This section uses an example from engineering to illustrate the adaptability and accuracy of the method proposed in this article. First, the finite element method is used to establish a fire model for prestressed steel–concrete structures. Based on this, the uniform design method is used to extract structural fire resistance performance samples and identify the optimal probability model via overall goodness-of-fit testing. Then, the probability method is deployed to analyze the structural fire resistance reliability performance and perform parameter sensitivity analysis.
5.4. Statistical Analysis of Structural Resistance of Prestressed Concrete Beam Bridges after Fire
The statistical parameters of various calculated random variables affecting the structural performance of prestressed concrete beam bridges after a fire are shown in
Table 7.
For the purpose of analyzing the statistical characteristics of the resistance of prestressed concrete beam bridges after a fire, a uniform design method was used to randomly generate sample points based on the determination of the statistical characteristics of the principal parameters affecting the structure’s fire resistance performance. Then, statistical analysis was conducted on the structure of prestressed concrete beam bridges post-fire via goodness-of-fit testing. All told, just six factors affect the prestressed concrete beam bridge following a fire, and 30 samples were randomly generated within a range of three times the standard deviation of each factor. The uniform design table is shown in
Table 8.
After conducting finite element random analysis, we assessed the bending bearing capacity samples of the maximum prestressed concrete beam bridge after 15 min, 30 min, and 60 min of fire exposure.
- (1)
Sample of prestressed concrete after 15 min of fire
The flexural capacities (KN·m) of samples were as follows: 102,130, 106,127, 98,723, 103,120, 110,203, 104,298, 105,267, 108,272, 109,172, 104,152, 108,279, 103,728, 110,289, 109,821, 106,672, 106,827, 104,263, 105,527, 104,263, 107,723, 109,283, 105,637, 102,891, 106,374, 108,273, 105,273, 108,374, 106,627, 107,263, and 108,273.
The statistical characteristics of the flexural bearing capacity of prestressed concrete beams following 15 min of exposure to fire were analyzed using goodness-of-fit testing (see
Table 9). Flexural bearing capacity followed a logarithmic normal distribution, with statistical characteristics, including a mean of 106,237 KN·m, a standard deviation of 2558 KN·m, and a coefficient of variation of 0.024.
- (2)
Sample of prestressed concrete after 30 min of fire
The flexural capacities of samples (KN·m) were as follows: 99,283, 95,637, 92,891, 96,374, 98,273, 95,273, 9837, 96,627, 97,263, 98,273, 92,130, 96,127, 98,723, 93,120, 100,203, 94,298, 95,267, 98,272, 99,172, 94,152, 98,279, 93,728, 100,289, 99,821, 96,672, 96,827, 104,263, 95,527, 94,263, and 97,723.
Following 15 min of fire treatment, we analyzed the statistical characteristics of the flexural bearing capacity of prestressed concrete beams using goodness-of-fit testing (see
Table 10). It can be seen that the flexural bearing capacity followed a logarithmic normal distribution, and the statistical characteristics were a mean of 96,904 KN·m, a standard deviation of 2608 KN·m, and coefficient of variation of 0.027.
- (3)
Sample of prestressed concrete after 60 min of fire
The flexural capacities of the samples (KN·m) were as follows: 83,728, 90,289, 89,821, 86,672, 86,827, 84,263, 85,527, 84,263, 87,723, 89,283, 85,637, 82,130, 86,127, 78,723, 83,120, 90,203, 84,298, 85,267, 86,627, 87,263, 88,273, 88,272, 89,172, 84,152, 88,279, 82,891, 86,374, 88,273, 85,273, and 88,374.
The statistical characteristics of the flexural bearing capacity of prestressed concrete beams after 15 min of fire exposure were analyzed using goodness-of-fit testing (see
Table 11). The flexural bearing capacity followed a logarithmic normal distribution, with statistical characteristics of a mean of 86,238 KN·m, a standard deviation of 2628 KN·m, and a coefficient of variation of 0.030.
5.6. Parameter Sensitivity Analysis
The principal factors that affect the reliability index and probability safety coefficient of the structural performance of prestressed concrete beam bridges following a fire are the following: (1) the mean value of variables, (2) the coefficient of variation of variables, and (3) target reliability indicators.
- (1)
The Influence of Random Variable Mean on Reliability Index and Probability Safety Factor
The control variable method is adopted in order to study the influence of the mean value of random variables on the reliability indicators and probability safety factors of prestressed concrete beam bridges following fire. Each analysis only alters the mean value of a certain variable. The change plan involves taking 0.9, 1.0, and 1.1 times the original value, respectively, with the mean values of other random variables taken as the original value. The specific calculation results of the influence of the mean of each random variable on the reliability index and probability safety coefficient of the performance of prestressed concrete beam bridges after fire are shown in
Table 13,
Table 14,
Table 15,
Table 16,
Table 17,
Table 18,
Table 19 and
Table 20.
By analyzing the contents of
Table 13,
Table 14,
Table 15,
Table 16,
Table 17,
Table 18,
Table 19 and
Table 20, it can be concluded that the reliability index of prestressed concrete beam bridges following a fire increases with the increase in the main beam width, main beam height, concrete strength, calculation mode uncertainty, prestressed steel area, and the average of prestressed steel strength while decreasing with the increase in the average of dead and live loads. The reliability index is a functional relationship between the mean and variability of random variables. As such, an increase in the mean of random variables that helps to improve resistance will enhance the reliability index, while an increase in the mean load will decrease the reliability index. Among the random variables that affect resistance, the influence of section height is more significant than that of section width, and the influence of steel reinforcement is greater than that of concrete.
The results in
Table 13,
Table 14,
Table 15,
Table 16,
Table 17,
Table 18,
Table 19 and
Table 20 show that the probability safety factor of prestressed concrete beam bridges after a fire increases with the increase in the main beam width, main beam height, concrete strength, calculation mode uncertainty, prestressed steel bar area, and the mean value of prestressed steel bar strength, decreasing with the increase in the mean value of dead and live loads. The probability safety factor has a similar functional relationship to that of the reliability index, meaning that an increase in the mean value of the random variable that helps to improve resistance will increase the safety factor, whereas an increase in the mean value of the load will decrease the safety factor. Among the random variables related to safety factors, the influence of section height is more significant than that of section width, and the influence of steel reinforcement is more significant than that of concrete.
Overall, the mean of random variables exerts a significant impact on the reliability index and probability safety factor of prestressed concrete beam bridges after exposure to fire. In specific engineering practice, attention should be paid to monitoring and the use of statistics to reduce structural safety risks and ensure the normal operation of prestressed concrete beam bridges after a fire. Special attention should be paid to the impact of fire on the thickness of concrete protective layers and steel reinforcement. These are important factors affecting the fire resistance performance of prestressed steel-reinforced concrete. Corresponding risk sources should be strictly controlled in structural design and construction processes.
- (2)
The Influence of Random Variable Variation Coefficient on Reliability Index and Probabilistic Safety Factor
The control variable method is adopted to study the influence of the coefficient of variation of random variables on the reliability indicators and probability safety factors of prestressed concrete beam bridges after exposure to fire. Each analysis only alters the coefficient of variation of a certain variable. The change plan involves taking values 0.5, 1.0, and 2.0 times the original value, respectively, while also ensuring that the coefficient of variation remains unaltered for other variables. The specific calculation results of the influence of the coefficient of variation of each random variable on the reliability index and probability safety coefficient of the structural performance of prestressed concrete beam bridges after exposure to fire are shown in
Table 21,
Table 22,
Table 23,
Table 24,
Table 25,
Table 26,
Table 27 and
Table 28.
By analyzing the contents of
Table 21,
Table 22,
Table 23,
Table 24,
Table 25,
Table 26,
Table 27 and
Table 28, it can be concluded that after a fire, the reliability indicators of prestressed concrete beam bridges decrease with the increase in the main beam width, main beam height, concrete strength, calculation mode uncertainty, prestressed steel bar area, prestressed steel bar strength, constant, and live load variation coefficients. The variability of random variables has a significant impact on structural reliability indicators, as an increase in parameters related to variability can lead to an increase in the discreteness of the structure, resulting in a decrease in reliability. Among all relevant random parameters, the variability of concrete section height, steel reinforcement area, and live load exert the most significant impact.
As with the results in
Table 21,
Table 22,
Table 23,
Table 24,
Table 25,
Table 26,
Table 27 and
Table 28, it can be shown that the probability safety factor of prestressed concrete beam bridges after a fire decreases with the increase in main beam width, main beam height, concrete strength, calculation mode uncertainty, prestressed steel bar area, prestressed steel bar strength, and variation coefficients of dead and live loads. Similarly, the safety factor is closely related to the discreteness of structural parameters, and an increase in variability-related parameters will lead to an increase in the discreteness of the structure, resulting in a decrease in the safety factor. Among the random parameters that affect structural safety, the variability of concrete section height, steel reinforcement area, and live load exerts the most significant impacts.
Overall, the coefficient of variation of random variables has a significant impact on the reliability index and probability safety factor of prestressed concrete beam bridges after a fire. In specific engineering practice, attention should be paid to the discreteness of monitoring and statistical parameters in order to reduce structural safety risks and ensure the normal operation of prestressed concrete beam bridges following a fire. Therefore, during the construction process of prestressed steel–concrete structures, it is necessary to strictly monitor the relevant indicators of the structure to prevent the dispersion of structural parameters from increasing and improve the reliability and safety of the structure. Special attention should be paid to the control of the thickness of the protective layer, the area of the steel reinforcement, and the live load, as these parameters possess the most obvious discreteness.
- (3)
The Influence of the Target Reliability Index on the Probability Safety Factor
In order to study the impact of target reliability indicators on the probability safety factor of performance of prestressed concrete beam bridges after exposure to a fire, the target reliability indicators were modified each time; that is, the changed target reliability indicators were 3.2, 3.7, 4.2, 4.7, and 5.2. The specific calculation results of the impact of target reliability indicators on the probability safety coefficient of performance of prestressed concrete beam bridges after a fire are shown in
Table 29.
According to the analysis in
Table 29, as the target reliability index increases, the probability safety coefficient of prestressed concrete beam bridges shows a decreasing trend after fire exposure. This indicates that, with the increase in the target reliability index, the probability safety coefficient of prestressed concrete beam bridges after fire gradually decreases, the actual required safety performance of prestressed concrete beam bridges after fire gradually increases, and the safety reserve of prestressed concrete beam bridges after fire gradually decreases. The probability safety coefficients, calculated based on the reliability back analysis method under each target reliability index, are all smaller than the safety coefficients calculated based on the deterministic model, indicating that parameter uncertainty has a significant impact on the probability safety coefficient of prestressed concrete beam bridges after fires. Ignoring parameter uncertainty will lead to the overestimation of the safety coefficient of prestressed concrete beam bridges after fires.