SDSS Rio STABILITY AND DUCTILITY OF STEEL STRUCTURES Nobutaka Shimizu et al. (Eds.) Rio de Janeiro, Brazil, September 8 -, STRENGTH ESTIMATION OF END FAILURES IN CORRUGATED STEEL SHEAR DIAPHRAGMS Nobutaka Shimizu*, Kikuo Ikarashi** * Nippon Steel Corporation e-mail: shimizu.nobutaka@nsc.co.jp ** Tokyo Institute o Technoloy e-mail: ikarashi.k.aa@m.titech.ac.jp Keywords: Corruated Steel Sheet, Diaphram, Shear Strenth, End Failure. Abstract. Shear tests on corruated steel sheets were carried out to examine the eects o shape and thickness o proiles on the end ailure action in lateral sway that may occur at the ends o lutes in shear diaphrams. Based on the test results, the previous shear deormation model or stiness estimation was reviewed, and the deormation model was developed into a new shear strenth model with application o yield-line hines. The strenth desin ormula was derived rom the strenth model throuh a limit analysis technique. Comparison with the test results showed that the derived ormula provided appropriate strenth estimation. 1 INTRODUCTION Structural desin methods or corruated steel sheets under in-plane shear orce are systematized as diaphram desin and summarized in uidelines such as ECCS [1] and SDI []. These desin uidelines allow rational structural systems, such as roo decks that resist shear orce due to wind pressure or seismic loadin without an additional bracin structure. In the structural desin, lobal and local bucklin o sheets, ailure at the seam between sheets, and ailure in sheet-to-rame asteners are commonly estimated. Besides these ailure modes, end ailures, which may occur at the ends o lutes in corruated shear diaphrams, are occasionally critical in deeper roo decks. To use deeper decks as shear diaphrams as shown in Fi. 1a, Davies [3] proposed the strenth desin ormula o end ailures based on test results, in which the ends o lutes ailed in lateral sway mode (Fi. 1b) or in web cripplin mode (Fi. 1c). But the ormula leans to ive too conservative estimation especially or lateral sway, because the ormula uses a actor based on test results in which almost all the specimens ailed in web cripplin mode. In this study, end ailure actions in lateral sway are examined throuh shear panel tests, and the eects o shape and thickness o proiles on lateral sway actions are clariied. Based on the test results, the previous shear deormation model or stiness estimation is developed into a yield mechanics model or strenth estimation. The strenth desin ormula is derived rom the model to improve the precision o strenth estimation or deeper roo decks where end ailure in lateral sway can occur. (a) Shear diaphram (b) Lateral sway (c) Web cripplin Fiure 1: End ailure in a corruated steel shear diaphram 43
PREVIOUS STRENGTH DESIGN FORMULA OF END FAILURE Davies [3] proposed a strenth ormula o end ailure based on shear diaphram tests with actual deeper roo decks. Fiure expresses the relationships between the shape o the proile and the end ailure mode o each specimen, in which specimens with astener ailure are excluded. The horizontal axis in Fi. indicates the ratio o proile-heiht h to proile-pitch q, and the vertical axis indicates the ratio o proile-heiht h to proile-thickness t. The circles represent lateral sway, and the trianles represent web cripplin. Specimens o Davies study are plotted with open symbols, and in this study, they are plotted with closed symbols. Fiure shows that many specimens o Davies test collapsed in web cripplin, and this trend is clearer in the rane o h/t over 0 and h/q over 0.5. Althouh Davies investiated theoretical approaches to lateral sway mode and web cripplin mode, identical strenth desin ormula c P 0 was inally applied or both end ailure modes as ollows: c 0 y 1.5 0.5 P t b q (1) where y is the yield stress o the proile material, b is the lenth o the diaphram, and is a nondimensional actor deined as 0.5 in reerence [3] and as 0.9 in reerence [1] to ive conservative strenth estimation or the proiles astened to the rame in every trouh o corruation. h / t h / q Lateral sway [3] Web cripplin [3] Lateral sway [This study] t q Shape o proile Fiure : Relationship between proile shape and ailure mode h 3 SHEAR DIAPHRAGM TESTS 3.1 Outline o tests The end ailure actions o corruated steel sheets with various shapes o proile were investiated by shear diaphram tests usin the setup shown in Fi. 3a. Every tauht o the sheets was connected by a bolt (M6) on a steel rame (Fi. 3b) to make a riction-bolted joint. Static-monotonous shear orce P was loaded to the rame by a hydraulic cylinder, and shear deormation was measured. 800mm 800mm Pin Fixed Beam Hydraulic Cylinder P Support Specime M6 Mechanical Pin Bush 60mm 95mm Loadin Beam Support M6 (a) Loadin system Fiure 3: Test setup (b) Steel rame The test specimens were corruated steel sheets deined by the shapes accordin to the size symbols shown in Fi. 4a: a is the width o the diaphram, b is the lenth o the diaphram, q is the pitch o the 44
proile, h is the heiht o the proile, t is the thickness o the proile, e is hal o the lower lane width, is the upper lane width, is the web projection width, and l is addition o the upper lane width and the web projection width. The sheets were bolted on the rame with a washer, whose width was j as shown in Fi. 4b, to secure a riction surace. A bolted connection is adopted to avoid connection ailures, and a sheet without seams is used to avoid seam astener ailures in this test. 16mm b = 800mm 16mm Washer Bolt (M6) Washer Bolt (M6) R=t t h l q j j e e e e (a) Diaphram size and shape o proile (b) Connection between sheet and rame Fiure 4: Specimen Table 1 shows a list o the test parameters. The aim o Series I is to identiy the eect o the shape o the proile: the upper lane width, the heiht o proile h, and the thickness o proile t. The aim o Series II is to identiy the eect o the width o washer j. Typical shapes o the specimens, like t04-h- 5 or t04-h-3, correspond with one-third scale roo deck in Japan. As shown in Fi., proile shape ratios h/t and h/q or this test lie in the rane o 5 0 and 0.5 0.5, respectively. Series I II Table 1: Test parameters Specimen name a b t q e h l e j (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) t04-h-40 800 800 0.4 80 40 0.0 40 40 34 t04-h-5 800 800 0.4 80 5 7.5 40 40 34 t04-h-15 800 800 0.4 80 15 1.5 40 40 34 t04-h-0 800 800 0.4 80 0 40 40 34 t04-h-5 800 800 0.4 80 5 7.5 40 40 34 t04-h40-5 800 800 0.4 80 5 7.5 40 40 40 34 t06-h-5 800 800 0.6 80 5 7.5 40 40 34 t08-h-5 800 800 0.8 80 5 7.5 40 40 34 t04-h-3 800 800 0.4 80 31.6 15.8 8.4 19.6 48.4 31.6 8 t04-h-3j 800 800 0.4 80 31.6 15.8 8.4 19.6 48.4 31.6 1 The steel plates used or abrication o the specimens are common between Series I and II, and their mechanical properties are shown in Table. Table : Mechanical properties o steel plates Thickness(mm) Yield strenth Tensile strenth Elonation Nominal Measured (N/mm ) (N/mm ) (%) 0.4 0.41 67 366 37.9 0.6 0.58 77 373 38.4 0.8 0.77 51 353 41. 3. Test results Almost all the specimens deormed elastically under shear orce with elastic distortion o the lute cross sections at initial loadin, and as shown in Fi. 5a, the distortion then moved to plastic end ailure action 45
with lateral sway, which determines the ultimate strenth. Viewed rom a vertical anle as shown in Fi. 5b, every lute deormed linearly alon the ride lines and symmetrically around the center point o the upper lane. Thereore, the distortional deormation o the cross sections linearly decreased as they approached the centerline o the diaphram lenth (A-A section). (a) Lateral sway in a horizontal anle (b) Lateral sway in a vertical anle (c) Ater lateral sway Fiure 5: Observation o end ailure (t04-h-5) Load-deormation (P- relationships or all the specimens are shown in Fi. 6. Ater showin linear deormation behavior durin the early loadin stae, almost all the specimens showed radual yieldin and reached the ultimate strenth determined by end ailure in lateral sway. When loadin advanced urther rom the status o Fi. 5a and 5b, the lateral sway chaned into web cripplin at the inclinin sides; urthermore, the local tension ield action mode o the steel plate around the bolted connection as shown in Fi. 5c, so that the specimens maintained strenth in the load-decreasin rane ater reachin the ultimate strenth. The experimental values shown in Table 3 are obtained by the deinition in Fi. 6e: The initial stiness e K is deined as secant riidity at shear deormation anle 1/800, the ultimate strenth e P u is deined as the maximum strenth determined by lateral sway, and the yield strenth e P y is deined as the load correspondin to displacement o the intersectin point between a line drawn parallel to the horizontal axis passin throuh the point o e P u and a line indicatin e K. 40 P(kN) t04-h-0 t04-h-15 t04-h-5 t04-h-40 (mm) 0 5 15 5 P(kN) t04-h-5 t04-h40-5 t04-h-5 (mm) 0 5 15 5 60 P(kN) 50 40 t06-h-5 t04-h-5 t08-h-5 (mm) 0 5 15 5 P(kN) t04-h-3 t04-h-3j (mm) 0 5 15 5 (a) Upper lane (b) Heiht h (c) Thickness t (d) Washer j (e) Deinition Fiure 6: Load-deormation (P- relationship Fiure 6 shows that obvious dierences due to the eect o the test parameters were identiied in the initial stiness, the yield strenth, the ultimate strenth, and load reduction ater reachin ultimate strenth. Furthermore, the P-relationship o t08-h-5 shows luctuation o the load because o slip action in the riction-bolted connections. Fiure 6 and Table 3 show the eects o the experimental parameters as ollows: Eect o upper lane width (Fi. 6a): The narrower the upper lane width is, the more the initial stiness increases. The yield and ultimate strenth rise as the upper lane width decreases, but the ultimate strenth shows a sliht dierence between a width o 15 mm and a width o 0 mm. Furthermore, the load reduction tends to be rapid as the ultimate strenth becomes hiher. P ek =1mm epu 46
Eect o proile heiht h (Fi. 6b): The hiher the proile heiht is, the more the initial stiness decreases. The yield and ultimate strenth decrease as the proile heiht increases. In addition, the load reduction tends to be moderate as the ultimate strenth becomes lower. Eect o plate thickness t (Fi. 6c): The thicker the plate thickness is, the more the initial stiness increases. The yield and ultimate strenth rise as the thickness increases. Additionally, the load reduction tends to be somewhat rapid as the ultimate strenth becomes hiher. Eect o the washer width j (Fi. 6d): The larer the washer width is, the more the initial stiness increases, while there is no evident dierence in the yield and ultimate strenth as the washer width increases. 4 STRENGTH ESTIMATION OF END FAILURES IN LATERAL SWAY 4.1 Stiness ormula o distortional deormation The previous stiness ormula o the corruated shear diaphrams is reviewed to develop its deormation model into a strenth model or a strenth ormula o end ailure in lateral sway. When corruated sheets receive shear orce, distortional deormation o the lute o the cross section, as well as in-plane shear deormation o the plate elements, causes deormation o shear diaphrams. This distortional action is modeled as the deormed mechanics shown in Fi. 7a by Davies [4], in which it is assumed that the cross section receives bendin moment and deorms linearly alon the ride lines and symmetrically around the center point o the upper lane. Based on equilibrium between the internal enery o the deormed mechanics and the external enery produced by the shear orce, shear stiness d K or the distortional deormation is obtained as ollows: d 3 3 Et b K 144Dh 3 () where E is Youn s modulus and D is the actor o the proile shape that is expressed in ollowin equation: 3 3 8e 8e w4e w w D 1 hl ( e) (3) Then, shear stiness s K or in-plane shear deormation o the plate elements is iven by the ollowin equation: s bte K ( w e)(1 ) (4) where is Poisson s ratio. Usin stiness values d K and s K per lute, the estimation ormula o shear stiness ck o the corruated steel diaphram, which has n number o lutes, is obtained as ollows: c K 1 1 1 1 K sk n d (5) 4. Strenth ormula o end ailure in lateral sway The yield mechanics model shown in Fi. 7b is assumed by deinin the yield-line hines on the ride lines o the proile, at which the bendin moment o the cross section is maximized in the deormed mechanics model shown in Fi. 7a. The yield mechanics model, as well as the deormed mechanics model, is symmetrical around the center point o the upper lane. A new strenth ormula is derived rom equilibrium between internal enery c U i absorbed in the yield-line hines and external enery c U e 47
produced by shear orce c P and shear deormation c. In calculatin the internal enery, the steel plate is considered to have the material properties o riid plastic on the basis o the limit analysis technique. The internal enery c U i is iven by plastic moment per unit lenth m p ( t y /4) and the hine rotation anles o web w and upper lane at both ends o diaphram lenth b as Equation (6), considerin the eometrical condition o the mechanics whereby the hine rotation anles linearly vary so as to all to zero at the center o diaphram lenth in which no cross section deormation occurs. w b U m (4 ) (6) c i p w and are associated in Equation (7) considerin displacement dierence v between both ends on the upper lane and the continuous web. cu i is obtained by substitutin Equation (7) or Equation (6) as ollows: w v w cos (7) w w b U 4m 1 cos c i p Moreover, c and w are associated by the assumption that the upper lane rotates as a riid body, so that the external enery c U e is iven as ollows: w c e c c c U P w sin b P (9) Consequently, c P is obtained rom the equilibrium between c U i and c U e as ollows, correspondin to ultimate strenth c P u : (8) b w P 1 cosm P wsin c p c u () In addition, yield strenth c P y is iven by replacin the plastic moment m p in Equation () with yield moment m y ( t /6) as ollows: y b w P 1 cosm (11) wsin c y y Yield line y dy e l e dp d A point o symmetry m x ph l+e p ph(l-) (l+e) p/ x phe p/ l+e ph l+e w w sin (a) Assumed deormed mechanics (b) Assumed yield mechanics Fiure 7: Estimation model w w m p cp c 48
4.3 Correspondence between test results and ormulas Table 3 shows a comparison between test values ( e K, e P y, and e P u ) and calculated values ( c K, c P y, c P u, and c P 0 with =0.9). Fiure 8 and Fi. 9 indicate the relationships between the proile shape and the estimation result or stiness and strenth, respectively, or Series I. In these iures, the closed diamond, the closed square, and the closed circle indicate e K, e P y, and e P u, respectively, and the open diamond, the open square, the open circle, and the open trianle indicate ck, c P y, c P u, and c P 0, respectively. Fiure 8 shows that the previous stiness ormula c K can provide appropriate estimation o the experimental initial stiness e K when the proile shape varies, excludin the specimen with the upper lane width set to zero (t04-h-0). Furthermore, Fiure 9 shows that the previous strenth ormula c P 0 has low sensibility toward the test parameters and provides too conservative estimation o the experimental yield and ultimate strenth ( e P y and e P u ). By contrast, the proposed strenth ormulas c P y and c P u have avorable sensibility toward the test parameters and provide better estimation o e P y and e P u than o c P 0. Reardin Series II, Table 3 shows that calculated values ive suitable estimation or the test results, but the specimens with small washer width (t04-h-3j) tend to ive slihtly hiher calculated stiness ck than experimental stiness ek. K (kn/mm) t04-h-0 t04-h-15 t04-h-5 t04-h-40 (mm) 0 40 50 15 K (kn/mm) 5 t04-h-5 t04-h-5 t04-h40-5 0 40 50 h (mm) 60 40 K (kn/mm) t04-h-5 0 0. t06-h-5 t08-h-5 t (mm) 0.4 0.6 0.8 1.0 1. (a) Upper lane (b) Heiht h (c) Thickness t Fiure 8: Estimation o stiness e K c K 40 P (kn) t04-h-0 t04-h-15 t04-h-5 t04-h-40 (mm) 0 40 50 P (kn) 0 t04-h-5 t04-h-5 t04-h40-5 h (mm) 40 50 60 70 P (kn) 60 t08-h-5 50 t06-h-5 40 t04-h-5 t (mm) 0 0. 0.4 0.6 0.8 1.0 1. (a) Upper lane (b) Heiht h (c) Thickness t Fiure 9: Estimation o strenth e Py e Pu cpy cpu cp0 Name Table 3: Comparison between test result and calculation Test result Calculation Test result / Calculation e K e P y e P u c K c P y c P u c P 0 e K e P y e P y e P u e P u I II (kn/mm) (kn) (kn) (kn/mm) (kn) (kn) (kn) c K c P y c P 0 c P u c P 0 t04-h-40 3.93 9.1 13.1.74 5.98 8.98 5.64 1.43 1.5 1.6 1.46.3 t04-h-5 8.17 13.9 18.7 7.86 1.4 18.7 5.64 1.04 1.11.45 1.00 3.31 t04-h-15 1.4 1.9 5.5 14.6 9.3 43.9 5.64 0.85 0.75 3.88 0.58 4.51 t04-h-0 16. 8. 9. 6.8 - - 5.64 0.61-5.00-5.18 t04-h-5 4.4.0 14.3 3.63 8.3 1.4 5.64 1. 1. 1.77 1.15.53 t04-h40-5 3.14 7.5.9 1.9 6. 9.34 5.64 1.64 1.1 1.33 1.17 1.93 t06-h-5 15.1 6.4 35.8 16.8 5.8 38.8 9.85 0.90 1.0.68 0.9 3.63 t08-h-5 3.1 38.8 53.7 8.5 41.3 61.9 13.7 0.81 0.94.84 0.87 3.93 t04-h-3 7.7 13. 18.8 6.34 9.78 14.7 5.64 1.15 1.35.33 1.8 3.34 t04-h-3j 4.47 1.7 18.4 6.34 9.78 14.7 5.64 0.71 1..5 1.5 3.6 49
The precision o estimation o stiness and strenth seems to be lower as the upper lane width decreases because the deormed mechanics model and the yield mechanics model tend to be determined by not only bendin moment across the section but also in-plane axis orce due to truss action. Improvin the estimation accuracy o the mechanics model controlled by the truss action is a problem to be solved hereater. Furthermore, the eect o the ratio o proile-pitch q to diaphram-lenth b, which may inluence the bendin action across the section, ouht to be investiated. 5 CONCLUSION The eects o shape and thickness o the proile on end ailure in lateral sway were examined throuh shear diaphram tests on corruated steel sheets. Based on the test results, the previous shear stiness ormula was reviewed and developed into a strenth ormula or end ailure in lateral sway. The conclusions and indins are summarized as ollows: 1. Almost all the specimens deormed elastically under shear orce with elastic distortion o the lute cross section at initial loadin, and the distortion moved to plastic end ailure action with lateral sway, in which every lute deormed linearly alon the ride lines and symmetrically around the center point o the upper lane. Obvious dierences due to the eect o shape and thickness o the proile were identiied in the initial stiness, the yield strenth, and the ultimate strenth o end ailure action with lateral sway.. Based on the test observation o end ailure in lateral sway, the yield mechanics model is assumed by deinin the yield-line hines on the ride line o the proile, at which the bendin moment o the cross section is maximized in the previous deormed mechanics model. The strenth ormula was derived rom equilibrium between the internal enery o the yield-line hines and the external enery o the shear orce usin a limit analysis technique. Comparison between the test results and the ormulas shows that the previous stiness ormula and the proposed strenth ormula provide appropriate estimation in contrast to the excessive conservativeness o the previous strenth ormula. 3. The precision o estimation o stiness and strenth tends to be lower as the upper lane width becomes smaller because the deormed mechanics and the yield mechanics tend to be determined by not only bendin moment across the section but also in-plane axis orce due to truss action. Improvin the estimation accuracy o the resistance mechanism determined by both bendin and truss action is a problem let or uture study. REFERENCES [1] European Convention or Construction Steelwork (ECCS), European Recommendations or the Application o Metal Sheetin actin as a Diaphram, 1995. [] Steel Deck Institute (SDI), Diaphram Desin Manual Third Edition, 04. [3] J.M. Davies, J. Fisher, End ailure in stressed skin diaphrams, Proc. Instn Civ. Enrs, Part, 75-93, 1987. [4] J.M. Davies, R.M. Lawson, The Shear Deormation o Proiled Metal Sheetin, International Journal or Methods in Enineerin, Vol.1, 1507-1541, 1978 [5] J.M. Davies, A General Solution or the Shear Flexibility o Proiled Sheets. I & II, Thin-Walled Structures 4, 41-68 and 141-161, 1986 50