Evaluation of live-load distribution factors for high-performance prestressed concrete girder bridges

1 Introduction

The early use of high-performance concrete (HPC) in building highway bridges was in Japan in the early 1970’s for Japan National Railway. Germany, Canada, Norway, France, Denmark, and USA followed the same track. The early uses of HPC in bridges construction was aimed at designing a bridge with decreased dead load, decreased deflection, decreased vibration, as well as, noise [1]. With the use of chemical admixtures, pozzolanic materials, and optimization of mix proportions, HPC with high compressive strengths between 60 MPa and 90 MPa have been successfully produced with conventional methods. The use of HPC in bridges is becoming widely spread among bridge construction industry. However, the use of such HPC material with high compressive strengths allowed design engineers to exceed the range of applicability of current design code provisions.

Barr et al. [2] evaluated the procedure for computing live-load distribution factors (LLDF’s) proposed by international codes based upon finite element parametric study calibrated to match live-load test data. The parametric study considered the effect of end and intermediate diaphragms, lift, continuity, skew angle, and load-type on distribution factors. In their study they reported decreased distribution factors due to lifts, skew angle, load-type, and diaphragms whereas continuity and intermediate diaphragm had insignificant effect. Eom and Nowak [3] carried out live load testing of 20 steel girder bridges. The test results were compared to finite element modelling which unveiled that LLDF’s measured by tests were lower than values proposed by Association of State Highway and Transportation Officials (AASHTO) LRFD design specifications. They also pointed out the importance of considering support partial fixity when applying load-rating of exiting bridges utilizing AASHTO LRFD distribution factors.

Live-load distribution factors for girder bridges loaded with oversized trucks were studies by Tabsh and Tabatabai [4]. Based upon their analysis, correction factors for AASHTO LRFD LLDF’s were proposed to account for such up-normal loading. Live-load test results of HPC Bridge in Missouri were reported by Yan and Myers [5]. Test results showed that LLDF’s computed based on AASHTO standard specification are too conservative, however, LLDF’s computed based on AASHTO LRFD appears more appropriate. Zokaei et al [6] studied LLDF’s for high-strength concrete bridges through finite element modelling. As the use of high-strength concrete results in larger spans exceeding the code limits for LLDF’s, the study concluded that current LLDF’s could be extended to longer span bridges without sacrificing bridge safety.

Zhang and Cai [7] considered girder bridges with fibre reinforced polymer (FRP) decks in evaluating LLDF’s. They found that LLDF’s for FRP bridge decks are higher than similar bridges with reinforced concrete decks. Dicleli and Erhan [8] studied LLDF’s for girder bridges with integrally built abutments. They concluded that LLDF’s calculated by AASHTO LRFD specification are significantly higher than those obtained through analysis. Dwairi et al. [9] reported live-load test and health monitoring of high-performance, high-strength concrete girder bridge. In their study they provided details of standardized testing of the concrete properties and field instrumentation of the bridge girders, as well as, a discussion of service level monitoring and controlled load testing. Girder strains were measured for instrumented girders under normal traffic condition and controlled truck loading.

This paper reports the live-load test data for high-performance, high-strength prestressed concrete girder bridge tested by Dwairi et al. [9]. Live-load distribution factors for moment obtained from controlled load testing were computed and compared to AASHTO LRFD distribution factors in order to evaluate the factors applicability. LLDF’s for shear were not covered in this paper as they were not measured in the live-load test. A FEM was developed and calibrated to match test results, where several variations of the model were developed in order to study the effect of various parameters on the LLDF’s as compared to current AASHTO LRFD specifications.

2 Research significance

Current AASHTO LRFD code didn’t account for high-strength concrete when developing LLDF’s for highway bridges. Current code LLDF’s are based on single-span finite element model (FEM) developed by Zokaie et al. [10] for normal-weight concrete which didn’t account for lane loading, end and intermediate diaphragms, and bridge continuity as well. In an attempt to evaluate LRFD code LLDF, Chen and Aswad [11] concluded that LRFD LLDF’s yield uneconomical conservative designs for bridges with large span-to-depth ratios. They quantified this conservatism as much as 23% for interior girders and 12% for exterior girders. The current study utilizes live-load test data for high-strength concrete girder bridge to evaluate the applicability of AASHTO LRFD [12] distribution factors. Based upon the live-load test data, a parametric study was developed using a calibrated FEM to evaluate the effect of skew angles, intermediate and end diaphragms, continuity and type of loading on LLDF’s for such bridges.

3 Overview of the current AASHTO LRFD code design provisions

Live-load distribution factors (LLDF’s) were used as early as 1931 through the AASHTO Standard Specifications for Highway Bridges. These factors were originally developed by Westergaard [13], and Newmark et al. [14]; since then these factors evolved as new research results became available. AASHTO Standard Specifications distribution factors are based on girder spacing only and were developed for non-skewed, simply-supported bridges. However, the LRFD formulas for LLDF’s are developed based on parametric study by Zokaie et al. [10] and Zokaie [15], which accounted for girder spacing, composite girder bending stiffness, bridge length, skew, and slab stiffness. These formulas were adopted by AASHTO as the guide specifications for distribution of live loads on highway bridges in 1994.

The AASHTO LRFD (2017) specifications proposed LLDF’s for interior girders for one-lane loaded as shown in Equation (1) and for two or more lanes loaded as given by Equation (2): 0.06 + ( S 4300 ) 0.4 ( S L ) 0.3 ( K g Lt s 3 ) 0.1(1)0.075 + ( S 2900 ) 0.6 ( S L ) 0.2 ( K g Lt s 3 ) 0.1(2)

The above equations have limited applicability where girder spacing (S) ranges from 1.1m to 4.9m, slab thickness (t_s) ranges from 0.11m to 0.30m, and girder span length (L) ranges from 6.0m to 73.0m. In the case of exterior girders, a modification factor (e) is applied to Equation (1) and Equation (2) as shown below: e = 0.77 + ( d e 2800 )(3)

The North Carolina Department of Transportation (NCDOT) used HPC in the superstructures of two parallel bridges located over the Neuse River in Wake County for the divided highway US401. HPC concrete was used in the girders and the deck and has the strength requirements given in Table 1. In addition, the HPC mix was formulated to achieve resistance to internal chemical attack, resistance to Chloride ion intrusion, and freeze-thaw durability. Each bridge has four spans with length of 28.0, 28.0, 17.5, and 17.5 m (91.9, 91.9, 57.4, and 57.4 ft). The width of the bridge is 14.1 m (47.1 ft), carriageway is 12.0 m (39.4 ft), and sidewalk is 1.9 m (6.2 ft). The bridges were designed using simple-span prestressed concrete I-girders for dead loads, made continuous for live-load. AASHTO Type IV and Type III prestressed girders were used for the 28.0 m (91.9 ft) and 17.5 m (57.4 ft) spans, respectively. Due to the use of high-strength concrete, five girders were used per deck instead of six typically used for normal-strength concrete. Girders were spaced at 3.12 m (10.25 ft) on centres, and the deck thickness was 215 mm (8.5 in.).

Figure 1 shows the framing layout for the southbound US401 Bridge and Fig. 2 shows AASHTO Type III & IV girder cross-sections. Girder number four was the instrumented line of girders and each of the four spans was designated a letter (A - D). Material testing was conducted using numerous specimens taken from concrete batches used in casting the instrumented girders. During curing, the specimens were kept next to the girders in order to maintain curing temperatures similar to that of the girders. At 28 days, AASHTO Type IV and Type III girders had average concrete compressive strength of 72.9 MPa and 73.2 MPa, respectively. Silica fume was used in the concrete mix which caused the concrete to gain strength rapidly at early age and after about two months there was virtually no gain of strength.

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Fig.1

US401 southbound bridge framing layout.

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Fig.2

AASHTO type III and type IV girder cross sections.

Since HPC generates high temperatures during hydration, this could provide in-accurate measure of strength utilizing standard cylinder tests. Thus for the determination of modulus of elasticity (MOE), four cylinders for each girder were match-cured with the help of FHWA Mobile Concrete Laboratory. As a result, MOE for girders A-4, B-4, C-4, and D-4 were 29.8 GPa, 30.3 GPa, 31.4 GPa and 34.3 GPa, respectively. For further material testing results including flexural strength, modulus of rupture, permeability, creep, shrinkage, and thermal properties refer to Dwairi et al. [9].

Static live-load tests were carried out for the bridge using a five-axle truck (3S2 AASHTO truck), fully loaded in the first run and partially loaded in the second. The weight of the tuck fully loaded was 367 kN and partially loaded weight was 250 kN as portrayed in Fig. 3. The truck was placed on 10 different positions selected to maximize bridge responses at midspan and supports as shown in Fig. 4. In order to obtain LLDF’s from the test, the driver-side wheel was positioned on Girder no. 4 in one run and the passenger-side wheel was positioned on adjacent Girder no. 3 in another. Vibrating Wire Gauges (VWG’s) embedded within Girder no. 4 were used to obtain girder strains across its cross-section depth. Table 3 shows typical strains recorded in girder no. 4 located 0.15m from the bottom of the girder for various loading positions. Initial reading of all instrument were recorded while the truck is off the bridge and before running the test. Afterwards the truck was moved into desired loading position and as the truck came to rest, instruments’ readings were recorded for 30 seconds. Without unloading the bridge, the truck was moved to the next loading position, and instruments’ readings were recorded as the truck came to rest again. The measured moments in Girder no. 4 were calculated using VWG’s readings recorded at various truck positions.

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Fig.3

AASHTO type 3S2 vehicle and weights used in live-load test.

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Fig.4

Truck loading positions used in live-load test.

Figure 5 shows measured moments in Girder no. 4 due to midspan loading positions, while Fig. 6 presents a comparison between measured midspan moments in Girder no. 4 when truck is positioned at line-3 and when truck is positioned at line-4. As expected, maximum midspan moments of any girder occur when truck was located on midspan position of the girder in question. As the truck was moved transversely from line-4 to line-3, a reduction in girder no. 4 moments occurred. The reduction in the moments for spans A-4, B-4, C-4, and D-4 were 15.5%, 11.0%, 18.4% and 20.8%, respectively. Computed LLDF’s from measured strains for girder no. 4 are portrayed in Table 2, where the calculated distribution factors are on average 21% less than the AASHTO LRFD distribution factors. Measured LLDF’s were calculated as the ratio of maximum moment in a single span due to loading positions, to the maximum moment in the simple span due to truck 3S2 positioned on the girder. The AASHTO LRFD factors are based upon a parametric study by Zokaie et al. [10] for a single span FEM which didn’t consider continuity of girders and presence of intermediate and end diaphragms. This conservatism was pointed out as well by Chen and Aswad [11] based upon finite element analyses.

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Fig.5

Measured and calculated mid-span moments due to mid-span loading.

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Fig.6

Measured mid-span moments in girder-4 due to truck on line-3 and line-4.

Accurate finite element model must take into consideration the exact geometry of the bridge, material properties, loading conditions, and if needed response of foundations. A finite element model (FEM) was developed for the US 401 Bridge which accounted for bridge geometry, diaphragms, continuity, and material properties. Foundation response was not modelled since the focus of the study is on the superstructure response. AASHTO 3S2 truck was simulated in the model as point loading located at the desired positions. In order for this to be achieved the deck was modelled as linear, four-node shell elements with six degrees of freedom at each node. After several meshing trials the node spacing selected was 0.52m in transverse direction and 0.5 m in longitudinal. As a result, the total number of nodes used to simulate the deck and girders in the model was approximately 6000 nodes.

In order to achieve precise modelling of torsional and flexural stiffness for the girders, eccentric girder model was used. In such a model, the vertical placement of girders, deck and supports must imitate that of the actual bridge pattern as pointed out by Nutt et al. [16]. Therefore, girders are modelled as frame elements located at the centre of gravity of each girder. The supports are modelled at the soffit of the girder and linked to the frame element through a rigid-link. Rigid-links were used to link the girders to supports and deck elements, as a result plain sections before bending remain plain after bending. This modelling technique was investigated and recommended by several researchers (Barr et al. [2]; Bapat [17]; Chen and Aswad [11]). Fig. 7 shows a schematic of the bridge FEM developed for this study, as this model was implemented using CSI-Bridge software [18]. Frame elements used in the model are a general, three-dimensional, beam-column formulation, two-node elements with six degrees of freedom for each node. Cubic variation of flexural stiffness over element length was adopted for the frame element. Bapat [17] compared various girders modelling techniques, including linear, quadratic, and cubic formulation. Based on his comparative study results, quadratic and cubic formulations converged to the analytical solution regardless of mesh refinement, while linear formulation needed more refinement.

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Fig.7

Transverse section in finite element model for two girders.

The US 401 Bridge substructure includes three multicolumn piers and two closed abutments. Multicolumn piers and pier caps were modelled as 0.3m long frame elements while abutments were constrained against longitudinal, transverse, and rotation movements. Diaphragms were modelled as linear, four-node shell elements with six degrees of freedom at each node similar to Wolek et al. [19]. Elastomeric bearings of the bridge were simulated as linear springs connected to the bottom flange of the girders in order to model partial-fixity. Springs stiffness were adjusted to obtain the same movements of bridge ends recorded during the live-load test (Eom and Nowak [3]). Various material properties including Young’s modulus and Poisson’s ratio were adopted in the model from material tests reported by Dwairi et al. [9].

In order to verify the accuracy of the finite element model developed, the US 401 bridge response was obtained for AASHTO 3S2 truck loading positions shown in Fig. 4. Midspan moments obtained from the FEM analysis for Girder-4 were compared to the measured moments calculated from the strains recorded by VWG’s for the same loading position as depicted in Fig. 5. As the truck position was shifted from girder line-4 to girder line -3, the difference in the results between midspan measured and calculated moments for two analogous loading positions (i.e. position 6 & position 10) is about the same magnitude. Fig. 8 shows the relationship between measured and FEM moments at midspan of Girder no.4 due to loading positions shown in Fig. 4. Each symbol represents midspan moments in a single span due to four loading positions; two positions in span under consideration and another two positions in adjacent span. A perfect correlation between measured and FEM moments is shown through the diagonal line on the figure. A good correlation between measured and FEM moments is clearly presented, where FEM analyses slightly overestimated the moment for majority of loading positions.

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Fig.8

Comparison of measured and calculated mid-span moments.

In order to evaluate the effect of various parameters on LLDF’s, several variations of the finite element model were created. Model 1 matches exactly the model utilized by Zokaie et al. [10] in their original study. The model uses single-span of the bridge (i.e. span length of 28.0 m), developed for US 401 bridge without cross diaphragms. Other models were created to study the effect of various members on LLDF’s by adding different members for each model. Members considered include end and intermediate diaphragms, skew angle, and type of loading. Model 2 is similar to model 1 with added end diaphragms, while model 3 has end and one intermediate diaphragms. In model 4 two intermediate diaphragms were added and in model 5 three intermediate diaphragm were added. For all models skew angle was varied between 0° and 60° to evaluate the effect of skewness on the LLDF’s. Model 6 considered bridge continuity through modelling two adjacent spans A and B of US401 Bridge.

8.1 Evaluation of code live-load distribution factors

For all six models, LLDF’s specified by AASHTO LRFD (2017) specifications were compared to those calculated using finite element models. For each model, HS-20 AASHTO truck was placed on the deck and its position was varied longitudinally and transversally in 0.3m increments within a lane pattern, in order to produce maximum midspan moments in the girders. Analogues to AASHTO LRFD LLDF’s definition, one-lane loaded and two and more lanes loaded cases were considered. Wheels of adjacent trucks were never closer than 1.2m and truck wheel distance to curb edge was never less than 0.6m. Maximum moments of exterior and interior girders were obtained separately considering one-lane loaded and two and more lane loaded cases. AASHTO LRFD multi-presence factors of 1.2 to 0.85 were applied to all moments computed due to all cases of loading. LLDF’s were then calculated through dividing those obtained maximum moments by maximum moment in a single isolated simply-supported girder due to HS-20 ASSHTO truck.

Figure 9 depicts a comparison between LLDF’s computed based on FEM analysis of a single span of the US 401 bridge and AASHTO LFRD formulae. In all five models, LLDF’s for interior girders were higher than those for exterior girders which conforms to AASHTO LRFD formulae. AASHTO LRFD assumes LLDF’s for exterior girders may be computed by reducing interior girders distribution factors using a reduction factor given by Eq. (3). Similar behaviour has been noticed as presented in Fig. 9a for model 1 which is the same model used by Zokaie et al. [10]. The LLDF’s from model 1 are on average 4% less than the AASHTO LRFD specification factors which is close to the 5% difference reported by Zokae et al. [10]. In subsequent models 2 to 5 the difference between FEM LLDF’s and LRFD code factors is on average 8% which is mainly attributed to the presence of end diaphragms.

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Fig.9

Comparison of AASHTO LRFD live-load distribution factors and FEM computed factors.

8.2 Effect of end diaphragms

AASHTO LRFD accounted for the effect of bridge skewness however it didn’t account for the presence of end diaphragms. The addition of end diaphragms in model 2 resulted in reduced LLDF’s for exterior and interior girders as shown in Fig. 9b. This could be attributed to the fact that generally end diaphragms used in practice are relatively torsionally stiff, which will cause end rotations for loaded and unloaded girders. This introduces end-negative moments which consequently reduces girder midspan moments and as a results reduces LLDF’s. Similar behaviour has also been reported by Barr et al. [2]. Fig. 10 shows a comparison between LLDF’s and AASHTO LRFD specification factors for bridge with no skew. For both exterior and interior girder the distribution factors reduced on average 2% due to the addition of end diaphragms. This difference increased with the presence of skewness and was about 6% for skew angle of 60°.

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Fig.10

Effect of end and intermediate diaphragms in LLFD’s of exterior and interior girders.

8.4 Effect of type of loading

In practice the design of a girder is based on the moment due to the superposition of truck and lane loads. Lane loading used in this study is a uniformly distributed load of 9.3 kN/m distributed over the width of the lanes for each model as specified by AASHTO LRFD specifications. Although the AASHTO LRFD specifications LLDF’s were developed based on truck loading, they are still being used for lane loading as well. Fig. 11 shows that LLDF’s due to lane loading are less than those due to truck loading. Distribution factors due to lane loading are on average 4% less than distribution factors due to truck loading. In practice girder moments due to lane loading could be as high as half the moments due to truck loading, therefore a reduction in LLDF’s for lane loading used in the design should results in economic savings.

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Fig.11

Effect of type of load on LLDF’s for exterior and interior girders.

8.5 Effect of skew angle

The effect of skew angle on LLDF’s obtained from FEM is shown in Fig. 12. The figure shows the ratio of distribution factors of skewed bridge to the factors from non-skewed bridges. The results agree well with the AASHTO LRFD reduction factors due to skew angle applied to non-skew bridge distribution factors. The reduction in distribution factors due to skewness is more evident in models 3, 4 and 5 which comprise the presence of intermediate diaphragms. Skewness effect is more evident for angles larger than 20°, at larger angles the LLDF’s decreased as the skew angle increased. The exterior girder was affected less by the skewness as compared to the interior girder. Model 1 was affected the least among all six models, and model 2 which comprises end diaphragms was affected the most.

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Fig.12

Effect of skew angle on LLDF’s for exterior and interior girders.

8.6 Effect of continuity

In order to study the effect of continuous spans on the LLDF’s, model 6 was developed with additional span as compared to model 1. Continuity of spans develops negative moment at interior supports which reduces midspan moments and consequently reduces distribution factors. Fig. 13 shows the effect of continuous spans on both exterior and interior girders. Distribution factors were on average 21% less than AASHTO LRFD specification factors. Adding one continuous span to model 1 reduced LLDF’s on average 17%. This reduction of LLDF’s due to continuity is not accounted for in AASHTO LRFD.

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Fig.13

Effect of bridge continuity on LLDF’s for exterior and interior girders.

AASHTO LRFD accounted for the influence of compressive strength of concrete through the inclusion of modular ratio in load distribution factors as given by Eq. 1 and Eq. 2. The effect of various modular ratios on LLDF’s is shown in Fig. 14 for exterior and interior girders for bridges with zero skew angle. The effect of the variance in the compressive strength of concrete girders is none for exterior girders and insignificant for interior girders. The results of FEM analysis agree well with the AASHTO LRFD distribution factors. AASHTO LRFD factors were on average 8% higher than FEM factors for exterior girder and at most 3% higher in the case of interior girders.

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Fig.14

Effect of concrete compressive strength on LLDF’s for exterior and interior girders.

Four-span high-performance, high-strength prestressed concrete highway bridge was tested to truck loading. Live-load distribution factors for the bridge girders were computed from measured stains of instrumented girders. A detailed finite element model was developed and calibrated to match the live-load test midspan moments of instrumented girders. Sixty variations of the calibrated FEM were developed to study the effects on LLDF’s due to: end and intermediate diaphragms, type of loading, skew angle, and continuity. The following conclusions are listed based on the test and FEM analyses results:

Live-load distribution factors calculated based on AASHTO LRFD (2017) specifications were significantly less than computed distribution factors based on measured girder strains due to live-loading test. Distribution factors from live-load test were on average 21% less than AASHTO LRFD factors.

Conflict of interest

None to report.