In-plane shear stiffness of bare steel deck through shell finite element models. G. Bian, B.W. Schafer. June 2017

Transcription

1 In-pane shear stiffness of bare stee deck through she finite eement modes G. Bian, B.W. Schafer June 7 COLD-FORMED STEEL RESEARCH CONSORTIUM REPORT SERIES CFSRC R-7- SDII Stee Diaphragm Innovation Initiative

2 CFSRC Information The Cod-Formed Stee Research Consortium (CFSRC) is a muti-institute consortium of university researchers dedicated to providing word-eading research that enabes structura engineers and manufacturers to reaize the fu potentia of structures utiizing cod-formed stee. More information can be found at A CFSRC reports are hosted permanenty by the Johns Hopkins University ibrary in the DSpace coection: SDII Information The Stee Diaphragm Innovation Initiative (SDII) is a muti-year industry-academic partnership to advance the seismic performance of stee foor and roof diaphragms utiized in stee buidings through better understanding of diaphragm-structure interaction, new design approaches, and new three-dimensiona modeing toos that provided enhanced capabiities to designers utiizing stee diaphragms in their buiding systems. SDII was created through coaboration between the American Iron and Stee Institute and the American Institute of Stee Construction with contributions from the Stee Deck Institute, the Meta Buiding Manufacturers Association, and the Stee Joist Institute in partnership with the Cod-Formed Stee Research Consortium; incuding, researchers from Johns Hopkins University, Virginia Tech, Northeastern University, and Water P Moore. Project updates are avaiabe at Nationa Science Foundation Acknowedgment This report was prepared with partia support from the U.S. Nationa Science Foundation: NSF-CMMI-568: NEESR-CR: Transforming Buiding Structura Resiience through Innovation in Stee Diaphragms. Any opinions, findings, and concusions or recommendations expressed in this pubication are those of the author(s) and do not necessariy refect the views of the Nationa Science Foundation.

3 Abstract The objective of this report is to expore the in-pane eastic shear stiffness of bare stee deck diaphragms through high fideity she finite eement modeing. Profied stee panes, i.e, meta deck, often serve as a key distribution eement in buiding atera force resisting systems. Acting argey as an in-pane shear diaphragm, meta deck as empoyed in was, roofs, and foors pays a key roe in creating and driving three-dimensiona buiding response. Accurate prediction of shear stiffness is needed to better understand the shear behavior of bare stee deck. According to the shear stiffness expression in AISI S or SDI s Diaphragm Design Manua (DDM), tota shear deformation can be separated into three parts: pure shear deformation, warping deformation, and connection sip. In this report, bare deck finite eement modes were buit with typica fastener ayouts from DDM. The shear condition was performed on the modes by imposing deformations at the perimeter. By changing boundary conditions, these shear deformation terms can be separated and the vaues of shear stiffness were compared with DDM expressions. The resuts showed that the prediction of shear stiffness from pure shear deformation and connection sip from finite eement modeing agree we with the DDM prediction; however, the warping stiffness predicted from DDM is smaer than she finite eement predictions. DDM uses a simpified mode to predict warping deformation and this mode may need to be revised in the future.

4 . Introduction Profied stee panes, i.e., meta deck, are ro-formed from thin stee sheet and can resut in simpe corrugated shapes or reativey compex ongitudina profies with additiona transverse features such as embossments (Figure a). These panes serve as the was and roof in many meta buidings, see Figure b, and form an integra component of common foor systems in a wide variety of buidings. Under atera oads the panes pay a particuary important roe as a distribution eement, one in which the in-pane shear behavior of the pane is paramount. A typica profied stee pane roof is iustrated in Figure. When distributing atera oad this system acts as a diaphragm, with a eements in the system contributing: pane, pane inter-connections, joists, joist-to-pane connections, primary framing, and framing-to-pane connections. (a) Bare stee deck on joists [] (b) Meta buiding with deck [] Figure Meta deck in buidings Figure shows the deformed shape of bare stee deck under shear. Δ P a pane joist Pa Pa Figure. Stee deck pane under pure shear and its dispacement

5 From AISI S or the Diaphragm Design Manua (DDM, DDM was pubished in 6 and used herein), shear stiffness G of the bare deck diaphragm is defined as: G' = Gt = P / Δ / a = Pa / Δ () Where, P is the appied shear oad, and, and a are the pane ength and width respectivey, and t is the thickness. Δ is the tota pane deformation under shear oading. Pane deformation comes from three terms: shear deformation of the pane Δ s, warping deformation Δ d, and connection sip Δ c. Therefore, Eq. can be written as foowing: G' = Pa / () Δ s + Δ d + Δ c. Description of Bare Deck Modes To understand the shear behavior of a bare deck diaphragm, she finite eement (FE) modes were buit and compared with DDM s prediction. Pure shear boundary conditions were enforced by imposing shear dispacement at the pane edge and any interior purin/joist nodes (the same as shown in Fig. ). In the modes, using Eq., by dividing the tota reaction force by the rotation ange one can obtain the bare deck shear stiffness. Tabe. Modeing matrix across corrugation aong corrugation designation spacing (in.) (in.) 6 (c) (d) 6/ / /7 6-6/ bottom (a) every (b) (a) singe row of a FE nodes in the bottom of the fute are connected and experience imposed shear dispacements (b) entire end cross-section are connected in a rigid pane and experience imposed shear dispacements (warping restricted) (c) no sideap, mode uses continuous deck equivaent to perfect fastening (idea case) (d) every node aong a ine connected driven to imposed shear dispacement In stee deck testing or in rea construction, a shear forces are transferred to the neighboring deck or frame through sideap or deck-to-frame (structura) connections. In our FE modes, each diaphragm is made of four separate pieces of deck. In our FE modes a imposed shear deformation act at connection ocations. The considered fastener ayouts in both directions aong the deck are shown in Tabe. These fastener ayouts are typica cases from DDM Chapter 9, augmented with some additiona idea cases. For the parae-to-purin/joist direction, there are four different cases whie for the perpendicuar-to-purin/joist direction there are five cases, as shown in Fig..

6 6/9 6/7 6/5 6/ a fastener ayout parae to purin 6 fastener ayout perpendicuar to purin Figure. Fastener pattern in the modes Some specia fastener ayout cases (Tabe a, b, c, and d) can affect the deformation term. By imposing dispacement at a edge nodes in the parae-to-purin/joist direction, warping can be eiminated, whie by imposing dispacement at a edge nodes in the perpendicuar-to-purin direction connection sip can be eiminated. By modeing a whoe pane board, instead of four different panes, the effect of the sideap connection can be examined. A typica B type WR deck cross section is seected as shown in Fig. with a thickness of.9mm (.58 ). And the Young s moduus is set to be 5 N/mm (95 ksi) with Poisson s ratio at.. f = 9 mm f =.5 in. D d = 8 mm D d =.5 in. e = 9 mm d = 5 mm e =.7 in. d = 6. in. (a) unit: mm (b) unit: inch Figure. Stee deck pane cross section According to the three terms contributing to the shear deformation of the diaphragm (see Eq. ), modes can be divided into four categories: first: shear deformation from Δ s, Δ d and Δ c ; second: shear deformation from Δ s and Δ d (no connection sip); third: shear deformation from Δ s and Δ c (no edge warping);

7 fourth: shear deformation from Δ s ony (no warping and connection sip) Pane dimension and ength-to-width ratio is another factor that can potentiay affect the shear behavior of the bare deck. Therefore, for each mode in Tabe, the dimension and ength-to-width ratio were aso changed. The ength-to-width ratio of each mode changes from.5 to, as shown in Fig. 5 (Note that here ength means the direction perpendicuar to the purin and width is the direction parae to the purin). The minimum pane ength or width dimension is (658mm) in a the modes. This ength-to-width ratio can affect the warping behavior and connection behavior of the pane. a :a = : purin : edge perpendicuar to purin a: edge parae to purin a a :a = : :a = : a :a = : a :a = : (a) Figure 5. Stee deck pane dimension ratios (b). Evauation of Shear Stiffness from Diaphragm Design Manua (DDM). Mechanism of shear deformation As discussed in the ast section, in Eq. the denominator is the summation of three terms according to the shear deformation mechanism. If the shear stiffness is treated as three separate terms, Eq. can be written as: G' = G s ' + G c ' + G d ' () Expicit expressions for the Δ terms in Eq. are provided in DDM, and resut in the foowing: Et G' = (+ν) s d + D n + C (a)

8 or G' = (b) (+ν) s d Et + D n Et + C Et Detais of the parameters in Eq. (a) and (b) are expained beow.. from pane shear deformation G s The first term in the denominator of Eq. (b) represents the deformation caused by pane shear deformation. If we define the shear stiffness for pure pane deformation, the expression for G s wi be: Et G s ' = (+ν) s d or Gt s d (5) For our modes, the deveoped fute width per width: s = (e + w) + f (6) Where e, w and f are a cross section dimensions (See Fig. ). Pane corrugation pitch d = 5 mm (6 in.). Substituting into Eq. 5, G s becomes 59. N/mm (98. kip/in.), which is the DDM prediction for pane shear stiffness from pure pane deformation for the studied profie.. from warping deformation G d The second term in the denominator of Eq. (b) represents the deformation caused by warping deformation. Per DDM the shear stiffness for warping deformation is: Where D n is the warping coefficient shown beow. G d = Et (7) D n D n = D L (8) Where D is the warping constant given in DDM Tabe.- and L is the pane ength. The effect of D n is often reduced to ρd n depending on the number of equa spans within the pane. G d is defined in the tabe beow.

9 across corrugation 6/ 6/5 6/7 6/9 across corrugation 6/ 6/5 6/7 6/9 Tabe. G d cacuated from Diaphragm Design Manua (a) in unit N/mm aong corrugation : : : : : (b):in unit kip/inch aong corrugation : : : : : The origins of the G d prediction can be found in DDM (98). The mode assumes the top fange warping movement is restrained by an eastic foundation coming from the transverse stiffness of the web. Further work investigating this mode is now underway.. from connection sip G c The ast term in the denominator of Eq. (b) represents the deformation caused by connection sip. If we define the shear stiffness for connection sip, it is: G c = Et C (9) In Eq. 9, C is the sip coefficient and is defined in AISI S or DDM as:

10 C = E t w S f S α + n p α + n f s S s Where S f is the structura connection fexibiity and S s is the sideap connector fexibiity. For an idea case with equa fasteners empoyed for sideap and from connections, noting that the sideap has two sides of panes and pane-to-framing has ony one side of pane, then: () Eq. then becomes: S f = S s () G c = S f w α + n p α + n s () In which: α is end distribution factor. α is the same as α. n p is the number of purins. n p = for a modes here. n s is the number of stitch connectors within the ength L. α, α, n p and n s can be obtained from DDM Section.. The resut of G c cacuated from Eq. is provided in Tabe. across corrugation 6/ 6/5 6/7 6/9 Tabe. G c cacuated from Diaphragm Design Manua (a): in unit N/mm aong corrugation : : : : :

11 (b): in unit kip/inch across aong corrugation corrugation : : : : : 6/ / / / Note: S f is estimated from isoated finite eement modes. Detais in Section.. Description of finite eement modes in ABAQUS In ABAQUS a series of modes are deveoped. Linear eastic isotropic materia is used for the cod-formed stee. The SR eement is used. Singe step inear eastic anaysis (inear perturbation) was conducted under the shear boundary condition. To simuate the pure shear behavior, shear dispacement was imposed at the edge fasteners and purin fasteners. Out of pane movement and rotationa degrees of freedom are constrained at edge nodes. Boundary condition detais are iustrated in Fig. 6. x edge B.C.: u x and u y foow shear dispacement and u z = at fastener ocation y deformed shape undeformed shape γ=/5 purin/joist pane u x =u y =u z =θ x =θ y = at fastener ocation Figure 6. Boundary condition in ABAQUS modes The undeformed mode (at ) in ABAQUS and detais of the boundary conditions are shown in Fig. 7.

12 panes purin distance: SR eement sideap: MPC pin in. (5 mm) fastener Figure 7. Undeformed shape of she FE mode (fastener ayout: 6/-6) Fig. 8 (a), (b) and (c) provide three representative modes at :, : and : with deformed shape. Fig. 8 (d) iustrated the oca deformation at fastener ocations. (a) deformed shape of FE mode with : sideap pane (b) deformed shape of FE mode with :

13 (c) deformed shape of FE mode with : (d) oca deformation in mode Figure 8. Deformed shape of seected finite eement modes To estimate S f and S s in Eq., representative FE modes with the same boundary condition as in pure shear were buit, and a unit oad was imposed at a singe fastener ocation. The finite eement modes are iustrated in Fig. 9 (a) and (b). Since connections in a modes are idea and their stiffness is infinity, S f and S s here incude oca pate

14 deformation around connections in the she finite eement modes. The resuting fexibiity are S f =. -5 mm/n and S s =6.6-5 mm/n. Note in this case S f = S s /. In actua diaphragms connectors with discrete stiffness are used this ideaization is that of a perfect fastener. Note, in typica fastener testing the recorded fexibiity incudes the fastener fexibiity and the oca deformations of the pate therefore the reported vaues are essentiay ower bound on the fexibiity. (a) mode to evauate sideap S s (b) mode to evauate perimeter S f Figure 9. Finite eement modes to evauate S f and S s 5. Evauation of shear stiffness from finite eement modeing (FEM) 5. Overa resut In Eq., Δ/a (shear ange) =/5 for a FE modes. Therefore, shear stiffness of the pane deck can be obtained from the foowing: G' = P / Δ / a = 5P Where P is the reaction force and is pane ength. () With finite eement resuts and Eq., one can cacuate eastic shear stiffness of a the modes. Resuts are shown in Tabe. From the resuts, we can see that as the of the modes changes from : to :, the shear stiffness doesn't change. However, when the ratio is arger than, as the ratio increases, the eastic stiffness increases. In a discrete fastener ayout cases, the deformation is contributed to by the three terms in Eq.. For the specia boundary conditions in the ast two rows of Tabe, we can separate either shear deformation caused by warping or by connection sip. The effect of fastener ayout on shear stiffness is provided in the next sections. To iustrate how the eastic shear stiffness increases and converges as the increases, additiona modes were competed with from. to. From the resuts, as shown in Fig., we can see that the shear stiffness asymptoticay converges to a constant at arge. A mode with of 5 is deemed arge enough to estimate the asymptote.

15 Tabe. from finite eement modes (a): in unit N/mm perimeter fastener spacing fied (interior) fastener spacing pane dimensions and across corrugation aong corr. across corrugation aong corr. a(ft) 8 b(ft) 8 designation spacing spacing spacing spacing a/ (ft) (e) 6 designation (in.) (in.) (in.) (in.) (a/b) (c) / 6 6/ (c) (d) (d) (c) /5 6 6/ (c) (d) (d) (c) (c) /7 6 6/ (c) (c) (d) 6 (d) (c) /9 6 6/ (c) (d) 6 (d) (a) (a) bottom (a) (a) (c) bottom (a) (d) (a) (d) (a) (d) (a) (c) (b) (b) every (b) (b) (c) every (b) (d) (b) (d) (b) (d) (b) (c) (a) singe row of a FE nodes in the bottom of the fute are connected and experience imposed shear dispacements (b) entire end cross-section are connected in a rigid pane and experience imposed shear dispacements (warping restricted) (c) no sideap, mode uses continuous deck equivaent to perfect fastening (idea case) (d) every node aong a ine connected driven to imposed shear dispacement

16 (b): in unit kip/inch perimeter fastener spacing fied (interior) fastener spacing pane dimensions and across corrugation aong corr. across corrugation aong corr. a(ft) 8 b(ft) 8 designation spacing spacing spacing spacing a/ (ft) (e) 6 designation (in.) (in.) (in.) (in.) (a/b) (c) / 6 6/ (c) (d) (d) (c) /5 6 6/ (c) (d) (d) (c) (c) /7 6 6/ (c) (c) (d) 6 (d) (c) /9 6 6/ (c) (d) 6 (d) (a) (a) bottom (a) (a) (c) bottom (a) (d) (a) (d) (a) (d) (a) (c) (b) (b) every (b) (b) (c) every (b) (d) (b) (d) (b) (d) (b) (c) (a) singe row of a FE nodes in the bottom of the fute are connected and experience imposed shear dispacements (b) entire end cross-section are connected in a rigid pane and experience imposed shear dispacements (warping restricted) (c) no sideap, mode uses continuous deck equivaent to perfect fastening (idea case) (d) every node aong a ine connected driven to imposed shear dispacement 5 (N/mm) 5 (N/mm) at 5: 5 5 at 5: : : : : : : 5: 6: 8: : : : : : : : 5: 6: 8: : (a) 6/-6 (b) 6/-

17 5 (N/mm) 5 (N/mm) at 5: 5 5 at 5: : : : : : : 5: 6: 8: : : : : : : : 5: 6: 8: : (c) 6/5-6 (d) 6/5-5 (N/mm) at 5: (N/mm) at 5: : : : : : : 5: 6: 8: : : : : : : : 5: 6: 8: : (e) 6/7- (f) 6/7-5 (N/mm) 5 (N/mm) at 5: 5 5 at 5: : : : : : : 5: 6: 8: : : : : : : : 5: 6: 8: : (g) 6/7- (h) 6/7-6

18 5 (N/mm) 5 (N/mm) at 5: 5 5 at 5: : : : : : : 5: 6: 8: : : : : : : : 5: 6: 8: : (i) 6/9- (j) 6/9-6 Figure. G of different modes, designated by fastener ayout across and aong corrugation 5. Effect of sideap connection Sideap connections decrease shear stiffness of the bare deck by connection sip. For any mode with a given fastener ayout, there are two sets of modes: the first incudes discrete panes and sideaps, the second has one whoe pane with no sideap. Tabe 5 provides the ratio of shear stiffness G between the two modes with the same fastener ayout and dimension. Tabe 5 shows that the sideap effect is not significant for most modes with ess than. As mode ength increases there are more sideap connectors and the effect becomes more significant. Generay, the factor decreases as the dimension ratio increases, but sti not ess than.7. The resuts of Tabe 5 depend on perfect sideap connectors it ony considers pane oca deformation instead of connection faiure. Thus, this gives ony an idea of the infuence of oca panes fexibiity on overa stiffness. It is known that if the sideap fasteners are highy fexibe this can dominate the fu response. Tabe 5. Ratio of shear stiffness between modes w/o and w/ sideap across aong corrugation corrugation : : : : : 6/ / / / bottom every..... every every.....

19 5. Effect of connections parae to purins From Figure we can see that as the number of fasteners is increased at the edge and purins, shear stiffness of the bare stee deck increases. For different dimension aspect ratios, the increase in shear stiffness foows a simiar magnitude. As in the Diaphragm Design Manua (DDM), there is amost doube the shear stiffness if we change the fastener ayout from 6/ to 6/9 at 6 in. connector spacing at the other direction. 5 6/9-6 6/7-6 6/5-6 6/-6 (N/mm) : : : : : : 5: 6: 8: : Aspect ratio Figure. Effect of connections parae to purin in she FE modes 5. Effect of connections perpendicuar to purins With a fixed fastener ayout in the parae-to-purin direction (6/7 in the figure beow), as fasteners increase in the other direction, shear stiffness changes. In Fig., as the fastener ayout changes from 6/7-6 to 6/7-, shear stiffness changes from. N/mm ( kip/inch) to N/mm (7 kip/inch) at ow. The magnitude of increase for higher wi increase, e.g.: from 5 N/mm ( kip/inch) to 7 N/mm (7 kip/inch).

20 5 (N/mm) 6/7-6/7-6/7-6/ : : : : : : 5: 6: 8: : Aspect ratio Figure. Effect of connections perpendicuar to purin 6. Comparison of Shear Stiffness between FEM and DDM 6. Comparison of G s and G c In one of the FE modes (in Tabe ) we assume the connectors are at a FE mesh nodes aong the pane edge and sideaps (footnote b/d and b/c in Tabe ). This boundary condition can eiminate the effect of warping deformation and connection sip. In this case shear stiffness G = G s and it is observed as 5758 N/mm (.5 kip/in.) from the FEM mode, whie it is 59 N/mm (98. kip/inch) in the DDM prediction. By making the ength of the finite eement modes (the dimension perpendicuar to purin) as arge as 5 times the width of the modes, the warping stiffness wi increase to infinity (see Eq. 7). For these specia cases, shear stiffness of the stee deck from connection sip can be obtained from the equation beow: G c ' = G' - G s ' () G c from DDM are from Eq. (9)-(). G c for different panes is provided and compared in Figure. From the pot we can see that for modes 6/ and 6/5, FE resuts can we predict the shear stiffness from connection sip. However, as the number of connectors in

21 the mode increases, the difference between the FE prediction and the DDM prediction increases and the FEM prediction is aways arger than the DDM prediction. 5 5 (N/mm) Gc from FEM Gc from DDM fastener ayout (in.): 6/- 6/5-6/-6 6/5-6 6/7-6 6/7-6/7-6/9-6 6/7-6/9- # of fasteners: Figure. from connection sip G c of FEM and DDM This FE mode does not use discrete springs, but rather the oca fexibiity that derives from appying a point oad to a pate eement. For the extreme cases with many fasteners, each oading point in the mesh is not independent. (See the infuence area in Figure 9.) In the rea word it aso doesn t act independenty and it is not cear whether the FE or DDM predictions are more accurate in this case. Experiments or more detaied modes woud be required. 6. Warping stiffness G d For seected finite eement modes (Tabe footnote d), a fasteners are coincident with the FE mesh nodes in the perpendicuar-to-purin direction. The arge amount of fasteners can eiminate connection sip effect, and the shear stiffness from edge warping can be expressed as foowing: G d ' = G' G s ' (5) The FE modeing prediction of shear stiffness caused by warping is provided in Tabe 6. By comparing the resuts in Tabe we can see that DDM and FEM predictions are quite different. In DDM (98), the expression for warping stiffness is cacuated from the assumption of a cantiever beam with springs distributed aong the pane. Apparenty this

22 assumption varies significanty from the warping behavior obtained in the FE mode. It is aso possibe the SR in shear is providing artificiay high resuts. Additiona study is needed. across corrugation 6/ 6/5 6/7 6/9 across corrugation 6/ 6/5 6/7 6/9 Tabe 6. G d cacuated from finite eement modeing (a) in unit N/mm aong corrugation : : : : : (b) in unit kip/inch aong corrugation : : : : :

23 7. Discussion and Concusion In this report, the eastic shear diaphragm stiffness of bare stee deck is expored through she finite eement modeing. The effect of different fastener ayoust and diaphragm dimensions is considered. She FE modes are buit in ABAQUS and modeing resuts are compared with AISI /Diaphragm Design Manua (DDM) predictions. The diaphragm stiffness (G ) term can be written in a series of three stiffness terms: pure shear (G s ), warping (G d ) and connection sip (G c ). The finite eement prediction for G s and G c agrees we with the DDM prediction. However, prediction for the G d term attributed to warping is stiffer than DDM prediction. This work shows that she finite eement modes are capabe of capturing the compex deformation that are inherent in the shear behavior of diaphragm composed of inter-connected stee deck. Differences in the mode and methods in DDM suggest that additiona investigation into the warping deformation prediction of DDM is warranted. 8. Reference [] [] Bian, G., Torabian, S, and Schafer, B. W. (6). Reduced Order Modes for Profied Stee Diaphragm Panes. Proceedings of Wei-Wen Yu Internationa Speciaty Conference on Cod-Formed Stee Structures 6, Batimore, MD, USA. [] Stee Deck Institute. Diaphragm Design Manua, first edition, 98. [] Stee Deck Institute. Diaphragm Design Manua, fourth edition, 5. [5] AISI-S-6. North American Standard for Design of Profied Stee Diaphragm Panes. [6]