GEOMETRIC SOLUTION IN PROGRESSIVE COLLAPSE ANALYSIS OF HULL GIRDER

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1 Journal of Marne Scence and Technology DOI: 0.69/JMST Ths artcle has been peer revewed and accepted for publcaton n JMST but has not yet been copyedtng, typesettng, pagnaton and proofreadng process. Please note that the publcaton verson of ths artcle may be dfferent from ths verson. Ths artcle now can be cted as do: 0.69/JMST GEOMETRIC SOLUTIO I PROGRESSIVE COLLAPSE AALYSIS OF HULL GIRDER Ertekn Bayraktarkatal and Gökhan Tansel Tayyar Key words: ultmate strength, progressve collapse, curvature, effectve wdth. ABSTRACT Ths paper presents a calculaton model for stffened plates to determne the ultmate strength of shp hull grders from ther curvatures usng a geometrcal approach. The present study employed Smth s method, n whch the cross-secton s dvded nto smaller elements consstng of a stffener(s) and attached platng. The strength of beam-columns and stffened plates was obtaned usng an teratve numercal approach. The deflecton curve was evaluated usng the curvature values drectly nstead of by solvng dfferental equatons. The deflecton curve was taken as an assembly of chans of crcular arcs. The ultmate strength of the hull grder of a /3 scaled frgate model was analyed usng the proposed method through an teratve approach. The present method produced reasonably accurate solutons wth low modelng and computatonal tmes. I. ITRODUCTIO A new type of calculaton model was presented for stffened plates to determne the ultmate strength of shp hull grders from the curvatures usng a geometrcal approach; curvature-based deflecton method [0]. Smth s method, n whch the cross-secton s dvded nto smaller elements composed of a stffener(s) and attached platng, s the preferred method for utlng the results of the present approach. The shape of the load-shortenng relatonshp was reported to affect the ultmate hull grder strength sgnfcantly when Smth s method was appled [7]. Hence, the load-shortenng relatonshp of the element forms the bass of the proposed method. The load-carryng capacty of a stffened plate beyond ts ultmate strength generally decreases rapdly due to the progress of bucklng deformaton, whch s normally accompaned Paper submtted 07/5/; revsed 0//3; accepted 05/3/3. Author for correspondence: Ertekn Bayraktarkatal (e-mal: bayrak@tu.edu.tr). Faculty of aval Archtecture and Ocean Engneerng, Istanbul Techncal Unversty, Maslak, Istanbul, Turkey. by localed plastc deformaton ether n the plate or stffener. Therefore, t s mportant to assess the progressve collapse behavor of structural systems, such as hull grder collapse, by eamnng both the ultmate strength and post-ultmate strength behavor of stffened panels [8]. The accuracy of the ultmate strength computatons of a shp cross secton s closely related to the strength analyss of stffened plates, whch n turn depends on the accuracy of the deflecton curve [,]. Therefore, the method proposed n ths study focused on mathematcal modelng of the deflecton curve. In accordance wth second order theory, the aal/n-plane stresses rely on accurate modelng of the deflecton curve. In addton to usng the curvature value n dfferental equatons of the deflecton curve, the curvature also represents the deflecton geometrcally. In ths paper, the radus of curvature was used drectly n deflecton geometry, and the deflecton curve was obtaned numercally usng an teratve method [0-]. The deflecton curve of the structure was modeled easly usng the curvature values, even f the materal or geometrcal nonlneartes occur [0-3]. Furthermore, the ultmate strength can be obtaned usng a sngle numercal procedure, regardless of the structure ehbtng elastc or nelastc behavor [0]. The man motvaton of ths approach was to reduce the computatonal and modelng tmes, as well as obtan a more precse soluton. The present paper does not nclude an evaluaton of the curvature values from the moment values. In ths paper, the effectve wdth calculaton method was etended to the curvature based deflecton method [], and an applcaton to column bucklng analyss of a perfect beam-column was evaluated by a comparson wth the analytcal results. In addton, the Dow s Frgate eperment model was assessed and compared wth the results reported n the lterature. II. METHOD Consderng the equlbrum of a cross-secton of a beam-column or stffened plate, the nternal forces dstrbuted over the cross secton were statcally equvalent to the eternal forces. The structure was modeled wth a one-dmensonal regular curve called the deflecton curve, whch was generated by the centrods of the cross secton of the stffened plate or

2 beam-column: C moves along ts major prncpal plane normal, as shown n fgure, where s s the curve length of the deflecton curve []. The deflecton curve s an arc length parametered contnuous and dfferentable curve that can be defned by the poston vector, α(s)(s) (s)k []. Fg.. Deflecton curve. The curvature of varous ponts on a deflecton curve could be obtaned usng the reacton forces and the materal propertes of the cross secton. In ths case, the deflecton curve wll be composed of arcs. The curve s represented by a sequence of crcular arcs wthn a user-specfed tolerance, and the relatonshp between the crcular arcs s establshed usng the proposed method.. Deflecton curve modelng The rad of the arcs are nsuffcent for deflecton calculatons because ther centers are unknown. To appont the poston of the center ponts of the arcs, adjacent arcs are needed to jon wth G contnuty, whch has a common unt tangent vector on common ponts []. Consequently, the arcs have the same unt normal vector at a common pont of the deflecton curve, whch means that a lne connectng the center of the adjacent osculatng arcs always passes through a common pont, as shown n fgure 3 []. Therefore, t s possble to determne the poston of the adjacent arc from the prevous poston of the arc, curve length and radus of curvature [0-]. Fg. 3. Curve Modelng []. Fg.. Curvature. Consder a suffcently small segment between ponts and on the deflecton curve: the length between these ponts s ds. The tangent angles at ponts and between the -as are θ and θ, respectvely (Fg. ). By takng the segment, ds, as beng suffcently small, t was assumed that the change n curvature across the segment would be neglgble. Therefore, the shape of the segment ndcates the arc of a crcle wth a unform curvature dstrbuton [,]. Ths constant curvature also equals the rato of the dfference between the ntal and termnal pont of the tangent angle of the segment, and dθ s the segment length, as epressed n Eq., where r represents the radus of curvature or the radus of the crcle of the arc (fgure ) and K represents the curvature of the segment [4]. The chord length between these ponts was dc (Fg. ) []. The curvature equaton also makes a connecton between knetc and knematc analyss. / r K dθ ds () The procedure needs to start from the precse pont where tangent angle s known. These ponts can be selected as the orgn pont of the deflecton curve and procedure. A procedure can be developed where the frst pont slope s predcted and revsed teratvely accordng to the deflectons and slopes at the boundares or supports. If the frst pont s between the span of the structure, the procedure should progress separately for the left and rght sde of the frst pont unless there s symmetry. For smplcty, ero tangent angle ponts can be selected as clamped edges or symmetry as. Fg. 4. Dsplacement vectors of a deflecton curve [].

3 Fgure 4 shows the total dsplacement between ponts and 3, whch s the vectoral summaton of the two dsplacement vectors. Therefore, the general dsplacement vector between the startng pont and any pont of the deflecton curve can be obtaned by a smple vectoral summaton of the dsplacement vector of all prevous segments epressed as follows, where and k represents the unt vectors of the -as and -as n the rectangular Cartesan coordnate system, respectvely []: have a sgnfcant effect on the bucklng collapse behavor of the plate [8]. The concept of the effectve wdth has been used for practcal desgn. The pure formula wthout the emprcal formulatons of the ntal mperfecton from von Karman was used for the effectve wdth calculatons, as epressed n Eq. 6, where ma and cr represent appled mamum edge stress of the structure and crtcal local plate bucklng stress, respectvely [3,7,5]. dc..., dc, dc,3 dc3,4 dc, k α () ρ ρ ρ cr b e ma b (6) ρ > The tangent angle at pont on the deflecton curve can be epressed as follows: θ θ K n, n dsn, n θ n θ K, ds, dθ, where K, and ds, represents the curvature value and curve length between pont and pont on the deflecton curve, respectvely []. The dsplacement vector from ntal pont can be epressed as follows: ( snθ snθ ) d, (4) K, ( cosθ θ ) (5) d, cos K, (3) The elastc crtcal local plate bucklng stress can be epressed as follows, where t and b are thckness and wdth of the correspondng plate, respectvely; E s the Young modulus; ν s the Posson rato and k s the bucklng coeffcent. kπ E ( ν ) b ( t cr ) (7) The value of k for the attached platng at bottom or deck can be taken as 4 for pure bendng [7,9]. The other type of bucklng coeffcents for the stffener web or stffener flange and local bucklng stress [8] can be epressed as. Fgure 5 presents the typcal stffened plate cross secton composed of the attached effectve plate, web and flange. As a consequence of the effectve wdth calculaton, effectve dmensons or locatons represented by the subscrpts, e, can be recalculated. where, and θ were taken as ero []. The same approach can be used nversely to calculate the ntal curvatures. Therefore, there s no need to defne the ntal deflecton wth a trgonometrc seres [0]. III. STIFFEED PLATE MEMBERS When estmatng the ultmate strength of a stffened panel usng smplfed methods, t s often necessary to accurately determne the post-bucklng effectve wdth of a local plate panel. The nteracton between the platng and stffener n the bucklng behavor must also be consdered carefully [8]. The effectve wdth of the platng between stffeners was formulated, accountng for the appled compressve loads, ntal mperfectons and weld nduced resdual stresses. The effectve wdth of the buckled platng vares wth the compressve loads because t s a functon of the appled compressve stress [6]. On the other hand, most smplfed methods assume that the effectve wdth of platng does not depend on the appled compressve load, and the ultmate effectve wdth of platng s typcally constant. An equaton characterng the ultmate lmt condton for a stffened panel showed a much hgher degree of nonlnearty than would otherwse be derved by treatng the effectve wdth of the platng as a varable [4]. The ntal shape mperfecton and weldng resdual stresses Fg. 5. Stffened plate cross-secton wth stran stress dstrbutons. The stress dstrbuton over the cross secton can be defned easly usng the curvature values, eternal loads and materal propertes. The stran dstrbuton can be acheved usng Eq. 8, where represents the shft between the deflecton curve as or between the effectve centrod of the cross secton, C e, to the neutral as (Fg.5). ε K( ) (8) The stress dstrbuton can be derved usng the materal propertes of the materal. The soluton for the elastc materal can be epressed as follows:

4 EK( ) The stress dstrbuton for the elasto-plastc materal type can be obtaned as follows: EK( ) K( ) ε yeld K( ) (0) > yeld K( ) ε yeld K( ) where yeld and ε yeld are the yeldng stress and stran of the materal, respectvely. The nternal and eternal forces need to be n equlbrum over the cross secton. Therefore, the stress dstrbuton needs to satsfy Eq., where F s the aal normal forces actng on the cross secton area, A. The neutral as poston can be defned by Eq.. Determnaton of the curvature value s obtaned from Eq. by substtutng Eq.9 or Eq. 0 for elastc type or for elasto plastc type of materals. (9) IV. PROCEDURE The teraton procedure of the numercal method s based on the assumpton that the curvature of a segment s constant and equals the curvature obtaned from the mdpont or average moment of the segment. The slope of the frst pont s ero. Therefore, a symmetrcal deflecton s epected for the segments of the left and rght sdes of the start pont. The mean moment of the start pont can be accepted nstead of the mdpont moment value of the frst segment. The mdpont coordnates of the followng segment were evaluated usng the prevous segment curvature by assumng that the curvature does not change from the start pont of the segment to the mdpont of the followng segment. Fgure 6 presents the notatons, and values are the dstance from reference pont. The teraton only needs to be performed from the left or rght sdes of the startng pont. F dyd () A M dyd () A The stffener plate stress along the longtudnal drecton has a dstrbuton that s dependent on the curvature values. The concentrated loads, pressure loads and ntal mperfectons were consdered n the knetc calculatons of the moment curvature. Therefore, there s no need to use the emprcal formulatons to model the ntal mperfecton of the deflecton curve or add addtonal formulatons to model the eternal loads. The stress of the plate was taken as the mamum compressve stress of the attached plate, ma-plate from Eqs. 0 and to obtan the effectve wdth of the stffener plate n Eq. 3 where cr-plate was obtaned from Eq. 7. The effect of weldng resdual stress, R, was ncluded by addng the resdual stress drectly to the mamum compressve stress of the plate as follows accordng the studes summared n reference [7]: ρ ρ ρ ρ > cr plate b e plate (3) ma plate R b The collapse modes for stffened plates reported elsewhere can be classfed nto s types [7]. Column bucklng, yeldng or local bucklng nduced types collapses can be obtaned easly usng the proposed method. Therefore, the applcaton n secton IV llustrates the ablty of the method to determne column bucklng for a perfect beam-column. On the other hand, the proposed method cannot eamne the response to lateral torsonal bucklng because the method s based on planar dsplacements of the deflecton curve. Ths one degree of freedom restrcts the torson of the cross secton. The method bascally models the one dmensonal dsplacements, and t s possble to etend ths developng method wth two dmensonal dsplacements to determne the torson behavor. Fg. 6. Dsplacements of the deflecton curve. The procedure for the load-shortenng calculatons s summared as follows []:. Defne n, total number of the segments,. Calculate dl, shortenng of the structure due to compressve aal load, 3. Calculate segment length; ds (L 0 dl) /( n), 4. Predct du, total horontal shortenng of the boundares and predct δ ma, mamum deflecton; set n L 0 du and, n δ ma, 5. Set; 0, 0, 0, dl 0, θ 0, 6. Obtan the average nternal forces for the segment, 7. Calculate dl, shortenng of the segment due to the average normal force, and set dldldl 8. Calculate r,, radus of curvature value due to nternal forces usng Eqs. and. Consder the ntal curvature values f avalable, 9. Obtan the locaton of pont usng Eqs. 4 and 5 wth the ds segment length, 0. Obtan the followng segment mdpont locaton, usng Eqs. 4 and 5 wth the.5 ds segment length. Take and repeat steps 6 to0 untl the fnal segment.. Go to step 4 and modfy du and δ ma untl the predcted and calculated values are converged, 3. Calculate the effectve wdth usng Eqs. 3. Repeat the teraton from step wth the modfed cross-secton untl the result converges. 4. Go to step and ncrease the number of the segments to check the accuracy and convergence of the result.

5 . Applcaton of the Procedure A very smple and basc applcaton of the procedure, whch s a perfect centrc aally loaded, perfectly straght, smply supported elastc beam-column, was studed (Fg. 7). The crtcal bucklng load was determned to compare wth classcal methods. Fg. 7. Euler-column. For non-bfurcaton bucklng solutons, there are two equlbra, one n pre-bucklng wth a small dsplacement and the other n post-bucklng wth a large dsplacement. At the end of the teraton procedure, the equlbrum value of the mdpont dsplacement can be determned usng the gven aal load. For bfurcaton bucklng problems, a pre-bucklng pattern was stabled usng a straght as. Append presents the calculatons of the frst teraton and man propertes of the beam-column. Fgure 8 presents the results of the frst and subsequent teratons for a 3000 aal load. For the subsequent teratons, the mdpont dsplacement was modfed usng a lnear correcton. Through several teratons, the mdpont deflecton converged rapdly to ero at a gven aal load. Fgure 9 shows the converged mdpont dsplacements as a functon of the aal loads; the crtcal bucklng load was ~4336 wth four segments. Table. Crtcal Bucklng Loads. n P cr (du 0, dl 0 ) P cr (du0, dl0) V. FRIGATE AALYSIS The applcablty of the present method was tested by performng progressve collapse analyss on a large /3-scaled frgate model, whch had been studed epermentally []. The propertes of the frgate model were reported by Dow []. The weldng resdual stress was taken as one tenth of the yeldng stress, as shown n Eq. 4 [4,5]. The mamum ntal deflecton of the stffened plates was calculated usng the unsupported length, as shown n Eq. 5, where a s the dstance between the frame spacng [4]. 0. (4) R yeld δ 0.005a ma (5) The cross-secton was dvded nto stffened plates. In addton to the stffened plate and plate members, there were corner members. The calculatons were performed usng present method for a saggng stuaton. Fg. 8. Deflecton curve aganst the teratons for a 3000 aal load. Fg. 0. Moment curvature dagram. Fg. 9. Converged mdpont dsplacements aganst aal loads. The accuracy was eamned by ncreasng the number of segments, and the crtcal bucklng load was If the secondary effect s neglected and the assumed shortenng value was taken to be ero, the mdpont dsplacement wll converge to appromately 444, as the Euler bucklng soluton. Table lsts the crtcal bucklng aal load values, P cr consderng the segment number and shortenng value. When the ultmate strength was reached by performng the load shortenng values nto Smth s method [7], the locaton of the neutral as was 40 mm, whereas the curvature was l/m due to the predcted 9.74 Mm moment. Fgure 0 shows the moment-curvature dagram. The ultmate moment was measured to be 9.64 Mm [6]. The followng data was obtaned from the range of numercal and analytcal methods reported elsewhere: 0.6 Mm by non-lnear fnte element analyss usng ASYS software [4,9]; 9.83 Mm usng the Idealed Structural Unt Method (ISUM) [4]; 9.94 Mm usng the Intellgent Superse Fnte Element Method (ISFEM) usng ALPS/HULL software; 5.59 Mm usng the classcal beam method [4]; 9.54 Mm reported by Chen usng the ISUM [6]; 9.67 Mm reported by Dow usng Smth s method ncludng the shear and lateral load effects [6]; and 8.58 reported by Yao usng the computer code, HULLST

6 [6]. The proposed method yelded 9.74 Mm, whch s close to the epermental results. Fg. compares the moment curvature dagrams [4, 9]. Fg.. Comparsons of the moment curvature dagrams. VI. COCLUSIOS A new soluton method for beam-columns on a bfurcaton analyss was evaluated. The proposed method enabled the modelng of deflectons wth comple shapes and ther applcaton n the strength of stffened plates and beam-columns. A geometrcal representaton of the curvature not only provdes accuracy n deflecton calculatons but also has advantages n computaton tme. The rapd and accurate eecuton of progressve collapse analyss was realed. A new soluton was developed for ndvdual shp structure members by consderng the effects of the plate/stffener nteracton, ntal deflectons and weldng resdual stress usng the curvature geometrcally. Usng the developed method, progressve collapse analyss was performed usng the solutons of the stffened plate members. The proposed method produced reasonably accurate solutons wth low modelng and computatonal tmes. Ths applcaton can be etend easly to non-bfurcaton bucklng analyss of beam-columns wth ntal deflecton or lateral loads. In addton, strength calculatons of stffened plates can be performed by consderng the effectve wdth wth relable solutons. REFERECES. Barsky, B.A. and DeRose, T.D., Geometrc contnuty of parametrc curves: three equvalent characteratons, IEEE Computer Graphcs & Applcatons. Vol. 9, o. 6, pp (989).. Dow, S.R., Testng and analyss of a /3 Scale Welded Steel Frgate Model, Proc. Int. Conf. on Advances n Marne Structures. pp (99). 3. Faulkner, D., A revew of effectve platng for use n the analyss of stffened platng n bendng and compresson, Journal of Shp Research. Vol. 9, o., pp. -7 (975). 4. Huges, O. and Pak, J. K., Shp Structural Analyss and Desgn, The Socety of aval Archtects and Marne Engneers, (00). 5. Pak, J.K., Thayamball, A.K. and Km, D.H., An analytcal method for the ultmate compressve strength and effectve platng of stffened panels, Journal of Constructonal Steel Research. Vol. 49, pp (999). 6. Pak, J.K., Thamyamball, A.K., Lee, S.K. and Kang, S.J., A sem-analytcal method for the elastc-plastc large deflecton analyss of welded steel or alumnum platng under combned n-plane and lateral pressure loads, Thn-Walled Structures. Vol. 39, pp. 5 5 (00). 7. Pak, J.K., Thamyamball, A.K., Ultmate Lmt State Desgn of Steel-Plated Structures, John Wley & Sons, (003). 8. Pak, J., Branner, K., Choo, Y., Cujko, J., Fujkubo, M., Gordo, J., Parmenter, G., Iaccarno, R., O nel, S., Pasqualno, I., Wang, D., Wang, X. and Zhang, S. Ultmate Strength, n: Jang C.D. and S.Y. Hong (Eds), 7th Internatonal Shp and Offshore Structures Congress, Commttee III., Unversty of Seoul. Vol., pp (009). 9. Pak, J.K., Km, D.K., Park, D.H., Km, H.B., Mansour, A.E. and Caldwell, J.B., Modfed Pak-Mansour formula for ultmate strength calculatons of shp hulls, Shps and Offshore Structures, (0). 0. Tayyar, G.T., Determnaton of Ultmate Strength of Shp Hull Grder, (n Turksh) PhD thess, Graduate School of Scence Engneerng and Technology, Istanbul Techncal Unversty, Istanbul, Turkey (0). Avalable at: [Accessed 3..0].. Tayyar, G.T. and Bayraktarkatal, E., A ew Appromate Method to Evaluate the Ultmate Strength of Shp Hull Grder, n: E. Ruto and C.G. Soares, (Eds), Sustanable Martme Transportaton and Eplotaton of Sea Resources, Taylor & Francs. pp (0).. Tayyar, G.T. and Bayraktarkatal, E., Knematc Dsplacement Theory of Planar Structures. Internatonal Journal of Ocean System Engneerng, Vol., o., pp (0). 3. Tayyar, G.T., A ew Analytcal Method wth Curvature Based Knematc Deflecton Curve Theory, Internatonal Journal of Ocean System Engneerng, Vol., o. 3, pp (0). 4. Tmoshenko, S., Strength of materals part I elementary theory and problems. D.Van ostrand Company, (948). 5. von Karman, T., Sechler E.E. and Donnell, L.H., Strength of Thn Plates n Compresson, ASME Trans. Vol.54, o. 5, pp (93). 6. Yao, T., Astrup, O.C., Cards, P., Chen, Y.., Cho, S.R.,Dow, R.S., ho, O. and Rgo, P., Ultmate Hull Grder Strength, Specal Task Commttee VI., n: Ohtsubo, H. and Sum, Y., (Eds), 4th Internatonal Shp and Offshore Structures Congress, Vol., pp (000). 7. Yao, T., Hull grder strength, Marne Structures, Vol. 6, pp.-3 (003). 8. Yu, W., Cold-Formed Steel Desgn-3th edton, John Wley & Sons, (000) APPEDIX The man propertes of the problem are gven below. Only the half-length of the beam-column was consdered due to the symmetrc shape of the model. Total segment number s 4. 4 A 0mm 440mm I L 800mm E / mm F 3000 dl ( F L) /( EA) 0. mm ds ( L dl) / mm Calculatons for the Frst segment: du 0.mm mm δ mm mm ma 0, ( EI) / M 0 M F 3000mm r 96000mm 0 dθ ds r rad θ θ dθ rad 0, / 0, 0 0, 0. 08mm mm dθ.5ds / r rad 0,' 0, ' mm mm ' Calculatons for the Second segment: M F 594. mm r EI) / M mm ' ' dθ ds r rad, /,, ( ' mm mm du ( L / ) 0. 0mm Intals for the second teraton: du 0. 0mm δ ma mm