Development of Live Load Distribution Factor Equation for Concrete Multicell Box-Girder Bridges under Vehicle Loading
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Abstract
The evaluation and design of concrete bridges in large part depend on the transverse distribution characteristics of the live load carried and the service level. The live load distribution for continuous concrete multicell box-girder bridges varies according to bridge configuration, so when designing such bridges, it is important to determine the maximum negative stress at the piers, the midspan positive (tensile) stress and the deflection of the bridge when subjected to live loads. This paper reports an extensive parametric study to determine the maximum stress, deflection, and moment distribution factors for two span multicell box-girder bridges based on a finite element analysis of 120 representative numerical model bridges. Bridge parameters were selected to extend the parameters and ranges of current live load distribution factors defined by AASHTO LRFD specifications. The results indicate that the span length, number of boxes, and the number of lanes all significantly affect the positive (tensile) and the negative (compression) stress distribution factors. A set of equations proposed to describe the behavior of such bridges under AASHTO LRFD live loads yielded results that agreed closely with the numerically derived results for the stress and deflection distribution factors.
Keywords
finite element modelling distribution factor truck box bridgesAbbreviations
- \(a,b1,b2,b3\)
Constants for regression analyses
- AVG
Average
- \(B\)
Width of the boxes
- \(D\)
Depth of the boxes
- \(d^{\prime}\)
Thickness of the top flange
- \(d^{\prime\prime}\)
Thickness of the bottom flange
- \(DF\)
Distribution factor
- DLA
Dynamic load allowance
- \(D\delta_{s}\)
Distribution factor for maximum deflection
- \(D\sigma_{ne}\)
Distribution factors for the maximum negative stresses
- \(D\sigma_{n,sb}\)
Distribution factors for the maximum negative stresses for steel spread open box girder bridges
- \(D\sigma_{po}\)
Distribution factors for the maximum positive stresses
- \(D\sigma_{p,sb}\)
Distribution factors for the maximum positive stresses for steel spread open box girder bridges
- \(E_{c}\)
modulus of elasticity of the concrete
- \(E_{s}\)
Modulus of elasticity of the steel
- \(F_{N} ,F_{p}\)
Correction factor
- \(L\)
Length of the span
- \(L_{c}\)
Length of the cantilever
- ∑L_{i}
Sum of all girder actions
- \(LDF\)
Live-load distribution factor
- \(LDF_{i}\)
Live-load distribution factor of the ith girder
- \(LDF_{m}\)
Live-load distribution factor of multicell box girder bridges
- \(L_{i}\)
Moment or deflection of ith girder
- \(M^{ - }\)
Maximum negative moment
- \(M^{ + }\)
Maximum positive moment
- \(n\)
Number of bridge girders
- \(N_{B}\)
Number of boxes
- \(N_{C}\)
Number of boxes
- \(N_{L}\)
Lanes of traffic
- \(R1,R2\)
Ratio of the positive (tensile) stress distribution factor
- \(S\)
Width of each box
- STD
Standard deviation
- \(W_{r}\)
Width of the road way of the bridge
- \(W_{total}\)
Total width of the bridge
- \(\delta_{\text{max} }\)
Maximum deflections
- \(\delta_{s}\)
Maximum deflection at the midspan of simple ideal girder
- \(\sigma_{n}\)
Maximum negative (compression) stress for the bridges were then obtained for the three-dimensional bridges using FEA
- \(\sigma_{n,I}\)
Negative (compression) stress at the bottom fiber were calculated using a simple beam bending formula
- \(\sigma_{p}\)
Maximum positive (tensile) stress for the bridges were then obtained for the three-dimensional bridges using FEA
- \(\sigma_{p,I}\)
Maximum positive (tensile) stress at the bottom fiber were calculated using a simple beam bending formula
- \(\upsilon_{c}\)
Poisson’s ratio of elasticity of the concrete
- \(\upsilon_{s}\)
Poisson’s ratio of elasticity of the steel
1 Introduction
Concrete multicell box-girder (MCB) bridges are commonly used for highway bridges in road networks all over the world. Voids are created in girders to reduce their weight, creating bridges that combine excellent torsional stiffness with elegance (Song et al. 2003). Accurately calculating the design stress and deflection actions for a multicell box-girder bridge under service loads can be a complex task, however. The design stresses and deflection demands for an individual box depend on a number of parameters, including the position of the live loads, the web spacing, the span length, and the relative deck-to-girder stiffness. To simplify the design process, a long-standing methodology has evolved in which a multiple girder bridge deck is treated as a one-girder line or beam element (Semendary et al. 2017; Samaan et al. 2002b). Early live load distribution factors were obtained based on the method proposed by Newmark (1938), which over time has been updated as improved bridge analysis methods have become available. The concept of a live load distribution factor (LDF) was first used in the bridge specifications issued by the American Association of State Highway Officials (AASHTO) in 2002 through empirical S/D expressions (known as S-over equations), where S is the girder spacing and D is a constant that depends on the bridge’s superstructure and the type of lane loading. S-over equations were used for bridge design for over a decade until the 8th edition of AASHTO’s LRFD Bridge Design Specifications (2017) was published.
2 Objectives
The main objective of this study was to evaluate the LDFs for concrete MCB bridges with two equal spans under vehicle loads using finite element analysis (FEA). One-hundred twenty numerical models were analyzed to: (a) determine the influence of each of the parameters affecting the prototype bridge responses; (b) produce a database for negative (compression) and positive (tensile) distribution factors corresponding to the AASHTO (2014) live loads; and (c) develop a set of empirical equations for a bridge’s stress and deflection distribution factors under AASHTO-LRFD live loads. As previous sensitivity studies revealed that changing the slab thickness has an insignificant effect on the live load distribution factors (Huo et al. 2003; Huo and Zhang 2008), only the following parameters were investigated in this study: the span length, the number of lanes and the number of boxes. The superstructure is idealized using the following assumptions: (a) all materials are elastic and homogenous; (b) the slab has a constant thickness; (c) the slab and girder exhibit full composite action; (d) the effects of the curbs and web slope are ignored; and (e) the skew angle of the bridges is less than 30°.
3 Geometric and Structural Properties
Geometry of the bridges used in the parametric study (unit: m).
Set | L (m) | N _{B} | N _{ L} | W _{ r} | W _{ Total} | d′ | d″ | d _{ w} | L _{ c} |
---|---|---|---|---|---|---|---|---|---|
1 | (30.5, 45.75, 61, 76.25, 91.50) | 2, 3 | 1, 2 | 9.10 | 13.00 | 0.20 | 0.15 | 0.10 | 0.61 |
2 | 2, 3, 4 | 1, 2, 3 | 14.0 | 16.70 | 0.20 | 0.15 | 0.10 | 1.20 | |
3 | 3, 4, 5, 6 | 2, 3, 4 | 17.1 | 24.00 | 0.20 | 0.15 | 0.10 | 1.45 |
4 Bridge Prototype Modeling
4.1 Field-Testing Study on Tsing Yi South Bridge
Comparison between field test and FEA.
Mode number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|
Ashebo et al. (2007) | ||||||||
SAP2000 | 4.54 | 4.81 | 6.70 | 7.53 | 10.40 | 10.78 | 14.72 | 16.34 |
Field test | ||||||||
1st | 4.46 | * | 6.25 | 7.82 | * | 10.87 | 13.67 | 15.74 |
2nd | 4.61 | * | 6.22 | 7.73 | * | 10.81 | 13.30 | 15.74 |
3rd | 4.58 | * | 6.39 | 7.76 | * | 11.11 | 13.31 | 15.73 |
Control traffic | 4.58 | * | 6.15 | * | * | * | * | * |
CSIbridge | 4.47 | 4.88 | 6.45 | 7.62 | 10.22 | 11.07 | 14.33 | 16.03 |
4.2 Scaled Two-box Bridge Under Concentrated Loads
5 Loading Conditions
6 Live Load Distribution Factor (LDF)
7 Discussion of Results
Parametric studies were performed on straight MCB bridges (θ = 0) with two continuous spans. The live load stress and deflection distribution factors were obtained using FEA for various types of MCB bridges. The effects of various structural parameters on stress and the deflection distribution factors were investigated to identify the parameters affecting the load distribution factors under live loads. The following results were obtained.
7.1 Effect of the Number of Boxes
7.2 Effect of the Number of Lanes
7.3 Effect of Span Length
As the span length increases, the distribution factors for maximum positive (tensile) stresses increase but those for negative (compression) stresses decrease as shown in Figs. 10 through Fig. 12. For instance, it is shown from Fig. 12 that the increment of the span length from 30.5 to 91.5 m decreased LDFs for tensile and compressive stresses by up to 11% and 33%, respectively. Similarly, the LDFs for deflection also decreased as the span length of bridges increased. As shown in Fig. 11, LDFs for deflection decreased with increasing the span length from 45.75 to 91.5 m by 15%.
7.4 Comparison of Analytical Results with Current Specifications
8 Empirical Formulae for the Stresses and the Deflection
Since the key parameters determining the stress distribution factor for the two types of superstructure are similar, in this section appropriate correction factors are identified to modify Eqs. (5)–(7) in order to calculate \(D\sigma_{po}\), \(D\sigma_{ne}\) and \(D\delta_{s}\) for MCB bridges.
8.1 Positive Stress Distribution Factor
8.2 Negative (Compression) Stress Distribution Factor
8.3 Distribution Factor for Deflection
9 Verification of the Proposed Equations
Average, standard deviation and variance of the ratios for the DF equations.
10 Applicability of the Proposed Equations for Three and Four-Equal-Span Bridges
11 Conclusions
- 1.
The three-dimensional finite element modeling developed herein was verified with results of field and laboratory tests. It was concluded that the adopted modelling method are able to accurately estimate the elastic responses as well as free vibration characteristics such as mode shapes and natural frequencies for the bridges.
- 2.
The live load distribution factor for bending moment of AASHTO (2002) standard and AASHTO LRFD (2017) specifications were reviewed for applicability to MCB bridges. It was revealed that they obtain conservative values for tensile stresses and unconservative values for compressive stresses. Furthermore, these codes are unable to estimate the live load distribution factors for maximum deflection. The newly proposed live load distribution factor equations were developed for tensile and compressive stresses and deflection, which provided conservative results with respect to finite element analysis.
- 3.
Based on the results of parametric study, it was concluded that the span length, number of lanes and number of boxes are the most crucial parameters that could affect the load distribution factors of such bridges. The proposed parameters, therefore, were developed as a function of these key parameters.
- 4.
Empirical equations were derived for live load distribution factors of maximum tensile stresses at the bottom fiber of box-girders along the span, compressive stresses at the bottom fiber of box-girders at the pier, and deflection along the span of two-equal span MCB bridges. The proposed equations can be applied in the design of equal-span continuous bridges with number of spans up to four. They can be also applicable under AASHTO LRFD truck loading.
Notes
Authors’ contributions
IM performed a numerical study on MCB bridges; JK, WC and JP performed a theoretical study on data collected, and JK and IM suggested simplified methods using proposed equations to deduce proposed expressions for LDFs of MCB bridges. All authors read and approved the final manuscript.
Acknowledgements
The work reported here was supported by Grants (17CTAP-C132629-01, 17CTAP-C132633-01, 18CTAP-C144787-01) funded by the Ministry of Land, Infrastructure and Transport (MOLIT) of the Korean Agency for Infrastructure Technology Advancement (KAIA). This financial support is gratefully acknowledged.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The data used to support the findings of this study are available from the corresponding author upon request.
Ethics approval and consent to participate
For this type of study formal consent is not required.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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