COMPARISON OF MEASURED AND COMPUTED STRESSES IN A STEEL CURVED GIRDER BRIDGE

Transcription

1 COMPARISON OF MEASURED AND COMPUTED STRESSES IN A STEEL CURVED GIRDER BRIDGE By Theodore V. Galambos, 1 Honorary Member, ASCE, Jerome F. Hajjar, 2 Member, ASCE, Wen-Hsen Huang, 3 Associate Member, ASCE, Brian E. Pulver, 4 Roberto T. Leon, 5 Member, ASCE, and Brian J. Rudie, 6 Associate Member, ASCE ABSTRACT: Steel curved I-girder bridge systems may be more susceptible to instability during construction than bridges constructed of straight I-girders. The primary goal of this research is to study the behavior of the steel superstructure of a curved steel I-girder bridge system during all phases of construction and to ascertain whether the actual stresses in the bridge are represented well by linear elastic analysis software developed for this project and typical of that used for design. Sixty vibrating wire strain gauges were applied to a two-span, four-girder bridge, and elevation measurements were taken by a surveyor s level. The resulting stresses and deflections were compared to computed results for the full construction sequence of the bridge as well as for live loading from up to nine 50-kip trucks. The analyses correlated well with the field measurements, especially for the primary flexural stresses. Stresses due to lateral bending and restraint of warping induced in the girders and the stresses in the cross frames were more erratic but generally showed reasonable correlation. In addition, it is shown that, for the magnitude of live load applied to the bridge, analyses in which composite behavior is assumed in the negative moment region yield better correlation than analyses in which just the bare steel girders are used (no shear connectors were used on the bridge in the negative moment region). It is concluded that the curved girder analysis software captures the general behavior well for these types of curved girder bridge systems at or below the service load level, and that the stresses in these bridges may be relatively low if their design is controlled largely by stiffness. INTRODUCTION Composite, I-shaped, steel curved girder bridges are relatively stiff and strong when the structure is completely erected and subjected to service loading resulting from daily traffic. However, the structure may be quite flexible and potentially susceptible to stability problems during construction, prior to its stabilization after installation of all diaphragms and hardening of the concrete deck. When designing these types of bridges, linear elastic analysis software is typically used to determine the stresses and deflections: (1) Due to dead load just after the casting of the concrete deck, but prior to the deck hardening; and (2) due to live load on the finished bridge structure. To ensure safe design according to the American Association of State Highway and Transportation Officials (AASHTO) provisions [e.g., AASHTO (1994)], it is vital that the stresses and deflections resulting from such analyses be representative of the service-level stress state in the actual bridge structure. This ensures that appropriate stress distributions are used for member design according to either working stress or load and resistance factor design methodologies and that serviceability and fatigue limit states are evaluated accurately. In addition, it is important to know whether the stress state in this type of bridge, at other points in the construction 1 Prof. Emeritus, Dept. of Civ. Engrg., Univ. of Minnesota, Minneapolis, MN Assoc. Prof., Dept. of Civ. Engrg., Univ. of Minnesota, Minneapolis, MN. 3 Struct. Engr., Stanley D. Lindsay & Associates, Ltd., 2300 Windy Ridge Parkway SE, Ste. 200 South, Marietta, GA Struct. Engr., Wiss, Janney, Elstner Assoc., Northbrook, Illinois Prof., School of Civ. and Envir. Engrg., Georgia Inst. of Tech., Atlanta, GA Struct. Engr., Ofc. of Bridges and Struct., Minnesota Dept. of Transp., Roseville, MN Note. Discussion open until January 1, To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on December 10, This paper is part of the Journal of Bridge Engineering, Vol. 5, No. 3, August, ASCE, ISSN /00/ /$8.00 $.50 per page. Paper No process, may be represented computationally to ensure that no unusual stress states occur that are possibly being neglected in current design practice. Curved girder bridge systems exhibit special behavior as compared to bridge systems with straight girders (Hall 1994). Such unique behavior includes, for example, the effect of warping restraint on the behavior of the I-girders, the behavior of cross frames in a curved girder system, and the potential susceptibility of these bridges to lateral-torsional buckling during construction. These added considerations require special care regarding the accuracy of the computational simulations used to obtain forces for design. In spite of this, to date there have been few measurements of actual stresses in these girders recorded during construction (Zureick et al. 1993). In addition, assumptions must be made in all analysis programs regarding the idealization of steel curved I-girder bridge systems. The primary objective of this research was to monitor the strains in the steel superstructure of a two-span curved I-girder bridge during its entire construction process (Pulver 1996) and to compare these field measurements with results obtained from the University of Minnesota Steel Curved Girder Bridge System Analysis Program (referred to herein as the UM program ). This software was developed specifically for this research. Although it is similar to linear elastic software used commonly for design of curved girder bridges, the UM program permits detailed specification of loading and assessment of stress states (Huang 1996). Details of this research are presented in Galambos et al. (1996) and Hajjar and Boyer (1997). Figs. 1 and 2 show plan and section views, respectively, of Minnesota Department of Transportation (MNDOT) Bridge No (Galambos et al. 1996). This bridge includes four concentric I-girders, each of differing depth ranging from 50 to 72 in. In this work, the outside fascia girder is referred to as Beam 1, and the girders are numbered consecutively, with the inside fascia girder referred to as Beam 4. The bridge girders are divided into three segments over two spans, with one central support (Fig. 1). The spans range from 139 to 155 ft each, and the girders are continuous over the center support. The in-plane radius of curvature of the bridge is relatively small, varying from approximately ft. In addition, two of the three supports have substantial skews relative to JOURNAL OF BRIDGE ENGINEERING / AUGUST 2000 / 191

2 FIG. 1. Bridge Framing Plan of Two-Span Curved Steel I-Girder FIG. 2. Superstructure Profiles of Two-Span Curved Steel I- Girder Bridge the longitudinal axis of the bridge at the point of support. Cross frames consisting of a bottom chord (a WT section), a double angle top chord, and double angle X-brace diagonals are welded to gusset plates that are in turn bolted to the I- girder transverse stiffeners at the locations shown in Fig. 1. Stiff I-shaped diaphragms are used at the two end abutments of the bridge in lieu of cross frames. The nominal yield stress of the girders was 50 ksi. Because of the substantial coordination required to measure strains in an actual bridge system, from fabrication through opening of the bridge, only one curved I-girder bridge was included in this research. The conclusions resulting from this research must therefore be interpreted within the context of the specific type of bridge selected. The MNDOT Bridge No is an off-ramp from an interstate highway, and it has a general layout that is similar to that used throughout the country for these types of bridges. In addition, its design was controlled mainly by ensuring adequately small live load deflection (L/800). The construction process went relatively smoothly, as did the field measurements (i.e., there were no substantial anomalies that biased the results), and for these reasons it is felt that the scope of this research is justifiably broad, such that the conclusions are relevant for the type and scale of the curved I-girder bridge system studied in this work. Nevertheless, it is important to recognize that the conclusions drawn in this paper are based upon the results of studying only one bridge. FINITE-ELEMENT MODEL There are several commercial, proprietary, and in-house computer programs available for the analysis of curved girder bridges. Stegmann and Galambos (1976) and Zureick et al. (1993) summarized common methods of analysis for curved steel girder bridges. In addition, several texts present the theory for the development of many of these analytical and computational approaches [e.g., Heins (1975), and Nakai and Yoo (1988)]. Among the several methods available for analysis of curved bridge girder systems, the grillage method, a stiffnessbased finite-element formulation that uses a 2D planar grid model to simulate the 3D effects of bridge superstructures, is used by the UM program (Huang 1996). The curved I-girders were represented by a 3D, two-node curved beam element having 4 degrees of freedom (DOF) at each node. In a curved steel I-girder bridge system, the two translational displacements in the plane of curvature of the bridge and the rotational displacement about the axis perpendicular to the plane of curvature of the bridge are small compared to the out-of-plane displacements and may be neglected. This is the principal idea behind the grillage method, which generally assumes 4 DOF at each node, including a warping degree of freedom. Shear connectors were not used in the negative moment region of all four girders (i.e., in the region of the center pier). Once the concrete deck hardened, composite action of the concrete deck with the four girders was considered in the positive moment region (i.e., in regions with shear studs) during the analysis. For the negative moment region, there was some speculation that friction and adhesion may induce some level of composite action between the concrete deck and the four curved steel girders. Thus, analyses were conducted both with and without composite action in the negative moment region of the bridge (i.e., including the added stiffness due to the longitudinal reinforcing bars in the deck). For all composite action, the effective width of the composite slab was taken as 12 times the thickness of the structural slab (Fig. 2). The ratio of the steel to concrete modulus was taken as either N =6orN = 8. The value of N = 6 was felt to be an appropriate lower bound value of N to investigate and, in fact, generally yielded slightly better correlation with the field measurements (Hajjar and Boyer 1997). Results are presented for both modular ratios. For the curved girder finite element, displacement representations that provide the exact deformation of a planar curved beam, which include a combination of hyperbolic and trigonometric functions, are used in the UM program. The X-bracing cross frames were modeled as trusses consisting of four pinned-end truss elements, for which only axial force was assumed. End diaphragm and transverse grid elements were assumed to use the same type of straight beam finite-element formulation in this analysis. End diaphragms were assumed not to be integral with the concrete slab. Transverse grid elements, used to simulate the transverse behavior of the concrete deck, were assumed as straight beams with an effective width of concrete. They were assumed to be attached on the top of the cross frames, spanning from girder to girder. Although the transverse grid elements and the cross-frame top chords shared the same nodes in the analysis model, they were assumed to deform independently, except for the nodal connectivity at their ends. Four DOF at each joint are generally appropriate for the analysis of curved I-girder bridges, including one translational component normal to the plane of grillage, two rotational components about axes in the plane of grillage, and one additional component to account for the warping effect. However, a fifth and sixth degree of freedom were incorporated into these analyses. For consideration of thermal expansion, vulcanized expansion bearings were used on the bridge. The girders thus were allowed to displace in their longitudinal direction. To better track the structural response of the bridge, the axial de- 192 / JOURNAL OF BRIDGE ENGINEERING / AUGUST 2000

3 gree of freedom was included at each free joint in all analyses conducted with the UM program. Thus, the boundary conditions included pins at the center support and rollers at the abutments; twist was restrained at the abutments and center pier. In addition, the comprehensive restraint of lateral (transverse) displacements throughout the structure provided artificial support at the nodal joints, thus affecting the analysis results in the cross frames. To investigate better the stresses in the cross frames, a sixth degree of freedom was often included in the UM program analyses (i.e., a degree of freedom in the transverse direction of the bridge at each node), but not in the results reported herein (Galambos et al. 1996; Huang 1996). In addition to conducting linear elastic analysis, the UM program accurately assesses the stability of curved girder bridge systems during construction. The bridge modeled in this work was shown to have adequate strength and stiffness using second-order inelastic analysis techniques (Huang 1996). GAUGE PLACEMENT To monitor the strains and stresses induced during the successive construction phases, and to assess the ability of the UM program to model actual behavior, strains were measured during all phases of erection up to the completion of construction. In addition, strains were also measured during two field tests using up to nine trucks with known weight and axle configuration. Sixty gauges were attached to the steel superstructure of the bridge. Fig. 3 shows the position of these gauges. Twenty-four gauges, six per girder, were placed along a section near the midspan of Span 1 (i.e., the southern span) to determine the member stresses occurring in the positive moment region of that span. Span 1 was selected for instrumentation because it was to be erected first. This section along the midspan gauges was labeled as Gauge Line A (Fig. 1), and the gauges were numbered as 1A 24A starting with the outside fascia girder and progressing toward the inside fascia girder. For each girder, four gauges, oriented along the longitudinal axis of the girder, were placed close to the flange tips, with two gauges attached to the bottom surface of the top flange and two gauges attached to the top surface of the bottom flange. The last two gauges were affixed to the girder web, also oriented along the longitudinal axis of the girder approximately 1.5 in. away from each inside flange face. These web gauges are most appropriate for tracking the predominant strong axis flexural strains in the girders, and the gauges at the flange tips also track straining due to lateral bending and resistance to warping. Twentyfour gauges also were placed along a section parallel to the FIG. 3. Locations of Vibrating Wire Strain Gauges and Deflection Measurements skewed middle pier to determine the member stresses occurring in the negative moment region of the girders. This section was labeled as Gauge Line B, and these gauges were numbered from 1B to 24B in the same manner as described for Gauge Line A (Fig. 3). The final 12 gauges were placed on three cross frames spanning the width of the bridge approximately at the midspan of Span 1. A gauge was attached to each diagonal and chord member of the cross frame, oriented longitudinally along each member, to monitor the axial strain present in each of the members. The section through the instrumented cross frames was labeled Gauge Line C (Fig. 3), and the gauges were numbered 1C 12C, beginning with the top chord member connected to the outside fascia girder and increasing in value as one proceeds downward and toward the inside fascia girder. To obtain a set of baseline strain readings that effectively corresponded to a state of zero stress in the girders and cross frames, the strain gauges were attached at the fabrication plant, with the individual girders unassembled and supported on the ground with wood blocks. Following the gauge installation, a set of initial strain readings was recorded for each gauge before the members were lifted from the ground and transported to the construction site. Vibrating wire gauges were used because monitoring strains during fabrication and construction required connecting and disconnecting the gauges from the data acquisition instrument on the system. Electrical (resistance) strain gauges could not be used because large zero shifts would have been unavoidable. Once the girders were delivered to the construction site, a second set of zero readings was taken with the girders on the ground prior to their erection. A comparison of the two initial zero readings resulted in a difference that was generally within 2 3%. Gauges 1A and 11B (Fig. 3) were damaged during the erection and construction of the bridge. Both of these gauges were replaced. The readings after their replacement thus are used only to indicate change in strain from the time of replacement. Displacement measurements were obtained by taking the elevation with a surveying rod and level with an accuracy of 1/16 in. Rotation angles also were obtained at the midspan of Span 2 (Galambos et al. 1996); however, the values of the rotation were so small that they were within the tolerance of the measuring equipment and are not reported. COMPARISON OF FIELD MEASUREMENTS AND FINITE-ELEMENT ANALYSES Field measurements were taken at all key stages of construction [nomenclature relating to dead load stages and steps of construction is given in Galambos et al. (1996), and Huang (1996)]. Nine of the most important stages are reported here (refer to Figs. 1 and 2): 1. Step 1-1: After the erection of Span 1, with cross-frame bolts in place, but loose (shoring towers were in place under each girder of Span 1) 2. Step 1-2: After all bolts on Span 1 were tightened and half of Span 2 was erected (shoring towers were in place under each girder of Span 1) 3. Step 1-3: After all girders and cross frames were erected, with all cross-frame bolts in their tightened position (all shoring towers were permanently removed) 4. Step 2-2: After all deck formwork was in place 5. Step 2-3a: After all deck formwork and steel reinforcement were in place 6. Step 3: During and after casting of the concrete deck (continuous readings were taken during casting of the deck) 7. Step 4-1: After the parapet walls (traffic barriers) were cast JOURNAL OF BRIDGE ENGINEERING / AUGUST 2000 / 193

4 8. Step 4-2: After the deck overlay was cast 9. Step 5: During several different sets of live loading, the bridge was loaded with two 50-kip trucks approximately 1 month after the bridge opened to traffic, and with both three and nine 50-kip trucks approximately 2 years after the bridge opened; the trucks were located in several positions along the bridge for each reading Self-weight of all structural members and the concrete slab were modeled in detail. An additional 20% of the cross-frame self-weight was added to the girder self-weight to account for the stiffeners connecting the cross frames to the webs of the girders. Stresses due to fit up were not modeled, nor were residual stresses. For simplicity, the applied loads due to formwork, concrete casting, and equipment loads were defined in terms of uniformly distributed floor loads. In the grillage method, applied loads can be either concentrated loads input at nodal points or uniformly distributed linear loads input on the girder elements. Accordingly, an appropriate lateral transfer of uniform floor load to each girder of the bridge system must be considered before the structural analysis. Simple distribution factors based on tributary area may be satisfactory if the girders are uniform in their cross sections. However, more rigorous distribution factors are required for a curved I-girder system, because the depth of the I-girder varies from the innermost girder to the outermost girder to provide an adequate superelevation for the curved roadway alignment. The rigorous distribution factors consider the behavior of the bridge cross section as a whole. The distribution factors obtained using both approaches are outlined in Galambos et al. (1996) for the loads due to the bridge deck, the parapets, and the two trucks used for live load testing. The analysis at each stage was conducted sequentially, with stresses due to the incremental load added in a particular stage added to the stresses from the previous stage, so as to model properly the built-in stresses due to the construction sequencing. Stresses in Steel Superstructure Construction Stages 1 and 2 yielded little direct correlation. All measured stresses remained <3 ksi in the girders, with the exception of the stresses in the outside fascia girder at the center support, which had stresses in the flange tips ranging from 5.9 ksi in compression to 7.9 ksi in tension. In Gauge Line B, the bending normal stresses on the webs correlated better than the stresses on the flanges. The flange tip stresses, which are more affected by warping, did not correlate consistently. In Gauge Line C, stresses were generally <3 ksi, but one high stress of 5.9 ksi in Gauge 1C after construction Stage 1 probably resulted from a fit-up constraint. There are several reasons for the discrepancies between measured and computed stresses: The shoring towers were modeled as rigid vertical supports in the analysis, rather than as elastic supports; the loose bolting of the diaphragms at this stage of construction was not modeled; and, most importantly, the loading on the girders was so small that local eccentricities and minor fit-up stresses dominated the measured results. The third set of readings was taken when all the girders were erected in place and all the cross frames were fully tightened (rattled up). All the shoring towers were removed and the steel structure became self-supporting. Better correlations were found at this stage. In both Gauge Lines A and B, the primary flexural bending behavior showed moderate correlation in the girder webs, except for Gauge 15A, in which the values are 1.4 and 0.67 ksi for the computational and the field measurements, respectively (positive stresses are tensile). The stresses in the flanges obtained from the analysis indicated the direction of twist for each girder was in the same direction (i.e., clockwise looking north). However, the prediction of stresses due to warping restraint and lateral bending differed from the measurements. In Gauge Line A, the measured stresses indicated that warping restraint of Beam 1 was similar to the computed results but that the stresses in Beams 2 4 were affected more by lateral bending than by torsional stresses. In Gauge Line B, Beam 1 had stresses due to restraint of warping that were oriented in the same direction as the computed results, but the stresses due to restraint of warping in Beams 3 and 4 were oriented in the direction opposite from the computed prediction. The stresses in the cross frames correlated better in the diagonal members than in the top and bottom chords. Note, however, that the level of stress was still so small in all members that small eccentricities in the field may explain these variations. The significant finding from these readings is that, with shoring towers used, the stresses in the bridge during erection of the steel superstructure were minor, as would be expected. Stresses During Placing of Formwork and Reinforcing Steel The completed steel frame was used as the base structure for the analysis including formwork and deck reinforcing bars. The estimates of the uniform floor loads in this phase was 5 psf for formwork and 10 psf for reinforcing (Galambos et al. 1996). Results are compared here for the stage when all formwork and reinforcement were in place (i.e., the condition of the bridge just before casting of the concrete deck). Better correlation between the computations and the measurements may be found at this stage due to the increased load on the bridge (Fig. 4). However, the primary bending behavior of the girders, as seen in the web gauges, continued to correlate better than the stresses due to warping restraint and lateral bending, as seen in the flange gauges. Little correlation of stresses was found in the cross frames. Note that these stresses correlated best in an average sense, and local anomalies and fit-up stresses still dominate the results at several locations. Results During Casting of Concrete Deck During the casting of the concrete deck, complete readings were taken once every 30 min for the 4 h of the casting. The distributed floor loads during concreting ranged from 15 psf, where no deck had been placed but where nominal construction and/or equipment loads were included, to psf with the inclusion of the deck and a 0 30 psf surcharge to account for the construction and/or equipment loads. The base structure throughout this phase was assumed to be bare steel. FIG. 4. Computed versus Measured Stresses in All Gauges for Step 2-3a (N 8) 194 / JOURNAL OF BRIDGE ENGINEERING / AUGUST 2000

5 The stress comparison shown in Fig. 5 corresponds to readings taken when Span 2 (north span) concreting was just completed and Span 1 was yet to be cast (termed Step 3-1). The distributed floor load was taken as 15 psf on Span 1 and 110 psf on Span 2, where the wet concrete was in place. In this step, stresses were increased by several ksi from their values prior to the casting. Because only one span was loaded by the concrete weight, it may be seen that Gauge Line B picks up stress, and Gauge Line A tends to show some unloading behavior relative to the results in Fig. 4. The correlation of stresses was showing a clear improvement, as the effects of local fit-up stresses were starting to dissipate under the more substantial loading. The correlation continued to improve as the concrete of Span 1 was cast. A set of deflections was also obtained at this loading stage. The initial elevation was surveyed after all diaphragms were fully tightened. After the concrete deck was in place, another leveling was performed to get the vertical deflection near the midspan of each span. Results are tabulated in Table 1. As the concrete casting proceeded into Span 1, composite behavior was developing in Span 2, especially during the 30-min intermission immediately after the passing of the middle pier. Ignoring the composite behavior may have resulted in the overestimate of the deflections on Span 1 as compared with field data. However, with respect to Span 2, it was also found that a temperature gradient of F caused a change in deflection on the order of 0.02 ft in the girders (Rudie 1997). The deflections measured on Span 2, in turn, were taken 5 days later than those measured on Span 1. Thus, a temperature difference (as compared to the initial deflection measurements when the steel was rattled up) could cause these differences in deflection. The deflections of the bridge were studied further by Rudie (1997) for other stages of construction and due to truck live loading, and measured results were found to match analysis results reasonably well using a grillage analysis and exceptionally well using a 3D finite-element analysis of the bridge. Stresses from Parapets and Overlay For modeling the parapets, the estimated vertical loads were computed to be 443 plf for the inner parapet and 578 plf for the outer parapet. Loading of the 2-in. overlay was estimated as a floor pressure of 24 psf over all the bridge. The reading was taken with little construction loading on the bridge. Composite behavior was considered in the analysis. Fig. 6 shows the comparison between analysis results and field measurements using N equal to 8. Due to adding the parapets and overlay, an approximately 3.0-ksi stress increase resulted in the girders. Stress redistribution as compared to previous results also occurred. The outermost cross frame was previously the most highly stressed, but now the other two cross frames picked up more stress. Again, correlation in most of the web gauges was strong. Correlation in the flange gauges was also good, but not as consistent as the web gauges, again showing that the primary bending behavior is more predictable from analysis than the behavior due to warping restraint and minor axis bending behavior. Truck Loading Approximately one month after the bridge was opened for traffic, the bridge was closed for one night, and two loaded trucks weighing approximately 50 kips each were provided by MNDOT to permit live load measurements. A total of 15 static loading cases were conducted, including two trucks side by side at six locations along the bridge, two trucks in line at three locations along the bridge, and a single truck at six locations along the bridge. Results from this live loading were relatively small, on the order of 2 ksi. Consequently, approximately 2 years after the bridge opened, the bridge was again closed to traffic, and nine trucks weighing approximately 50 kips each were used for live load measurements to yield more significant stress measurements, thus allowing a clearer assessment of the correlation between measured and computed stresses. A total of 11 static loading cases were measured, including nine trucks (arranged in three back-to-back groups of three side-by-side trucks) at six locations along the bridge, three side-by-side trucks at four locations along the bridge, and four trucks placed simultaneously at midspan of each of the two spans (Hajjar and Boyer 1997). Results of the truck FIG. 5. Computed versus Measured Stresses in All Gauges for Step 3-1 (N 8) TABLE 1. Deck Deflection Comparison after Pouring of Concrete Deflection (1) Beam 1 (2) Beam 2 (3) Beam 3 (4) Beam 4 (5) (a) Span 1 (South) Composite displacement (ft) Measured displacement (ft) (b) Span 2 (North) Composite displacement (ft) Measured displacement (ft) FIG. 6. Computed versus Measured Stresses in All Gauges for Step 4-2 (N 8) JOURNAL OF BRIDGE ENGINEERING / AUGUST 2000 / 195

6 loading with three and nine trucks are discussed in the next section. Results of loading with two trucks showed similar trends but not as distinctly due to the lighter loads and the resulting influence of local loading anomalies on the correlation. Discussion of Results In Table 2, each gauge is assigned an overall rating indicating its correlation between measured and computed stresses. The table shows four different percent errors for each gauge, Gauge (1) TABLE 2. Gauge Correlation Rating for Measured versus Computer Results (Fully Composite Analysis with N 6) Step 3-3a (2) Step 4-2 (3) Percent Error Average 3 trucks (4) Average 9 trucks (5) Measured stress magnitude a (6) Correlation (7) 1A 71% 1% 3% 5% >5 ( 2.64) Moderate b 2A 27% 4% 4% 7% >5 Strong 3A 0% 3% 7% 11% >5 Strong 4A 49% 34% 25% 22% >5 (3.81) Moderate 5A 26% 15% 15% 17% >5 Strong 6A 33% 19% 5% 2% >5 Moderate 7A 255% 89% 78% 83% <5 Moderate 8A 17% 6% 6% 4% >5 Strong 9A 73% 104% 105% 94% <5 (6.23, 5.79) Moderate 10A 19% 11% 4% 4% >5 Strong 11A 58% 38% 37% 40% >5 Weak c 12A 346% 520% 360% 354% <5 (5.09) Moderate 13A 211% 48% 48% 51% <5 Moderate 14A 143% 56% 53% 58% <5 Moderate 15A 414% 324% 362% 340% <5 Moderate 16A 24% 16% 15% 16% >5 Strong 17A 12% 8% 8% 6% >5 Strong 18A 130% 113% 126% 118% <5 (5.81, 5.51) Moderate 19A 13% 33% 32% 29% >5 Moderate 20A 5% 18% 16% 14% >5 (4.88) Strong 21A 31% 28% 26% 21% <5 (6.46, 6.18) Moderate 22A 126% 77% 102% 87% <5 Moderate 23A 30% 16% 16% 19% >5 Strong 24A 48% 24% 17% 16% >5 (6 gauges < 5) Moderate 1B 7% 15% 18% 20% >5 Strong 2B 9% 17% 14% 13% >5 Strong 3B 26% 2% 6% 12% >5 Strong 4B 45% 10% 14% 20% >5 Moderate 5B 22% 7% 10% 11% >5 Strong 6B 48% 10% 13% 18% >5 Moderate 7B 40% 30% 34% 36% >5 Weak c 8B 56% 42% 46% 46% >5 Weak c 9B 9% 2% 1% 5% >5 Strong 10B 88% 57% 59% 67% >5 Weak c 11B 2% 22% 20% 19% >5 Strong b 12B 134% 58% 58% 63% >5 ( 4.18) Weak c 13B 37% 35% 33% 32% >5 Weak c 14B 23% 31% 34% 33% >5 Weak c 15B 4% 15% 13% 8% >5 Strong 16B 73% 32% 34% 37% >5 Weak c 17B 11% 20% 19% 19% >5 Strong 18B 89% 65% 67% 75% >5 ( 4.86) Weak c 19B 23% 25% 28% 27% >5 Strong 20B 2% 8% 5% 5% >5 Strong 21B 13% 15% 12% 6% >5 Strong 22B 212% 35% 35% 40% >5 ( 3.08) Weak c 23B 17% 20% 19% 19% >5 Strong 24B 11% 15% 13% 9% >5 Strong 1C 63% 62% 62% 64% >5 Weak 2C 175% 879% 1,443% 122% <5 Moderate 3C 380% 1,008% 109% 112% <5 ( 6.02, 5.76) Moderate 4C 57% 65% 65% 66% >5 Weak 5C 68% 72% 73% 74% >5 Weak 6C 67% 27% 39% 35% <5 (5.61, 5.41, 5.15) Moderate 7C 9% 33% 35% 34% >5 ( 4.47) Weak 8C 56% 72% 72% 72% >5 ( 4.14) Weak c 9C 22% 63% 65% 56% <5 Moderate 10C 1% 56% 72% 65% <5 Moderate 11C 71% 43% 42% 40% >5 ( 1.75) Weak 12C 131% 118% 121% <5 Moderate a Numbers in parentheses are stresses (in ksi) that deviate from indicated general stress range magnitude. b Gauge was damaged during erection. It was replaced after Step 1-3. c Dead load up to Step 3-3a introduced possible fit-up stresses; correlation due to any change in stress was moderate to strong thereafter. 196 / JOURNAL OF BRIDGE ENGINEERING / AUGUST 2000

7 including Step 3-3a (the bare steel structure, including wet weight of the concrete deck after the deck casting, with superimposed loading of 95 psf on each span), Step 4-2 (the self-weight of the completed composite bridge), the average percent error for the six locations of the nine trucks, and the average percent error for the four locations of the three trucks. In the table, the percent error is computed as {[measured stress computed stress]/[measured stress]}. Typically, if the analysis underpredicts the stress, the result may be considered unconservative; the corresponding percent error is shown as positive. The percent errors for the nine and three truck cases are for the stresses due to the total load, not just the change in stresses from the trucks relative to Step 4-2 (Stage 8). Note that no account was taken of the change in stress due to any temperature variation between the readings. The table was constructed based upon percent errors computed by comparing field results to analyses in which N was taken as 6, and fully composite action was assumed across the entire length of the bridge. Table 2 also displays whether the stress magnitude for the gauge was above or below 5 ksi for the readings at Step 3-3a, Step 4-2, and with three and nine trucks. Finally, the overall assessment of the correlation was included for each gauge. The overall correlation rating was determined as follows: (1) Strong = all of the percent errors for the four cases shown must be below 30%; (2) Moderate = all gauges that do not fit within the criteria for a strong or weak correlation; and (3) Weak = the measured stresses must be above 5 ksi, and the percent errors must be >30% for at least three of the four cases shown. These criteria help eliminate percent error skews from early dead load stages or skews due to the overall stress in the gauge being very low (i.e., below 5 ksi, which corresponds approximately to the fatigue threshold of the most severe fatigue details in steel bridge structures), which in turn can cause huge percent errors that are not necessarily representative of the overall correlation of the gauge. This is discussed later in the Conclusions. Hajjar and Boyer (1997) tabulated the percent error of measured versus computed results for each gauge at every major stage of loading. In addition, a series of graphs were constructed that highlighted the correlation of the gauges at all different stages of construction and live loading. Figs. 7 9 show plots of the stress during the eight stages of dead load construction, identified earlier, for three sets of gauges, including the inside fascia girder (Beam 4) near midspan of Span 1 (Gauge Line A), Beam 2 over the center pier (Gauge Line B), and the cross frame between Beams 2 and 3 near midspan of Span 1 (Gauge Line C) (Fig. 3). Field measurements are shown with solid lines, and analysis results are shown with dotted lines. In the legend, the suffix c to the gauge nomenclature refers to the fact that composite action was used for the full length of the bridge when analyzing Stages 7 and 8. These figures identify how the correlation of the gauges changed as the loading progressed and provide an excellent assessment of the trends in the behavior. In Table 2, the percent error for Step 3-3a corresponds to the stresses shown in Step 6 in these figures; similarly, the percent error for Step 4-2 corresponds to Step 8 in the figures. With respect to the truck loading, Figs show representative results. In these graphs, six data points are plotted per gauge, corresponding to the location of the center of gravity of the nine trucks, arranged in three back-to-back groups of three side-by-side trucks, at six locations (approximately uniformly spaced) along the length of the bridge. Fig. 10 shows the total stress in the outside fascia girder near midspan of Span 1 (Gauge Line A). Figs. 11 and 12 show the total stress in Beam 2 over the center pier (Gauge Line B); Fig. 11 shows results from an analysis in which composite action was assumed in the negative moment region, and Fig. 12 assumes only the bare steel section resists load in the negative moment FIG. 8. Computed versus Measured Stresses for Gauges 7B 12B for Primary Dead Load Stages FIG. 9. Computed versus Measured Stresses for Gauges 5C 8C for Primary Dead Load Stages FIG. 7. Computed versus Measured Stresses for Gauges 19A 24A for Primary Dead Load Stages FIG. 10. Computed versus Measured Stresses for Gauges 1A 6A for Loading Due to Nine Trucks at Six Different Locations JOURNAL OF BRIDGE ENGINEERING / AUGUST 2000 / 197

8 FIG. 11. Computed versus Measured Stresses for Gauges 7B 12B for Loading Due to Nine Trucks at Six Different Locations (Composite Action Assumed in Negative Moment Region) FIG. 12. Computed versus Measured Stresses for Gauges 7B 12B for Loading Due to Nine Trucks at Six Different Locations (Noncomposite Action Assumed in Negative Moment Region) FIG. 13. Computed versus Measured Stresses for Gauges 13B 18B for Loading Due to Nine Trucks at Six Different Locations region (denoted with suffix nc in the legend). Fig. 13 shows the change in stress due to the nine trucks, in this case for Beam 3 over the center pier (Gauge Line B). Fig. 14 shows the total stress due to the nine trucks in the cross frame between Beams 1 and 2 near midspan of Span 1 (Gauge Line C). The analyses for Figs all assumed N = 6 after the deck solidified. A summary of the correlation seen in Table 2 and Figs. 4 14, whose data are representative of the results as a whole, follows. First, the percent errors generally reduce greatly as the magnitude of the strain in each gauge increases. Thus, the percent errors due to initial dead loading are often excessive, although the overall correlation of the gauge is acceptable once loading is applied. Consequently, to assess the correlation of FIG. 14. Computed versus Measured Stresses for Gauges 1C 4C for Loading Due to Nine Trucks at Six Different Locations the total stress due to self-weight plus truck loading, this correlation must be compared to the correlation for the gauge due to self-weight alone (Steps 3-3a and 4-2). Generally, if the gauge correlated well at the end of the dead load analysis (e.g., as seen for Beam 4 at Gauge Line A in Fig. 7), the total stresses also correlated well during the truck loading. In contrast, if the correlation of the total stresses was moderate to weak (e.g., as seen for Beam 2 at Gauge Line B in Fig. 8), it was often largely because initial fit-up stresses immediately established a difference in stress between measured and computed results that is on the order of several ksi. These cases are marked in the footnotes of Table 2. In addition, some gauges saw <5 ksi for their entire loading history. For these gauges, a small difference in stress (which may be caused by local irregularities in the loading, small fit-up stresses, etc.) causes a large percent error. These cases are highlighted in Table 2 both in Column 6 ( Measured Stress Magnitude ) and in the correlation classification. Second, as a whole, the 24 gauges attached to the girders near midspan correlated very well with the analyses (Table 2, Figs. 4 7, and Fig. 10). All of the gauges received ratings of strong or moderate, except gauge 11A, which was the only gauge with a weak correlation. However, the weak correlation of this gauge (e.g., Fig. 6) was not excessive, and it was due largely to errors introduced in the middle stages of dead loading. In addition, the gauges that received a moderate rating that had stress magnitudes >5 ksi were all very close to fitting into the strong gauge criteria. They generally missed being rated strong because one of the dead load errors was >30%. Of the gauges that had stresses >5 ksi, the gauges attached to the web tended to correlate better, probably due to being less influenced by warping restraint and lateral bending behavior. The 24 gauges attached to the girders near the middle pier did not have as strong a correlation with the analysis as did the midspan gauges (Table 2, Figs. 4 6, Fig. 8, and Figs ). As seen in Table 2, there were nine gauges in this region of the bridge that received weak ratings, compared to only one in the midspan. However, 13 gauges had strong correlation, and all weak correlations were readily explainable: The differences in the magnitudes of the measured and computed stresses largely occurred in the early or middle stages of dead loading, due perhaps to fit-up stresses. This is readily evident if one compares Figs. 8 and 11; little additional error in stress resulted from the substantial truck loading. Also, the analysis results were generally conservative. All of these nine gauges except one (Gauge 22B) were on Beams 2 and 3, the center two girders. The two outer girders generally correlated well. In addition, of the gauges that had stresses >5 ksi, the gauges attached to the web tended to correlate better, probably due to being less influenced by warping restraint and lateral bending behavior. 198 / JOURNAL OF BRIDGE ENGINEERING / AUGUST 2000

9 The cross frames correlated worse than the girders (Table 2, Figs. 4 6, Fig. 9, and Fig. 14), with half of the gauges having weak correlation and the other half having moderate correlation. In addition, the moderate correlation was always due to the stresses in the cross frames being very small. Also, the analysis results were often not conservatively predicated the cross frames often had forces up to several times higher than those predicted in any of the analyses. The figures show that the behavior of the gauges generally followed patterns seen in the measured results. However, the measured stresses in the cross frames were not readily predictable, and further research would be required to determine how best to increase the accuracy of the analysis predictions. Third, the gauges near the center pier showed a marked change in correlation depending on whether the behavior in this region of the bridge was assumed to be fully composite or only the bare steel bridge was modeled in the negative moment region. The differences based upon the modeling of composite action were most noticeable in the gauges that were nearest to the concrete deck in the negative moment region (i.e., Gauges 1B, 2B, 5B, 7B, 8B, 11B, 13B, 14B, 17B, 19B, 20B, 23B), as may be expected due to the shift in the neutral axis toward the top flange of the girders. The correlation of each group of three gauges near the bottom flange of each girder over the interior pier changed little on the whole. The correlation of the gauges in the positive moment region (both on the girders and the cross frames) changed little as well. The improvement in correlation when assuming fully composite action was generally both consistent and dramatic. It is for this reason that analyses assuming composite action across the entire bridge were used in Table 2. CONCLUSIONS This paper has outlined (1) the development of software at the University of Minnesota to analyze steel curved girder bridge systems; and (2) the comparison of this software with field measurements taken on MNDOT Bridge No , a two-span curved steel I-girder bridge system having skew supports. Conclusions drawn from this study include This bridge was shored in the early stages of construction of the steel superstructure, and the bridge design was controlled by stiffness, not strength. The stresses were well below the yield stress throughout construction. Computational results generally matched qualitatively and often quantitatively with measured results, both for stresses and deflections. The bridge behavior was generally predictable at all stages, particularly in the girders. The primary difference between measured and computed results was due mainly to the erratic effects of warping restraint and minor axis bending on the measured results and to the less predictable behavior seen in the measured results of the cross frames. In addition, the correlation between measured and computed results in the negative moment region due to live truck loading improved substantially if composite action was assumed over the center pier. As there were no shear connectors in the negative moment region of the bridge, composite action in this region was due to friction and adhesion. The unreliable nature of these mechanisms of force transfer justify the traditional practice of neglecting composite action in the negative moment region of these types of bridges. Nevertheless, the magnitude of stresses predicted in the analyses affects design calculations, in particular for fatigue design. Fit-up stresses were seen in the measured results, especially in the cross frames, but they dissipated as the construction progressed, and they remained <6 ksi. Further research is required to determine how best to model the bridge over interior piers, where the effects of composite action are more ambiguous, and also in the cross frames, which show less predictable behavior at all stages of dead and live loading and often produce unconservative results in the analyses. ACKNOWLEDGMENTS The writers would like to thank the Offices of Bridges and Structures, Materials, and Research Administration at the Minnesota Department of Transportation, including D. Flemming, G. Peterson, P. Rowekamp, S. Ellis, K. Anderson, T. Nieman, J. Southward, D. Reinsch, L. Lillie, P. Koff, and J. Michaels, for support of this research on steel curved girder bridges. The writers would also like to give special thanks to J. Bates and R. Cisco of the PDM Bridge Company, fabricators of the curved girder bridge; D. Davick of the Lunda Construction Company, the general contractors on the bridge construction; B. Theis of High Five Erectors, the erectors of the steel superstructure of the bridge; and P. Bergson, T. Boyer, J. Hanus, A. Lawver, J. Millman, T. Plueth, and A. Staples of the University of Minnesota. Without their extensive cooperation, this research would not have been possible. This project was conducted with funding provided by the Minnesota Department of Transportation and the Center for Transportation Studies under MNDOT Agreement Nos and The writers gratefully acknowledge this support. The findings and conclusions expressed in this publication are those of the writers and not necessarily the Minnesota Department of Transportation or the Center for Transportation Studies. APPENDIX. REFERENCES American Association of State Highway and Transportation Officials (AASHTO). (1994). AASHTO LRFD bridge design specifications, 1st Ed., Washington, D.C. Galambos, T. V., Hajjar, J. F., Leon, R. T., Huang, W.-H., Pulver, B. E., and Rudie, B. J. (1996). Stresses in steel curved girder bridges. Rep. No. MN/RC-96/28, Minnesota Department of Transportation, St. Paul, Minn. Hajjar, J. F., and Boyer, T. A. (1997). Live load stresses in steel curved girder bridges. Mn/DOT Proj Task 1 Rep., Dept. of Civ. Engrg., University of Minnesota, Minneapolis, Minn. Hall, D. H. (1994). Curved girders are special. Link between research and practice, Structural Stability Research Council, Bethlehem, Pa., Heins, C. P. (1975). Bending and torsional design in structural members, Lexington Books, New York. Huang, W.-H. (1996). Curved I-girder systems. PhD dissertation, Dept. of Civ. Engrg., University of Minnesota, Minneapolis, Minn. Nakai, H., and Yoo, C. H. (1988). Analysis and design of curved steel bridges, McGraw-Hill, New York. Pulver, B. E. (1996). Measured stresses in a steel curved girder bridge system. MCE Rep., Dept. of Civ. Engrg., University of Minnesota, Minneapolis, Minn. Rudie, B. J. (1997). A study of the deflections of a curved steel girder bridge. MS thesis, Dept. of Civ. Engrg., University of Minnesota, Minneapolis, Minn. Stegmann, T. H., and Galambos, T. V. (1976). Load factor design criteria for curved steel bridges of open section. Rep. No. 43, Dept. of Civ. Engrg., Washington University, St. Louis, Mo. Zureick, A., Naqib, R., and Yadlosky, J. M. (1993). Curved steel bridge research project. Interim report I: Synthesis. Rep. No. FHWA-RD , Federal Highway Administration, McLean, Va. JOURNAL OF BRIDGE ENGINEERING / AUGUST 2000 / 199