Finite-Element Investigation And Design Recommendations For Perforated .

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Finite-Element Investigation and Design Recommendationsfor Perforated Steel Plate Shear WallsRonny Purba1 and Michel Bruneau, F.ASCE2Abstract: This paper presents results from an investigation of the behavior of unstiffened thin steel plate shear wall 共SPSW兲 having aregular pattern of openings 共a.k.a. perforated SPSW兲. Finite element monotonic pushover analyses were conducted, first on a series ofindividual perforated strips with variation in perforation diameter, to develop a fundamental understanding of the behavior of completeperforated SPSW, then on a corresponding series of complete perforated SPSW having various perforation diameters. Three different setsof wall boundary conditions are considered, namely: flexible beam laterally braced, rigid floor, and rigid beam. Though some differencesbetween the SPSW panel strips and the individual strip results are observed at large monitored strain, at lower monitored strain howeverthe two models are in a good agreement. Based on the analytical results design recommendations of these perforated SPSWs arepresented. The shear strength of a SPSW infill plate having a pattern of multiple regularly spaced circular perforations can be calculatedas a function of the shear strength of a solid panel, perforation diameter, and distance between perforations.DOI: 10.1061/共ASCE兲ST.1943-541X.0000061CE Database subject headings: Steel plates; Shear walls; Finite element method; Design.IntroductionSteel plate shear walls 共SPSW兲 have been rapidly gaining interestin recent years as an effective lateral force resisting system 共Sabelli and Bruneau 2007兲. A key feature of SPSW systems is thesignificant stiffness and strength they can provide to buildingscompared to other lateral force resisting systems. However, insome SPSW applications, the minimum available thickness ofinfill plate might be thicker such than required by design. Percapacity design principles, at development of the system’s plasticmechanism, yielding of the SPSW infill plates will induce relatively large forces to the surrounding frames and consequentlywill increase the sizes of horizontal and vertical boundary members to which the infill plates are connected. A number of solutions have been proposed to alleviate this concern, either bychanging properties of the infill plate via using thin light-gaugecold-rolled 共Berman and Bruneau 2003,2005兲, using low yieldstrength steel 共Vian and Bruneau 2005兲, introducing vertical slits共Hitaka and Matsui 2003兲, or introducing multiple regularlyspaced perforations, also known as perforated SPSW 共Vian andBruneau 2005兲. The later solution is appealing as it can at thesame time accommodate the need for utility systems to passthrough the infill plate, without detrimental effects to the SPSW.This paper presents the results of an investigation to betterunderstand one aspect of the behavior of unstiffened thin perfo1Assistant Professor, Dept. of Civil Engineering, Univ. of BandarLampung, Lampung 35142, Indonesia 共corresponding author兲. E-mail:ronnypurba@yahoo.com2Professor, Dept. of CSEE, Univ. at Buffalo, Amherst, NY 14260.E-mail: bruneau@buffalo.eduNote. This manuscript was submitted on March 12, 2008; approvedon April 14, 2009; published online on April 16, 2009. Discussion periodopen until April 1, 2010; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Structural Engineering, Vol. 135, No. 11, November 1, 2009. ASCE, ISSN 0733-9445/2009/11-1367–1376/ 25.00.rated SPSW, more specifically the distribution of yielding aroundthe regularly spaced openings on the infill plate and related requirements to achieve adequate ductile performance, as drift demands relate to infill plate elongations demands. Finite-element共FE兲 monotonic pushover analysis of subelement 共strips兲 and fullspecimens are conducted. Based on the analytical results designrecommendations and consideration of these perforated SPSW arepresented.Previous Research on Steel Plate Shear WallsMuch research has been conducted since the mid 1980’s onSPSW, as summarized in Sabelli and Bruneau 共2007兲. Early studies by Thorburn et al. 共1983兲 introduced the relatively simpleStrip model to represent the behavior of unstiffened thin SPSW.Using this procedure, each infill plate is replaced by at least 10tension strips 共of equal width兲, pin-ended and inclined in the direction of the tension field. This procedure has been demonstratedto generally provide good results 共Timler and Kulak 1983; Elgaaly and Liu 1997; Driver et al. 1997兲. More recently, someresearchers have used FE to investigate issues related to SPSWs.Driver et al. 共1997兲 developed FE models to investigate a largescale, four-storey, single bay unstiffened SPSW having momentresisting beam-to-column connections and tested by quasistaticcyclic loading. Eight-node quadratic shell elements 共S8R5兲 wereused for infill plates directly connected to three-node quadraticbeam element 共B32兲 for the beams and columns, omitted the “fishplate” in the model. Initial imperfections of 10 mm based on thefirst buckling mode of the plate and residual stresses were alsoincorporated in the FE model. It was found that omitting geometric nonlinearity and second order effects in the FE model causeddiscrepancy between the cyclic experimental and the analyticalresults, as pinching of the hysteretic loops was not replicated.Behbahanifard et al. 共2003兲 investigated a three storey specimencreated by removing the lower storey of the four-storey specimentested by Driver et al. 共1997兲. The specimen was tested underJOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009 / 1367Downloaded 21 Oct 2009 to Redistribution subject to ASCE license or copyright; see

Fig. 1. Perforated SPSW specimen 共Vian and Bruneau 2005兲lateral quasistatic cyclic loading in the presence of gravity loads.A nonlinear FE model was developed using ABAQUS/Explicit v.6.2 to accurately simulate the monotonic and cyclic behaviors ofthe test specimen. Four-node shell elements with reduced integration 共S4R兲 were used for the entire SPSW. Material and geometricnonlinearity were considered in the FE model. However, residualstresses and plastic deformations from the previous test were notconsidered due to their complexity. Good agreement was obtained, although the analytical strength underestimated the experimental strength. Both the Driver et al. 共1997兲 and theBehbahanifard et al. 共2003兲 studies focused on SPSW with solidinfill plates.A study on SPSW having single perforations was performedby Roberts and Sabouri-Ghomi 共1992兲. The researchers performed a series of quasistatic cyclic loading tests on unstiffenedsteel plate shear panels with centrally placed circular openingsand recommended that the ultimate strength and stiffness of aperforated panel can be conservatively approximated by applyinga linear reduction factor 共1 D / d兲 to the strength and stiffness ofa similar solid panel, where D is the hole diameter and d is thespecimen width.Vian and Bruneau 共2005兲 conducted analytical and experimental work on three SPSW to investigate the behavior of speciallydetailed ductile perforated SPSWs of the type considered in thispaper. A first “reference” specimen consisted of a solid infill plateof 2.6-mm thick made of low yield steel with yield stress of 165MPa framed by 4 , 000 mm 2 , 000 mm centerline dimensionswith I-shaped sections W18X65 共beams兲 and W18X71 共columns兲,and reduced beam sections 共RBS兲 connections. The other twospecimens had the same boundary frame, but had staggered holesof diameter 200 mm arranged at a 45 angle with 300 mm centerto-center spacing along both the vertical and horizontal directionson the infill plate, as shown in Fig. 1. A third had reinforcedquartercircle cutouts of 500 mm radius at the upper corners of theotherwise solid infill plate. Perforated SPSWs conceptually identical to Vian and Bruneau’s second specimen are the subject ofthis paper. Vian and Bruneau 共2005兲 conducted FE analyses onsimplified models to replicate the experimental results. Thesesimplified analytical models were extended to consider variousperforation diameters using steel typically specified in NorthAmerican construction projects. Results illustrated general trendsbut some erratic jaggedness in the results—Vian and Bruneau’sstudy called for further parametric studies to investigate thecauses of the observed variability. This paper undertakes such aninvestigation, comparing results obtained from individual perforated strip models and full SPSW in terms of structural behavioras well for monitored maximum strain as a function of total stripelongation.Finite-Element Analysis of Individual PerforatedStripFE models of individual perforated strips were developed to provide an understanding of their behavior as a fundamental buildingblock in understanding the behavior of complete perforatedSPSW. The commercially available software ABAQUS/Standard关Hibbitt, Karlsson, and Sorenson, Inc. 共HKS兲 2004a,2004b兴 wasused for all analyses in this study. Perforation layout is schemedin Fig. 2 with perforations of diameter D are equally spaced ofdiagonal width Sdiag, arranged at an angle with respect to thebeam axis. A “typical” panel strip defines as the region within atributary width of 共1 / 2兲Sdiag on either side of a perforation layoutline 共Vian and Bruneau 2005兲, in the figure the region is shadeddifferently. Typical perforated strips of length L equal to 2,000mm, diagonal width Sdiag equal to 400 mm, perforation diameterD ranging from 10–300 mm 共corresponding to a perforation ratioD / Sdiag varying from 0.025–0.75兲, number of perforations alongthe diagonal strip Nr equal to 4, and plate thickness t p equal to 5mm were investigated 共Fig. 3兲. A perforation diameter incrementof 10 mm was chosen for analyses between the limit values of 101368 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009Downloaded 21 Oct 2009 to Redistribution subject to ASCE license or copyright; see

Sdiag“Typical”diagonal stripFig. 2. Schematic detail of perforated SPSW 共Vian and Bruneau2005兲and 300 mm, to obtain a relatively large number of data pointsand thus relatively smooth curves in the plots that express thevariation of behavior for various perforation diameters.Preliminary studies indicated that, due to buckling of theSPSW infill plate, adjacent strips do not affect the stress distribution within an individual strip. Each strip therefore behaves as anindependent strip. Results for the full SPSW model presented in alater section will further confirm this postulate. Because the stripgeometry and loading are symmetrical about horizontal and vertical axes through the center of the strip, a quadrant of the fullstrip is modeled with proper constraints along the symmetricboundaries 关Fig. 3共b兲兴. A monotonic incremental displacement was applied to the strip models uniformly along their right-edgeuntil the strips reached a displacement equal to 50 mm, or atotal uniform strip elongation un共 2 · / L兲 of 5%. During theanalysis, total uniform strip elongations were noted when themaximum principal local strain max reached values of 1%, 5%,10%, 15%, and 20% somewhere in the strips.Isoparametric general-purpose 4-node shell element 共S4兲 wasused in the FE models. To investigate the effects of meshingFig. 4. Typical strip analysis results at max 20%, D 100 mm共D / Sdiag 0.25兲algorithm 共transition zones close to the perforations兲 and meshrefinement on the stress-strain distribution throughout the stripsand to ensure a smooth continuity of results, several FE meshmodels were developed and analyzed 共Purba and Bruneau 2007兲.As a result, and on the basis of computation time efficiency, a 5 5 mm mesh size without any transition zone close to the perforations was selected for the parametric study 共except that forrelatively small and big perforation diameter, i.e., D ⱕ 60 and Dⱖ 250 mm, a rectangular transition zone was used as needed byABAQUS to mesh the regions close to the perforations correctly,without element distortion兲.ASTM A572 Gr. 50 共Fy 345 MPa兲 steel was selected. Toinvestigate how the response of perforated SPSW could be affected by the model assumed for the selected grade of steel, threematerial models were defined to represent various ways to expressthe constitutive stress-strain relationship, namely: the idealizedtrilinear stress-strain model used by Vian and Bruneau 共2005兲, amonotonic uniaxial noncyclic stress-strain, and an elastoperfectlyplastic bilinear stress-strain model. The results confirmed the importance of duly modeling strain hardening in the material modelto properly capture the spread of yielding needed in this system toallow the strips to reach the total ductile elongation necessary toaccommodate the drift demands in perforated SPSW. On the basisof the results, the behavior of steel used was represented by anidealized trilinear stress-strain model.Behavior of Perforated Strip ModelFig. 3. Individual perforated strip 共Vian and Bruneau 2005兲Fig. 4 shows strip deformations and maximum in-plane principalstress and strain contours at the surface of the shell element forthe case having a 100 mm perforation diameter when maxreached a value of 20% somewhere in the strip. Note that sincethe high stresses of interest here are tension stresses, results forthe center and surface membranes of the shell elements used 共S4兲were found to be practically identical. As shown in the figure, thein-plane principal stress and strain contours are uniform at theright edge of the strip. However, holes in the strip disturbed the“regularity” of the stress and strain flows and high stress andstrain concentrations developed at the perforation edge and zonesof yielding radiate out from this location at approximately 45 angles to the left and right of the perforations. In combinationwith Poisson’s-ratio effect, these concentrations also accountedfor inward 共in addition to rightward兲 movement of the unre-JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009 / 1369Downloaded 21 Oct 2009 to Redistribution subject to ASCE license or copyright; see

Total Uniform Strip Elongation, un (%)Max Principal Local Strain, max (%)50D/Sdiag 0.2340 L/L 5%30 L/L 4% L/L 3%20 L/L 2%10 L/L 1% Ratio, D/Sdiag0.6 max 20%3.0 max 15%2.0 max 10%1.0 max 5% max 1% Ratio, D/Sdiag0.60.70.25Fig. 5. Maximum local strain max versus perforation ratio D / Sdiagstrained top edge 共the interface edge to the adjacent strip兲 adjacent to the perforations. In an actual SPSW, the interface betweenadjacent strips correspond to a buckle “ridge,” and this inwardpull toward the hole due to Poisson’s-ratio effect would locallyreduce the amplitude of the ridge.As a simple tension member, a more ductile behavior is expected when yielding on gross section precedes net section fracture as the applied axial load is increased 共Dexter et al. 2002兲, i.e.Fu · An ⱖ F y · Ag(a)4.00.7Normalized Strip Elongation[ un]/[NrD/L] max 20%0.10 max 15% max 10%0.05 max 5% max 1%0.000.0共1兲where An net strip area; Ag gross strip area; Fu ultimate tensilestrength; Fy yield strength. Substituting An 共Sdiag D兲 · t p, Ag Sdiag · t p, and the properties of A572 Gr. 50 steel used 共Fy 50 ksi, Fu 65 ksi兲 into Eq. 共1兲, the equation can be simplifiedto D / Sdiag ⱕ 0.23. This value is used to examine the trend in maximum local strain in the strip for various perforation ratios asplotted in Fig. 5. Note that the increasing and decreasing parts inthe figure are roughly separated by the D / Sdiag 0.23 limit.For the zone where perforation ratio D / Sdiag ⱕ 0.23, yieldingoriginally occurred in the elements close to the perforation edgeand progressively distributed to the gross section as the tensileload increased. As a result and because of strain hardening, thenet section has a significant capacity to stretch beyond the pointfor which the strip has reached the monitored total strips elongation; as the perforation ratio increases, the decreased net sectionobviously has to stretch more to reach the same monitored elongation. However, for the zone where perforation ratio D / Sdiagⱖ 0.23, yielding will be localized to the region close to the perforation while the gross section remains progressively more elastic. By the time the monitored total strip elongation is reached, theshell element close to the perforation edges has reached higherstrain and plastic deformation. As perforation diameter furtherincreases, this target limit strain is reached earlier, correspondingto a lesser magnitude of total member elongation. However, notethat in SPSW applications even though gross section yieldingcannot develop in some cases, the spread of localized yielding,and repetition of it at multiple holes for the perforated plate configuration considered, still make it possible for the perforatedplate to reach target total elongations adequate to meet the maximum drift demands for actual SPSW.Fig. 6共a兲 presents the effect of holes on strip global deformation where uniform distributed strip elongation un versus perforation ratio D / Sdiag are plotted at 1%, 5%, 10%, 15%, and 20%maximum principal local strain. At higher monitored strain max(b) Ratio, D/SdiagFig. 6. Strip elongation 共a兲 real value; 共b兲 normalized valueequal to 10%–20%, the total strip elongation decreases significantly at small perforation ratios 共i.e., D / Sdiag 0.025 to 0.1 orD 10– 40 mm兲, and then gradually increases between D / Sdiag 0.1 and 0.6 共D 40– 240 mm兲 before slightly decreasing againfor D / Sdiag 0.6. At the lower monitored local strain levels 共i.e., max 1% and 5%兲, the total strip elongation remains almost constant for the entire range of perforation diameters.One might argue that an increase in perforation diameter 共forD / Sdiag ⱖ 0.23兲 leading to an increase in total strip elongation 共forthe same monitored local strain兲 is counterintuitive. For example,to reach a 20% maximum local strain, the strip having a 100 mmperforation diameter 共D / Sdiag 0.25兲 has elongated 21.0 mm共 un 2.10%兲 but the strip with 200 mm perforation diameter strip共D / Sdiag 0.50兲 elongated even more 共as much as 30.7 mm or un 3.07%兲 before this local strain limit was reached. To explainthis behavior, it is first useful to compare the respective area ofstrip stressed beyond the yield point 共 y 1.725 10 3兲 for thestrips having 100 and 200 mm diameter holes. While it was originally suspected that the greater elongation of the strip having 200mm diameter holes might have been attributed to the longerlength over which yielding spread 共as a percentage of total platelength兲, the results actually shows that this is not the case. Thearea over which inelastic behavior develops 共i.e., inelastic area兲for the strip having 100 mm diameter holes is larger than that forthe strip having 200 mm diameter holes. The percentage of inelastic area over strip net area is about 60% and 43% for the striphaving 100 and 200 mm perforations, respectively. Note thatthese percentages become 58% and 36% if the inelastic area isdivided by the gross strip area 共i.e., a constant value of1 , 000 mm 200 mm 200.000 mm2 in this case兲. However, the1370 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009Downloaded 21 Oct 2009 to Redistribution subject to ASCE license or copyright; see

magnitude of the inelastic strain develop within these areas ofinelastic deformations differs very significantly. One way to capture this difference is by comparing the energy dissipated by plastic deformation for both plates. Even though the inelastic area ofthe 100 mm perforated strip are bigger than that of the 200 mmperforated strip, its plastic deformation energy 共5,530 kN/mm兲 issmaller than for the 200 mm perforated strip 共5,734 kN/mm兲. Thisconfirms that shell elements close to a bigger perforation edgestretched more than those close to a smaller perforation edge.To provide additional insight into this behavior, a variation ofFig. 6共a兲 is plotted in Fig. 6共b兲 by normalizing the total stripelongation by the factor Nr . D / L, which is the ratio of perforatedlength to overall length in a strip 共Vian and Bruneau 2005兲. Simultaneously, the vertical axis is expressed as 2 · / Nr . D, whichis the total strip displacement divided by a total length of perforations over the entire strip. As shown in the figure, for all casesthe normalized strip elongation gradually decreases as the perforation ratio increases.Finite-Element Analysis of Complete PerforatedSteel Plate Shear Walls „SPSW FE models of complete perforated SPSW 共panel models兲 weredeveloped to verify the appropriateness and accuracy of the individual perforated strip model results and to investigate why priorresults from panel analysis in Vian and Bruneau 共2005兲 did notsupport the predictions from individual strip model analysis.Hence, the same specimen Vian and Bruneau 共2005兲 previouslyinvestigated was studied for this objective.ABAQUS/CAE, a graphical preprocessor program, was usedto define the model of the described specimen. Each “plate” of thespecimen was modeled independently in its own coordinate system, and using the Assembly Module tools, the parts were thenpositioned, relative to each other in a global coordinate system,thus creating one final assembly. The fish plate, used in the testspecimens to connect the infill plate to the surrounding frame,was not considered in the FE model. Instead, the infill plates wereconnected directly to the beams and columns, the effects of thisassumption to the overall behavior of SPSWs were found to besmall 共Driver et al. 1997兲. To avoid distortion, the frame members共beams and columns兲 were first meshed 共using Structured Meshing technique兲, followed by the infill plate 共using Free Meshingtechnique兲. As in the strip models, no transition zone close to theperforations was applied except for a small perforation diametermodel 共i.e., D 50 and 100 mm兲. The meshes started with 50 50 mm shell elements near the boundary elements and gradually reducing to an average dimension of 35 35 mm per shellelement adjacent to the perforations. The entire infill plate andboundary elements 共beams and columns兲 were meshed using theS4R shell elements isoparametric general-purpose 4-node shellelement with reduced integration and hourglass control. ASTMA572 Gr. 50 steel 共Es 200000 MPa, Fy 345 MPa, Fu 450 MPa, and 0.3兲 was selected for boundary elements andinfill plate. From case study on material definitions conducted, theunidirectional idealized trilinear stress-strain was appropriate tomodel the infill plate steel, which can only yield in tension, andimmediately buckles in compression. The cyclic stabilized backbone stress-strain curve 关equivalent to “Steel A” in Kaufmann etal. 共2001兲 and comparable to the Ramberg-Osgood hysteresis兴was used in the boundary elements for the same steel grade.To help initiate panel buckling and development of tensionfield action, an initial imperfection was applied to the modelsFig. 7. FE model of perforated SPSWanalyzed. The shape of the imperfection was a summation of thefirst 20 modes, each mode having a different amplitude of maximum out of plane deformation, varying linearly from 1 mm forthe fundamental mode to 0.01 mm for the twentieth mode.CONN3D2 connector elements were used to model the hinges atthe base of the Vian and Bruneau 共2005兲 specimen in theABAQUS model. This connector links reference nodes at the location of the hinges center, 850 mm below the centerline of bottom beam, to the corner nodes at the tip of each column flangeand at the intersection of the flanges and web, and provides effectively a rigid beam connection between two nodes 关Hibbitt,Karlsson, and Sorenson, Inc. 共HKS兲 共2004b兲兴. At the two reference nodes, only rotation about the axis perpendicular to the planeof the wall is allowed, to replicate the hinge rotation in Vian andBruneau test specimen. The exterior nodes of the flange elementsaround the perimeter of the panel zones at the top of columnswere restrained against out-of-plane movement to replicate theexperimental setting of Vian and Bruneau tests. To achieve convergence results without unstable responses due to the higher degree of nonlinearity in the system 共infill plate buckling兲, Stabilizeoption in ABAQUS/Standard was activated 关Hibbitt, Karlsson,and Sorenson, Inc. 共HKS兲 共2004b兲兴. A monotonic pushover displacement was applied to a reference node located at the middlecenterline of the top beam. Fig. 7 shows the resulting FE model.During the analysis, frame drifts and strip elongations were measured 共as annotated in Fig. 8兲 when the maximum principal localstrain max somewhere in the infill plate reached values of 1%,5%, 10%, 15%, and 20%.Behavior of Perforated Steel Plate Shear Walls„SPSW Considering Alternative ModelsAt large in-plane drifts, a first model having the boundary conditions described above experienced lateral torsional buckling, pri-JOURNAL OF STRUCTURAL ENGINEERING ASCE / NOVEMBER 2009 / 1371Downloaded 21 Oct 2009 to Redistribution subject to ASCE license or copyright; see

STRIP 1STRIP 2STRIP 3STRIP 4Fig. 9. Infill plate deformation shaped 共RF model, deformation scalefactor 2.0兲locations to measurestrip elongationlocations to measureframe driftFig. 8. Strip annotation on perforated SPSWmarily at the top beam and slightly at the bottom beam 共notshown here兲. This phenomenon also affected the columns displacement as the left column deformed in a manner not parallel tothe right column. The model was then revised to have lateralsupports restraining the out-of-plane movement of the boundarynodes 共nodes at the tip of the beams flanges兲, and called theFlexible Beam Laterally Braced 共FLTB兲 model. In addition, finemeshes were used in this model, starting with 25 25 mm shellelements near the boundary elements and gradually reducing to anaverage dimension of 15 15 mm per shell element adjacent tothe perforations.For the FLTB model, analysis results showed that every stripreached a different strip elongation. In particular, only Strip 1matched the individual strip results, while the elongation observed for the other strips in the SPSW panel was less than thatfor the corresponding stand-alone strip, by as much as 22% forthe Strip 4 at 20% maximum principal local strain in the infillplate. The nonsymmetrical beam deflections under the applieddiagonal tension from the infill plate caused these deflections andresulting unequal strip axial deformations.To investigate the significance of beam deformations on stripelongations, the FLTB model was modified by adding verticalconstraints at the boundary nodes at the beams flanges 共to approximate a rigid-body motion兲, while all other model propertiesremained the same, and called the Rigid Floor 共RF兲 model.For the RF model, the analysis results showed that all stripsreached about the same elongation and matched the results obtained for the individual strip model—for example, at monitoredstrain max ⱖ 5%, the difference between the results for the twomodels is less than 2%. Note that the total uniform strip elongation un on the tension field strips of a SPSW having rigid pinended frame members theoretically can be related to interstoreydrift through compatibility relations 共Vian and Bruneau 2005兲 by un F · sin 2 2共2兲where F frame 共interstorey兲 drift and is the tension field inclination angle which is typically forced to be 45 in perforatedSPSW. The RF model was developed as a way to approximate共using FE analysis兲 that theoretical case. The equivalent stripelongation calculated using Eq. 共2兲 for example at 20% max consistently closely matched the SPSW strip elongations, i.e., lessthan 1% difference from the SPSW strips average elongation.The regularity of the observed deformed shape 共Fig. 9兲, withpeaks at the strips lines and valleys between them, illustrates howeach strip reached the same elongation. Note that all points alonga given peak “ridge” do not reach the same maximum out ofplane deformation. Indeed, the magnified deformations shown inFig. 9 illustrates that the maximum value for the peaks occuralong the ridge at the location furthest away from holes, whereassome reduction in the amplitude of the out-of-plane buckle occursat the point closest to two adjacent holes. This behavior illustratesthat the boundary conditions between each individual strip is unrestrained by adjacent strips. As such, each strip behaves asshown in Fig. 3, and the variation of amplitude of out-of-planedeformation along a buckling fold occurs as a result of the reduction of effective width due to Poisson’s-ratio effect that developsas the strip elongates.Note that in the RF model, plastic hinges were constrained tooccur in the columns by artificially making the beams infinitelyrigid across the entire width of the SPSW. This was done as aninterim measure to establish the linkages between full plate behavior and the simplified individual strips. As demonstrated, sucha match exists and difference between results for actual unconstrained SPSW and individual strips are primarily due to flexibility of the top and bottom beams, and not some of the other factors共e.g., plate buckling, initial imperfections, etc兲. To further the understanding of how strip elongations in actual SPSW relate to theindividual strip model, an alternative Rigid Beam 共RB兲 modelwas considered. In this model, a very stiff beam between the RBSis modeled by increasing the thickness of the flanges and webs tobe 10 times thicker than for the actual beam. The RBS segmentsremained at their actual thickness and unconstrained. This allowsthe rigid-body motions of the beams

Finite-Element Analysis of Individual Perforated Strip FE models of individual perforated strips were developed to pro-vide an understanding of their behavior as a fundamental building block in understanding the behavior of complete perforated SPSW. The commercially available software ABAQUS/Standard

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