoreabThe International Centre for Advanced Safety Studies (The Lloyds Register Foundation Research Centre of Excellence), Pusan National University, Busan 46241, Republicof KoreacPOSCO Engineering & Construction, Incheon 22009, Republic of KoreadDepartment of Mechanical Engineering, University College London, London WC1E 7JE, UKA R T I C L E I N F OKeywords:Box girder crane structuresUltimate limit statesServiceability limit statesNonlinear finite element methodAllowable working stressA B S T R A C TIt is now well recognized that limit states are a much better basis than allowable working stresses for economical,yet safe design of structures. The objective of the present paper is to apply the ultimate limit states as thestructural design criteria of a box girder crane. For this purpose, a reference box girder crane structure which wasoriginally designed and constructed based on the allowable working stress criteria and still in operation isselected. The structure is then redesigned applying the ultimate limit state criteria, where other types of limitstates such as serviceability limit states and fatigue limit states are also considered. Nonlinear finite elementmethod is applied to analyze the progressive collapse behaviour of the structure until and after the ultimate limitstate is reached. While the ultimate strength or maximum load-carrying capacity of the structure has never beenrealized as far as the allowable working stress based design method is applied, this study starts with identifying itby the nonlinear finite element method. The structural weight was then attempted to minimize by changingstructural member dimensions while the ultimate strength remains the same. It is concluded that the ultimatelimit state based design method provides more economical, yet safe structures, compared with the allowableworking stress based design method. Considerations for the serviceability limit states are addressed in the paper,while those for the fatigue limit states are presented in a separate article.1. IntroductionExamples of steel plated structure include ships, offshore installa-tions, bunkers, box girder bridges and box girder cranes 1,2. In thepast, steel plated structures have typically been designed using the al-lowable working stress based design method (AWSDM). In the AWSDM,the structural design criterion is that the working stress under the ap-plied design loads must be smaller than an allowable stress. On theother hand, the ultimate limit state based design method (ULSDM) isbased on the criterion of the ultimate limit state or ultimate strength ormaximum load-carrying capacity which is represented by point B inFig. 1, where the ultimate strength should not be smaller than the ap-plied design loads with an adequate margin of safety 2. In theAWSDM, point B remains unknown although a simplified bucklingstrength check represented by point A may be applied. This means thatthe AWSDM is unable to define the true margin of structural safety thatcan be determined only by a comparison between the ultimate strengthand the applied design loads.For ships and offshore structures, a number of contributions towardthe application of the ULSDM has been undertaken 38. Also, in thecivil engineering field for land-based structures, the ULSDM has beenapplied 913. The determination of the ultimate strength for struc-tural components such as plates and stiffened panels under variousconditions has been undertaken by the experimental test and the non-linear finite element method 1426. Similar contributions have alsobeen performed in box-shaped structures 2743. Koteko et al. 44carried out a sensitivity analysis of thin-walled box girders subjected topure bending. They concluded that a redundancy of the load-carryingcapacity of box-section girder is sensitive to the yield stress deviation,implying that the AWSDM based on the yield stress of material has the/10.1016/j.tws.2018.10.029Received 25 January 2018; Received in revised form 17 October 2018; Accepted 18 October 2018Corresponding author at: Department of Naval Architecture and Ocean Engineering, Pusan National University, Busan 46241, Republic of Korea.E-mail address: jeompaikpusan.ac.kr (J.K. Paik).Thin-Walled Structures 134 (2019) 491507Available online 15 November 20180263-8231/ 2018 Published by Elsevier Ltd.Tuncertainty to structural safety.The main objective of the present study is to apply the ULSDM forbox girder crane structures which have originally been designed usingthe AWSDM. Additionally, the serviceability limit state is considered tosecure the operability and functionality of the crane structures. Thereference structure is part of a real crane system and it has been suc-cessfully operated without structural safety issues. The structure ismade of SM490 steel but the manufacturer of the crane wants to use ahigh tensile steel of HSB500 which is a very modern material. This is amotive of the study. The manufacturer wants to keep the same amountof maximum load-carrying capacity as the reference structure whichwas originally designed using the ASWDM, while the structural weightor building cost must be minimized or at least be similar to the originalstructure because of using a high tensile steel.For this purpose, the progressive collapse behaviour of the structureuntil and after it reaches the ultimate strength was first determined byusing the nonlinear finite element method. Case studies were then un-dertaken with varying the structural member dimensions as well asmaterial while keeping the same (similar) maximum load-carrying ca-pacity and minimizing the structural weight. It is obvious that theULSDM is useful to achieve the goals and it is concluded that theULSDM is much better than the ASWDM in terms of designing moreeconomical, yet safe structures.2. The procedure for the limit state based structural designoptimizationIn the traditional ASWDM, the focus is on keeping the workingstresses resulting from the design loads under a certain working stresslevel that is usually based on successful past experience:a(1)whereis the working stress andais the allowable stress which isusually specified by regulatory bodies or classification societies as somefraction of the mechanical properties of materials, e.g., uniaxial yield orultimate tensile strength.On the other hand, the limit state approach is based on limit statesas described in Fig. 2 2. It starts with the definition of target struc-tures to be designed where the conditions of functionality and oper-ability as well as geometric and material properties are identified. In thenext step, the objective function of the structural design is defined.LoadDisplacementDesign load levelBuckling strengthUltimate strengthLinearelasticresponseABFig. 1. Ultimate limit state of structures 2.Fig. 2. Flowchart of the limit state based structural design optimization appliedin the present study.Fig. 3. The target box girder crane structures.D.H. Lee et al.Thin-Walled Structures 134 (2019) 491507492Typically, the structural optimization is aiming at minimizing structuralweight or building costs while keeping the structural safety.In the next step, the design demand or applied design loads aredefined as follows:=DDddk(2)whereDdis the design demand,Dkis the characteristic value of thedemand which is the mean value of applied loads, anddis the partialsafety factor associated with the uncertainties of applied loads.Also, the design capacity or the maximum load-carrying capacity isdefined as follows:=CCdck(3)whereCdis the design capacity,Ckis the characteristic value of thedesign capacity which is the mean value of the limit states, andcis thepartial safety factor associated with the uncertainties of the limit states.In Eq. (3), four types of limit states are relevant, namely the ultimatelimit states (ULS), the serviceability limit states (SLS), the fatigue limitstates (FLS) and the accidental limit states (ALS). In the present study,both the ULS and the SLS are considered. The characteristic value of theULS or the ultimate strength or the maximum load-carrying capacitycan be determined by experiments or nonlinear finite element methods.This study applies the nonlinear finite element method to determine theultimate strength of the box girder crane structures.Once both the design demand and the design capacity are de-termined, the structural constraints are applied to be safe as follows:=CD1dd(4)whereis the safety factor of a structure.With varying the structural design parameters, the above-mentionedprocess is repeated until the requirement of the object function is met.In the present study, the ULSDM is applied for the structural design of abox girder crane.3. Applied exampleThe target box girder crane structure made of SM490 was originallydesigned applying the ASWDM. The manufacturer of the crane struc-tures now wants to use a high tensile steel of HSB500. To redesign thestructure using the HSB 500 steel, the USLDM is now applied.3.1. Characteristics of target structuresFig. 3 shows the target box girder crane structures in real appear-ance of operation and three-dimensional CAD modelling where the boxgirder structures of the electric overhead travelling (EOT) system is thetarget structures to be designed in the present study. The crane systemis composed of main and auxiliary girders together with hooks, trolleysand saddle girders. Fig. 4 shows the plan view of the target structurewhich is driven by the travelling truck at the ends of main and saddlegirders. In the present study, the ultimate strength is analysed sepa-rately for the main girder and the combined auxiliary and saddle gir-ders. Fig. 5 shows the cross section of the main girders and the auxiliarygirders.Fig. 4. Plan view of the target box girder crane structures.D.H. Lee et al.Thin-Walled Structures 134 (2019) 4915074933.2. Objective function and limit state criteriaThe objective of the structural optimization is to minimize thestructural weight of the main and auxiliary girders, keeping the safetyat a required level. Table 1 indicates the limit state criteria applied forthe present design in association with the ULS and the SLS. The FLS wasalso considered but it will be reported in a separate paper.3.3. Nonlinear finite element modellingIn order to determine the ultimate strength of the target structures,the elastic plastic large deflection finite element method is appliedusing ANSYS computer program 46.3.3.1. Nonlinear finite element modelsFig. 6 shows the FE models of the target structures, where a half partof the combined auxiliary and saddle girders is taken as the extent ofthe analysis because of the symmetric condition with regard to thecenter line, although a full structure of the main girder is included inthe FE model.In industry practice, stiffeners are often modelled by beam elements.However, beam element models for stiffeners may not give accuratecomputations in terms of local buckling, e.g., local web buckling andtripping, and plasticity. In this regard, only 4-node plate-shell elements areused to model not only plating and stiffener webs but also stiffener flangesIn order to ensure the mesh size, convergence studies for the main girderand auxiliary girder are performed by varying the element size. Fig. 7Fig. 5. Cross section of the main girders and auxiliary girders.Table 1Limit state criteria applied for the structural scantlings in the present study.Constraint parameterType of limit statesDescriptionUltimate strength (maximum load-carryingcapacity)ULSUltimate strength of the structure with new design at least should be maintained at the equivalentultimate strength level with the existing design.Maximum deflectionSLSMaximum defection at design load level should not exceed L/1000 45.Natural frequencySLSThe minimum requirement of natural frequency for crane structure is 1.7 Hz 45.Plate thicknessSLS (Fabrication)Minimum plate thickness is limited by 9 mm, considering welding.Fig. 6. The nonlinear FE model with applied boundary conditions.D.H. Lee et al.Thin-Walled Structures 134 (2019) 491507494shows the results of the convergence studies in terms of ultimate strengthand maximum deflection. It is obvious that a convergence is achieved atthe element size of 50mm where all element sizes are identical.3.3.2. Boundary conditionsFig. 6 presents the boundary conditions for the nonlinear finiteelement analysis. The crane structural system is considered to be simplysupported at the ends because the main girder is supported on thetravelling truck which is positioned on rigid rails. In this regard, theshaded areas with a rectangular shape shown in Fig. 6 are assigned bythe corresponding boundary conditions. In the ANSYS code application,the boundary conditions with the “Coupling” option was adopted.3.3.3. Material propertiesTable 2 indicates the material properties of existing SM490 steel andnew HSB500 steel. In the present study, an elastic perfectly-plasticmaterial model is applied without considering the strain-hardeningeffect.3.3.4. Loading conditionsThe live loads from the trolley are transferred through the wheels tothe box girder. In this study, two loading positions are considered ac-cording to the position of trolley. Figs. 8 and 9 present the loadingpositions which are considered to be the most severe loading scenariosin operation. Table 3 summarizes the structural weight of the main andauxiliary girders in the original design.The loads applied to the crane structure are classified into two cate-gories. The first category embodies dead loads of built-in machinery andequipment, and the machinery-induced moment. In the second category,Fig. 7. The convergence study to determine the best element size using identical meshes.Table 2Material properties of SM490 steel and HSB500 steel.Material propertyUsed materialNew materialSM490 steelHSB500 steelDensity, (ton/m3)7.857.85Youngs modulus, E (GPa)205.8205.8Poissons ratio,0.30.3Yield strength,Y(MPa)318380Ultimate tensile strength,T(MPa)490500(a) Loadin ng position 1: The troll ley locates at mid span n of the main n girder (b) Loading position 2: The trolley locates at end of the main girder Fig. 8. Loading positions of the main girder considered in the FE analysis.(a) Loading(b) Load position 1ing position: The trolley 2: The troy locates at olley locatesmid span o at end of thof the auxiliae auxiliaryry girder y girderFig. 9. Loading positions of the auxiliary girder considered for the present FEanalysis.D.H. Lee et al.Thin-Walled Structures 134 (2019) 491507495all loads transferred from the wheels of the trolley are included. Theauxiliary girder does not facilitate machinery and equipment. Fig. 10 andTable 4 summarize the load cases for the main girder, while Fig. 11 andTable 5 present the load cases for the auxiliary girder. It is noted that theload cases have been decided by the design specification of the electronicoverhead travelling crane as presented by AIST 47.The design demandDdof the target structure can be defined con-sidering the partial safety factors for loads as follows 48:=+DDLLL()dd cd dd l,(5)whered c ,is the partial safety factor associated with the uncertainties ofload combination,d d,is the partial safety factor associated with theuncertainties of dead loads, andd,is the partial safety factor asso-ciated with the uncertainties of live loads. The partial safety factors areindicated in Table 6.3.4. Results of the nonlinear finite element analysisFig. 12 shows the results of the ultimate strength behaviour for themain girder with different loading positions, and Table 7 summarizesthe safety margin of the main girder which is determined as the ratio ofthe ultimate strength to the design loads. Fig. 13 shows the results ofthe ultimate strength for the auxiliary girder structures with differentloading positions where the structural scantlings are the same as theoriginal design and Table 8 indicates the safety margin of the auxiliarygirder.It is obvious that the ultimate strengths of the main and axillarygirders can be increased by using high tensile steel with the samestructural scantlings as those of the original structure. This means thatthe structural scantlings can be reduced in the use of high tensile steel ifthe ultimate strength of the girder structures is kept at the similar levelto the original design using the SM490 steel.3.5. Structural optimizationIt is now attempted to reduce the structural scantlings (platethickness) using high tensile steel when the ultimate strength under thecorresponding design loading conditions is kept at the similar level toTable 3Structural weight of the main and auxiliary girders in the original design.CategoryMagnitude (tonf)Main girderAuxiliary girderGirder weight54.6614.35Attachment weight36.247.45Total self-weight (WM)90.9021.80Fig. 10. Presentation of the load cases for the main girder with the nomenclature indicated in Table 4.Table 4The load cases for the main girder.Case No.LabelMagnitudeDescriptionRef. Table 3Self-weight0.2 WMGirder inertia force1A0.495 tonf/mUpper walkway loadA0.669 tonf/mLower walkway loadB, B0.227 tonfEccentric loadC, C3.5 tonfCabin loadD3.2 tonfDriving motor weightE1.82 tonf-mMotor-induced moment2F220 tonfDead load of trolley (DLT)G450 tonfLifted load (LL)H0.2 LLVertical impact loadI0.2 (DLT+LL)Bridging inertia forceJ0.2 (DLT+LL)Skewing forceK0.15 (DLT+LL)Inertia force from drive (IFD)FFig. 11. Presentation of the load cases for the auxiliary girder with the no-menclature indicated in Table 5.Table 5The load cases for the auxiliary girder.Case No.LabelMagnitudeDescriptionRef. Table 3Self-weight0.2 WAGirder inertia force1A49.0 tonfDead load of trolley (DLT)B50 tonfLifted load (LL)C0.2 LLVertical impact loadD0.2 (DLT+LL)Bridging inertia force fromE0.2 (DLT+LL)Skewing forceF0.15 (DLT+LL)Inertia force from drive (IFD)D.H. Lee et al.Thin-Walled Structures 134 (2019) 491507496the original design. As structural design variables, only the thickness ofstructural members is considered where the stiffener spacing is kept thesame as the original design. The target plates were selected consideringthe efficiency of structural optimization. Figs. 14 and 15 show thetarget plates to be reduced for this purpose, while support members areunchanged in their scantlings.The optimal thickness of target plates are modified following thecase studies using trial and error method. The ultimate strength, max-imum displacement, stress distributions, and local buckling strength areevaluated for each analysis case. Each analysis case is defined withreference to the structural effect of reduction of each plate thickness inprevious analysis case.3.5.1. Main girder structuresA total of 7 case studies were undertaken as indicated in Table 9.The ultimate strength and the Z-displacement (deflection) were calcu-lated for the cases as shown in Figs. 16 and 17. It is found that the maingirder reaches the ULS after local bucking takes place in the inner webwhich is positioned below the trolley wheels followed by the extensionof plasticity. It is considered that the collapse of the box girder cranesystem is triggered by the local buckling in the inner web, and thus amore detailed investigation is made in association with the localbucking strength of the inner web.According to Table 9, the most successful achievement of thestructural weight reduction is case 7. However, it is recognized that thesafety margin of the main girder structures for case 7 is only 1.89 inassociation with the local buckling in the inner web as shown in Fig. 18and Table 10, implying that the inner web buckling occurs earlier. Totake a greater safety margin value, therefore, the structural scantlings ofcase 6 are selected in the present study as the final design of the maingirder structures using high tensile steel. Fig. 19 shows the von-Misesstress distribution of the final design at ultimate limit state.3.5.2. Auxiliary girder structuresA total of 6 case studies were undertaken to reduce the structuralscantlings as indicted in Table 11. Figs. 20 and 21 show the results ofthe ultimate strength computations. It is found that the auxiliary girderstructures also reach the ULS after local bucking takes place in the innerweb which is positioned below the trolley wheels. The local buckingstrength of the inner web is also checked for each of the analysis cases.In case of the auxiliary girder, the degradation of the bending ri-gidity caused by a reduction of the plate thickness makes it difficult tosatisfy the design constraint associated with the defection. In order tosecure the bending rigidity, the most economical way is to increase theweb height of the girder. Therefore, the web height of the auxiliarygirder is changed based on following conditions:To keep the bending rigidity at the same level of the existing designTo meet the limiting condition on the maximum web height of2050mm associated with structural clearance in operationIn order to obtain the best web height, the moment of inertia of theTable 6The partial safety factors associated with the uncertainties of loads.CategorySymbolValueLoad combination factord c ,1.2Dead load factor (for trolley, DLT)d d,1.5Live load factor (LL)d,1.5Fig. 12. Ultimate strength behaviour of the main girder with the structural scantlings in the original design.Table 7Safety margin of the main girder with the structural scantlings in the originaldesign.Loading positionSafety marginCD/ddB/AOriginal (SM490) design(A)New (HSB500) design(B)13.383.8814.8%24.315.0717.6%D.H. Lee et al.Thin-Walled Structures 134 (2019) 491507497box girder is investigated with varying the plate thickness and the webheight as shown in Fig. 22. With consideration of the bending rigidityand the clearance, case 6 is selected as the best design as indicted inTable 11. The web height of case 6 increased by 250 mm, i.e., from1700 mm in the original design to 1950mm in the new design.The safety margin of the new design of case 6 is 2.58 as shown inFig. 23 and Table 12. Fig. 24 shows the von-Mises stress distribution ofthe final design of the auxiliary box girder crane structures at ultimatelimit state.3.6. Eigenvalue analysisIn order to calculate the natural frequencies in terms of the SLScriteria, the eigenvalue analysis is carried out. The minimum require-ment of natural frequencies is 1.7Hz in association with the comfort ofan operator and the proper functioning of crane structures 45. Thefinal designs of the main and auxiliary girders satisfy the minimumFig. 13. Ultimate strength behaviour of the auxiliary girder with the structural scantlings in the original design.Table 8Safety margin of the auxiliary girder with the structural scantlings in the ori-ginal design.Loading positionSafety marginCD/ddB/AOriginal (SM490) design(A)New (HSB500) design(B)13.974.5213.9%24.465.1615.7%Fig. 14. Target plates to be reduced for the structural optimization (cross sec-tion).Fig. 15. Target plates to be reduced for the structural optimization (profileview).D.H. Lee et al.Thin-Walled Structures 134 (2019) 491507498Table 9Case studies for reducing the plate thickness of the main girder structures.CaseWeight (ton)Plate thickness (mm)WeightReductionTopBottomInner webOuter webDiaph-ragmOriginal54.6601416161214Case 150.843.82 (7.0%)14 (0)12 (4)14 (2)10 (2)12 (2)Case 249.754.91 (9.0%)14 (0)12 (4)13 (3)10 (2)10 (4)Case 349.385.28 (9.7%)14 (0)11 (5)13 (3)10 (2)10 (4)Case 449.345.32 (9.7%)14 (0)12 (4)13 (3)9 (3)10 (4)Case 550.164.48 (8.2%)14 (0)12 (4)14 (2)10 (2)10 (4)Case 649.065.60 (10.2%)14 (0)11 (5)14 (2)9 (3)9 (5)Case 747.427.24 (13.3%)11 (3)11 (5)14 (2)9 (3)9 (5)Note: The values in the parenthesis indicate the reduced thickness.Fig. 16. The results of the ultimate strength computations for the main girder structures at loading position 1.D.H. Lee et al.Thin-Walled Structures 134 (2019) 491507499requirements as indicted in Table 13. Fig. 25 show the natural mode I ofthe main and auxiliary box girders.3.7. Recommended practicesThe ULS based safety factor of the reference box girder cranestructures using SM490 which were originally designed using theAWSDM was found in the present study. Also, the safety factorin Eq.(4) for the new designs of the main and auxiliary box girder cranestructures using HSB500 are given by= 3.41 and= 4.09,respectively, wherec= 1.0 was taken in Eq. (3).As per the requirement of the manufacturer, the safety margins ofthe main and auxiliary box girder structures have been kept at the si-milar level to the original design. However, it is considered that thestructural weight of the new design can be further reduced, consideringthat the safety margin is quite large. In the industry practice, thestructural safety factor is often taken as=1.15 for stiffened panelsand girders 49.Fig. 17. The results of the ultimate strength computations for the main girder structures at loading position 2.D.H. Lee et al.Thin-Walled Structures 134 (2019) 4915075004. Concluding remarksThe objective of the present study has been to redesign an existingbox girder crane structures using the limit state approach with the focuson the ultimate limit states and the serviceability limit states. The ori-ginal structures were made of SM490 steel and designed using the al-lowable working stress based method. The manufacturer of the cranewanted to make the structures using HSB500 steel and to redesign themapplying an advanced design technology.For this purpose, the ultimate strength behaviour and the safetyfactor of the original structures was identified by the nonlinear finiteelement method. By keeping the safety factor of the crane structures atthe same (or similar) level as the original design, new structural designsusing HSB500 steel were developed by minimizing the structuralweight.It is found that the reduction of the structural weight in the newdesign of the main box girder crane structures is 5.60 tonf or 10.2% ofthe original structures, where the safety factor based on the ultimatestrength is 3.41, while the safety factor of the original design is 3.38.Similarly, the reduction of the structural weight in the new design ofthe auxiliary box girder crane structures 1.56 tonf or 10.8% of theoriginal structures, where the safety factor based on the ultimatestrength is 4.09, while the safety factor of the original design is 3.97.Based on the studies, it is concluded that the ultimate limit statedesign method is much better than the allowable working stress designmethod in terms of designing more economical, yet safe structures. Thisbenefit is obviously due to the fact the former method is unable todetermine a realistic factor of the structural safety which is determinedas a ratio of the ultimate strength or maximum load-carrying capacityto the design loads, although it remains unknown in the latter method.It is hoped and believed that the insights and practices developed inthe present study will be useful for the optimum design of box girdercrane structures using the limit state approach. While the present studypresents the design results associated with the ultimate limit states andthe serviceability limit states, those for the fatigue limit states will bepresented in a separate paper.Table 10Local buckling strength checks of the inner web of the main girder structures.CaseLocal bucklingstrength (kN)Rate, divided byoriginal designSafety margin forlocal buckingOriginal16,693.141.002.88Case 614,839.610.872.56Case 710,941.570.641.89Fig. 18. Local buckling strength of the inner web of the main girder structures.D.H. Lee et al.Thin-Walled Structures 134 (2019) 491507501Fig. 19. von-Mises stress distribution of the main girder at ultimate limit state.Table 11Case studies for reducing the plate thickness of the auxiliary girder structures.CaseWeight (ton)Plate thickness (mm)WeightReductionTopBottomInner webOuter webOriginal14.35018181010Case113.201.15 (8.0%)18 (0)11 (7)10 (0)10 (0)Case212.561.79 (12.4%)18 (0)10 (8)9 (1)9 (1)Case313.390.96 (6.7%)18 (0)14 (4)9 (1)9 (1)Case413.061.29 (9.0%)18 (0)13 (5)9 (1)9 (1)Case512.901.45 (10.2%)18 (0)12 (6)9 (1)9 (1)Case 6a12.791.56 (10.8%)11 (7)11 (7)9 (1)9 (1)Note: The values in the parenthesis indicate the reduced thickness.aCase 6, Girder height 1950mm (+250mm).D.H. Lee et al.Thin-Walled Structures 134 (2019) 491507502(a) Ultimate strength behaviour(b) Ultimate strength(c) Z-displacement at the design load levelDesign load levelDd: 0.252Allowable Z-displacement, 21mmFig. 20. The results of the ultimate strength computations for the auxiliary girder structures at loading position 1.D.H. Lee et al.Thin-Walled Structures 134 (2019) 491507503Fig. 21. The results of the ultimate strength computations for the auxiliary girder structures at loading position 2.D.H. Lee et al.Thin-Walled Structures 134 (2019) 491507504Original IzzFig. 22. Variation of the moment of inertia with varying the plate thickness and the web height.Design load levelFig. 23. Local buckling strength of the inner web in the auxiliary girder structures.Table 12Local buckling strength checks of the inner web in the auxiliary girder structures.CaseLocal buckling strength (kN)Rate, divided by original designSafety margin for local buckingOriginal3202.911.003.31Case 52921.200.913.02Case 62494.520.782.58D.H. Lee et al.Thin-Walled Structures 134 (2019) 491507505Fig. 24. von-Mises stress distribution of the auxiliary box girder crane structures at ultimate limit state.Table 13The results of the natural frequencies computations for the main and auxiliary box girders.Natural modeNatural frequency (Hz)Main girderAuxiliary girderI14.900111.0365II15.260113.5584III16.174016.5402IV21.711118.2878V23.175430.5637Fig. 25. The natural mode I of the main and auxiliary box girders.D.H. Lee et al.Thin-Walled Structures 134 (2019) 491507506AcknowledgementsThis study was undertaken in the International Centre for AdvancedSafety Studies of the Korea Ship and Offshore Research Institute atPusan National University, South Korea which has been a LloydsRegister Foundation Research Centre of Excellence since 2008. Theauthors acknowledge the support of POSCO Engineering & Constructionin South Korea.References1 B. Johansson, M. Velijkovic, Steel plated structures, Progress. Struct. Eng. 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