Ultimate Shear Resistance of Plate Girders Part 2- Höglund Theory Ahmed S. Elamary

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1 World Academy o Science, Engineering and Technology International Journal o Ciil and Enironmental Engineering Vol:7, No:, 03 Ultimate Shear Resistance o Plate Girders Part - Höglund Theory Ahmed S. Elamary Digital Open Science Index, Ciil and Enironmental Engineering Vol:7, No:, 03 aset.org/publication/ Abstract Ultimate shear resistance (USR) o slender plate girders can be predicted theoretically using Cardi theory or Höglund theory. This paper ill be concerned ith predicting the USR using Höglund theory and. To main actors can aect the USR, the panel idth b and the eb depth d, consequently, the panel aspect ratio (b/d) has to be identiied by limits. In most o the preious study, there is no limit or panel aspect ratio indicated. In this paper theoretical analysis has been conducted to study the eect o (b/d) on the USR. The analysis based on ninety six test results o steel plate girders subjected to shear executed and collected by others. Ne ormula proposed to predict the percentage o the distance beteen the plastic hinges orm in the langes c to panel idth b. Conseratie limits o () hae been suggested to get a consistent alue o USR. Keyords Ultimate shear resistance, Plate Girder, Höglund s theory,. F I. INTRODUCTION OR a plate girder subjected to a small shear load, bending theory can be used to determine ho the internal orces are carried by the eb and the langes. When the applied load is increased, the ailure mode o a plate girder ill depend largely on the panel aspect ratio (b/d) and the eb slenderness ratio (d/t). The ultimate shear resistance o steel plate girders has been studied extensiely, both experimentally and theoretically []-[6]. Experimental studies o the ultimate shear resistance o steel plate girders hae indicated that at ailure, the girders exhibit the characteristic diagonal shear buckling o the eb and deeloped the plastic hinges in the langes. Hglund s theory is based on a system o perpendicular bars in compression and tension, hich is assumed to represent the eb panel [7]. The procedures incorporated in are diided in to methods o design; the irst method is the simple post critical design, hich is applicable to either stiened or unstiened girders; here the second design method called tension-ield method is applicable to stiened girder [8]. The rotating- stress-ield theory deeloped by Hglund orms is the basis o the irst method o design stated in Euro code 3. The second design procedure in based on Cardi tension-ield theory and it is applied to girders ith intermediate transerse stieners. This method is intended to produce more economical designs or a limited range o girder conigurations. Theoretical predictions o the ultimate shear resistance o plate girders Ahmed S. Elamary is Associated Proessor at Structural Ciil Engineering Department, Al-Azhar uniersity, Qena-Egypt Currently, Tai uniersity, Ciil Engineering Department, Faculty o Engineering K.S.A ( zshamary@gmail.com). based on simple post-critical design procedure appear inconsistent and unduly conseratie hen compared ith the aailable test data []. Theoretical predictions based on tension-ield design procedure (the second method), taking into account the limited range o eb panel aspect ratios, are less conseratie. Analysis o design methods made by Nethercot and Field [] indicated that, the existing procedures do not achiee the speciied reliability by predicting the ultimate shear resistance o plate girders, the partial saety actor m ould hae to be increased rom.05 to.35, hich may reduce the competitieness o many aspects o steel construction. In this paper theoretical analysis using Hglund theory and nd design method ill be conducted to determine the limits o panel aspect ratio hich can be applicable in each case. II. SHEAR STRENGTH OF PLATE GIRDER WEB PANEL USING HGLUND THEORY Hglund rotating- stress-ield theory is based on a system o perpendicular bars in compression and tension, hich are assumed to represent the eb panel. Originally it as deeloped or girders ith eb stieners at support only. For ebs in shear, there is a substantial post-buckling strength proided ater buckling by the anchoring system ith the surrounding langes and transerse stieners [7]. In pure shear the absolute alue o the principal membrane stresses and are the same as long as no buckling occurred ( < cr ). Ater buckling load reached (V cr = d * t * cr ), the eb plate ould buckle and redistribution o stresses ould start. Any increase in the applied load ill increase the tensile stress associated ith slight increase in compressie stress may be occurred as shon in Fig.. From Fig. ; Fig. State o stress in eb sin *cos 0.50* *sin () * International Scholarly and Scientiic Research & Innoation 7() 03 98

2 World Academy o Science, Engineering and Technology International Journal o Ciil and Enironmental Engineering Vol:7, No:, 03 here the direction o the tensile stresses chose to gie maximum. When is equal to the yield strength o the eb, y, then h * tan tan (6) u 0.5* y 3 or 45 () The total longitudinal orce in the eb is less than N * d * t (7) h h Digital Open Science Index, Ciil and Enironmental Engineering Vol:7, No:, 03 aset.org/publication/ here y (3) 3 This theory is called ideal tension ield theory, and is alid only i the langes are preented rom moing toards each other by an external structure [7]. Figs. (a) and (b) sho the total longitudinal orce and shear orces in the eb panel due to the externally applied load, here (c) and (d) sho the stress distribution through the beam cross section corresponding to these orces. In long beam, ith transerse stieners at the end only, the eb preents the langes to moe toards each other, that is hy the membrane stresses in the transerse direction are zero. From the triangle shon in Fig. (g) gies: (4) tan * tan (5) here constitutes the direction o the principal stress. This state o stress has a stress component h in the longitudinal direction. This component can be expressed as ollo: This orce has to be anchored at the ends o the beam by a transerse short beam called rigid end post, in order to ully deelop the rotated stress ield as shon in Fig.. The ultimate shear strength o the beam can be deried using the Von Mises yield theory criterion as ollo: (8) * y Assuming that the compression stress remain equal to the shear buckling stress ater buckling, and acting in a smaller angle than 45º: cr (9) Furthermore, the slenderness parameter is introduced here E cr K cr * t d y (e) Shear Stress Only () Shear and membrane stresses (g) Principal Stresses Fig. State o stress in eb o a beam ith transerse stieners at the ends only [9] (0) From (3)-(5), (8)-(0) the ultimate strength u = can be deried as a unction o as ollo: The square-root in () is close to i.5; this leads to u 4 3 * or () u. 3 () International Scholarly and Scientiic Research & Innoation 7() 03 99

3 World Academy o Science, Engineering and Technology International Journal o Ciil and Enironmental Engineering Vol:7, No:, 03 The inclination o the tension stress, deined by the angle, is decreased hen the ratio u is increased. For this reason cr the theory is called Rotated Stress Field Theory. The shear buckling capacity can be obtained rom V * * d * t (3) y Hoeer, it is assumed that the shear resistance o the eb, V is not changed by the ormation o the tension ield beteen langes. Then the shear resistance o the girder, V u, is the sum o the shear resistance o the eb, V, and the shear resistance contributed by the langes V. V u = V + V. (6) Digital Open Science Index, Ciil and Enironmental Engineering Vol:7, No:, 03 aset.org/publication/ here can be deined as shear buckling reduction actor hich is gien in Table I: TABLE I REDUCTION FACTOR FOR SHEAR BUCKLING Rigid end Post (Steel) Non-Rigid end Post (Steel) The alue o is a reduced alue related to scatter in test results such as initial imperections [7]. For small slenderness ratios, 0. 48, strain hardening in shear can take place, this produce larger strength than the corresponding to initial yielding A. Transersely Stiened Web Transerse stieners elded to the eb hae to main eects on the behaior and strength o a girder in shear; irst, increase the elastic buckling strength by preenting the eb rom out-o-plane delection, and second, they preent the langes rom coming closer to each other. The shear buckling coeicient K, can be expressed as ollo [9]: d K *, or b (4) b d d K 5.35* b 4, In ailure stage, our hinges denoted E, H, G, and K, orm in the top and bottom langes respectiely as shon in Fig. 3 (b). A tension stress ield, EHGK, deelops in the eb as illustrated in Fig. 3 (b). The ultimate shear orce, V, hich is transmitted by the tension stress ield is obtained rom the equation o equilibrium o the langes portion c as ollo: V b d 4 * Z * y (5) c (a) Shear orce carried by eb (b) Shear orce carried by truss action (c) Cross Section Fig. 3 Model o eb in post buckling range The distance c is estimated or steel plate girders as ollo:.6* b * t * y c 0.5 * b (7) t * d * y III. EUROCODE 3 To methods o design are proposed in, the irst is the simple post critical design here the second is the tension ield design method. The simple post critical design procedure incorporated in is based on theory proposed by Hglund and is applicable to either stiened or unstiened girders. In this method o design the ultimate design shear resistance o a plate girders reerred to it by V ba,rd and is gien as: V d * t * ba ba. Rd or (M M ) (8) m International Scholarly and Scientiic Research & Innoation 7() 03 90

4 World Academy o Science, Engineering and Technology International Journal o Ciil and Enironmental Engineering Vol:7, No:, 03 Digital Open Science Index, Ciil and Enironmental Engineering Vol:7, No:, 03 aset.org/publication/ here m is the partial material saety actor and ba is the simple post-critical shear stress, hich depends on the eb slenderness parameter. For ebs ith transerse stieners at the supports and intermediate transerse stieners cr is calculated using K as stated in (4). For ebs ith transerse stieners at the supports but ithout intermediate transerse stieners, cr is calculated assuming K = 5.3 [4]. For stocky ebs ( 0.8) ba y (9) For ebs o intermediate slenderness (0.8.) * 0.65 * ( 0.80) (0) ba y For slender ebs (.) ba y 0.9 * () here the second method is knon as tension-ield design method. tension-ield design shear resistance V,Rd is based on the Cardi tension-ield theory and is expressed as: V * d * t 0.9* g * t * * sin. Rd or (M M ) () m here is the shear buckling stress, is the tension-ield stress, g is the idth o the tension ield and is the inclination o the tension ield. or stocky ebs ( 0.8) y (3) For ebs o intermediate slenderness (0.8 < <.5) * 0.8* ( 0.80) (4) y For slender ebs (.5) * (5) y here the tension-ield stress is gien by * 3 (6) y b S c S * sin g d * cos t (8) here S c and S t are distances at hich plastic hinges orm in the compression and tension langes, respectiely, gien by S * sin M N, Rk * t (9) In hich M N. Rk is the reduced plastic moment o the lange alloing or a longitudinal orce N F.Sd in the lange and can be expressed as: M N. Rk 0.5* * b y N F. Sd * t * y * b * t m (30) The angle can be either determined by iteration to gie the maximum alue o V,Rd or approximated as d * tan (3) 3 b IV. EXPERIMENTAL RESULTS Extensie Experimental studies hae been conducted in Cardi on the ultimate shear resistance o steel plate girders, summary o hich hae been presented by Hglund, Nethercot and Neark [] presented in Table II. Hoeer, ninety six test results collected by Hglund, Nethercot and Byield and listed by Daies et al [] ill be used in this study. A summary o the test results (girder dimensions, material properties and ailure loads) is presented in Table II. in hich.5* *sin (7) The idth o the tension-ield g is gien by International Scholarly and Scientiic Research & Innoation 7() 03 9

5 World Academy o Science, Engineering and Technology International Journal o Ciil and Enironmental Engineering Vol:7, No:, 03 Digital Open Science Index, Ciil and Enironmental Engineering Vol:7, No:, 03 aset.org/publication/ Girder reerence TABLE II DETAILS OF TEST GIRDERS AND TEST RESULTS b d t b t E y y Vu Vexp./VS Hglund Vexp./VS C G6-T G6-T G6-T G7-T G7-T G8-T G8-T G8-T G9-T G9-T G9-T HT HT C-AC C-AC C-AC B S S TG TG TG TG TG TG US/ US3/ TG TG TG TG TG TG TG TG TG TG STG STG STG RTG RTG RTG TS/ TS/ MSO SD SD TGV TGV TGV TGV TGV TGV TGV International Scholarly and Scientiic Research & Innoation 7() 03 9

6 World Academy o Science, Engineering and Technology International Journal o Ciil and Enironmental Engineering Vol:7, No:, 03 Digital Open Science Index, Ciil and Enironmental Engineering Vol:7, No:, 03 aset.org/publication/ TGV TGV TGV TGV S3/ S4/ S5/ S/ S3/ S4/ ls-ba LS3-BA MCS-PB PA PA PA PB PB PC PC PD PD PD PC PB PB B B K A B A B A B A B CP/ RCP/ V. RELATION BETWEEN PANEL ASPECT RATIO (B/D) AND ULTIMATE SHEAR BY HGLUND OR Based on the extensie experimental study conducted by Hglund, Nethercot and Byield and collected by Daies et al, [] the olloing theoretical analysis is perormed. For each test rom the ninety six girders, to dierent alues o the ultimate shear ere calculated theoretically, the irst using Hglund theory, here the second using the second method o design stated in. Comparison beteen the shear alues obtained rom both cases ith dierent ranges o b/d are shon in Figs. rom 4 to 7; and listed in Tables rom III to VI HOGLUND Fig. 4 Comparison beteen V exp./v S predicted using Hglund Fig. 4 illustrated the relation beteen V exp / V S ith panel aspect ratio b/d, aries rom 0.5 to 0.94, these results related to 7 samples out o 96 samples. Table III also summarized the International Scholarly and Scientiic Research & Innoation 7() 03 93

7 World Academy o Science, Engineering and Technology International Journal o Ciil and Enironmental Engineering Vol:7, No:, 03 Digital Open Science Index, Ciil and Enironmental Engineering Vol:7, No:, 03 aset.org/publication/ mean, standard deiation and coeicient o ariation or the results o the ratio beteen ultimate shear resistance obtained experimentally compared ith the alue o ultimate shear resistance obtained using Hglund theory or ( nd method). From Fig. 4 and Table III, it can be summarized that the results obtained or ultimate shear resistance o plate girder using ( nd method) more consistent than Hglund theory in this range o b/d. Fig. 5 shos the relation beteen Vexp / VS ith panel aspect ratio b/d, aries rom 0.98 to, these results related to 39 samples out o 96 samples. Table IV also summarized the mean, standard deiation and coeicient o ariation or the results o the ratio beteen ultimate shear resistance obtained experimentally compared ith the alue o ultimate shear resistance obtained using Hglund theory or ( nd method). From Fig. 5 and Table IV, it can be summarized that the results obtained or ultimate shear resistance o plate girder using it can be summarized that the results obtained or ultimate shear resistance o plate girder using (nd method) more consistent than Hglund theory in this range o b/d. Fig. 6 shos the relation beteen V exp / V S ith panel aspect ratio b/d, aries rom. to 3, these results related to 6 samples out o 96 samples. Table V also summarized the mean, standard deiation and coeicient o ariation or the results o the ratio beteen ultimate shear resistance obtained experimentally compared ith the alue o ultimate shear resistance obtained using H glund theory or ( nd method). From Fig. 6 and Table V, it can be summarized that the results obtained or ultimate shear resistance o plate girder using it can be summarized that the results obtained or ultimate shear resistance o plate girder using ( nd method) more consistent than H glund theory in this range o b/d. TABLE III MEAN,STANDARD DEVIATION AND COEFFICIENT OF VARIATION FOR 7 SAMPLES WITH 0.5 B/D < Hglund and Data Sample Size (S.Size) = 7 From b/d To b/d< Standard deiation Coeicient o ariation 0.5 Mean Hglund Fig. 7 shos the relation beteen V exp / V S ith panel aspect ratio b/d, aries rom 3 to 5, these results related to 4 samples out o 96 samples. Table VI also summarized the mean, standard deiation and coeicient o ariation or the results o the ratio beteen ultimate shear resistance obtained experimentally compared ith the alue o ultimate shear resistance obtained using Hglund theory or ( nd method). From Fig. 7 and Table VI, it can be summarized that the results obtained or ultimate shear resistance o plate girder using it can be summarized that the results obtained or ultimate shear resistance o plate girder using Hglund theory more consistent than ( nd method) in this range o b/d Fig. 5 Comparison beteen V exp./v S predicted using Hglund and or 0.98b/d Fig. 6 Comparison beteen V exp./v S predicted using Hglund TABLE IV MEAN,STANDARD DEVIATION AND COEFFICIENT OF VARIATION FOR 39 SAMPLES WITH 0.98 B/D Hglund and Data Sample Size (S.Size) = 6 From b/d To b/d Standard deiation Coeicient o ariation. 3 Mean Hglund TABLE V MEAN,STANDARD DEVIATION AND COEFFICIENT OF VARIATION FOR 6 SAMPLES WITH. B/D.6 Hglund and Data Sample Size (S.Size) = 39 From b/d To b/d HOGLUND HOGLUND Standard deiation Coeicient o ariation 0.98 Mean Hglund International Scholarly and Scientiic Research & Innoation 7() 03 94

8 World Academy o Science, Engineering and Technology International Journal o Ciil and Enironmental Engineering Vol:7, No:, HOGLUND TABLE VI MEAN,STANDARD DEVIATION AND COEFFICIENT OF VARIATION FOR 4 SAMPLES WITH 3 B/D 5 Hglund and Data Sample Size (S.Size) = 4 From b/d To b/d Standard deiation Coeicient o ariation More than 3 up to 5 Mean Hglund Digital Open Science Index, Ciil and Enironmental Engineering Vol:7, No:, 03 aset.org/publication/ Fig. 7 Comparison beteen V exp./v S predicted using Hglund b/d Hglund TABLE VII RELATION BETWEEN B/D VALUE AND C/B Notes VI. RELATION BETWEEN PANEL ASPECT RATIO (B/D) AND (C/B) Table VII shos the relation beteen panel aspect ratio (b/d) and the corresponding alue o the ratio (); here c is the distance beteen the plastic hinges orm in the langes and b is the panel idth. b/d Hglund Hoglund alue is Constant The eect o the distance beteen the plastic hinges c on ultimate shear resistance alue has been studied using the corresponding equations No. (7) and (9). This parametric study based on the data o G8-T girder hich has percentage o experimental ultimate shear resistance to theoretical.0 and.0 as obtained using Hglund and ( nd design method) respectiely. The only ariable ill be apply is the panel idth b here the rest o the data ill be constant (eb thickness, lange thickness, eb depth,.). By comparing the results obtained rom (b/d) equal 0.5 to (b/d) equal 3 in both cases o design methods ith the releant alue o ; it can be summarized that; the conenient alue o ultimate shear resistance can be obtained in case o () ranged beteen 0. and Using these limits one chart plotted to represent the relationship beteen b/d and in case o here the results obtained rom Hglund theory ere constant or all dierent alue o b/d as shon in Fig Maximum Limit o = 0.5 Minmum Limit o = 0. Notes Constant in both Hoglund and y=0.046x x x 4 6.7x x 5.93x+.00 (nd method o design) Hoglund b/d 3.0 Fig. 8 Relation beteen and panel aspect ratio b/d From Fig. 8, a ormula proposed to predict the alue o using the assumed panel aspect ratio b/d. This ormula as ollo: In case o ( nd method o design - tension-ield design method) c b b 0.046* d 6 b 0.538* d 5 b.54* d 4 b 6.7 * d 3 b 8.356* d b 5.93* d International Scholarly and Scientiic Research & Innoation 7() 03 95

9 World Academy o Science, Engineering and Technology International Journal o Ciil and Enironmental Engineering Vol:7, No:, 03 Digital Open Science Index, Ciil and Enironmental Engineering Vol:7, No:, 03 aset.org/publication/ VII. CONCLUSION Hglund theory can be used to predict the ultimate shear resistance o plate girder haing intermediate transerse stieners in case o eb panel aspect ratios b/d (idth o eb panel/depth o eb panel) more than 3. In case o b/d aries rom 0.5 to 3, the ultimate shear resistance o plate girder using ( nd method o design- tension-ield design method) more consistent than Hglund theory. With panel aspect ratio b/d, more than 3, the ultimate shear resistance o plate girder using Hglund theory more consistent than ( nd method o design- tensionield design method). Ne ormula based on ( nd method o designtension-ield design method) proposed to predict the alue o in the begging o design by using the assumed distance beteen transerse stieners and eb depth. The consistent alues o () ranged beteen 0. and 0.50 to predict a conenient alue o ultimate shear resistance using ( nd method o design- tension-ield design method). The panel aspect ratio (b/d) on Hglund theory is not the main actor here the eb slenderness and lange rigidity are the main actors hich aect theory results. REFERENCES [] A.W. Daies, D. S. Griith Shear Strength o steel plate girder Prog. Instn Ci. Engrs Structs & Bldgs, 999, 34 May, PP [] M. Sulyok, T.V. Galambos Ealuation o eb buckling test results on elded beams and plate girders subjected to shear J Engineering Structures June 996, pp [3] S.C. Lee, J.S. Daidson and C.H. Yoo, Shear buckling coeicients o plate girder eb panels, Computers and Structures 59 (5) (996), pp [4] S.C. Lee and C.H. Yoo, Strength o plate girder eb panels under pure shear, Journal o Structural Engineering ASCE 4 () (998), pp [5] F. Shahabian and T.M. Roberts Combined Shear and Patch Loading o plate girders Journal o Structural engineering, March 000. [6] M.A. Bradord, Improed shear strength o ebs designed in accordance ith the LRFD speciication, Engineering Journal 33 (3) (996), pp [7] Torsten Hglund, Shear Buckling Resistance o Steel and Aluminium Plate Girders, Journal o Thin Walled Structures Vol pp [8] ENV , Eurocode 3: Design o steel structures: Part.. General rules and rules or buildings, 99, and Amendment A o Eurocode 3: Annex N Openings in ebs. British Standards Institution, 998. [9] S.P. Timoshenko and J.M. Gere (Int. Student edn, nd ed.), Theory o elastic stability ol. 54, McGra-Hill (985). International Scholarly and Scientiic Research & Innoation 7() 03 96