Structural Analysis and Response of Curved Prestressed Concrete Box Girder Bridges

Transcription

1 72 TRANSPORTATON RESEARCH RECORD 118 Strctral Analysis and Response of Crved Prestressed Concrete Box Girder Bridges DEEPAK CHOUDHURY AND ALEX C. ScoRDELS A nmerical procedre for linear-elastl.c analysis and nonlinear material analysis of crved prestressed concrete box girder bridges s demonstrated throgl) two examples. A crved nonprlsmatk thin-walled box beam element s sed to model the bridges. The cross section of the element s a rectanglar single-cell box with side cantilevers. Elght djsplacement degrees of freedom, lncldlng transverse distortion and longltdlnal warping of the cross section, are considered at each of the three element nodes. Prestresslng, conslstlng of posttensloned bonded tendons n the longitdinal direction, fs considered. For nonlinear materjal analysis, the nlaxjal stress-strain crves of concrete, reinforcing steel, and prestressing steel are modeled. The shear and the transverse ftexral responses of the box beam cro.ss section are modeled sing trlllnear constittive relationships based on cracking, yielding, and ltimate stages. The first example demonstrates the versatility of the nmerical method in determining the linear-elastic distribtion of forces in a three-span prestressed box girder bridge of crved plan geometry and variable cross section. Dead load, live load, and prestressing load cases are analyzed. n the second example, overload behavior and ltimate strength of a three-span crved prestressed concrete box girder bridge nder 1.ncreaslng vehiclar load are nvestigated. The dlfferent response characteristics or the bridge indced by different transverse locations of the overload vehicle are presented. Prestressed concrete box girder bridges have gained importance as economic and esthetic soltions for the overpasses, nderpasses, separation strctres, and viadcts fond in today's highway system. These bridges are often continosspan strctres (Figre la). Transverse diaphragms are placed at the end and interior spport sections, and additional diaphragms are sometimes sed between the spports. The typical cross section of sch a box girder bridge consists of a top slab and a bottom slab connected monolithically by vertical webs to form a celllar or boxlike strctre (Figre lb). Design and esthetic considerations often call for longitdinal variations in the cross-sectional dimensions. n plan, the bridge can have a straight or crved geometry. Sometimes part of the bridge may be straight and part of it may be crved (Figre le). The complex spatial natre of a crved box girder bridge with a variable cross section (nonprismatic) makes it difficlt to predict the strctral response to a general loading D. Chodhry, T. Y. Lin nternational, 315 Bay Street, San Francisco, Calif A. C. Scordelis, Department of Civil Engineering, University of California, Berkeley, Calif END DAPHRAGM NTEROR DAPHRAGM (a) ELEVATON () (2) (3) (4) (5) (b) CROSS SECTONS (c) PLAN FGURE 1 Crved nonprismatk box girder bridge. case accrately. The presence of prestressing frther complicates the analysis. Even with the assmption of homogeneos linear-elastic material, the accrate analysis of sch a strctre remains a formidable challenge to the engineer. Highway bridges are being sbjected to increasing vehiclar loads and traffic densities. A better nderstanding of the overload behavior of these bridges beyond the service load range is necessary. Also, in order to assess the degree of safety against failre, an accrate estimate of the ltimate load has to be made. A nonlinear analysis procedre incorporating the effects of nonhomogeneity of the material, concrete cracking, and nonlinearities in the stress-strain relationships of concrete reinforcing steel and prestressing steel is ths reqired. The objective of this paper is to demonstrate the capabilities of a nmerical procedre for linear-elastic analysis and nonlinear material (5)

2 Chodhry and Scor.Ulis analysis of crved nonprismatic prestressed concrete box girder bridges. The overall response of the strctre, rather than the local behavior, is stdied METHOD OF ANALYSS The method of analysis employed in the present stdy ses a finite-element formlation based on thin-walled beam theory (1, 2). The theoretical details, smmarized briefly in this paper, have been presented elsewhere (3). Crved Nonprlsmatlc Box Beam Element Thin-walled box beam elements have been sed by several investigators (4-6) for linear-elastic analysis of box girder bridges. These elements are capable of captring the dominant strctral actions, bt at considerably redced comptational effort. The simplicity and redced comptational effort inherent in a beam-type element make it particlarly sitable for nonlinear analysis, which reqires mch greater central processing nit time and storage space in the compter than linear-elastic analysis. Ths in the present stdy, certain aspects of the formlations sed by Bazant and El Nimeiri (4) and by Zhang and Lyons (5) are combined to develop a crved nonprismatic thin-walled box beam element that can be easily extended to nonlinear analysis. The three-node element is shown in Figre 2. ts axis lies in the global X-Y plane and may be crved The cross section of the element perpendiclar to its axis is a rectanglar, singlecell, thin-walled box with side cantilevers. The dimensions of the box cross section, indicated in Figre 2b, can all vary along the length of the element. The geometry of the element is defined sing an isoparametric mapping between the three nodes. The shape fnctions associated with the three nodes are second-order Lagrangian polynomials. At each element node, eight generalized displacement degrees of freedom are considered. These are the sal six degrees of freedom associated with the rigid body translations and rotations of the cross section and, in addition, a longitdinal warping mode and a transverse distortional mode of the cross section (Figre 3). A generalized strain and its associated generalized stress are sed to represent the strain energy contribtion from transverse bending of the walls of the box cross section cased by the transverse distortional mode (Figre 3b). Also, the longitdinal normal strains and stresses and the shear strains and stresses acting in the plane of the cross section along the walls of the box are monitored. 73 b = = + l/\j z NODEi = - LEFT (a) SOMETRC VEW CANTLEVER TOP FLANGE RGHT CANTLEVER \ t, / bf 1 -' b, )'. -., / ,.--l LEFT WEB 1, \ RGHT WEB J a 12 BOTTOM FLANGE a / 2 f '" ".J..:..:=.= H f 1-= - t \ (a) UNT LONGTUDNAL WARPNG \ ' \' - r \' t. " -- \ h, H ' ' ' : ' "' t H -----;.'b 1-f h, h: a 2 h, a 2 o, (b) CROSS SECTON FGURE 2 Crved nonprlsmatlc thin-walled box beam element. lb) UNT TRANSVERSE DSTORTON FGURE 3 Longitdinal warping and transverse dlstortlonal modes.

3 74 The element property matrices are obtained nmerically sing two-point Gassian qadratre along the element axis. This is fond to eliminate the sprios shear stiffness sally associated with beam and shell formlations, inclding shear deformations. Sprios shear effects relative to the torsional, distortional, and warping degrees of freedom are also eliminated by this redced order of integration. Prestresslng Prestressing, by means of posttensioned tendons in the longitdinal direction, is considered. The tendons are idealized as straight prestressing steel segments between the nodes of the box beam elements. Friction and anchorage slip losses are considered. For linear-elastic analysis of the strctre nder prestressing, the effect of prestress is represented by a set of eqivalent loads applied at the nodes of the box beam elements. For the analysis at transfer of prestress in a nonlinear analysis, the prestressing is similarly represented by a set of CONCRETE FLAMENT i; d H j *"... EEHY tst"flum<n' CONCRETE LAYER - -p F fr4. g. + 1g FGURE 4 Box cross section dlscretl:zed nto concrete and longitdinal reinforcing steel filaments. TRANSPORTATON RESEARCH RECORD 118 eqivalent nodal loads, and the strctre is analyzed as if it were of ordinary reinforced concrete. The contribtion of the prestressing steel to the overall strctral stiffness is neglected becase at this stage the steel is nbonded. For the sbseqent application of external loads, the prestressing steel is assmed to be bonded to the concrete (i.e., groted), and the prestressing steel stiffnesses are inclded in the overall strctral stiffness. Nonlinear Analysis Procedre For nonlinear material analysis, the box beam elements are considered to be concrete elements reinforced with steel in the longitdinal and transverse directions. n order to model the response of the reinforced concrete box beam element nder longitdinal normal strains and stresses, the element cross section is idealized as concrete and longitdinal steel filaments that are assmed to be in a state of niaxial stress (Figre 4). The niaxial stress-strain crve adopted for the concrete filaments is based on the widely sed relationship in compression sggested by Hognestad (7), together with cracking in tension at the tensile strength. With additional assmptions for load reversal, a total of 11 possible material states are obtained (Figre Sa). For the longitdinal steel filaments, a bilinear stress-strain relationship with load reversal is sed (Figre Sb). For different material states are possible. n addition, the tension-stiffening effect between the concrete and the steel filaments is modeled with a "steel-referred" method. The trilinear constittive relationship shown in Figre Sc is sed to represent the shear response of the walls of the box section. The model incorporates the cracking, yielding, and ltimate stages. The postcracking behavior is based on the 4Sdegree trss analogy. The possibility of load reversal is considered, which gives a total of five different material states., (a CONCRETE ( bl RENFORCNG STEEL ' t;. \ J.. ' J ) -y,. / 'Y,LJG. ",,, 2 a ( c l SHEAR FGURE S Nonlinear constittive models. ( d i PRESTRESSNG STEEL

4 Clwdhry and Scordelis The constittive relationship for transverse flexre is obtained from a flexral analysis of the box beam cross section as a transverse reinforced concrete frame of nit width in the longitdinal direction. The reslting response is modeled with a trilinear relationship similar to the one sed for shear (Figre 5c). The niaxial stress-strain relationship of the prestressing steel segments is modeled by sing a mltilinear idealization with load reversal (Figre 5d). Becase of the nonlinearities in the constittive relationships, the eqilibrim eqations of the strctre are nonlinear. These nonlinear eqilibrim eqations are solved in increments. ncrements of either prescribed loads or displacements are applied, and for each increment nbalanced load iterations are performed ntil certain predefined convergence criteria are satisfied. With this nonlinear analysis procedre, the strctral response can be traced throghot the elastic, inelastic, and ltimate load ranges. COMPUTER PROGRAMS The method of analysis has been incorporated into two compter programs: LAPBOX for linear-elastic analysis, and NAPBOX for nonlinear material analysis. The basic inpt for both programs consists of strctre geometry and bondary conditions, material properties, prestressing data, strctre loading, and locations for stress otpt. n addition, NAPBOX reqires data on concrete and longitdinal steel filaments, transverse steel, and convergence tolerances. The varios generation schemes implemented in the programs allow accrate modeling of complex bridge geometries, loadings, and pre- stressing tendon profiles on the basis of a few simple inpt parameters. The compter programs LAPBOX and NAPBOX have been verified (3). Linear-elastic soltions from LAPBOX were compared with the folded-plate elasticity method (8), the finite-strip method (9), and the experimental reslts from a tapered box girder (1). NAPBOX reslts were compared with the compter program PCFRAME (J) for the special case of planar loading. PCFRAME, which is capable of performing nonlinear analysis of planar prestressed concrete frames, has been extensively tested (JJ, 12). Detailed information on the theoretical basis, inpt and otpt, and a compter tape containing the sorce listings for LAPBOX and NAPBOX is available by writing to the second athor. EXAMPLE 1: LNEAR-ELASTC ANALYSS Example 1 demonstrates the applicability of the proposed method and the capabilities of the compter program LAP BOX in determining the linear-elastic distribtion of forces in a complex prestressed concrete box girder bridge of crved plan geometry and variable cross section. Strctre Details and Analytical Modeling A hypothetical three-span continos prestressed concrete box girder bridge (Figre 6) is analyzed. n plan (Figre 6a), the bridge has a straight span of 14 ft between spports Sl and S2, and two circlarly crved spans of 24 ft between spports S2 and S3 and 18 ft between spports S3 and S4. The radis 75 (STRAGHT) 14 S S2 (a) PLAN!' ' lo 25' r '.< " a. f. h, ; vary (b) CROSS SECTON 1.25 FGURE 6 Example 1: three-span crved nonprismatlc prestressed concrete bridge. (a) Plan view, (b) cross section.

5 76 TRANSPORTATON RESEARCH RECORD 118 of the circlarly crved spans is R = 5 ft to the centerline. For convenience, 11 coordinate.v is defined along the centerline with its origin at S 1 and directed toward S4. At each of the for spports of the bridge, vertical movement, twist, and transverse distortion of the cross section are prevented. The cross section of the bridge is shown in Figre 6b. The following eqations define the variation of the cross-sectional dimensions a, f, b, and 1,, (all in feet) in terms of the s-<:oordinate. A modls of elasticity of 68,256 kips/ft2 and a Poisson's ratio of.18 are assmed for the concrete. a = 12 + s/1 = 4 + 9s/l8 b = 6 + s{to = 52/3 - s/15, =4 = s/15, = 11/9 - s/9 = 3/5 + s/loo = 67 /3 - s/15 = -37/3 + s/15 = 25/9-7s/18 for < s < 56 ft for < s < 56 ft for < s < 14 ft for 14 ft < s < 2 ft for 2 ft < s < 32 ft for 32 ft < s < 38 ft for 38 ft < s < 56 ft for < s < 14 ft for 14 ft < s < 26 ft for 26 ft < s < 38 ft for 38 ft < s < 56 ft The bridge is prestressed with 14 different tendons, 7 in each web (Figre 7). The jacking forces and the geometry data of each tendon are given in Table 1. Each tendon nmbered 1 throgh 6 has a vertical profile along its span consisting of one parabolic segment. Each of the other 8 tendons has two parabolic segments co111ected by a linear segment in the middle. All the tendons are stressed simltaneosly from both ends. A wobble friction coefficient of.2/ft, a crvatre friction coefficient of.2/radian, and no anchorage slip are assmed. Three different load cases are considered They may be smmarized as follows: Case 1: Dead load de to a nit weight density of 16 pcf. Case 2: Prestressing. Case 3: Uniform live load of.15 kip/ft2 over the fll width of the the top deck and between s = 18 ft and 34 ft. TABLE 1 EXAMPLE 1: PRESTRESSNG TENDON DATA TENDON NO. TArKJNr. nrat(w()f oraton OF LOCATON!Sl OF FORCE ENDA END B ZERO TENDON SLOPE (kip) s (ft) z (ft) s(ft) z (ft) ' (ft) z (ft), 2 3-2, ,357 3, , , ; , ; , O 8; 2 --,5 13, ; 44 --,5 The bridge is analyzed by sing 56 crved nonprismatic box beam elements, each 1 ft long. All load cases are represented by eqivalent nodal loads generated atomatically within the program LAPBOX. An additional analysis is performed for load case 3 with the compter program SAP V (J 3) sing 56 one-dimensional (straight) beam elements, each spanning 1 ft (measred along the actal axis of the bridge). Each element is prismatic, so the nonprismatic bridge can only be modeled approximately by sing different cross-sectional properties for each element. Reslts Table 2 gives a smmary of the spport reactions obtained from LAPBOX for all load cases. n the last two colmns, the total of all the vertical reactions is compared with the total applied load calclated by hand from the actal geometry and loading of the strctre. The agreement is perfect, which is indicative of the high accracy with which the box beam elements can be sed to model the geometry and loading of a crved nonprismatic bridge. The spport reactions obtained from the SAP V analysis for load case 3 are also shown in Table 2. The agreement between the LAPBOX and the SAP V reslts. is good, particlarly for the vertical reactions. z t TENDON FGURE 7 Example 1: three-span crved nonprismatic prestressed concrete bridgelongitdinal section along centerline.

6 Chodhry and Scordelis 77 TABLE 2 EXAMPLE 1: SUMMARY OF SUPPORT REACTONS TOTAL TOTAL LOAD FORCE SUPPORT SUPPORT SUPPORT SUPPORT VER11CAL APPLED CASE QUANTTY S S2 SJ S4 REACTON LOAD (kip) (kip) Distonional Moment M, (kip fi) DEAD Vertical LOAD Force ,7 476,7 v, (kip) PRE- Torqe M, (kip ft) Distonioal Moment M, (kip ft) Vertical STRESS Force , v, (kip) Torqe 1.6 -! ,3 - - M, (kip ft) LVE LOAD (LAPBOX) LVE LOAD (SAP JV) Dis1ortional Moment M, (kip fl) Vertical Force -116, v, (kip) Torqe M, (kip ft) Venical For M, (kip) Torqe M, (kip ft) The longitdinal vanallons of the vertical centerline displacements are shown in Figre 8. The plotted vales for the live load case represent both the LAPBOX and the SAP V analyses, which predicted practically the same vertical displacements. The transverse distribtions of the vertical displacements at the middle of the interior span (s = 26 ft) are shown in Figre 9. The vertical displacements are larger at the oter web than at the inner web, with almost a linear variation across the bridge width, which indicates a twist of the cross section with little transverse distortion. Longitdinal variations of bending moments for dead load, prestressing, and live load are shown in Figres 1 throgh 12, a DEAD LOAD.4 PRES TRESSNG "'. 3.2?''\ f "' LVE LOAD ; \ f \.::::. 1 f- z.; \\, UJ ::< UJ <!'....J. -., VJ Ci \ \ J (ft) FGURE 8 Example 1: longitdinal distribtions of vertical centerline displacements.

7 78 respectively. t shold be noted that the moments are with respect to the local y-axes of the cross sections (Figre 6h). When axial loads are present, which is the case with prestressing, the plotted moments will differ from the moments abot the netral axes of the cross sections. The locations of R , TRANSPORTATON RESEARCH RECORD 18 the discontinities in the prestressing moment diagram (Figre 11) corresponct to the: te:ncton encts. The: live: loact mome:nts (Figre 12) obtained from the LAPBOX and the SAP V analyses are fond to agree closely. The close agreement between the LAPBOX and the SAP V reslts for the live load case indicates that the longitdinal flexral behavior of the bridge was not affected significantly by transverse distortion and longitdinal warping of the cross sections. However, ordinary beam theory (SAP V) cannot predict the transverse distribtion of stresses and the transverse flexral moments (not reported), which can be obtained from thin-walled beam theory (LAPBOX). (a) DEAD LOAD (),42 ().4 15 (b) PRESTRESSNG EXAMPLE 2: NONLNEAR ANALYSS Example 2 demonstrates how the proposed nonlinear analysis procedre can be sed to trace the response of a crved prestressed concrete box girder bridge throghot the elastic, inelastic, and ltimate load ranges. Overload behavior and ltimate strength of the bridge nder increasing vehiclar load are investigated. The different response characteristics of the bridge, indced by different transverse locations of the overload vehicle, are presented A , : i [ , (c) LVE LOAD FGURE 9 Example 1: transverse distribtions or vertical displacements (s = 26 ft). R Strctre Detalls and Analytical Modeling The three-span, crved, continos, prestressed concrete box girder bridge considered in this example is shown in Figre 13. ts span arrangement is symmetric, with spans of 16, 2, and 16 ft. The axis of the bridge is circlarly cived in plan with a radis of R = ft to the centerline. Fll restraint against vertical translation, twist, and transverse distortion of the cross section at the for spports is assmed. For the sake of simplicity, the cross section (Figre 13b) is assmoo lo be constant from one end of the strctre Lo the c. f- z -1 UJ ::< ::< (ft) FGURE 1 Example 1: longitdinal distribtion or bending moment M 1 de to dead load.

8 Clwdhry and Scordelis ::: b. -16 "" z "' :::; -2 :::; Z 3 5.\(fl) FGURE 11 Example 1: longitdinal distribtion of bending moment My de to prestressing (7166) LAPBOX xxx.x SAP V (xxxxx).::: b. "" z :::; "' :::; -5-1 \ (-162) 1 Z 3 "(fl) (-1623) 6 FGURE 12 Example 1: longitdinal distribtion of bending moment MY de to live load. o.i1er, althogh design considerations often call for thickening of the bottom flange or the webs, or both, in regions near the spports. The total top deck width of 34 ft is typical of a twolane highway bridge. The prestressing in the bridge consists of two tendons in the longitdinal direction, one in each web. The prestress is achieved by posttensioning from both ends simltaneosly, after which the prestressing steel is groted and hence bonded to the concrete. The tendon profile along the crved webs is the same in each web, and is shown in Figre 13c. The following conditions are assmed: concrete compressive strength of 4, psi, mild steel yield strength of 6 ksi, prestressing steel ltimate strength of 27 ksi, wobble friction coefficient of.2/ft, crvatre friction coefficient of.25/ radian, anchorage slip at each jacking end of.25 in., and nit weight of composite strctre of 155 pcf.

9 8 TRANSPORTATON RESEARCH RECORD 118 y + SUppORr -""t"---' (a) PLAN " < s (b) CROSS SECTON 36+y /+T/'. - - i '., PARABOLA, ' PARABOLA, / '-,.... j.... -<: - - -l. - _'>,., t! (c) PRESTRESSNG TENDON PROFLE FGURE 13 Example 2: three-span crved prestressed concrete bridge. Strctral design of the box girder is based on State of California standard criteria (14). The prestressing steel reqirements are calclated by considering HS2 lane loading in each of the two lanes, impact factors on the basis of span lengths, and an allowable tensile stress of 6 <fc ')112 psi in concrete after all prestressing losses have occrred. A total of 172 seven-wire strands with.5-in. diameter (86 strands per web), which gives a total prestressing steel area of in.2, is provided. The reqired jacking force is 2,66 kips per web. Ultimate strength nder factored loads is checked. n the load factor design, an additional overload vehicle, designated a P13 trck (Figre 14), is considered. Longitdinal mild steel reinforcement corresponding to.3 percent of the concrete area is provided in each wall of the box cross section. This steel is not reqired becase of strength considerations, bt is provided for constrction prposes. The transverse reinforcement consists of two legs of No. 5 stirrps at 4-in. longitdinal spacing in each web and two legs of No. 5 stirrps at 1-in. longitdinal spacing in each slab. The response of the bridge strctre is stdied nder increasing levels of vehiclar overload. The overload vehicle, the P13 trck (Figre 14), is typical of the heaviest vehicles fond on California's highways. n order to take advantage of the symmetry of the strctre, the P13 trck loading in Figre 14b is approximated with the symmetric loading in Figre 14d This symmetric trck loading is positioned in the middle of the center span of the bridge (Figre 15a), and lhi.: strctral load vector de to its weight is increased in increments ntil ltimate failre occrs. For convenience in presentation of reslts, P is sed to denote the factor of trck load applied. Ths P = 1 represents the overload de to one trck (Figre 15a).

10 Chodhry and Scordelis 81 1s 1s 1s 1s 1s 1s ' (al TRUCK ELEVATON r 6k t 48k t48k + 48k r 8k + 48k + 48k lb) TOTAL AXLE LOADS ON BRDGE 1 - o fr=; -Lh - LF. (c) TRUCK SECTON l 48k l 48k 1 48k 1 48k l 48k l 48k l 48k (d) SYMMETRC APPROXMATON USED N ANALYSES FGURE 14 Example 2: P13 trck loading. l P= 48k each 4 HHH 16. 6xl8 16. (a) LONGTUDNAL POSTON OF TRUCK LOAD FOR LOAD CASES. 2 AND 3 l Three different load cases are considered: Case 1: Trck load positioned over bridge centerline (Figre 15b). Case 2: Trck load positioned over oter web (Figre 15c). Case 3: Trck load positioned over inner web (Figre 15d). Becase of the symmetry of the strctre and the loadings, only half the length of the bridge is analyzed. Forty crved box beam elements of varying lengths are sed. The cross section is discretized into 52 concrete and 26 longitdinal steel filaments. (b) TRANSVERSE POSTON OF TRUCK LOAD FOR LOAD CASE Reslts! (c) TRANSVERSE POSTON OF TRUCK LOAD FOR LOAD CASE 2 t. 1---:R l (d) TRANSVERSE POSTON OF TRUCK LOAD FOR LOAD CASE 3 FGURE 15 Example 2: position of trck load for different load cases. Figres 16 throgh 18 show the vertical web displacements at the middle of the center span nder increasing trck loads for load cases 1, 2, and 3, respectively. t is evident from the loaddisplacement crves that the strctre responds qite differently depending on the transverse position of the trck load Load case 1 (Figre 16), with the trck load over the centerline, yielded the highest ltimate load of P = 5.6. Failre occrred becase of the yielding of concrete filaments in compression at the bottom flange over the interior spports. The small differences between the vertical displacements at the two webs indicate that the overall response of the bridge was governed by its longitdinal flexral behavior. n Figre 17, the differences between the vertical displacements at the two webs are seen to be qite significant for load case 2, in which the trck load is positioned over the oter

11 82 TRANSPOK'ATON RESEARCH RECORD '::-. Cl 4 < _, :..: a: -... "' 3 t:; 2 <... S' 1, S' ' DSPLACE MENT (in.) FGURE 16 Example 2: load verss vertical web displacements at middle of center span (load case 1). 6 5 '::-. Cl < _, :..: - "'... "' < s ' DSPLACEMENT (in.) FGURE 17 Example 2: load verss vertical web displacements at middle of center span (load case 2). web. The differential web displacements, which increase with increasing load levels, are primarily de to the twist of the cross section and were cased by rapid deterioration in the torsional stiffness of the bridge. The ltimate load of the bridge was redced to P = 5.2. Failre occrred when the shear stress reached its ltimate vale at the oter web of the sections jst to the inside of the interior spports. An ltimate load of only P = 4. was obtained for load case 3 (Figre 18), with the trck load positioned over.the inner web. This loading prodced rapid deterioration in the

12 Chodhry and Scortk/is <..J ::> 3 <>: 1-.. <>: 2 1- : OUTER WEB S DSPLACEMENT (in,) FGURE 18 (load case 3). Example 2: load verss vertical web displacements at middle of center span transverse flexral rigidity of the bridge. The transverse distortion of the cross section at the middle of the center span increased rapidly, and this is reflected in the increasingly large differences between the vertical displacements at the two webs. Failre was de to the ltimate transverse distortion of the cross section at the middle of the center span, which is governed by the plastic hinge rotation capacities at the for corners of the box section. The cross-section twists at the middle of the center span are shown in Figre 19 for all three load cases. The rapid deterioration in the torsional stiffness of the bridge for load case 2 is evident. Of the three load cases considered, load case 2 represents the most severe torsional loading. The oter web loading tends to twist the cross section in the same direction as the dead load and prestressing, and hence prodces the largest cross-section twists. Load case 1 prodces a less severe 6 5 <..J 4 ::> <>: <>: < 1--R EP ZS TWST B, ( 1o ' radi an ) FGURE 19 cases 1-3). Example 2: load verss cross-section twist at middle of center span (load

13 84 torsional response of the bridge. Becase of the crvatre of the bridge axis, the centerline loading does tend to twist the cross section, bt its effect is not as prononced as that of load case 2. Load case 3 represents the least severe torsional loading. The inner web loading tends to decrease the cross-section twist de to dead load and prestressing. The net reslt is that the cross-section twist remains almost constant with increasing trck load levels. The transverse distortions of the cross section at the middle of the center span are shown in Figre 2 for the three load cases. Load case 3 is now fond to prodce the most severe response. The inner web loading tends to distort the cross section in the same direction as the dead load and prestressing. The combined effect is to case rapid deterioration in the transverse flexral rigidity of the cross section. nitial cracking de to transverse flexre was observed at the middle of the center span at the very first load step of P = 1.. This was followed by transverse flexral yielding at the same location at P = 2.5. Beyond P = 2.5, the transverse distortions increased rapidly as the transverse flexral yielding spread longitdinally. At the ltimate load of P = 4., the yielding spread to abot 82.5 ft from the middle of the center span. The transverse distortions for load cases 1 and 2 are seen in Figre 2 to be mch smaller than those for load case 3. No serios deterioration in the transverse flexral rigidity was observed for either of these load cases. The prestressing steel segment stresses at two critical locations, one near the middle of the center span and the other near the interior spport, are shown in Figre 21 for load cases 1 and 2 and in Figre 22 for load case 3. For each location, the prestressing stresses in both webs are shown. The differences between the initial stresses in the two webs are very small becase the higher wobble friction losses in the oter web 1'RANSPOKl'ATON RESEARCH RECORD 118 more or less balance ot the higher crvatre friction losses in the inner web. The load levels at which initial cracking of concrete filaments was observed at the middle of the center span and at the interior spport are smmarized in Table 3. These cracking loads can be identified in Figres 21 and 22 as the points at which large increases in the tendon stresses occr. The differences between the tendon stresses in the two webs, evident in Figres 21 and 22, provide frther insight into the behavior of the bridge nder the three different loading conditions. A higher tendon stress in one or the other web indicates that a greater proportion of the load is carried by that web. Figre 21a shows that for load case 1, a greater proportion of the load is carried by the inner web near the middle of the center span. Near the interior spport, the oter web carries a greater proportion of the load. However, the small differences between the tendon stresses in the two webs at both locations also indicate that the centerline trck loading is distribted fairly niformly across the width of the bridge. n Figre 2lb, practically eqal proportions of the oter web loading are fond to be carried by the two webs near the middle of the center span. Near the interior spport, the oter web carries a considerably greater proportion of the load. Figre 22 shows that for load case 3, increasingly greater proportions of the inner web loading are carried by the inner web with increasing load levels. At both critical locations, the differences between the tendon stresses in the two webs increase rapidly after the initiation of concrete filament cracking. Near the middle of the center span, the oter web tendon stresses in fact start to decrease near the ltimate, whereas the inner web tendon stresses contine to increase rapidly. The increasing deterioration in the ability of the box section to distribte the eccentrically applied inner web loading transversely is de 6 5 <>. Cl 4 <(...J => 3 "' f-... "' f- <(... 2 R --r \:]'-.. '-> """LOAD CASE TRANSVERSE DSTORTON'" (lo-' radian) FGURE 2 Example 2: load verss transverse distortion or cross section at middle or center span (load cases 1-3).

14 NNER WEB --OUTER WEB NEAR MDDLE OF CENTER SPAN zoo Z1 ZZll Z3 Z'4 Z5 STRESS (ksi) (a) LOAD CASE 6 <C.....J :,;! 3 ::::> X -.. z X t; <C.. NEAR NTEROR SUPPORT NEAR MDDLE OF CENTER SPAN ----NNER WEB -- OUTER WEB Z Z1 ZZ Z3 Z'4 ZSll STRESS (ksi) (b) LOAD CASE 2 FGURE 21 Example 2: load verss prestresslng steel segment stresses at critical locations (load cases 1 and 2). NEAR MDDLE OF CENTER SPAN --OUTER WEB - NEAR NTEROR SUPPORT ii Slil il 2 Zllil ZZ 231il Z<flll STRESS (ksi) FGURE 22 Example 2: load verss prestresslng steel segment stresses at critical locations (load case 3).

15 86 TABLE 3 LOADS SUMMARY OF CRACKNG LUAU N AL t 'KACKN(J NTAL CRACKNG CASE AT MDDLE OF AT CENTER SPAN NTEROR SUPPORT CENTERLN E P 5 p J.4 LOAD OUTER WEB p '"'2 ' J.O LOAD NNER WEB p.) p J. LOAD to the rapid deterioration in the trllllllvcrsc flexral rigidity of the bridge. CONCLUSONS The capabilities and seflness of the proposed analytical method for crved prestressed concrete box girder bridges have been amply demonstrated. The nonlinear analysis procedre is shown to be capable of captring the dominant strctral behavior in the elastic, inelastic, and ltimate load ranges. nternal strains and stresses in concrete, reinforcing steel, and prestressing steel can be determined at all loading stages. Usefl information on overload behavior and ltimate strength can be obtained from sch a detailed analysis. Depending on the strctre and the loading, warping, distortional, and torsional effects can inflence strctral behavior significantly. They not only affect the internal force distribtion within the box strctre, bt also may govern the failre mode and the ltimate load. ACKNOWLEDGMENT This research was performed at the University of California, Berkeley, nder the sponsorship of the U.S.-Spain Joint Committee for Scientific and Technological Cooperation. REFERENCES 1. V. Z. Vlasov. Thin-Walled Elastic Beams. srael Program for Scientific Translations, Jersalem, srael, TRANSPORTATON RESEARCH RECORD R. Dabrowski. Crved Thin-Walled Girders: Theory and Analysis.Springer-Verlag, New York, D. Chodhry. Analysis of Crved Nonprismatic Reinfo1d a11d Prestressed Concrete Box Girder Bridges. Strctral Engineering, Mechanics and Materials Report UCB/SEMM-86/13. University of California, Berkeley, Dec Z. P. Bazant and M. El Nimei.ri. Stiffness Method for Crved Box Girders at nitial Stress. Jornal of the Strctral Division, ASCE, Vol. 1, No. STO, Oct. 1974, l'p S. H. Zhang and L. P. R. Lyons. A Thin-Walled Box Beam Finite Element for Crved Bridge Analysis. Compters and Strctres, Vol. 18, No. 6, 1984, pp M.. Mikkola and J. Paavola. Finite Element Analysis of Box Girders. Jornal of the Strctral Division, ASCE, Vol. 16, No. ST6, Jne 198, pp E. Hognestad. A Stdy of Combined Bending and Axial Load in Reinforced Concrete Members. Blletin Series 399, Blletin 1. University of llinois Engineering Experiment Station, Urbana Champaign, Nov A. DeFries-Skene and A.C. Scordelis. Direct Stiffness Soltion for Folded Plates. Jornal of the Strctral Division, ASCE, Vol. 9, No. ST3, Jne 1964, pp Y.K. Cheng. Finite Strip Method Analysis of Elastic Slabs. Jornal of the Engineering Mechanics Division, ASCE, Vol. 94, No. EM6, Dec. 1968, pp V. Kristek. Tapered Box Girders of Deformable Cross Section. Jornal of the Strctral Division, ASCE, Vol. 96, No. ST8, Ag. 197, pp Y. J. Kang and A. C. Scordelis. Nonlinear Analysis of Prestressed Concrete Frames. Jornal of the Strctral Division, ASCE, Vol. 16, No. ST2, Feb. 198, pp J. Hellesland, D. Chodhry, and A. C. Scordelis. Nonlinear Analysis and Design of RC Bridge Colmns Sbjected lo mposed Deformations. Strctral Engineering and Strctral Mechanics Report UCB/SESM-85/3. University of California, Berkeley, April K. Bathe, E. L. Wilson, and F. E. Peterson. SAP V: A Strctral Analysis Program for Static and Dynamic Response of Linear Systems. Earthqake Engineering Research Center Report EERC University of California, Berkeley, Jne Standard Specifications for Highway Bridges. American Association of State Highway and Tran.sportation Officials, Washington, D.C., 1983, 13th edition. Revised by California Department of Transportation. Pblication of this paper sponsored by Committee on Concrete Bridges.