# Reliability Assessment Approach for Fire Resistance Performance of Prestressed Steel–Concrete Box Girder Bridges

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Principles of Prestressed Concrete Fire

#### 2.1. Thermal Performance of Materials

#### 2.1.1. Concrete

- (1)
- Thermal conductivity

- (2)
- Specific heat capacity

- (3)
- Thermal expansion coefficient

#### 2.1.2. Thermal Performance of Prestressed Steel Bars

- (1)
- Thermal conductivity

- (2)
- Specific heat capacity and density

^{3}.

- (3)
- Thermal expansion coefficient

#### 2.2. Temperature Transient Analysis

_{0}is the starting point temperature, and T is the duration of the fire.

#### Fire Temperature Field of Box Girder Section

#### 2.3. ANSYS Finite Element Temperature Field Analysis

- (1)
- Define the model: Determine the geometric dimensions, physical properties, and boundary conditions in the model, such as the size of the small box girder model and the position and quantity of prestressed steel bars. The three-dimensional thermal solid SOLID70 element can be used to simulate concrete in ANSYS temperature field analysis, with eight nodes and temperature degrees of freedom assigned to each node.
- (2)
- Develop mathematical models and assumptions for the model: Determine the mathematical model required to calculate the temperature field of the small box girder model using physical equations, taking into consideration heat transfer mechanisms such as radiation, conduction, and convection. Simplify the model using assumptions, such as assuming that the physical property constant of the small box girder is constant.
- (3)
- Determine boundary conditions: Determine boundary conditions, including initial temperature, fire conditions, material properties, and environmental conditions. The initial temperature is 20 °C according to the international standard organization IS0834 heating function.
- (4)
- Choose a numerical method to solve the mathematical model: Usually, the finite element method is employed to numerically calculate the temperature field of the small box girder model.
- (5)
- Calculation: Based on the mathematical model and boundary conditions, perform numerical calculations to calculate the temperature field of the small box girder model.

## 3. Reliability Theory

**S**and

**X**can be expressed as follows:

**S**=

**S**(

**X**).

**g**(s, x) is obtained as follows:

**s**,

**x**) to

**s**; ${\nabla}_{x}g$ is the gradient of limit state function g (

**s**,

**x**) to

**x**; ${J}_{u,x}^{}$ is the Jacobian matrix for probability transformation; ${J}_{s,x}^{}$ is the Jacobian matrix for mechanical transformation.

- (1)
- Establishment of structural finite element model: First, it is necessary to establish a finite element model of the structure based on the geometric model and material characteristics of the actual structure, including nodes, elements, constraint conditions, loads, etc.
- (2)
- Analysis of parameter uncertainty: Second, the researcher must describe the probability distribution of structural design parameters, such as mean and standard deviation, as well as analyze the sources of uncertainty, including measurement errors, manufacturing errors, changes in material parameters, etc.
- (3)
- Selection of reliability indicators: Third, it is necessary to determine the reliability indicators of the structure based on engineering requirements and design specifications, such as reliability indicators, failure efficiency indicators, safety factors, etc.
- (4)
- Reliability calculation: Then, the scholar must apply reliability theory and finite element method to conduct structural reliability analysis, calculate the reliability indicators of the structure, and the probability distribution of other parameters in the reliability analysis.
- (5)
- Sensitivity analysis: Fifth, it is necessary to analyze the sensitivity of parameter uncertainty in relation to reliability indicators and determine the parameters that have the greatest impact on structural reliability.
- (6)
- Optimization design: Based on the sensitivity analysis results, researchers should optimize the design scheme of the structure to improve its reliability indicators.
- (7)
- Result evaluation: Evaluate the analysis results to determine whether the reliability indicators meet the design requirements. If not, it is necessary to perform repeated calculations and optimization.

## 4. Finite Element Reliability Fire Resistance Analysis

- (1)
- Determine the design load and fire scenario: Based on the design load and environment of the bridge, determine the fire scenario of the bridge during a fire and assess the size of the fire, thermal radiation intensity, temperature changes, etc.
- (2)
- Determine material properties: Based on the design drawings and component material information of the bridge, determine the basic mechanical properties and fire resistance parameters of materials such as concrete and steel bars, as well as the changes in material mechanical properties under fire conditions.
- (3)
- Establish a mechanical model: Based on the structural and mechanical characteristics of the bridge, establish a mechanical model of the bridge under fire conditions, taking into account factors such as temperature changes and nonlinear behavior of the structure, including load displacement and stress–strain relationships.
- (4)
- Establish limit state equation: Based on the design load and mechanical model under fire scenarios, establish the limit state equation for bridge fire resistance, including strength limit state and deformation limit state.
- (5)
- Verification and optimization: Verify the established bridge fire resistance limit state equation through numerical simulation, experimental verification, and other methods, and optimize and adjust parameters as required.

## 5. Application

#### 5.1. Project Overview

#### 5.2. Finite Element Model

^{3}. Meanwhile, the constitutive relationships of thermal parameters and thermal coupling constitutive for steel under high-temperature conditions are shown in Table 3, Table 4 and Table 5, with density taken as ρ = 7850 kg/m

^{3}. The temperature field model uses SOLID70 elements for concrete and LINK33 elements for steel bars and steel strands. The fire resistance calculation model uses SOLID65 elements for concrete and LINK8 elements for steel bars and steel strands. The grid size is set to 10 mm.

#### 5.3. Structural Response of Prestressed Concrete Beam Bridges after Fire

#### 5.4. Statistical Analysis of Structural Resistance of Prestressed Concrete Beam Bridges after Fire

- (1)
- Sample of prestressed concrete after 15 min of fire

- (2)
- Sample of prestressed concrete after 30 min of fire

- (3)
- Sample of prestressed concrete after 60 min of fire

#### 5.5. Reliability Evaluation of Prestressed Concrete Beam Bridges after Fire

#### 5.6. Parameter Sensitivity Analysis

- (1)
- The Influence of Random Variable Mean on Reliability Index and Probability Safety Factor

- (2)
- The Influence of Random Variable Variation Coefficient on Reliability Index and Probabilistic Safety Factor

- (3)
- The Influence of the Target Reliability Index on the Probability Safety Factor

## 6. Conclusions

- (1)
- We conducted a study into the fire response of prestressed concrete beam bridges. Based on the nonlinear finite element analysis of the temperature field of the box girder section during a fire and the high-temperature mechanical performance analysis of prestressed steel–concrete box girder bridges, a method for analyzing the fire resistance performance of prestressed concrete beam bridges was established, laying the foundation for the subsequent reliability evaluation of the fire resistance performance of prestressed steel–concrete beam bridges.
- (2)
- A fire resistance reliability model for prestressed concrete continuous beam bridges was established. The main influencing factors on the fire resistance performance of prestressed concrete beam bridges were summarized through statistical research, including the high-temperature characteristics of reinforced concrete components, the strength reduction of steel and concrete after fire, the bonding strength of steel and concrete after high-temperature exposure, and the resistance performance of prestressed concrete beams.
- (3)
- A reliability analysis method was proposed to assess the fire resistance performance of prestressed concrete beam bridges. On the basis of clarifying the factors that affect the fire resistance performance of prestressed concrete beam bridges, a reliability model for evaluating the fire resistance performance of prestressed concrete beam bridges after a fire was constructed. By combining reliability theory with the finite element method, a reliability analysis method for the fire resistance performance of prestressed concrete beam bridges was proposed.
- (4)
- Based on the analysis of the structural response after a fire in a specific engineering case of a simply supported to continuous prestressed concrete continuous beam bridge, a uniform design method was used to generate structural resistance samples of the prestressed concrete beam bridge and statistical analysis was conducted. Subsequently, probability methods were used to evaluate the safety of the prestressed concrete beam bridge after a fire.
- (5)
- Using parameter sensitivity analysis of the reliability index and probabilistic safety factors of mean value and coefficient of variation, we concluded that the randomness of parameters exerts a significant impact on the safety reserve of prestressed concrete beam bridges after exposure to fire. In particular, the section height and steel bar parameters affect the fire resistance performance of the structure. The related discreteness of these two parameters exerts a very significant impact on the reliability and safety of the structure.
- (6)
- By analyzing the time-varying characteristics of fire resistance, it was determined that the fire duration exerts a significant impact on the structural performance of prestressed concrete beam bridges. It is necessary to pay attention to this factor in specific engineering practices and strengthen the monitoring and statistics of structural random characteristics. The variability of parameters related to target reliability indicators has a significant impact on structural safety assessment, especially the thickness of concrete cover and the statistical characteristics of steel reinforcement. Both of these should be strictly controlled during the design and construction process.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Wang, G.; Wei, Y.; Zhang, Y.; Lin, L.; Lin, Y. Experimental behavior of concrete-filled double-skin tubular columns with outer galvanized corrugated steel tubes under axial compression. Eng. Struct.
**2023**, 295, 116856. [Google Scholar] [CrossRef] - Zhao, Z.; Wei, Y.; Wang, G.; Zhang, Y.; Lin, Y. Axial compression performance of square UHPC-filled stainless-steel tubular columns. Constr. Build. Mater.
**2023**, 408, 133622. [Google Scholar] [CrossRef] - Wei, B.; Wei, Y.; Lin Yu Wang, G.; Zhang, Y. Compressive performance of bamboo scrimber and concrete-filled steel tube columns. Eng. Struct.
**2023**, 300, 117192. [Google Scholar] [CrossRef] - Wu, F.; Wei, Y.; Lin, Y.; Zhao, K.; Huang, L. Experimental study of bamboo scrimber-filled steel tube columns under axial compression. Eng. Struct.
**2023**, 280, 115669. [Google Scholar] [CrossRef] - Li, H.; Wei, Y.; Yan, L.; Katherine, E.; Semple Dai, C. Characterization of local compressive behavior for bamboo scrimber loaded perpendicular to the grain. Constr. Build. Mater.
**2023**, 397, 132421. [Google Scholar] [CrossRef] - Wei, Y.; Nie, Y.; Lin, Y.; Shen, D.; Wang, G. Axial stress-strain behaviour of bamboo composite tube-confined recycled aggregate concrete with different aggregate replacement ratios. Wood Mater. Sci. Eng.
**2023**, 4, 2229275. [Google Scholar] [CrossRef] - Ellobody, E.; Bailey, C.G. Modelling of Unbonded Post-tensioned Concrete Slabs under Fire Conditions. Fire Saf. J.
**2009**, 44, 159–167. [Google Scholar] [CrossRef] - Dwaikat, M.B.; Kodur, V.K.R. Hydrothermal model for predicting fire-induced spalling in concrete structural systems. Fire Saf. J.
**2009**, 44, 425–434. [Google Scholar] [CrossRef] - Yu, H.; Burgess, I.W.; Davison, J.B.; Plank, R.J. Tying capacity of web cleat connections in fire, Part 1: Test and finite element simulation. Eng. Struct.
**2009**, 31, 651–663. [Google Scholar] [CrossRef] - Bailey, C.G.; Ellobody, E. Fire tests on bonded post-tensioned concrete slabs. Eng. Struct.
**2009**, 31, 686–696. [Google Scholar] [CrossRef] - Liu, C.; Huang, J.S. Fire performance of highly flowable re active powder concrete. Constr. Build. Mater.
**2009**, 23, 2072–2079. [Google Scholar] [CrossRef] - Bažant, Z.P.; Cusatis, G.; Cedolin, L. Temperature Effect on Concrete Creep Modeled by Microprestress-Solidification heory. J. Eng. Mech.
**2004**, 130, 691–699. [Google Scholar] [CrossRef] - Hasofer, A.M.; Thomas, I. Analysis of fatalities and injuries in building fire statistics. Fire Saf. J.
**2006**, 41, 2–14. [Google Scholar] [CrossRef] - Lin, Y.S. Estimations of the probability of fire occurrences in buildings. Fire Saf. J.
**2005**, 40, 728–735. [Google Scholar] [CrossRef] - Arockiasamy, M.; Sivakumar, P.E. Time-Dependent Behavior of Continuous Composite Integral Abutment Bridges. Pract. Period. Struct. Des. Constr.
**2005**, 10, 161–170. [Google Scholar] [CrossRef] - Felicetti, R. The Drilling Resistance Test for the Assessment of Fire Damaged Concrete. Cem. Concr. Compos.
**2006**, 28, 321329. [Google Scholar] [CrossRef] - Abbasi, A.; Hogg, P.J. Fire Testing of Concrete Beams with Fibre Reinforced Plastic Rebar. Compos. Part A
**2006**, 37, 11421150. [Google Scholar] [CrossRef] - Herrtz, K.D.; Srensen, L.S. Test Method for Spalling of Fire exposed Concrete. Fire Saf. J.
**2005**, 40, 466476. [Google Scholar] [CrossRef] - Li, M.; Qian, C.X.; Sun, W. Mechanical Properties of High-strength Concrete after Fire. Cem. Concr. Res.
**2004**, 34, 1001–1005. [Google Scholar] [CrossRef] - Abbasi, A.; Hogg, P.J. A Model for Predicting the Properties of the Constituents of a Glass Fibre Rebar Reinforced Concrete Beam at Elevated Temperatures Simulating a Fire Test. Compos. Part B
**2005**, 36, 384–393. [Google Scholar] [CrossRef] - Yin, J.; Zha, X.X.; Li, L.Y. Fire Resistance of Axially Loaded Concrete Filled Steel Tube Columns. J. Constr. Steel Res.
**2006**, 62, 723–729. [Google Scholar] [CrossRef] - Sakr, K.; EL-Hakim, E. Effect of High Temperature or Fire on Heavy Weight Concrete Properties. Cem. Concr. Res.
**2005**, 35, 590596. [Google Scholar] [CrossRef] - Han, L.H.; Huo, J.S.; Wang, Y.C. Compressive and Flexural Behaviour of Concrete Filled Steel Tubes after Exposure to Standard Fire. J. Constr. Steel Res.
**2005**, 61, 882901. [Google Scholar] [CrossRef] - Bratina, S.; Saje, M.; Planinc, I. Numerical Modelling of Behaviour of Reinforced Concrete Columns in Fire and Comparison with Eurocode 2. Int. J. Solids Struct.
**2005**, 42, 5715–5733. [Google Scholar] [CrossRef] - Williams, B.; Bisby, L.; Kodur, V.; Green, M.; Chowdhury, E. Fire Insulation Schemes for FRP-strengthened Concrete Slabs. Compos. Part A
**2006**, 37, 1151–1160. [Google Scholar] [CrossRef] - Usmani, A.S.; Cameron, N.J.K. Limit capacity of laterally restrained reinforced concrete floor slabs in fire. Cem. Concr. Compos.
**2004**, 26, 127140. [Google Scholar] [CrossRef] - Gustaferro, A.H. Design of prestressed concrete for fire resistance. J. Prestress. Concr. Inst.
**1973**, 18, 102116. [Google Scholar] [CrossRef] - Ashton, L.A.; Bate, S.C.C. Fire resistance of prestressed concrete beams. J. ACI
**1961**, 32, 1417–1440. [Google Scholar] [CrossRef] - Gustaferro, A.H.; Selvaggio, S.L. Fire endurance of simply-supported prestressed Concrete slabs. J. Prestress. Concr. Inst.
**1967**, 12, 3752. [Google Scholar] [CrossRef] - Abrams, M.S.; Gustaferro, A.H. Fire endurance of prestressed Concrete units coated with spray-applied insulation. J. Prestress. Concr. Inst.
**1972**, 17, 82103. [Google Scholar] [CrossRef] - Krishnamoorthy, S.; England, G.L.; Yu, C.W. The behaviour of prestressed Concrete Portal Frames Influenced by Creep and Temperature. Mag. Concr. Res.
**1971**, 23, 23–36. [Google Scholar] [CrossRef] - Ellingwood, B.; Lin, T.D. Flexure and Shear Behavior of Concrete Beams during Fires. J. Struct. Eng. ASCE
**1991**, 117, 44045814. [Google Scholar] [CrossRef] - Cox, G. Fire research in the 21 st Century. Fire Saf. J.
**1999**, 32, 203–219. [Google Scholar] [CrossRef] - Roylesandp, R.; Morley, D. Futher response of the bondin reinforced concrete to high temperatures. Concr. Res.
**1983**, 35, 157–163. [Google Scholar] - Poon, C.S.; Shui, Z.H.; Lam, L. Compressive behavior of fiber reinforeed high-performance conerete subjeeted to elevated temperatures. Cem. Conerete Res.
**2004**, 34, 2215–2222. [Google Scholar] [CrossRef] - Chiang, C.H.; Tsai, C.L. Time-temperatures analysis of bond strength of are bar after fire exposure. Cem. Conerete Res.
**2003**, 33, 1651–1654. [Google Scholar] [CrossRef] - Bailey, C.G.; Lennonand, T.; Moore, D.B. The behavior rof full-wale steel-frame building subjected to compatment. Struct. Eng.
**1999**, 77, 15-2IP. [Google Scholar] - Gambhlr, M.L.; Singh, J. Behavior of axially-loaded RC colunms during fire. Indian Concr. J.
**1995**, 5, 241–246. [Google Scholar] - Lie, T.T.; Irwin, R.J. Fire resistance of retangular steel columns filled with bar-reinforced concrete. J. Struct. Eng.
**1995**, 121, 797–805. [Google Scholar] [CrossRef] - Commission of European Communities. Design of Concrete Structure, Eurocode 2 Part 10: Structure Fire Design; The Concrete Societies of the UK: Berkshire, UK, 1996.
- BS5950-8-2003; Structural Use of Steelwork in Building, Part 8: Code of Practice for Fire Resistance Design. Steel Construction Institution: Chicago, IL, USA, 2003.
- Ellingwood, B.; Lin, T.D. Simplified design of fire exposed concrete beams and columns, Flexure and Shear Behavior of Concrete Beam-s During Fire. ASCE J. Struct. Eng.
**1991**, 117, 440–457. [Google Scholar] [CrossRef] - Shi, X.; Tan, T.H.; Tan, K.H.; Guo, Z. Effect of force-temperature paths on behaviors of reinforced concrete flexural members. ASCE J. Struct. Eng.
**2002**, 108, 365–373. [Google Scholar] [CrossRef] - Kodur, V.K.R.; Dwaikat, M. A numerical model for predicting the fire resistance of concrete beams. Cem. Concr. Compos.
**2008**, 30, 431–443. [Google Scholar] [CrossRef]

Temperature/°C | 20 | 100 | 200 | 300 | 400 | 500 | 600 | 800 | 1000 | 1200 |

Thermal conductivity | 1.62 | 1.53 | 1.43 | 1.34 | 1.22 | 1.11 | 1.02 | 0.86 | 0.72 | 0.64 |

**Table 2.**The value of thermal expansion coefficient of concrete (unit: [$\mathrm{m}/(\mathrm{m}\xb7\xb0\mathrm{C})$]).

Temperature/°C | 20 | 100 | 200 | 300 | 400 | 500 | 600 | 800 | 1000 | 1200 |

Thermal expansion coefficient | 5.6 | 6.5 | 7.7 | 8.9 | 10.1 | 11.3 | 12.5 | 14.9 | 17.3 | 19.7 |

**Table 3.**Value of thermal conductivity of steel bar (unit: [$\mathrm{W}/(\mathrm{m}\xb7\xb0\mathrm{C})$]).

Temperature/°C | 20 | 100 | 200 | 300 | 400 | 500 | 600 | 800 | 1000 | 1200 |

Thermal conductivity | 49 | 47 | 45 | 43 | 41 | 38 | 35 | 29 | 22 | 19 |

**Table 4.**Specific heat capacity of steel bar (unit: [$\mathrm{J}/(\mathrm{k}\mathrm{g}\xb7\xb0\mathrm{C})$]).

Temperature/°C | 20 | 100 | 200 | 300 | 400 | 500 | 600 | 800 | 1000 | 1200 |

Thermal conductivity | 520 | 527 | 541 | 561 | 586 | 618 | 656 | 748 | 865 | 1005 |

**Table 5.**The value of thermal expansion coefficient of prestressed steel bar (unit: [$\mathrm{m}/(\mathrm{m}\xb7\xb0\mathrm{C})$]).

Temperature/$\xb0\mathrm{C}$ | 20 | 100 | 200 | 300 | 400 | 500 | 600 | 800 | 1000 | 1200 |

Thermal expansion coefficient | 11.4 | 12.0 | 12.8 | 13.6 | 14.4 | 15.2 | 15.3 | 16.1 | 17.0 | 17.8 |

**Table 6.**The value of comprehensive heat transfer coefficient (unit: [$\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}/(\mathrm{m}\xb7\mathrm{h}\xb7\xb0\mathrm{C})$]).

Flame temperature/$\xb0\mathrm{C}$ | 60–200 | 400 | 500 | 600 | 800 | 1000 | 1200 |

Thermal expansion coefficient | 10 | 15 | 20 | 30 | 55 | 90 | 150 |

**Table 7.**Statistical parameters for structural resistance calculation of prestressed concrete beam bridge.

Random Variables | Distribution Type | Mean Value | Standard Deviation | Coefficient of Variation |
---|---|---|---|---|

Section width | Normal distribution | 1.00 | 0.01 | 0.01 |

Section height | Normal distribution | 1.01 | 0.02 | 0.02 |

Concrete strength | Normal distribution | 1.39 | 0.19 | 0.14 |

Calculation mode | Normal distribution | 1.10 | 0.08 | 0.07 |

Area of prestressed steel bars | Normal distribution | 1.00 | 0.01 | 0.01 |

Strength of prestressed steel bars | Normal distribution | 1.08 | 0.13 | 0.12 |

Dead load effect | Normal distribution | 1.00 | 0.04 | 0.04 |

Live load | Gumbel distribution | 1.00 | 0.18 | 0.18 |

1 | 8 | 2 | 17 | 16 | 19 |

2 | 12 | 21 | 8 | 4 | 7 |

3 | 21 | 20 | 27 | 20 | 26 |

4 | 24 | 15 | 5 | 27 | 10 |

5 | 3 | 29 | 21 | 11 | 17 |

6 | 15 | 11 | 20 | 24 | 2 |

7 | 18 | 7 | 2 | 7 | 24 |

8 | 27 | 24 | 24 | 6 | 14 |

9 | 4 | 5 | 11 | 21 | 8 |

10 | 29 | 13 | 13 | 12 | 30 |

11 | 14 | 26 | 14 | 30 | 12 |

12 | 22 | 9 | 28 | 10 | 6 |

13 | 6 | 23 | 6 | 18 | 29 |

14 | 10 | 16 | 29 | 1 | 20 |

15 | 28 | 6 | 22 | 29 | 22 |

16 | 19 | 30 | 12 | 15 | 1 |

17 | 2 | 12 | 4 | 5 | 15 |

18 | 1 | 18 | 16 | 25 | 23 |

19 | 25 | 4 | 15 | 2 | 4 |

20 | 26 | 28 | 3 | 23 | 18 |

21 | 13 | 3 | 25 | 14 | 18 |

22 | 7 | 25 | 26 | 26 | 5 |

23 | 20 | 1 | 7 | 19 | 13 |

24 | 16 | 27 | 18 | 3 | 27 |

25 | 9 | 17 | 1 | 13 | 3 |

26 | 30 | 19 | 19 | 17 | 9 |

27 | 11 | 10 | 9 | 28 | 25 |

28 | 5 | 8 | 23 | 8 | 11 |

29 | 23 | 22 | 10 | 9 | 21 |

30 | 17 | 14 | 30 | 22 | 16 |

$\mathit{\alpha}=0.05$ | Normal Distribution | Log-Normal Distribution | Gumbel Distribution |
---|---|---|---|

${D}_{n}$ | 0.1503 | 0.1366 | 0.2063 |

${D}_{n}^{\alpha}$ | 0.242 | 0.242 | 0.163 |

Accept/reject | Accept | Accept | Reject |

${k}_{i}$ | 0.6217 | 0.5591 | - |

$\mathit{\alpha}=0.05$ | Normal Distribution | Log-Normal Distribution | Gumbel Distribution |
---|---|---|---|

${D}_{n}$ | 0.1321 | 0.1105 | 0.1899 |

${D}_{n}^{\alpha}$ | 0.242 | 0.242 | 0.163 |

Accept/reject | Accept | Accept | Reject |

${k}_{i}$ | 0.5869 | 0.4921 | - |

$\mathit{\alpha}=0.05$ | Normal Distribution | Log-Normal Distribution | Gumbel Distribution |
---|---|---|---|

${D}_{n}$ | 0.1321 | 0.0988 | 0.1799 |

${D}_{n}^{\alpha}$ | 0.242 | 0.242 | 0.163 |

Accept/reject | Accept | Accept | Reject |

${k}_{i}$ | 0.6314 | 0.5736 | - |

Parameter | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|

Reliability index | 5.2772 | 5.1031 | 4.9917 | 4.4732 |

Deterministic safety factor | 4.2901 | 3.9982 | 3.8871 | 3.6728 |

Probabilistic safety factor | 3.9827 | 3.7872 | 3.6279 | 3.4821 |

Parameter | Mean Value | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.9 | 5.1928 | 5.0018 | 4.7829 | 4.0192 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

1.1 | 5.3817 | 5.2938 | 5.1029 | 4.8372 | |

Probabilistic safety factor | 0.9 | 3.7182 | 3.6728 | 3.5782 | 3.3928 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

1.1 | 4.1029 | 3.8172 | 3.7292 | 3.6172 |

Parameter | Mean Value | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.9 | 5.1928 | 5.0018 | 4.8982 | 4.3827 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

1.1 | 5.2932 | 5.1827 | 5.1019 | 4.5627 | |

Probabilistic safety factor | 0.9 | 3.8272 | 3.6729 | 3.5827 | 3.3928 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

1.1 | 4.1029 | 3.8472 | 3.7182 | 3.5728 |

Parameter | Mean Value | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.9 | 4.9182 | 4.8271 | 4.8271 | 4.1029 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

1.1 | 5.4982 | 5.3919 | 5.2109 | 4.9828 | |

Probabilistic safety factor | 0.9 | 3.8271 | 3.6279 | 3.5826 | 3.2647 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

1.1 | 4.1029 | 3.8271 | 3.7463 | 3.6274 |

**Table 16.**Effect of the mean uncertainty of the calculation mode on reliability indicators and probabilistic safety factors.

Parameter | Mean Value | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.9 | 5.1716 | 5.0187 | 4.8721 | 4.2761 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

1.1 | 5.3928 | 5.1982 | 5.2817 | 4.5261 | |

Probabilistic safety factor | 0.9 | 3.7261 | 3.6251 | 3.5627 | 3.2817 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

1.1 | 4.2817 | 3.8172 | 5.7162 | 3.6581 |

**Table 17.**Effect of mean area of prestressed reinforcement on reliability index and probabilistic safety factor.

Parameter | Mean Value | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.9 | 5.1722 | 5.0018 | 4.8271 | 4.1029 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

1.1 | 5.3627 | 5.2171 | 5.0271 | 4.7182 | |

Probabilistic safety factor | 0.9 | 3.8172 | 3.6273 | 3.5182 | 3.2019 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

1.1 | 4.1029 | 3.7298 | 3.7182 | 3.6271 |

**Table 18.**Effect of average strength of prestressed steel bars on reliability indicators and probabilistic safety factors.

Parameter | Mean Value | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.9 | 4.9182 | 4.8172 | 4.6172 | 4.2018 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

1.1 | 5.4817 | 5.4716 | 5.3716 | 4.7162 | |

Probabilistic safety factor | 0.9 | 3.7162 | 3.6172 | 3.5827 | 3.3928 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

1.1 | 4.1028 | 3.8172 | 3.7162 | 3.5102 |

Parameter | Mean Value | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.9 | 5.3817 | 5.2817 | 5.1829 | 4.8172 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

1.1 | 5.1928 | 5.0182 | 4.8271 | 4.0192 | |

Probabilistic safety factor | 0.9 | 4.1029 | 3.9182 | 3.7172 | 3.6571 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

1.1 | 3.7162 | 3.6721 | 3.5817 | 3.2492 |

Parameter | Mean Value | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.9 | 5.4271 | 5.3281 | 5.1082 | 4.7168 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

1.1 | 5.2091 | 4.9281 | 4.7821 | 4.2918 | |

Probabilistic safety factor | 0.9 | 4.1029 | 3.8719 | 3.7162 | 3.6152 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

1.1 | 3.8721 | 3.6271 | 3.5721 | 3.2481 |

**Table 21.**Effect of section width variation coefficient on reliability index and probabilistic safety factor.

Parameter | Coefficient of Variation | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.5 | 5.3726 | 5.2918 | 5.1928 | 4.6872 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

2.0 | 5.1827 | 5.0182 | 4.8271 | 4.2817 | |

Probabilistic safety factor | 0.5 | 4.1028 | 3.8232 | 3.8172 | 3.6716 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

2.0 | 3.8271 | 3.6251 | 3.4726 | 3.2817 |

**Table 22.**Impact of section height variation coefficient on reliability index and probabilistic safety factor.

Parameter | Coefficient of Variation | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.5 | 5.3716 | 5.1726 | 5.1029 | 4.7162 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

2.0 | 5.0192 | 4.9181 | 4.8172 | 4.3716 | |

Probabilistic safety factor | 0.5 | 4.1028 | 3.8172 | 3.7164 | 3.7162 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

2.0 | 3.7172 | 3.6527 | 3.5627 | 3.1726 |

**Table 23.**Effect of concrete strength variation coefficient on reliability index and probabilistic safety factor.

Parameter | Coefficient of Variation | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.5 | 5.3918 | 5.1928 | 5.0182 | 4.5263 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

2.0 | 5.1928 | 5.0819 | 4.7182 | 4.3617 | |

Probabilistic safety factor | 0.5 | 4.1827 | 3.6172 | 3.7179 | 3.7162 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

2.0 | 3.7162 | 3.6521 | 3.5728 | 3.1928 |

**Table 24.**Effect of calculation mode uncertainty variation coefficient on reliability index and probabilistic safety factor.

Parameter | Coefficient of Variation | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.5 | 5.3198 | 5.2183 | 5.1029 | 4.7162 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

2.0 | 5.1827 | 5.2771 | 4.8172 | 4.5162 | |

Probabilistic safety factor | 0.5 | 4.1928 | 3.8172 | 3.7861 | 3.6172 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

2.0 | 3.8172 | 3.6173 | 3.5617 | 3.1874 |

**Table 25.**Effect of area variation coefficient of prestressed steel bars on reliability index and probabilistic safety factor.

Parameter | Coefficient of Variation | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire 60 min |
---|---|---|---|---|---|

Reliability index | 0.5 | 5.3817 | 5.2615 | 5.1823 | 4.6257 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

2.0 | 5.1726 | 4.9182 | 4.7456 | 4.2736 | |

Probabilistic safety factor | 0.5 | 4.2716 | 3.9182 | 3.7584 | 3.6474 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

2.0 | 3.6172 | 3.6153 | 3.8745 | 3.3162 |

**Table 26.**Effect of strength variation coefficient of prestressed steel bars on reliability index and probabilistic safety factor.

Parameter | Coefficient of Variation | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire 60 min |
---|---|---|---|---|---|

Reliability index | 0.5 | 5.4827 | 5.2737 | 5.2183 | 4.6517 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

2.0 | 5.2617 | 4.9827 | 4.7264 | 4.4726 | |

Probabilistic safety factor | 0.5 | 4.1725 | 3.9183 | 3.7261 | 3.6253 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

2.0 | 3.7163 | 3.5726 | 3.4516 | 3.2617 |

**Table 27.**Effect of constant load variation coefficient on reliability index and probabilistic safety factor.

Parameter | Coefficient of Variation | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.5 | 5.3716 | 5.2716 | 5.2172 | 4.7263 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

2.0 | 5.1827 | 5.0182 | 4.6735 | 4.3627 | |

Probabilistic safety factor | 0.5 | 4.2182 | 3.9271 | 3.8721 | 3.3627 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

2.0 | 3.8271 | 3.5262 | 3.5287 | 3.2617 |

**Table 28.**Effect of live load variation coefficient on reliability index and probabilistic safety factor.

Parameter | Coefficient of Variation | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Reliability index | 0.5 | 5.4638 | 5.2716 | 5.1028 | 4.6274 |

1.0 | 5.2772 | 5.1031 | 4.9917 | 4.4732 | |

2.0 | 5.0281 | 4.8927 | 4.8109 | 4.2817 | |

Probabilistic safety factor | 0.5 | 4.1726 | 3.8172 | 3.7263 | 3.5162 |

1.0 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

2.0 | 3.7162 | 3.6573 | 3.5267 | 3.3627 |

Parameter | Target Reliability Index | Before Fire | After Fire: 15 min | After Fire: 30 min | After Fire: 60 min |
---|---|---|---|---|---|

Probabilistic safety factor | 3.2 | 4.3627 | 3.9827 | 3.8172 | 3.6172 |

3.7 | 4.1827 | 3.8172 | 3.7162 | 3.5162 | |

4.2 | 3.9827 | 3.7872 | 3.6279 | 3.4821 | |

4.7 | 3.7182 | 3.6172 | 3.5263 | 3.3617 | |

5.2 | 3.5162 | 3.5018 | 3.4561 | 3.2817 |

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## Share and Cite

**MDPI and ACS Style**

Duan, M.; Miao, J.; Wu, J.; Dong, F.
Reliability Assessment Approach for Fire Resistance Performance of Prestressed Steel–Concrete Box Girder Bridges. *Fire* **2023**, *6*, 472.
https://doi.org/10.3390/fire6120472

**AMA Style**

Duan M, Miao J, Wu J, Dong F.
Reliability Assessment Approach for Fire Resistance Performance of Prestressed Steel–Concrete Box Girder Bridges. *Fire*. 2023; 6(12):472.
https://doi.org/10.3390/fire6120472

**Chicago/Turabian Style**

Duan, Maojun, Jianbao Miao, Jiahong Wu, and Fenghui Dong.
2023. "Reliability Assessment Approach for Fire Resistance Performance of Prestressed Steel–Concrete Box Girder Bridges" *Fire* 6, no. 12: 472.
https://doi.org/10.3390/fire6120472