Article Finite Element Analysis f Distrtin-Induced Web Gap Stresses in Multi-I Girder Steel Bridges Akhrawat Lenwari 1,*, and Huating Chen 1 Department f Civil Engineering, Faculty f Engineering, Chulalngkrn University, Phayathai Rd., Pathumwan, Bangkk 133, Thailand Institute f Rad and Bridge Engineering, Department f Civil Engineering, Beijing University f Technlgy, #1 PingLeYuan, ChaYang District, Beijing 114, China * E-mail: akhrawat.l@chula.ac.th (Crrespnding authr) Abstract. Unstiffened girder web gaps at the ends f transverse stiffeners that als serve as diaphragm cnnectin plates are subjected t high lcal stresses during cyclic ut-fplane distrtin. The ut-f-plane distrtin is mainly caused by the differential deflectin between adjacent girders. The purpse f the paper is t investigate the effects f bridge parameters including span length, girder spacing, slab thickness, and girder stiffness n the differential deflectin and distrtin-induced web gap stresses. Dual-level finite element analyses that cnsist f bth glbal mdel and sub-mdel were perfrmed. The glbal mdel was used t investigate the critical truck psitin and maximum differential deflectin between adjacent girders, while the sub-mdel was used fr the critical web gap vertical stress. A base case bridge was a simply supprted cmpsite superstructure with three steel I-girders that supprt tw traffic lanes, which is typical fr steel bridges ver intersectins in Bangkk, Thailand. A parametric study was cnducted by varying ne bridge parameter at a time. The analysis results shw that the maximum differential deflectins and web gap stresses caused by ne-truck lading are higher than thse caused by tw-truck lading (ne truck n each lane). Under ne-truck lading, the maximum web gap stress ccurs at the interir girder. In additin, bth the differential deflectins and web gap stresses are primarily dependent n the bridge span length. Keywrds: Distrtin-induced stress, web gap cracking, parametric study, cmpsite I- girder bridge, finite element methd. ENGINEERING JOURNAL Vlume 17 Issue 1 Received 8 June 1 Accepted 1 September 1 Published 1 January 13 Online at http://www.engj.rg/ DOI:1.4186/ej.13.17.1.95
DOI:1.4186/ej.13.17.1.95 1. Intrductin Steel-cncrete cmpsite girders are typically tied tgether by diaphragms r bracings. These secndary elements are intended t resist lateral lad, help distribute traffic lading, and stabilize the girders during cnstructin. The transverse stiffeners cnnecting the diaphragm and girder nrmally terminate a few inches frm the girder tensin flange. The web prtin between the transverse stiffener end and girder flange, called the web gap, is relatively flexible and susceptible t ut-f-plane distrtin. The web gap cracking has been mainly caused by the secndary stresses due t differential deflectin f main girders [1]. Figure 1 shws a typical fatigue crack that initiates at the weld te at the end f transverse stiffener. Fig. 1. Distrtin-induced fatigue cracking at stiffener end schematic clse-up view f crack. The AASHTO LRFD Specificatins [] categrizes this type f fatigue cracking as distrtin-induced fatigue in which the frce effect, nrmally transmitted by a secndary member, may tend t change the shape f, r distrt, the crss sectin f a primary member. Hwever, the ut-f-plane distrtin-induced stresses are nt quantified in the AASHTO design cde. The field measurement data shwed that the stress ccurring at the cnnectin plate t the web welds was large enugh t cause fatigue cracking [3, 4]. In Japan, mst cracks ccur at the tp end f vertical stiffener welded t the tensin tp flange [5]. These cracks were caused by the secndary stresses due t differential deflectin f the main girders and deflectin f the reinfrced cncrete deck by the wheel lad f heavy traffic. In the United States, Cnnr and Fisher [6] reprted that 9 percent f the fatigue cracking in steel bridges is related t distrtin-induced fatigue. Analytical mdels have been prpsed t predict the distrtin-induced stresses. Based n the linear elasticity, Jajich and Schultz [7] prpsed a simple mdel fr predicting the web gap stress, as illustrated in Fig.. The web gap was idealized as a beam that is fixed at bth ends and underges rtatin at ne end, the maximum mment at the base f web gap is btained as M 4 EI / g, where E = mdulus f elasticity; I wg = mment f inertia f web gap sectin resisting ut-f-plane bending; and g = web gap length. Hence, the maximum stress at the stiffener end f the web gap is wg.5 Mwgtw / Iwg, where t w = web thickness. Cmbining these expressins and substituting the diaphragm rtatin / S, where S = length f diaphragm r girder spacing and = differential r relative deflectin between adjacent girders, gives the maximum web gap stress as a functin f girder differential deflectin as fllws Et / g / S wg (1) wg w wg 96 ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/)
DOI:1.4186/ej.13.17.1.95 θ = Δ/S h Δ S Fig.. Schematic representatin f web gap and diaphragm rtatins [7]. In anther mdel [1, 8], the web gap was pstulated t behave like shrt, fixed-fixed beams underging a lateral deflectin, δ. Figure 3 shws the relatinship between the web gap lateral deflectin and diaphragm rtatin. Therefre, the maximum web gap stress can be calculated as where h = diaphragm depth. wg 3 Etw / g h / S () Fig. 3. Lateral web gap deflectin caused by diaphragm rtatin [1]. Accrding t bth analytical mdels, the differential deflectin between adjacent girders is a key variable affecting the distrtin-induced stress at web gap. Hwever, the amunt f differential deflectin cannt be easily predicted due t the interactins between cncrete decks, girders, and diaphragms. In additin, the secndary stress at the intersectin f stiffener, girder flange, and web is under cmplex, three dimensinal interactins. The lcal gemetry and relative stiffness f the detail is different fr individual bridge [9]. Berglund and Schultz [1] investigated the parameters that significantly affect the girder differential deflectin using finite element analysis. The results shwed that the differential deflectin generally increases as the bridge span length, girder spacing, and angle f skew increases. Jajich et al. [7, 11] perfrmed a linear finite element analysis f the diaphragm-stiffener-web cnnectin. A three dimensin finite element mdel was created using SAP. The mdel cnsisted f tw adjacent girders cnnected by the diaphragm. All prtins f steel (web, flanges, diaphragm, and stiffeners) were mdeled using the shell element. All dimensins were taken either directly frm the bridge r frm the ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/) 97
DOI:1.4186/ej.13.17.1.95 design plan. The cncrete deck was excluded in the mdel, hwever the rtatinal fixity f girder tp flanges due t the presence f the deck was assumed. The truck lads reprted in the field tests [7] were applied t a s-called macr mdel by Li and Schultz [8]. The macr mdel included the entire bridge and a prtin f the bridge surrunding the diaphragm t determine the defrmatins t be impsed n the bridge micr-mdel. Deck rtatin and diaphragm deflectins frm macr-mdel finite element analyses were applied as bundary cnditins fr the micrmdel finite element analyses at the lcatins where these members were discnnected frm the rest f the bridge. The cnnectins between cncrete deck and girder tp flanges were mdeled as rigid elements at clsely spaced intervals. The deck bundary was fixed against translatin alng the edges parallel t the girders. Girder differential deflectins and deck rtatins frm macr-mdel were applied t the micrmdel t determine the web gap mvement and stress field. Since distrtin-induced stress is related with differential deflectin between adjacent girders due t uneven distributin f live lad, the bridge parameters that affect live lad distributin are believed t affect the distrtin-induced stress. The AASHTO LRFD Specificatins [] define the live lad distributin factr fr beam and slab bridges with tw r mre lanes as.6. S S Kg LDF.75 3 9 L Lts.1 (3) where S = girder spacing (mm); L = span length (mm); K n( I Ae ) = lngitudinal stiffness (mm 4 ); t s = slab thickness (mm); n = mdular rati between steel and cncrete; I = girder stiffness (mm 4 ); A = girder area (mm ); and e = eccentricity between centrids f girder and slab (mm). In Eq. (3), the span length, girder spacing, slab thickness and girder stiffness are the key bridge parameters that affect live lad distributin. The purpse f the paper is t numerically investigate the effects f bridge parameters n the differential deflectin and distrtin-induced web gap stresses in cmpsite I-girder bridges. The bridge parameters include span length, girder spacing, slab thickness, and girder stiffness. Dual-level finite element analyses which include glbal mdel and sub-mdel are perfrmed. The glbal mdel is used t study the differential deflectin between adjacent girders, while the sub-mdel is used fr the web gap vertical stress. A base case bridge is idealized frm a typical three-girder cmpsite bridge in Bangkk, Thailand. In a parametric study, ne bridge parameter is varied, while ther parameters have the same values as thse in the base case.. Finite Element Mdel (SAP) A base case bridge is idealized frm an actual simply supprted cmpsite I-girder bridge fr which the field test data [1] are available fr the validatin f the finite element mdel. The base case bridge cnsists f three cmpsite I-girders equally spaced at 3. m, which spans 4 m and supprts tw traffic lanes with lane width f 3. m. The cncrete slab has a thickness f 4 mm. All three girders are built-up wide flange members with dimensins f 145 45 1 mm 4. Diaphragms are idealized as I-beam sectins f 678 53 1 16 mm 4. In rder t capture the distrtin-induced stress at web gap accurately while at the same time minimizing cmputatinal cst, tw levels f linear elastic finite element analysis which cnsists f glbal mdel and sub-mdel are perfrmed. SAP sftware package [13] is used t create bth glbal mdel and sub-mdel. The glbal mdel includes the entire superstructure as shwn in Fig. 4, which is aimed t investigate the glbal behavir f the bridge and differential deflectin between adjacent girders. The submdel is a mre detailed mdel f a prtin f the bridge near diaphragm, as shwn in Fig. 4, where the web gap stresses will be studied. g 98 ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/)
DOI:1.4186/ej.13.17.1.95 Prtin f sub-mdel Fig. 4. Finite element mdels fr base case bridge glbal mdel sub-mdel. The glbal mdel cnsists f cncrete slab, three main girders, and diaphragms at midspan, quarter spans and end spans f the bridge. The mduli f elasticity fr structural steel and cncrete are taken as MPa and 48 MPa, respectively. The Pissn ratis fr structural steel and cncrete are assumed t be.3 and., respectively. Main girders and diaphragms are idealized as frame elements and the cncrete slab is mdeled with quadrilateral shell elements. Cmpsite actin between steel girders and cncrete slab is mdeled using rigid links that cnnect the girder tp flange and cncrete deck by cupling all degrees f freedm between crrespnding ndes. The mdel is made up f 5 frame elements,,4 shell elements, and 43 rigid links. As a detailed finite element mdel f the diaphragm-stiffener-web cnnectin, the sub-mdel in Fig. 4 cnsists f all shell elements. Tp flange f girders is free t mve in the vertical directin and rtate in the lngitudinal directin. In rder t simulate the restraining effect f cncrete slab, the transverse rtatin f tp flange ndes is restrained. At the supprt, the bttm flanges ndes are restrained in the lngitudinal and transverse axes f the girder and free t mve in lngitudinal directin at the rller supprt. The vertical displacements f girders frm glbal mdel are applied t the girder tp flanges in the submdel. In rder t btain lcal distrtin-induced stresses, the mesh in web gap area is refined and a transitin mesh is used t maximize mdeling efficiency. A sufficiently refined sub-mdel cnsists f 8,638 shell elements. Figure 5 cmpares the lngitudinal bttm flange stresses frm the glbal mdel with thse measured frm strain gages at quarter span, midspan, and three quarter span f a base case bridge. The data were btained under tw test trucks that mved alng bth lanes simultaneusly. At midspan, the maximum strain differences are fund t be 8, 1 and 13 percent fr girder G1, G, and G3, respectively. As described in Sectin 1, the main bridge parameters chsen in this study include the bridge span length, girder spacing, slab thickness, and girder stiffness. The range f each parameter refers t the applicability range specified in the AASHTO LDF equatin. The bridge span length varies frm 18 m t 36 m, girder spacing varies frm.1 m t 3. m, and slab thickness varies frm 1 mm t 3 mm. The range f girder stiffness is based n practical range f actual bridges and is varied frm 1. 1 1 mm 4 t 1.7 1 1 mm 4. The actual web gap length, i.e., 35 mm, is used fr all mdels in the study. In rder t study the effect f a specific bridge parameter, a parametric study is cnducted by varying ne parameter while fixing all ther parameters at their base bridge values. The superstructure is assumed t remain tw lanes and the lane width varies with the girder spacing. Variatin in the girder stiffness is accmplished by varying the thickness f web and flange plates while keeping the girder height and flange width the same as the base case. Fr the fur cases f varying girder stiffness, the girder sectins are 145 45 1 18 mm 4, 145 45 1 mm 4, 145 45 14 6 mm 4, and 145 45 16 3 mm 4, respectively. Sixteen variatins f bridge parameters are mdeled in the parametric study as shwn in Table 1. ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/) 99
DOI:1.4186/ej.13.17.1.95 Bttmg flange stress (MPa) 3 1 G1 strain gage G strain gage G3 strain gage G1 FEM G FEM G3 FEM Bttm flange stress (MPa) 3 1 G1 strain gage G strain gage G3 strain gage G1 FEM G FEM G3 FEM 5 1 15 5 Truck psitin (m) 5 1 15 5 Truck psitin (m) Bttm flange stress (MPa) 3 1 G1 strain gage G3 strain gage G1 FEM G FEM G3 FEM 5 1 15 5 Truck psitin (m) Fig. 5. Cmparisn between glbal mdel results and measured stain data fr lngitudinal bttm flange stresses at quarter span midspan, and (c) three quarter span. Table 1. Variatins f bridge parameters in the parametric study. (c) Parameter Effect f bridge span length Effect f girder spacing Effect f slab thickness Effect f girder stiffness Span length (m) Girder Spacing (m) Slab Thickness (mm) Girder Stiffness (mm 4 ) 18 3. 4 1. 1 1 4 3. 4 1. 1 1 3 3. 4 1. 1 1 36 3. 4 1. 1 1 4.1 4 1. 1 1 4.4 4 1. 1 1 4.7 4 1. 1 1 4 3. 4 1. 1 1 4 3. 1 1. 1 1 4 3. 4 1. 1 1 4 3. 7 1. 1 1 4 3. 3 1. 1 1 4 3. 4 1. 1 1 4 3. 4 1. 1 1 4 3. 4 1.5 1 1 4 3. 4 1.7 1 1 The AASHTO standard truck HS was applied in the multistep analysis in the study. The truck lad cnfiguratin is shwn in Fig. 6. 1 ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/)
DOI:1.4186/ej.13.17.1.95 4.7 m V 3.63 T 14.5 T 14.5 T V = variable spacing (frm 4.7 m t 9.15 m) Fig. 6. Cnfiguratin f standard AASHTO truck (HS-44). Since the bridge is transversally symmetric with tw lanes, tw general lading types including ne truck n the left lane and tw trucks n bth lanes are cnsidered. Figure 7 shws pssible transverse truck lading cmbinatins fr tw-truck ladings. The uter-uter lading case is defined by ne truck mving in the left edge f left lane and anther truck n the right edge f right lane, as shwn in Fig. 7, the center-center lading case is when bth trucks mve in the center f the left and right lanes, as shwn in Fig. 7, and the inner-inner lading case is when ne truck mves in the right edge f left lane and anther truck n left edge f the right lane, as shwn in Fig. 7(c). Other transverse lading cases including the uter-center, uter-inner, and center-inner are shwn in Figs. 7(d), (e), and (f), respectively. (c) (d) (e) (f) Fig. 7. Transverse truck lading cmbinatins uter-uter center-center (c) inner-inner (d) utercenter (e) uter-inner (f) center-inner. 3. Effects f Bridge Parameters n Differential Girder Deflectin In the parametric study, the truck lading mves alng the bridge at speed f 1 mm/s, and fr every.1 m the bridge respnses frm the finite element mdel are recrded. After examining the vertical displacement data, the critical lngitudinal truck psitin causing the largest girder deflectin is fund t be arund midspan, e.g., the frnt wheels lcate at 17.9 m frm the supprt fr the base case bridge. The differential deflectins between adjacent girders, i.e., the difference between midspan deflectins f girder G1 and girder G, and the difference between midspan deflectins f girder G and girder G3, are als largest at this critical truck psitin fr bth ne-truck and tw-truck ladings. Fr the ne-truck lading, the largest differential deflectin between adjacent girders ccurs when the truck mves alng the left edge f left lane (uter). Fr the tw-truck lading, the critical transversal truck psitin is the uter-inner cnfiguratin, as shwn in Fig. 7(e). The differential deflectin between girders ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/) 11
DOI:1.4186/ej.13.17.1.95 G and G3 is higher than that between girders G1 and G. The effect f each bridge parameter will be discussed in the fllwing sectins. 3.1. Effects f Span Length Figure 8 shws the maximum differential deflectins between adjacent girders as a functin f bridge span length. Obviusly, the differential girder deflectin increases as the bridge span length increases. Figure 8 shws the results fr the case f ne-truck lading. At any bridge span length, the maximum differential deflectin between girders G1 and G (under laded lane) is almst identical t that between girders G and G3. The maximum differential deflectin ccurs when the truck is n the left edge f the left lane (uter f the lane). Figure 8 shws the results fr the maximum differential deflectin in the case f tw-truck lading. The critical transverse truck psitin is uter-inner cmbinatin. In the case f tw trucks, the differential deflectin between girders G and G3 is higher than that between girders G1 and G. Cmparing Fig. 8 with Fig. 8, it is apparent that the differential displacement under ne-truck lading is mre critical than that under tw-truck lading. Relative displacement between adjacent girders (mm) 1 8 6 4 16 18 4 6 8 3 3 34 36 38 Bridge length (m) G1 - G Relative displacement between adjacent girders (mm) 1 G1 - G 8 6 4 16 18 4 6 8 3 3 34 36 38 Bridge length (m) Fig. 8. Effect f bridge span length n differential deflectin ne truck tw trucks. 3.. Effects f Girder Spacing Figure 9 shws that the maximum differential deflectins between adjacent girders as a functin f the girder spacing. As the girder spacing increases, the length f the diaphragm increases, thus, reducing the diaphragm stiffness and resulting in the increase f differential deflectin. Fr the case f ne-truck lading, the difference between differential deflectins f girders G1 and G (under laded lane) and girders G and G3 is insignificant, as shwn in Fig. 9. Relative displacement between adjacent girders (mm) 5 4 3 1...4.6.8 3. 3. Girder spacing (m) G1 - G Relative displacement between adjacent girders (mm) 5 4 3 1...4.6.8 3. 3. Girder spacing (m) Fig. 9. Effect f girder spacing n differential deflectin ne truck tw trucks. G1 - G 1 ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/)
DOI:1.4186/ej.13.17.1.95 In the case f tw trucks, as shwn in Fig. 9, the differential displacement between G and G3 is higher than that between G1 and G. Again, the maximum differential deflectin under tw-truck lading is lwer than that under ne-truck lading. 3.3. Effects f Slab Thickness Figure 1 shws that the differential deflectin decreases as slab thickness increases and cmparisn f bth lading cases shws that ne truck-lading is mre critical than tw-truck lading. Increasing the slab thickness increases the rigidity f slab; therefre decreasing the deflectin f the girders. The slab imprves the lateral lad distributin and integrity f the girder system. Figure 1 shws that the differential deflectins between girders under laded and unladed lanes are similar in the case f ne-truck lading. In the case f tw trucks, the differential deflectin between girders G and G3 is higher than that between girders G1 and G, as shwn in Fig. 1. Relative displacement between adjacent girders (mm) 5 4 3 1 4 6 8 3 3 Slab thickness (mm) G1 - G Relative displacement between adjacent girders (mm) 5 4 3 1 4 6 8 3 3 Slab thickness (mm) Fig. 1. Effect f slab thickness n differential deflectin ne truck tw trucks. 3.4. Effects f Girder Stiffness G1 - G As shwn in Fig. 11, the differential deflectin between adjacent girders decreases as the girder stiffness increases. In the case f ne-truck lading, Fig. 11 shws that the differential deflectins between adjacent girders are similar. Fr the tw-truck lading, Fig. 11 shws that the uter-inner case is a critical case, and the differential deflectin between girders G and G3 is higher than that between girders G1 and G. One-truck lading causes higher differential deflectin between adjacent girders than tw truck-lading. Relative displacement between adjacent girders (mm) 5 4 3 G1 - G 1 1.e+1 1.1e+1 1.e+1 1.3e+1 1.4e+1 1.5e+1 1.6e+1 1.7e+1 Girder stiffness (mm 4 ) Relative displacement between adjacent girders (mm) 5 4 3 1 1.e+1 1.1e+1 1.e+1 1.3e+1 1.4e+1 1.5e+1 1.6e+1 1.7e+1 Girder stiffness (mm 4 ) Fig. 11. Effect f girder stiffness n differential deflectin ne truck tw trucks. G1 - G ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/) 13
DOI:1.4186/ej.13.17.1.95 3.5. Ranking f Influencing Bridge Parameters Figure 1 plts the percentage change f differential deflectin between adjacent girders with the change f each bridge parameter relative t the base case value. This facilitates ranking bridge parameters n their influence n the differential girder deflectin. Psitive slpe means the differential deflectin increases with the increase f bridge parameter and vice versa. Steeper line (larger abslute value f slpe) means that the differential deflectin between adjacent girders is mre sensitive t the change f the parameter thus the effect f the crrespnding parameter is larger, and vice versa. Fr ne-truck lading case shwn in Fig. 1, the slpes frm linear regressin analysis shw that the mst influential bridge parameter is the bridge span length fllwed by girder spacing and slab thickness, while girder stiffness is the least influential ne amng the fur bridge parameters. Similarly, fr tw-truck lading case shwn in Fig. 1, the differential deflectin between adjacent girders is mst sensitive t the change in bridge span length fllwed by girder spacing, slab thickness, and girder stiffness. In Figs. 1 and 1, similar influence f bridge parameters fr bth truck lading cases is bserved. The effects f bridge parameters n differential deflectin between girders G1 and G and that between girders G and G3 are almst identical, except fr the effect f bridge span length under tw truck-lading in which the slpe crrespnding t G1-G is slightly larger than that crrespnding t G-G3. Fr the effect f girder spacing, the regressin line slpe in Fig. 1 is slightly shallwer than that in Fig. 1. Therefre, the effect f girder spacing n differential deflectin is mre ntable fr tw-truck lading case than fr ne truck-lading case. Cnsequently, the bridge parameters affecting differential deflectin between adjacent girders in descending rder are the bridge span length, girder spacing, slab thickness, and girder stiffness. Change f relative girder deflectins (%) 15 1 5-5 G1 - G effect f girder spacing effect f bridge length effect f girder stiffness effect f slab thickness -1-3 -1 1 3 5 Change f bridge parameters (%) Change f relative girder deflectins (%) 15 1 5-5 G1 - G effect f bridge length effect f girder stiffness effect f girder spacing effect f slab thickness -1-3 -1 1 3 5 Change f bridge parameters (%) Fig. 1. Ranking f bridge parameters fr differential deflectin ne truck tw trucks. 4. Effects f Bridge Parameters n Distrtin-Induced Web Gap Stresses By applying the differential deflectin calculated frm the glbal mdel t the refined finite element mdel, the sub-mdel, the peak web gap stress can be investigated. Figure 13 shws the midspan transverse sectin f the bridge befre and after defrmatin under ne-truck lading. A typical vertical stress cntur in the web gap area, as shwn in Fig. 14, demnstrates that the stress field decays rapidly in bth lngitudinal and vertical directins away frm the end f stiffener. It is bserved frm stress distributin in the web gap area fr all sub-mdels that the critical vertical stress ccurs either at the tp f web gap near the end f stiffener r at the bttm f web gap near bttm flange, as shwn in Fig. 14. The web gap stress is then taken as the larger value frm these tw psitins. 14 ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/)
DOI:1.4186/ej.13.17.1.95 G3 G G1 Fig. 13. Midspan transverse sectin undefrmed shape defrmed shape. Girder tp flange Critical web gap stress psitins Stiffener Fig. 14. Web gap vertical stress distributin typical stress cntur critical psitins. 4.1. Effects f Span Length Girder bttm flange Figure 15 shws the critical web gap stress as a functin f bridge span length. Obviusly, the web gap stress increases as the bridge span length increases. In the case f ne-truck lading, the critical web gap stress ccurs at the end f stiffener f girder G, while in the case f tw-truck lading, the critical web gap stress ccurs in the area near the bttm flange f girder G3. Similar findings are bserved fr ther bridge parameters. Fr the case f ne truck, the maximum web gap stresses in girders G and G1 are higher than that in girder G3. Because the differential deflectin between girders G and G3 is larger than that between girders G1 and G, the web gap stresses in G and G3 are higher than that in G1 fr the case f tw trucks. Similar t the differential deflectin, the critical vertical web gap stress resulted frm ne-truck lading is higher than that frm tw-truck lading. Critical web gap vertical stress (MPa) 14 1 1 8 6 4 1 Girder 1 Girder Girder 3 Critical web gap vertical stress (MPa) 14 1 8 6 4 Girder 1 Girder Girder 3 - - 16 18 4 6 8 3 3 34 36 38 16 18 4 6 8 3 3 34 36 38 Bridge length (m) Bridge length (m) Fig. 15. Effect f bridge span length n distrtin-induced stress ne truck tw trucks. ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/) 15
DOI:1.4186/ej.13.17.1.95 4.. Effects f Girder Spacing Figure 16 shws the maximum web gap stress as a functin f girder spacing. Figure 16 shws that the increase f maximum web gap stress is minimal as the girder spacing increases. This implies that the rtatin f diaphragm, i.e., the differential deflectin divided by girder spacing, remains almst the same fr different girder spacing values under ne-truck lading. This bservatin is in accrdance with Jajich et al. [7, 11] where the web gap stress is determined by the rtatin f diaphragm instead f differential deflectin. Fr the tw-truck lading case, the maximum web gap stress increases as the girder spacing increases, as shwn in Fig. 16, and the critical web gap stress in G and G3 are higher than that in G1. Overall, the critical web gap stress caused by ne-truck lading is higher than that caused by tw-truck lading. 5 5 Critical web gap vertical stress (MPa) 4 3 1-1...4.6.8 3. 3. Girder spacing (m) Girder 1 Girder Girder 3 Critical web gap vertical stress (MPa) 4 3 1...4.6.8 3. 3. Girder spacing (m) Fig. 16. Effect f girder spacing n distrtin-induced stress ne truck tw trucks. 4.3. Effects f Slab Thickness Girder 1 Girder Girder 3 As shwn in Fig. 17, the maximum web gap stress decreases as slab thickness increases. Fr ne-truck lading, Fig. 17 shws that the maximum web gap ccurs in G. As the differential deflectin between G and G3 is higher than that between G1 and G, the web gap stress in the G and G3 are higher than that in G1, as shwn in Fig. 17 fr tw-truck lading. Cmparisn between tw lading cases shws that the vertical web gap stress under ne-truck lading is mre critical than tw-truck lading. Critical web gap vertical stress (MPa) 6 5 4 3 1 Girder 1 Girder Girder 3 Critical web gap vertical stress (MPa) 6 5 4 3 1 Girder 1 Girder Girder 3 4 6 8 3 3 4 6 8 3 3 Slab thickness (mm) Slab thickness (mm) Fig. 17. Effect f slab thickness n distrtin-induced stress ne truck tw trucks. 16 ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/)
DOI:1.4186/ej.13.17.1.95 4.4. Effects f Girder Stiffness As the main girders becme stiffer, bth differential deflectin between adjacent girders and maximum web gap stress decrease, as shwn in Fig. 18. The web gap stress is further decreased as the web gap area becmes stiffer since the thickness f bth web and flanges are increased with the increase f girder stiffness in the parametric study. Fr ne-truck lading case, the maximum web gap stress in G is the largest fllwed by G1 and G3. Figure 18 shws the results fr the case f tw-truck lading. The maximum web gap stresses in G and G3 are identical and higher than G1. This can be attributed t the fact that the differential deflectin between G and G3 is larger than that between G1 and G. Mrever, the critical web gap stress resulted frm ne-truck lading is higher than that frm tw-truck lading. Critical web gap vertical stress (MPa) 8 6 4 Girder 1 Girder Girder 3 Critical web gap vertical stress (MPa) 8 6 4 Girder 1 Girder Girder 3 1.e+1 1.1e+1 1.e+1 1.3e+1 1.4e+1 1.5e+1 1.6e+1 1.7e+1 1.e+1 1.1e+1 1.e+1 1.3e+1 1.4e+1 1.5e+1 1.6e+1 1.7e+1 Girder stiffness (mm 4 ) Girder stiffness (mm 4 ) Fig. 18. Effect f girder stiffness n distrtin-induced stress ne truck tw trucks. 4.5. Ranking f Influencing Bridge Parameters Similar t the ranking f bridge parameters n the differential girder deflectin described in Sectin 3.5, the parameters are nrmalized by relative percentage f change frm the base case value. The effects f changing these parameters n the change f maximum web gap stresses are shwn in Fig. 19 and Fig. 19 fr ne-truck and tw-truck lading cases, respectively. Change f maximum web gap stress (%) 15 1 5-5 effect f girder spacing G1 G G3 effect f bridge length effect f slab thickness effect f girder stiffness -1-3 -1 1 3 5 Change f bridge parameters (%) Change f maximum web gap stress (%) 15 1 5-5 effect f girder spacing G1 G G3 effect f bridge length effect f slab thickness effect f girder stiffness -1-3 -1 1 3 5 Change f bridge parameters (%) Fig. 19. Ranking f bridge parameters fr distrtin-induced web gap stress ne truck tw trucks. Sme lines in Fig. 19 can be best fitted by a quadratic plynmial, hwever, fr the purpse f ranking the sensitivity f web gap stress t the bridge parameters, linear regressin analysis (with R nt less than.9) are adpted. Fr the case f ne-truck lading, the mst influential parameter affecting web gap stress ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/) 17
DOI:1.4186/ej.13.17.1.95 is the bridge span length, fllwed by girder stiffness, slab thickness, and girder spacing. Unlike the differential girder deflectin, the maximum web gap stress is mre sensitive t the change in girder stiffness. In the finite element analysis, the variatin in girder stiffness is accmplished by varying thickness f girder flanges and web. Fr the case f tw-truck lading, the mst influencing bridge parameter is the bridge span length fllwed by girder spacing, girder stiffness, and slab thickness. Table summarizes the bridge parameter rankings fr their influence n the differential girder deflectin and critical web gap stress fr ne-truck and tw-truck lading cases. Table. Summary f bridge parameter rankings n differential girder deflectin and web gap stress. Degree f Differential Deflectin Critical Web Gap Stress Influence One Truck Tw Trucks One Truck Tw Trucks mst influential Span length Span length Span length Span length Girder spacing Girder spacing Girder stiffness Girder spacing Slab thickness Slab thickness Slab thickness Girder stiffness least influential Girder stiffness Girder stiffness Girder spacing Slab thickness Distrtin-induced stress is influenced by differential deflectin between adjacent girders [1]. Frm Sectin 3 and 4, bth differential deflectin and web gap stress increase with increasing span length and girder spacing, while decrease with increasing slab thickness and girder stiffness. Hwever, the effects f each bridge parameter n differential deflectin and web gap stress are nt cnsistent. Fr differential deflectin, the bridge span length is the mst influential parameter fllwed by the girder spacing, slab thickness and girder stiffness. While girder stiffness is the least influential parameter fr differential deflectin, it plays a mre imprtant rle n the critical web gap stress as the web thickness is changed implicitly in the finite element mdel. In additin, the girder spacing may nt affect the critical web gap stress under ne-truck lading as lng as the rtatin f diaphragm remains almst cnstant. Results frm sub-mdel finite element analysis shw that the web gap defrms as a duble curvature shape. Bth rtatin and lateral displacement ccur at bth web gap ends. Therefre, Eqs. (1) and () that cnsider nly rtatin r lateral displacement f the web gap ends can be versimplified. 5. Cnclusins Dual-level finite element analyses incrprating bth glbal mdel, which encmpasses the entire bridge superstructure, and sub-mdel, which encmpasses a prtin f bridge superstructure surrunding the studied stiffener, are perfrmed t investigate the effects f varius bridge parameters n the differential girder deflectins and distrtin-induced web gap stresses. The chsen bridge parameters include the bridge span length, girder spacing, slab thickness, and girder stiffness. Frm the parametric study, it can be cncluded as fllws: The differential girder deflectin increases as the bridge span length and girder spacing increase. In cntrast, it decreases with increasing slab thickness and girder stiffness. Fr ne-truck lading, the maximum differential deflectins between adjacent girders under laded lane and unladed lane are similar. Fr tw-truck lading, the critical truck transverse psitin is the uter-inner cmbinatin and the critical differential deflectin ccurs between girders G and G3. The differential deflectin between adjacent girders caused by ne-truck lading is mre critical than that caused by tw-truck lading. The mst influential bridge parameter affecting the differential girder deflectin is the bridge span length fllwed by girder spacing and slab thickness. The girder stiffness is the least influential ne amng the fur bridge parameters. The maximum vertical web gap stress increases as the bridge span length and girder spacing increase, while it decreases with increasing slab thickness and girder stiffness. Under ne-truck lading, the maximum web gap stress ccurs at the interir girder (G). Under twtruck lading, the critical girders are G and G3. The web gap stress caused by ne-truck lading is mre critical than that caused by tw-truck lading. 18 ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/)
DOI:1.4186/ej.13.17.1.95 The mst influential parameter affecting the web gap stress is the bridge span length fllwed by girder stiffness, slab thickness, and girder spacing. Unlike the differential deflectin, the maximum web gap stress is mre sensitive t the change in girder stiffness. The differential deflectin is an imprtant factr affecting the distrtin-induced stress but the effect is nnlinear. Other parameters such as web thickness and web gap length shuld be cnsidered explicitly in rder t establish the relatinship between the distrtin-induced stress and differential girder deflectin. Acknwledgements The authrs wuld like t acknwledge the AUN/SEED-Net prgram fr the financial supprt t the study. The secnd authr wuld als like t acknwledge the China schlarship cuncil fr the supprt during the visit at Department f Civil Engineering, Chulalngkrn University. References [1] J. W. Fisher, J. Jin, D. C. Wagner, and B. T. Yen, Distrtin-induced fatigue cracking in steel bridges, NCHRP Rep. 336, Transprtatin Research Bard, Washingtn, D.C., 199. [] American Assciatin f State Highway and Transprtatin Officials, AASHTO LRFD Bridge Design Specificatins, 4th ed., Washingtn DC: AASHTO, 7. [3] T. E. Cusins, J. M. Stallings, D. A. Lwer, and T. E. Staffrd, Field evaluatin f fatigue cracking in diaphragm-girder cnnectins, J. Perfrm. Cnstr. Fac., vl. 1, n. 1, pp. 5-3, Feb, 1998. [4] Y. Zha, and W. M. K. Rddis, Fatigue crack investigatin fr the Arkansas river bridge in Hutchinsn, Kansas, Cnstr. Build. Mater., vl. 14, n. 5, pp. 87-95, Jul,. [5] K. Nishikawa, J. Murakshi, and T. Matsuki, Study n the fatigue f steel highway bridges in Japan, Cnstr. Build. Mater., vl. 1, n. -3, pp. 133-141, Mar, 1998. [6] R. J. Cnnr, J. W. Fisher, Identifying effective and ineffective retrfits fr distrtin fatigue cracking in steel bridges using field instrumentatin, J. Bridge Eng., vl. 11, n. 6, pp. 745-75, Jun, 6. [7] D. Jajich, and A. E. Schultz, Measurement and analysis f distrtin-induced fatigue in multi-girder steel bridges, J. Bridge Eng., vl. 8, n., pp. 84-91, Mar, 3. [8] H. Li, A. E. Schultz, Analysis f girder differential deflectin and web gap stress fr rapid assessment f distrtinal fatigue in multigirder steel bridges, Rep. Mn/RC-5-38, Department f Civil Engineering, University f Minnesta, Minneaplis, MN, 5. [9] W. M. K. Rddis, and Y. Zha, Finite element analysis f steel bridge distrtin-induced fatigue, J. Bridge Eng., vl. 8, n. 5, pp. 59-66, Sep, 3. [1] E. M. Berglund, and A. E. Schultz, Girder differential deflectin and distrtin-induced fatigue in skewed steel bridges, J. Bridge Eng., vl. 11, n., pp. 169-177, Mar, 6. [11] D. Jajich, A. E. Schultz, P. M. Bergsn, and T. V. Galambs, Distrtin-induced fatigue in multigirder steel bridges, Rep. MnDOT -16, Department f Civil Engineering, University f Minnesta, Minneaplis, MN,. [1] A. Lenwari, T. Senjuntichai, T. Pinkaew, T. Thepchatri, and E. Limsuwan., Field investigatin n slab-n-gider steel bridges ver intersectins in Thailand, Prc. The 33 rd IABSE Sympsium, Bangkk, Thailand, September 9-11, 9, pp. 1-9. [13] Cmputers and Structures, SAP Getting Started, versin 7.4, Berkeley, CA: Cmputer and Structure Inc.,. ENGINEERING JOURNAL Vlume 17 Issue 1, ISSN 15-881 (http://www.engj.rg/) 19