A NEW SIMPLIFIED CLOSED-FORM INTERACTION FORMULATIONS FOR EVALUATION OF STRUCTURAL RESPONSE OF STIFFENED PLATES

Transcription

1 A NW SIMPLIFID CLOSD-FORM INTRACTION FORMULATIONS 41 FOR VALUATION OF STRUCTURAL RSPONS OF STIFFND PLATS A NW SIMPLIFID CLOSD-FORM INTRACTION FORMULATIONS FOR VALUATION OF STRUCTURAL RSPONS OF STIFFND PLATS Özgür ÖZGÜÇ* *Istanbul Technical University, Dept. Of Naval Arch. and Ocean ngineering ABSTRACT A semi-analytical mdel fr ultimate strength capacity assessment f stiffened plates has been develped based n ANSYS nn-linear elast-plastic buckling analyses f a wide range f typical ship panel gemetries. The primary aim f the present study is t investigate the ultimate strength interactin relatinship f a stiffened plate subject t cmbine lads with imperfectins in the frm f gemetric deflectins and welding induced residual stresses. The accuracy f the interactin relatinship is cnfirmed by use f inelastic finite element calculatins. Cmparisn is perfrmed with existing ship rules used by Classificatin Scieties as well. The results and insights derived frm the present wrk are summarized in great detail. Keywrds: Finite lement Analysis; Initial Imperfectin; Interactin Frmulas; Ship Hull; Ultimate Strength 1. Intrductin Stiffened plates is the main structural building blck in ship hulls and their structural respnse subject t cmbine lads is a tpic f significant practical interest in ship design. Figure 1 shws an example f such cnstructin where the stiffened plate spans between girders. Fr the real ship structural stiffened plates, the mst general lading case is a cmbinatin f lngitudinal stress, transverse stress, shear stress and lateral pressure. Due t the presence f the cmbine lads, stiffened panels are susceptible t failure by instability. Instability f stiffened plates can take ne f furs frms [1,4,5,7] such as plate induced verall buckling (PI), stiffener induced verall buckling (SI), plate buckling (PB) and stiffener tripping (ST). The typical buckling mdes are demnstrated in Figure. p la tin g p a n e l p la te tr a n s v. g ir d e r s s tiffe n e r s l n g. g ir d e r s Figure 1. A stiffened steel plate in ship hull R y s P a n e l i p ³ y t a w k n s t r u k c j i k r ê t w e j Precise mdelling f stiffened panels can be achieved by means f analysis tls and cmputing pwer. Initial imperfectins such as welding induced residual stress and initial deflectins f the crss sectin can be explicitly incrprated int numerical mdels. In a series f recent papers, Sayı 11, 018 GiDB DRGi

2 4 Ö. ÖZGÜÇ Grndin [3,4] cnsidered the behaviur f these elements under axial cmpressin, bth experimentally and numerically. The gal f that study was t investigate the tripping failure mde and validate with experiments, a sphisticated nn-linear finite element mdel that wuld allw a mre extensive study f the behaviur t be cnducted numerically. (b) (a) (d) (c) Figure. Typical buckling mdes, (a) Overall buckling (plate induced); (b) verall buckling (stiffener induced); (c) plate buckling; and (d) stiffener tripping Hughes, Ghsh and Chen [6] derived mdified expressins fr elastic lcal plate buckling and verall panel buckling expressins frm 55 Abaqus eigenvalue buckling analyses. Inelastic RISK analysis fr the ultimate cllapse stress and pst cllapse behaviur using Abaqus Fem was cnducted n their mdels. Ultimate stress was als calculated using Orthtrpic methds. It was fund that fr panels having crssver prprtins, Orthtrpic based methds are unsatisfactry. A direct calculatin mdel (PULS) fr determinatin f ultimate capacity f stiffened panels was develped using energy principles and nnlinear plate thery accrding t Marguerre and Vn Karman [9]. xtensive verificatins were carried ut by means f Abaqus F prgram. In general, very satisfactry crrespndence between PULS and mre advanced numerical prgrams were fund. Ozguc t al. [13] develped the new simple design equatins fr predicting the ultimate cmpressive strength f stiffened plates with initial imperfectins in the frm f weldinginduced residual stresses and gemetric deflectins were develped in this study. A nn-linear finite element methd was used t investigate n 60 ANSYS elastic plastic buckling analyses GiDB DRGi Sayı 11, 018

3 A NW SIMPLIFID CLOSD-FORM INTRACTION FORMULATIONS 43 FOR VALUATION OF STRUCTURAL RSPONS OF STIFFND PLATS f a wide range f typical ship panel gemetries. Reductin factrs f the ultimate strength are prduced frm the results f 60 ANSYS inelastic finite element analyses. The accuracy f the prpsed equatins was validated by the experimental results. Cmparisns shw that the adpted methd has sufficient accuracy fr practical applicatins in ship design. Paik t al. [14] cncentrated n methds fr the ultimate limit state assessment f stiffened plate structures under cmbined biaxial cmpressin and lateral pressure actins cnsidering the bttm part f an AFRAMAX-class hypthetical duble-hull il tanker structure. Three methds, namely ANSYS nnlinear finite element methd, DNV PULS methd, and ALPS/ULSAP methd were used. Chaithanya t al.. [11] evaluated the behavir f stiffened plates with different distrtin levels in rder t address a ratinal structural design prcedure, as pre-existing and fabricatin-related initial gemetrical distrtin frm a structural design pint f view. Nn-linear finite element (F) analysis using ABAQUS was carried ut under axial lading cnditin t predict the behavir and the buckling strength. Xu and Sares [1] simulated numerically the behavir f stiffened panels under uniaxial cmpressin until cllapse and beynd, and then cmpared with tests made t investigate the influence f the stiffener s gemetry and the bundary cnditins. The stiffened panel mdels have three lngitudinal bays t prduce reasnable bundary cnditins in the lngitudinal directin. The material and gemetric nnlinearities were accunted fr in the F analyses. The initial gemetric imperfectins, which affect significantly the cllapse behavir f stiffened panels, were assumed t have the shape f the linear buckling mde. Fur types f stiffeners were made f mild r high tensile steel fr bar stiffeners and mild steel fr L and U stiffeners t investigate different material and gemetry cnfiguratins, and fur bundary cnditins were analyzed. Tekgz t al. [15] analyzed the effect f different finite element mdels n the ultimate strength assessment f stiffened plates, where the effect f element size, and type, bundary cnditins, shape f initial imperfectin, thickness and net sectinal cnfiguratins were accunted fr. Fur different finite element mdels and different structural cnfiguratins were cmpared t the slutin described by the Cmmn Structural Rules (CSR). Ch t al. [16] prpsed ultimate strength frmulatin fr stiffened plates. The frmulatin was derived by a regressin study using the parametric study results. The accuracy and reliability f the prpsed frmulatin were cmpared with thse f cmmercial packages, such as ABAQUS and DNV PULS, and experimental results. Zhang [17] presented a review and study n ultimate strength analysis methds fr steel plates and stiffened panels in axial cmpressin. Buckling and cllapsing mechanisms f steel plates and stiffened panels are described. A study and further validatin n the authrs develped frmula fr ultimate strength f stiffened panels using a cmprehensive nn-linear finite element analysis, 110 mdels in ttal, and a wide range f mdel test results, 70 mdels in ttal, were carried ut. Finally, applicatins f the develped frmula t existing il tankers and bulk carriers were presented. The primary aim f the present study is t investigate the ultimate strength interactin relatinship f a stiffened plate subject t cmbine lads with imperfectins in the frm f Sayı 11, 018 GiDB DRGi

4 44 Ö. ÖZGÜÇ gemetric deflectins and welding-induced residual stresses. The accuracy f the interactin relatinship is cnfirmed by use f inelastic finite element calculatins. Cmparisn is perfrmed with existing ship rules used by Classificatin Scieties as well. The results and insights derived frm the present wrk are summarized.. Simplified Clsed-Frm Frmulatins fr Cllapse Strength A semi-analytical mdel fr calculating the values f the critical buckling stresses fr the plate, beam-clumn, trsinal-flexural (tripping) and lcal failure mdes are develped with reductin factrs which can describes initial imperfectins in the frm f gemetric deflectins and welding-induced residual stresses. Based n extensive numerical calculatins an interactin frmula is prpsed fr cmbine lading which invlves lngitudinal cmpressin, transverse cmpressin, shear lading and lateral pressure lading [8]..1 last-plastic Cllapse f the Structural lements The equatin describing the lad-end shrtening curve r the elastic-plastic cllapse f structural elements cmpsing the hull girder transverse sectin can be btained frm the fllwing frmula, valid fr bth psitive (shrtening) and negative (lengthening) strains. (1) where, is edge functin, is yield stress f element. 1 fr fr () 1 fr 1. Beam clumn Buckling Failure Mdel The equatin describing the lad-end shrtening curve CR 1 fr the beam-clumn buckling f the stiffeners cmpsing the hull girder transverse sectin can be btained frm the fllwing frmula: A S b t CR 1 C 1 A S bt (3) where, is edge functin defined in equatin 1, C 1 A is critical stress in MPa, S is net sectinal area f a stiffener and b is spacing f stiffeners. GiDB DRGi Sayı 11, 018

5 A NW SIMPLIFID CLOSD-FORM INTRACTION FORMULATIONS 45 FOR VALUATION OF STRUCTURAL RSPONS OF STIFFND PLATS 1 fr fr C (4) where C 1 is based n the Jhnsn-Ostenfeld frmulatin accunting fr inelastic effects n the clumn s buckling. In equatin (3) the secnd term cmputes the lss f efficiency f plate due t cmpressin lading. ffective width, b, based n the Frankland`s apprach develped t the plate strength and given by, b b fr b fr b fr (5) where, 1 is uler clumn buckling stress, which is calculated as belw, 1 I A a (6) where, I is net mment f inertia f rdinary stiffeners with attached shell plating f width b 1, A is net sectinal area f stiffeners with attached shell plating f effective width b, and a is length f stiffened plate. b fr 1 b 1 b fr 1 (7) Sayı 11, 018 GiDB DRGi

6 46 Ö. ÖZGÜÇ where, b t is defined..3 Plate induced buckling failure mde The equatin describing the lad-end shrtening curve CR fr the plate buckling cmpsing the hull girder transverse sectin can be btained the fllwing frmula: A S b t CR A S bt (8).4 Flexural trsinal (tripping) Buckling Failure Mde The equatin describing the lad-end shrtening curve CR 3 fr the flexural trsinal (tripping) buckling f stiffeners cmpsing the hull girder transverse can be btained accrding t fllwing frmula: CR A bt S C 3 CP (9) 3 A bt S where, C 3 is defined as critical stress. 3 fr fr C (10) where, 3 is uler trsinal buckling stress, defined as fllws. I W K C I t 3 m I m I P a P (11) GiDB DRGi Sayı 11, 018

7 A NW SIMPLIFID CLOSD-FORM INTRACTION FORMULATIONS 47 FOR VALUATION OF STRUCTURAL RSPONS OF STIFFND PLATS where, I W is net sectinal mment f inertia f the stiffener abut its cnnectin t the attached plating and is defined as fllws. I W 3 3 h w t w fr flat bars 36 3 t f b f h w fr T sec tins 1 3 b h f w t f b f b f h w 1 ( b h f w ) 4 h w 3t w b f h w fr angles and bulb sec tins (1) where, I P is net plar mment f inertia f the stiffener abut its cnnectin t the attached plating, defined as fllws. I P 3 h w t w 3 3 h w t w 3 fr flat bars h w b f t f fr stiffeners with face plate (13) where, I t is St. Venant s net mment f inertia f stiffener withut attached plating, defined as fllws: I t 3 h t w w 3 1 h t w 3 w 3 fr b f t flat bars f t b f f fr stiffeners with face plate (14) where, m is number f half waves, may be taken equal t the integer number and trsinal buckling f axially laded stiffeners, calculated by fllwing; K C is Sayı 11, 018 GiDB DRGi

8 48 Ö. ÖZGÜÇ m ( m 1) K m ( m 1) and C K C C a 0 4 I 4 W (15) where, C 0 is a spring stiffener f the attached plating and can be expressed as fllws, 3 t C (16) 0.73 b Table 1. Trsinal buckling f axially laded stiffeners Number f m half waves. K 0 K 4 4 K K 144 C C m 1 3 C C where, CP is buckling stress f attached plating, which can be determined by fllwing frmula. fr fr (17) CP 1.91 fr Web Lcal Buckling Failure Mde CR 4 The equating describing the lad-end shrtening curve fr the web lcal buckling f flanged stiffeners cmpsing the hull girder transverse sectin can be btained frm the fllwing frmula. b t h t b t we w f f (18) CR 4 bt h t b t w w f f h we where, is effective height f the web, which can be determined by fllwing frmula: GiDB DRGi Sayı 11, 018

9 A NW SIMPLIFID CLOSD-FORM INTRACTION FORMULATIONS 49 FOR VALUATION OF STRUCTURAL RSPONS OF STIFFND PLATS h we h fr 1.5 w W h fr w 1.91 h fr w W W 3.5 (19) h w is defined while is relative strain. W t w ffective width, b is multiplied by reductin factrs s as t intrduce initial imperfectins fr stiffened plates, namely, b ` b R d R r R y R R q (0) where b ` is the effective width f imperfect stiffened plate, b is the effective width f perfect stiffened plate, R d is a reductin factr due t initial deflectin, R r is a reductin factr due t welding-induced residual stress, R y is a reductin factr due t bi-axial cmpressin, R is a reductin factr due t shear stress present, and R q is a reductin factr due t lateral pressure lad. All reductin factrs prpsed are expressed by the fllwing equatins: g R d f (1) fr f ( ) () fr Sayı 11, 018 GiDB DRGi

10 50 Ö. ÖZGÜÇ fr g ( ) (3) fr.5 fr fr R 1.0 (4) r where r is defined as nrmalized welding residual stress. R y 1.0 y yu (5) which is prpsed by Faulkner [], where 0.5 y (6) yu R (7) q where is defined as nrmalized value f pressure. 0.5 R 1 (8) where 3 is given by Faulkner []. GiDB DRGi Sayı 11, 018

11 A NW SIMPLIFID CLOSD-FORM INTRACTION FORMULATIONS 51 FOR VALUATION OF STRUCTURAL RSPONS OF STIFFND PLATS Initial deflectin value is taken int accunt fr plating and stiffeners implicitly in this study. Fr clamped stiffened plates, b, effective width may be re-arranged by the fllwing simple equatins as well. b b fr b fr b fr (9) where is the slenderness rati, cmpressive welding-induced stress, is beam-clumn slenderness rati, w t r is nrmalized is nn-dimensinal initial deflectin, is shear stress, is transverse stress and is lateral pressure lad. Ship plates in decks and bttms are predminantly laded in lngitudinal cmpressin. Hwever, additinal lading systems may result in the simultaneus presence f transverse in plane and shear lads in additin t lateral lading f the plates. The influences f these lads n the cllapse strength f plates can be very significant. Based n extensive numerical results an interactin curve is suggested fr practical applicatins in ship design. y x xu y yu u 3.5 p p u (31) where p u is defined as critical (ultimate) lateral pressure f plating between stiffeners clamped at all edges frm rigid plastic thery prpsed by Wd [10] as belw: p p C, where u p 1 C p (3) 3 b / a b / a Sayı 11, 018 GiDB DRGi

12 5 Ö. ÖZGÜÇ 3. Finite lement Mdel fr Inelastic Buckling Analyses Authrs investigated the structural ultimate capacity f the gemetrical prperties f the 60 three-bay panels having three and five equally spaced T-stiffeners under cmbine lads using ANSYS Implicit nn-linear finite element cde [8]. All mdels were 3600 mm wide and it was intended that they cver the full range f prprtins f typical ship plates. A few mdels are addressed in this paper. An elastic perfectly plastic material mdel withut strain hardening may be cnsidered enugh fr pessimistic strength assessment f stiffened steel plates. Arc-length methd is applied t the slving f nn-linear finite element stiffness equatins. The material yielding stress,, is 35.8 MPa, Yung s mdulus, MPa and the pissn rati,, is assumed t be Fur-ndded shell elements are used t mdel stiffened plate, and a fine mesh is cnducted t adequately capture the stress and defrmatins. One f the examples f all investigated mdels is shwn in Figure 3., Figure 3. ANSYS slid mdel fr three-bay grillages in this study 3.1 Initial Imperfectins fr ANSYS F Mdel It is assumed that plating has the verall buckling mde initial deflectin, which crrespnds t 0 05 t.. The clumn-type r sideways initial deflectin f the stiffeners is taken int accunt t be a, where is the length f ne-bay. Initial deflectin f plating and stiffeners is autmatically accunted nce the scaling factr is applied which crrespnds t abve assumed equatins. The imperfectin patterns are determined frm an verall buckling mde shape f a linear eigenvalue buckling analysis. The cnsidered mde shape has an upward half wave deflectin in the full bay and a dwnward deflectin in the half bay, which is shwn with the lcal plate-buckling mde f a three-stiffener panel in Figure 4. a Figure 4. Overall buckling and lcal plate-buckling mde shapes f a three-stiffener panel, respectively GiDB DRGi Sayı 11, 018

13 A NW SIMPLIFID CLOSD-FORM INTRACTION FORMULATIONS 53 FOR VALUATION OF STRUCTURAL RSPONS OF STIFFND PLATS Fr residual stress distributin Faulkner s mdel is used t represent the distributin f the stresses, and is incrprated int ANSYS finite element mdel as a simple representatin f the actual residual stress present in the stiffened panels. The tensile regins arund the stiffeners represented as a tensin blck having base width prprtinal t the plate thickness ( where the value f this study. typically ranges 3.5 and 4 in a ship structures. It is cnsidered t be 3.5 in 3. Cmparisn between ANSYS FM and Simplified Clsed-frm Frmulatins Frm extensive F nn-linear numerical calculatins, fur cases are addressed in this paper. Typical ship panel gemetries studied are summarized in Table. Table. Gemetric prperties f stiffened panels analyzed in this paper Specime n n. (Plate Slenderness ) (Aspec t rati) (Beamclumn slenderness ) Plate initial deflecti n (mm) (0.05 t) Stiffener initial deflecti n (mm) (0.005a ) Residua l Stress fr stiffene d plate (MPa) p (MPa ) w (MPa ) x t plate ) f (MPa ) Case Case Case Case Cmputed F Results Figure 4 indicates Vn Misses stress distributins btained frm ANSYS, while Figure 5 shws stress-strain relatinships with cnsidering initial deflectin effects frm simple design equatins and ANSYS FM as well. Case 1 Case Sayı 11, 018 GiDB DRGi

14 54 Ö. ÖZGÜÇ Case 3 Case 4 Figure 4. Vn Misses stress distributin with ignring residual stress fr cnsidered all cases, Case1, Case, Case3 and Case 4, respectively A NS Y S FM A NS Y S FM A x ia l S tr e s s / Y ie ld S tr e s s Pres ent Methd A x ia l S tr e s s / Y ie ld S tr e s s Pres ent Methd A x ia l Dis p la c e m e n t / Y ie ld Dis p la c e m e n t A x ia l Dis p la c e m e n t / Y ie ld Dis p la c e m e n t Case 1 Case Case A x ia l S tr e s s / Y ie ld S tr e s s A N S Y S F M Pr e s e n t M e th d A x ia l S tr e s s / Y ie ld S tr e s s A N S Y S FM Pres ent Methd Case A x ia l D is p la c e m e n t / Y ie ld D is p la c e m e n t Figure 5. Cmparisn f ANSYS FM with apprximate frmulatin fr Case1, Case, Case3 and Case 4, respectively with initial deflectin effect A x ia l D is p la c e m e n t / Y ie ld D is p la c e m e n t GiDB DRGi Sayı 11, 018

15 A NW SIMPLIFID CLOSD-FORM INTRACTION FORMULATIONS 55 FOR VALUATION OF STRUCTURAL RSPONS OF STIFFND PLATS As demnstrated, cllapse behaviur f structural members cmpsing a crss sectin f a hull girder largely affects the cllapse behaviur f the crss sectin and its ultimate strength as whle. Frm this viewpint, it is very imprtant t knw hw accurately the applied methd simulates the cllapse respnse and predicts the ultimate strength f individual structural members as stiffened plates. Cmparisns f ultimate strength capacities using ANSYS FM and simplified clsed-frm frmulatins are very cnsistent fr all cases studied in this paper. 3.4 Interactin Capacity Curve New prpsed interactin frmula is als validated results reprted by DNV Research Team [6]. In this paper, capacity curves fr cmbined lads calculated by ABAQUS, DNV PULS, DNV and GL rules are presented with present methd. 3.5 Biaxial Cmpressin Capacity curves fr bi-axial cmpressin f bttm panel f a 173 m tanker are presented in Figure 6, while Table 3. summarizes main particulars f investigated mdel. Results fr the same panel under cmbined in-plane cmpressin and lateral pressure are presented in Figure 7. Table 3. The main particulars f the tanker bttm panel. Length f stiffened panel 400 mm Stiffener spacing 800 mm Plate thickness 13.5 mm Web height 40 mm Web thickness 11 mm Stiffeners 6 lngitudinal Bulb prfiles Yield stress 355 MPa Yung s mdulus MPa Pissn rati A B A Q U S -F M P re s e n t M e t h d D N V R u le s A B A Q U S - FM Pres ent Methd PU L S 0 G L R u le s 0 0 P U L S Figure 6. Tanker bttm panel, biaxial cmpressin withut lateral pressure Figure 7. Tanker bttm panel, biaxial cmpressin with lateral pressure, p MPa The cmparisns f ultimate strength capacities using ABAQUS, PULS and present methd are very cnsistent. Such deviatins are t be expected since the applied methds are very different. Typically, the largest deviatins are fr regins in lad space where the failure mde is nt unique and btained results depend strngly n hw the gemetrical imperfectins are mdelled Sayı 11, 018 GiDB DRGi

16 56 Ö. ÖZGÜÇ especially with respect t shape and definitin f bundary cnditin. It is seen that reductin in the in-plane capacity is nt very much reduced when the design lateral pressure is emplyed. The reductin is smewhat lesser fr present methd than ABAQUS and PULS. It is seen that present apprach predicts mre capacity than bth f the rule frmulatins in the bi-axial regin. Fr pure axial cmpressin, DNV Rules seem t be verly cnservative when cmpared with all predictin methds, while fr pure transverse cmpressin GL Rules seem t be nn-cnservative. The present methd curve is mre cnvex, which is als the case fr ABAQUS, PULS and DNV Rules, while the GL Rules interactin curve is clse t linear. 3.6 ffect f Shear Lad Capacity curves fr cmbined shear lad and transverse cmpressin f a bulk carrier side panel are presented in Figure 8, while Table 4. summarizes main particulars f investigated mdel. The lading is typically cmpressin perpendicular t the stiffener transverse directin acting simultaneusly with in-plane shear and lateral pressure frm the sea. Results fr the same panel under cmbined transverse cmpressin, shear and lateral pressure are presented in Figure 9. Table 4. The main particulars f the Bulk Carrier side panel. Length f stiffened panel 8800 mm Stiffener spacing 890 mm Plate thickness 14.5 mm Web height 700 mm Web thickness 13 mm Stiffeners 5 lngitudinal Tee prfiles Flange breadth 150 mm Flange thickness 18 mm Yield stress 355 MPa Yung s mdulus MPa Pissn rati A B A Q US - FM Pres ent Methd PUL S DNV Ru le s A B A Q US - FM Pres ent Methd PUL S G L Ru le s Figure 8. Bulk Carrier side panel, transverse cmpressin and shear lading withut lateral pressure Figure 9. Bulk Carrier side panel, transverse cmpressin, shear with lateral pressure, p=0.157 MPa It can be seen that present methd shws very reasnable results as cmpared t ABAQUS and PULS analyses cvering lad cmbinatins cvering lad cmbinatins dminated by shear GiDB DRGi Sayı 11, 018

17 A NW SIMPLIFID CLOSD-FORM INTRACTION FORMULATIONS 57 FOR VALUATION OF STRUCTURAL RSPONS OF STIFFND PLATS lading as well as lad cmbinatins dminated by transverse cmpressin. The presence f lateral pressure is nt very significant fr the in-plane capacity f this pane, thugh mre s fr transverse dminated lading than fr pure shear. It is seen that bth the rule frmulatins verpredict the capacity fr pure transverse cmpressin, while they significantly underestimate the capacity in the cmbined lad regin f the capacity curve. 3.7 ffect f Lateral Pressure Capacity curves fr the axial capacity fr a tanker bttm panel are presented as a functin f lateral pressure in Figure 10, while Table 5. summarizes main particulars f investigated mdel. The transverse capacity fr the same panel is presented as a functin f lateral pressure in Figure 11. Table 5. The main particulars f the tanker bttm panel. Length f each bay (mm) 510 Panel breadth (mm) 9100 Plate thickness (mm) 0 Web height (mm) Web thickness (mm) 1 Stiffeners 9 lngitudinal T-stiffeners Flange breadth (mm) 00 Flange thickness (mm) 0 Yield stress (MPa) 315 Yung s mdulus (MPa) Pissn rati A NS Y S FM Pr e s e n t s tu d y PUL S DNV Ru le s G L Ru le s Figure 10. ffect f lateral pressure n axial capacity fr tanker bttm panel Figure 11. ffect f lateral pressure n transverse capacity fr tanker bttm panel It is seen that present methd in the axial capacity predicts reasnable results as cmpared t ABAQUS, while the rule frmulatins verpredict the capacity f the panel since they are nt affected by influence f lateral pressure. Sayı 11, 018 GiDB DRGi

18 58 Ö. ÖZGÜÇ It is seen that present methd in the transverse capacity estimate slightly mre results up t 0.0 MPa since transverse capacity verpredicts when lateral pressure is zer, hwever, it estimates very gd at 0.30 MPa, where it underestimates at fixed pressure f 0.55 MPa. Rule frmulatins verpredict transverse capacity even fr zer lateral pressure and mre s fr increasing magnitude f pressure. The results indicate that lateral pressure has imprtant influence n the buckling capacity and shuld be taken int accunt. 4. Discussin and Cnclusin Simplified clsed-frm interactin frmulatins fr the ultimate capacity assessment f stiffened panels has been develped based n a large number f nn-linear finite element analyses using the cmmercial prgram ANSYS. It is believed that full nnlinear finite element cdes are able t predict buckling deflectin an accuracy which is sufficient fr advanced design purpses, n cnditin that the analyses are dne prperly such as bundary cnditins, mesh size, mdel extent, element types and imperfectins. Validatin f the prpsed mdel is cnducted by use f nn-linear finite element calculatins and by existing ship rules used by DNV and GL Rules. It is fund that present mdel is generally cnsistent with results btained frm by ABAQUS and PULS. The rules used by Classificatin Scieties are fund t be cnservative fr sme case and nn-cnservative fr ther cases as cmpared with ABAQUS and PULS. Therefre, it is difficult t assess the actual safety margin using these frmulatins. The main advantage f the apprximate methd relative t FM results frm the time cnsumptin bth in the creatin f mdel and in the CPU time, s it can be used fr practical applicatins in ship design. References [1] Bnell, M.A.; Chryssanthpuls, M.K.; Dwling, P.J. Ultimate Strength Design f Stiffened Plates under Axial Cmpressin and Bending, Marine Structure, 1993, Vl. 6, pp [] Faulkner, D. A review f ffective Plating fr use in the Analysis f Stiffened Plating in Bending and Cmpressin, Jurnal f Ship Research, 1975, Vl. 19, N. 1, pp [3] Grndin, G.Y.; Chen, Q.; lwi A..; Cheng J.J.R. Buckling f Stiffened Steel Plates- Validatin f a Numerical Mdel, Jurnal f Cnstructinal Steel Research, 1998, Vl. 45, N., pp [4] Grndin, G.Y.; Chen, Q.; lwi, A..; Cheng J.J.R. Buckling f Stiffened Steel Plates- A Parametric Study, Jurnal f Cnstructinal Steel Research, 1999, Vl. 50, N., pp [5] Hu, S.Z.; Chen, Q.; Pegg, N.; Zimmercman, T.J.. Ultimate Cllapse tests f Stiffened Plate Ship Structural Units, Marine Structure, 1997, Vl. 10, pp [6] Hughes, O.F.; Ghsh, B.; Chen, Y. Imprved Predictin f Simultaneus Lcal and Overall Buckling f Stiffened Panels, Thin-Walled Structures, Vl. 4, pp , 004. [7] Murray N.W., Buckling f Stiffened Panel Laded Axially and in Bending. Structure ngineer, 1973, Vl. 51, N. 8, pp [7] Steen,.; Byklum,.; Vilming, K.; Ostvlds, K. Cmputerized Buckling Mdels fr Ultimate Strength Assessment f Stiffened Ship Hull Panels, 9th Sympsium n Practical Design f Ships and Other Flating Structures, 004, Luebeck-Travemuende, Germany. [8] Wd, R.H. Plastic and Inelastic Design f Slabs and Plates, The Rnald Press, 1961, New Yrk. [9] Chaithanya, P.P.; Das, P.K.; Crw, A.; Hunt, S. The effect f distrtin n the buckling strength f stiffened panels. Ships and Offshre Structures, 010, Vl.5. GiDB DRGi Sayı 11, 018

19 A NW SIMPLIFID CLOSD-FORM INTRACTION FORMULATIONS 59 FOR VALUATION OF STRUCTURAL RSPONS OF STIFFND PLATS [10] Xu, M.C.; Sares, C.G. Numerical assessment f experiments n the ultimate strength f stiffened panels, ngineering Structures, 01, Vl.45, pp [11] Ozguc, O.; Das, P.K.; Barltrp, N. The new simple design equatins fr the ultimate cmpressive strength f imperfect stiffened plates. Ocean ngineering, 007, Vl. 34, Issue 7, pp [1] Paik, J.K.; Kim, B.J.; Se, J.K. Methds fr ultimate limit state assessment f ships and ship-shaped ffshre structures: Part II stiffened panels. Ocean ngineering, 008, Vlume 35, Issue, pp [13] Tekgz, M.; Garbatv, Y.; Sares, C.G. Ultimate strength assessment fr the effect f finite element mdelling. Maritime ngineering and Technlgy, 01. [14] Ch, S.R.; Kim, H.S.; Dh, H.M.; Chn, Y.K. Ultimate strength frmulatin fr stiffened plates subjected t cmbined axial cmpressin, transverse cmpressin, shear frce and lateral pressure ladings. Ships and Offshre Structures, 013, Vl.8. [15] Zhang, S. A. review and study n ultimate strength f steel plates and stiffened panels in axial cmpressin. Ships and Offshre Structures, 016, Vl.11. Sayı 11, 018 GiDB DRGi