# Live Load Distribution Factors for Skew Stringer Bridges with High-Performance-Steel Girders under Truck Loads

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## Abstract

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## 1. Introduction

_{y}= 345 MPa), HPS 70 W (F

_{y}= 485 MPa) and HPS 100 W (F

_{y}= 620 MPa), which became available commercially and globally in recent years, provide higher durability and strength, as well as boosted weldability. This can lead to considerable cost savings for big construction projects. For instance, Tennessee’s Department of Transportation reduced the cost of building a highway bridge by nearly 17% by using HPS [6]. HPS can also be highly beneficial when designing and constructing bridges with longer and shallower spans that exceed the range of applicability defined by the current specifications [1,7]. The critical issue limiting the wider use of HPS is our poor understanding of precisely how live loads are redistributed along a bridge’s transverse and longitudinal directions and in the girders of this type of bridge under realistic vehicle loads.

^{0}) for the obtuse corners but un-conservative values (approximately 19%) for the acute corners of skewed superstructures. They, therefore, developed a set of correct factor expressions to obtain LDFs for reactions from the corresponding shear LDFs for multicell box-girder bridges.

## 2. Finite Element Modeling and Verification

#### 2.1. Bridge Section

#### 2.2. Verification of Finite Element Models

#### 2.2.1. Laboratory Tests at the Turner-Fairbank Highway Research Center

#### 2.2.2. Laboratory Test of a Quarter Scale Model Bridge

## 3. Bridge Superstructure Database

_{g}). The skew angle (θ) at the abutment and piers increased across a range from 0° (non-skew bridges) to 60° at 15° intervals. Three deck widths were selected for this parameter study: 9.5, 13 and 15 m. The number of lane loads (N

_{L}) was taken to be 2 to 3 lanes for the bridges with total widths of 9 m and 13 m, and 2 to 4 lanes for bridges with a total width of 15 m, as shown in Table 2.

_{y}= 345 MPa), 70 W (F

_{y}= 485 MPa), and 100 W (F

_{y}= 620 MPa), were adopted for this study. Steel X-type diaphragms composed of L12 × 12 × 0.8 space at 5 m intervals along each span, and at the abutments and piers were modeled. The concrete deck had a modulus of elasticity of 28,000 MPa, a Poisson’s ratio of 0.2, and a density of 24 kN/m

^{3}.

## 4. Sensitivity Analysis

#### 4.1. Effect of Span Length

#### 4.2. Effect of Girder Spacing

#### 4.3. Effect of Number of Lane Loads

#### 4.4. Effect of the Skew Angle

## 5. Development of New Equations for the Live Load Distribution Factors

_{T}is the LDF equation deduced by Tarhini and Frederik [15] for a non-skewed I-girder bridge and is as follows;

## 6. Verification of the Proposed Equations

^{2}, ranged from 0.899 to 0.930, indicating that the variation in the data falls with the acceptable range. In particular, the low value of R

^{2}obtained for the ratio of AASHTO LRFD to FE results indicates that the current specification is unable to predict the LDFs for this type of bridge correctly. Table 3 shows the statistical results for the average (AVG.), standard division (SD.), and coefficient of variation (COV.) for the ratio of the results obtained using the proposed equations to those from the finite element analysis. The slightly-greater-than unity average obtained for the regression analysis reveals that the newly developed equations are indeed able to predict the LDFs for shear force and bending moment conservatively. The SDs listed in Table 3 range from 0.069 to 0.094 and the CVs are between 0.067 to 0.085, both of which confirm that the proposed equations can be used to determine LDFs for skewed bridges with HPS girders.

## 7. Conclusions and Recommendations

- A good agreement in the structural responses was achieved between the 3-dimensional modeling of the prototype bridges and the results obtained experimentally, confirming that numerical models can reliably predict the responses of slab-on-girder bridges.
- The discrepancy between the finite element results and those calculated using the codified AASHTO LRFD equations revealed that the current LRFD specifications are not suitable for predicting the live load distribution factors for both the bending moment and the shear force for skewed composite bridge with HPS girders. It was therefore necessary to develop a new set of LDF equations for both shear and moment.
- Based on the results of the parametric study on prototype bridges, the span length, girder spacing, number of lane loaded, and skew angle were identified as the key parameters affecting the LDFs of skewed composite bridges. The LDFs for both shear force and bending moment decreased with increasing span length and number of lanes, and increased with increased girder spacing. Increasing the skew angle of the bridge superstructures increased the LDFs for shear force but decreased those for bending moment.
- Based on the statistical analysis, conducted for this study, a set of simplified expressions were developed for the LDFs for both shear force and bending moment. The slightly greater than unity average and low standard deviation and coefficient of variation for each of the proposed expressions indicate high reliabilities for these proposed expressions in estimating the LDFs for shear force and bending moment of skewed composite bridge with HPS girders.
- More studies can be carried to assess the dynamic interaction of these type of bridge and moving load due to traffic conditions. The simplified equations can be derived to determine dynamic impact factor of bridges with HPS steel.
- Analytical and computational approaches to study the seismic response characteristics of bridges are the most economically feasible methods. It is particularly important to investigate the performance of skewed bridges with HPS steel due to vertical ground motions. The study, therefore, would provide comprehensive results through including all the parameters interacting for a wide range of skew angles.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Scaled bridge constructed for the laboratory tests at Turner–Fairbank Highway Research Center: (

**a**) plan view; (

**b**) cross-section; (

**c**) location of instruments; (

**d**) loading conditions.

**Figure 4.**Strain at flanges and web of G3 from Test and FE analyses: (

**a**) flange width; (

**b**) web height.

**Figure 5.**Deflection distribution at the girders along the mid-span cross section of the bridge: (

**a**) cross section and longitudinal view; (

**b**) deflection distribution factor.

**Figure 6.**Standard American Iron and Steel Institute (AISI) parapet used for the bridge models (in meter).

**Figure 7.**Effect of key parameters on bridge Load Distribution Factors (LDFs): (

**a**) span length; (

**b**) girder spacing.

**Figure 9.**Proposed LDF equations and American Association of State Highway and Transportation Officials (AASHTO) Load and Resistance Factor Design (LRFD)equations vs. Rigorous LDF. (

**a**) bending moment of external girder; (

**b**) bending moment of internal girder; (

**c**) shear force of external girder; (

**d**) shear force of internal girder.

Girder | Vertical Defl. (cm) | Web Rotat. (Degree) | ||||
---|---|---|---|---|---|---|

FE | Test | Error | FE | Test | Error | |

G1 | 0.47 | 0.51 | 6.2 | 0.18 | 0.19 | 5.5 |

G2 | 1.51 | 1.63 | 7.9 | 0.21 | 0.20 | 4.7 |

G3 | 2.83 | 2.69 | 4.9 | 0.33 | 0.31 | 6.10 |

Set | L (m) | HPS (w) | L/D | N_{L} | N_{g} | S (m) | W (m) | θ (Deg.) |
---|---|---|---|---|---|---|---|---|

1 | (30, 45, 60, 75, 90, 105) | (50, 70, 100) | 20 | 2,3 | 3,4,5 | (2, 2.5, 3.0, 3.5, 4) | 9.5 | (0, 15, 30, 45, 60, 75) |

2 | 25 | 2,3 | 3,4 | 13 | ||||

3 | 30 | 2,3 | 3,4 | 13 | ||||

4 | 25 | 2,3,4 | 3,4,5 | 15 |

LDF | Girder Type | AVG. | SD. | COV. |
---|---|---|---|---|

MDF_{ex} | External | 1.065 | 0.076 | 0.071 |

MDF_{in} | Internal | 1.045 | 0.069 | 0.067 |

CDF_{ex} | External | 1.072 | 0.082 | 0.076 |

CDF_{in} | Internal | 1.059 | 0.094 | 0.085 |

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**MDPI and ACS Style**

Mohseni, I.; Cho, Y.K.; Kang, J.
Live Load Distribution Factors for Skew Stringer Bridges with High-Performance-Steel Girders under Truck Loads. *Appl. Sci.* **2018**, *8*, 1717.
https://doi.org/10.3390/app8101717

**AMA Style**

Mohseni I, Cho YK, Kang J.
Live Load Distribution Factors for Skew Stringer Bridges with High-Performance-Steel Girders under Truck Loads. *Applied Sciences*. 2018; 8(10):1717.
https://doi.org/10.3390/app8101717

**Chicago/Turabian Style**

Mohseni, Iman, Yong Kwon Cho, and Junsuk Kang.
2018. "Live Load Distribution Factors for Skew Stringer Bridges with High-Performance-Steel Girders under Truck Loads" *Applied Sciences* 8, no. 10: 1717.
https://doi.org/10.3390/app8101717