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This paper proposes a reliability analysis framework for glass fiber-reinforced polymer- (GFRP-) reinforced concrete systems with uncertain capacities and demands over time. Unfortunately, there has been limited discussion or research done related to the potential change of failure modes over time. Therefore, a rational approach is needed to integrate multiple failure modes in a single analysis framework, considering uncertainties of time-variant demands and capacities. To account for multiple failure modes, this study proposes the limit state function to estimate the safety margin, based on strain values of GFRP-reinforcing bars. A proposed limit state function can capture the likelihood of both shear and flexural failure modes, simultaneously. In this study, seven typical bridge deck configurations (e.g., varied deck thickness, girder spacing, and bar size) were exposed to various ambient temperatures. Simulation results show that reliability indices of 100-year exposure exhibit significant variance, ranging from 2.35 to 0.93, with exposure temperatures ranging from 13 to 33°C. Exposure temperature and time are the dominant factors influencing the reliability indices, so are the ones that need to be changed. As exposure time and/or exposure temperature increase, the flexural capacity model plays an important role to determine the reliability indices. When flexural and shear failure modes are equally dominant, reliability indices can capture risks of both failures, using the proposed strain-based approach.

Engineers and researchers have made significant progress toward standardizing test methods and designs for structural elements, using composite reinforcement [

In conventional design practices, the design of a bridge deck primarily focuses on required flexural design in accordance with AASHTO specifications [

Additional research is needed on FRP-reinforced bridge decks. Several researchers have performed studies on bridge decks, using FRP reinforcement. El-Sayed et al. [

In the flexural design of steel-reinforced bridge decks (i.e., generally

In a recent article, the first author of this paper presented focused on the strength-based reliability analysis (Kim et al. [

Flowchart of framework.

This paper consists of seven parts. Section

Even though GFRP reinforcements do not exhibit the classical corrosion observed in steel, several researchers [

Recently, Park et al. [

To perform the time-variant reliability analysis, Gardoni et al. [

A Bayesian approach was used to estimate the statistics (means and covariance matrices) of the unknown parameters

Using the time-variant bar capacity model, Kim et al. [

Figure

Failure modes of a bridge deck: (a) bar failure, (b) concrete crushing failure, and (c) shear failure.

Capacities of bars in the unit-wide strip deck were assumed to be statistically independent and identically distributed random variables. The capacity of each bar was totaled to determine the moment capacity of the section.

The strain of GFRP reinforcement was determined by the two failure modes: concrete crushing failure and bar failure. When bar failure is dominant, the maximum strain in the concrete (which is defined here as

When the concrete crushing failure is dominant, the strain in the GFRP bar, ^{2}),

Hoult et al.’s [

Hoult et al.’s model [

Potential final failure mode can be determined based on the lowest strain value. When the strain of GFRP reaches the strain corresponding to failure modes, the bridge deck experiences either flexural or shear failure. The main reason to integrate both failure modes is to consider the interaction of shear and moment with respect to failure probability. Two distinct values of failure probability may provide limited information (i.e., no consideration of moment and shear interaction) and two different estimations (either flexural or shear failure). Therefore, it needs the unified reliability index that can be used as a single indicator of a representative value of failure probability. The mathematical expression of the governing strain to determine each failure mode at time,

Equation

This section presents the demand model for predicting the strain of a GFPR reinforcement at a critical section. This research adopted the demand model proposed by Akgul [

The unfactored demand moment, ^{3}), ^{3}),

The proposed formulation was used to estimate the probability of failure of various GFRP-reinforced bridge decks that were designed in accordance with AASHTO specifications [

Parameters of deck design and uncertainties of demand

Parameter | Mean | Standard deviation | COV | Distribution | Descriptions | Source |
---|---|---|---|---|---|---|

1.24 |
6.34 | 0.15 | Lognormal | Compressive strength | [ | |

38.1 | 1.9 | 0.05 | Lognormal | Clear cover | [ | |

Varied ( |
3.96 | 1/(6.4 |
Normal | Concrete deck depth | [ | |

0.99 |
— | 0.04 | Normal | Effective depth | [ | |

25 | N.A. | N.A. | Deterministic | Max. aggregate size | [ | |

N.A. | N.A. | Normal | Shear effective depth | [ | ||

1.0 | 0.25 | — | Lognormal | Uncertainty factor for asphalt weight | [ | |

1.05 | 0.11 | — | Lognormal | Uncertainty factor for concrete weight | ||

0.60 | 0.20 | — | Lognormal | Uncertainty factor for truck | ||

0.8 | — | — | Deterministic | Continuity factor | ||

50 | — | — | Deterministic | Asphalt thickness | ||

2.36 | — | — | Deterministic | Span length of slab | ||

71.17 | — | — | Deterministic | Wheel load of HS 20 truck | ||

0.9 | 0.56 | 0.61 | Lognormal | Shear span for girder spacing (1.8 m) | Authors’ assumption | |

1.2 | 0.72 | Shear span for girder spacing (2.4 m) | ||||

1.8 | 1.12 | Shear span for girder spacing (3.7 m) |

Note:

Table

Conventional deck design

Case I.D. | Girder spacing (m) | Deck thickness (mm) | Number of bars—bar size (metric) | Spacing, (mm) | Mean max. crack |
|||
---|---|---|---|---|---|---|---|---|

Mean | Std. | |||||||

1.8-180-13 | 1.8 | 180 | 3.13 - M13 | 97 | 0.43 | 1.82 | 12.97 | 3.72 |

1.8-200-13 | 200 | 3.73 - M13 | 82 | 0.27 | 1.82 | 13.18 | 3.68 | |

2.4-230-13 | 2.4 | 230 | 4.33 - M13 | 70 | 0.23 | 1.82 | 17.32 | 4.63 |

2.4-230-19 | 2.4 | 230 | 1.92 - M19 | 158 | 0.11 | 1.82 | 17.32 | 4.63 |

3.7-200-13 | 3.7 | 200 | 3.73 - M13 | 82 | 0.50 | 1.82 | 27.68 | 6.46 |

3.7-230-13 | 230 | 4.33 - M13 | 70 | 0.38 | 1.82 | 28.46 | 6.46 | |

3.7-250-13 | 250 | 4.92 - M13 | 62 | 0.27 | 1.82 | 29.50 | 6.59 |

Table

Parameters of GFRP reinforcement.

Bar size | Parameter | Mean | Standard deviation | Distribution | Descriptions | Source |
---|---|---|---|---|---|---|

M13 | 6.35 | 0.76 | Lognormal | Radius | [ | |

46,263 | 5212 | Lognormal | Modulus of elasticity | [ | ||

779 | Lognormal | Mean capacity at |
||||

^{2}/sec) |
Lognormal | Diffusion coeff. | [ | |||

M19 | 9.53 | 1.14 | Lognormal | Radius | [ | |

47,573 | 5357 | Lognormal | Modulus of elasticity | [ | ||

696 | 14 | Lognormal | Mean capacity at |
[ | ||

^{2}/sec) |
Lognormal | Diffusion coeff. | [ |

Note: asterisks indicate the source of standard deviation.

The temperature range is determined based on the mean values of annual temperatures in the US territories. The actual fluctuation of exposure temperatures is much greater than three constant values. In general, the lowest temperature (13 °C) represents the average exposure temperature of the cold regions, while the highest temperature (33 °C) represents the average temperature of the warm region. The reference temperature (23°C) were used to develop the model in the previous research of the author in Texas region [

It should be noted that the current study focuses on the application of GFRP reinforcement in bridge structures. When the data from carbon fiber reinforced polymer (CFRP) is available, the proposed methodology can be applied to the deterioration model of CFRP in concrete, in which CFRP is more durable and exhibited high strength and high stiffness in general. As the application of CFRP gains more popularity in the field of infrastructure construction, it is expected that the experimental data and reliability analysis would be readily available, which will lead to the application of the proposed approach to various composite materials for improvement of the system reliability in near future.

Using the probabilistic model for the time-variant strain capacity of the GFRP deck described in equation (

Typical plot of safety margin: (a) 1.8-180-13 and (b) 2.4-230-13.

The time-variant reliability index,

In this study, the initial reliability indices were calculated at

Figure

Time-variant reliability index: flexural, shear, flexural, and shear (total): (a, i) 13°C [1.8-180-13], (a, ii) 23°C [1.8-180-13], (a, iii) 23°C [1.8-180-13], (b, i) 13°C [3.7-250-13], (b, ii) 23°C [3.7-250-13], and (b, iii) 23°C [3.7-250-13].

Time-variant reliability indices of flexural failure, shear failure, and integrated two-failure modes (using equation (

The maximum service life is defined as the time required to reach the reliability index of 2.5 (black line, designated as LRFR [

Design parameters, such as girder spacing and deck thickness, influence the dominancy of failure mode at the initial design. For example, the design cases of 1.8-180-13 and 3.7-250-13 were compared with respect to failure dominancy. In one, the shear failure mode affected the overall failure probability (hereafter, reliability index); in the other, the flexural mode affected the overall failure probability. However, exposure temperature altered the dominancy of failure mode over time. In the design case of 1.8-180-13, shear failure (green line with star) dominantly controlled the failure probability, as shown in Figure

Integrated failure mode and its time-variant reliability are dependent not only on exposure time but also on exposure temperature. This tendency is clearly observed in the 1.8-180-13 design case, when flexural and shear failure modes yield closer values of reliability indices. This seems to capture both risks of failure, exhibiting lower reliability indices than the individual values obtained from each failure mode. When one typical failure (typically, the flexural failure) governs the overall failure mode, the reliability indices are determined by the governed failure mode. Therefore, the proposed limit state function provides the rational estimation, using Monte Carlo simulations.

Figure

Effect of exposure temperature of 13, 23, and 33°C: (a) 1.8-180-13, (b) 1.8-200-13, (c) 2.4-230-13, (d) 3.7-200-13, (e) 3.7-230-13, and (f) 3.7-250-13.

Table

Time-variant reliability indices,

Case | Girder spacing, |
Deck thickness mm (in.) | 13°C | 23°C | 33°C | |||
---|---|---|---|---|---|---|---|---|

1.8-180-13 | 1.8 | 180 | 3.86 | 1.82 | 3.88 | 1.43 | 3.84 | 1.04 |

1.8-200-13 | 200 | 4.83 | 2.29 | 4.97 | 1.72 | 4.61 | 1.26 | |

2.4-230-13 | 2.4 | 230 | >4.97 | 2.36 | >4.97 | 1.77 | 4.97 | 1.29 |

3.7-200-13 | 3.7 | 200 | 4.19 | 1.85 | 4.18 | 1.37 | 4.31 | 0.93 |

3.7-230-13 | 230 | 4.75 | 2.17 | 4.75 | 1.61 | 4.97 | 1.15 | |

3.7-250-13 | 250 | >4.97 | 2.33 | >4.97 | 1.74 | >4.97 | 1.27 | |

Avg. | 4.60 | 2.13 | 4.62 | 1.61 | 4.61 | 1.16 | ||

Std. | 0.42 | 0.22 | 0.43 | 0.16 | 0.42 | 0.13 | ||

COV | 0.09 | 0.10 | 0.09 | 0.10 | 0.09 | 0.12 |

Figures

Failure probability and reliability index (bar size effect).

They were exposed to a standard temperature of 23°C. As the size of the bar increased, the probability of failure decreased. In addition, the time to reach the reliability index of 2.5 extended from about 25 to 50 years. This concurs with a similar trend in analytical results [

In this study, a reliability analysis framework is proposed for glass fiber-reinforced polymer- (GFRP-) reinforced concrete systems with uncertain capacities and demands over time. This is the first attempt to integrate the flexural and shear failure modes for GFRP-reinforced concrete members in a single analysis framework. A total of seven typical configurations of bridge decks with a variety of deck thicknesses and girder spacing were investigated. These different configurations changed the demand as well. Six design cases with M13 bars were exposed to three different temperatures (13, 23, and 33°C). The ratio of the provided GFRP reinforcement,

The various combinations of deck thickness and girder spacing affect the reliability index. The safety margins and reliability indices dramatically decrease at an early age (i.e., 0 to 0.5 years). This is attributed to the increase in demand and the reduction in flexural capacity. The reliability index of shear failures is mainly affected by the change of demand, rather than exposure temperature

The proposed limit state function provides rational estimation, with the likelihood of capturing flexural and shear failure modes, simultaneously

The reliability indices of 100 years of exposure exhibited significant variances, ranging from 2.35 to 0.93 in the exposure temperatures ranging from 13°C to 33°C. The simulation results show that all the design cases exhibited lower than the target reliability index of 2.5 recommended by AASHTO LRFR after 100 years of exposure

In general, as exposure time and temperature increase, the flexural failure affects the determination of overall reliability indices. In a bridge deck having a thin thickness and short girder spacing, shear failure tends to affect the overall reliability index, resulting in a high probability of failure (i.e., low reliability index).

In the benchmark design case, the increase in the bar size could be an alternative for extending the service life. From these simulations, the M19 bar increased the service life approximately 100% to reach the target reliability index of 2.5

In summary, exposure temperatures dramatically affect the service life (e.g., expectancy of life to reach the specific target reliability index). The change in the reliability index is more sensitive (i.e., more changes) with exposure to higher temperatures. Therefore, further experimental and analytical research is needed on both deck configuration and exposure temperature of the GFRP-reinforced bridge deck exposed to high temperatures.

The area of the GFRP reinforcement in the given section (mm^{2})

Maximum aggregate size (mm)

The width of the cross section (mm)

The width of the web (mm), effective crack spacing

Continuity factor, and dynamic impact factor

The nominal moment capacity of concrete crushing failure

The nominal moment capacity of bar failure

The diffusion coefficient at a defined reference temperature (23

The distance from the extreme compression fiber to the centroid of tension reinforcement (mm)

The effective shear depth (mm) taken as

The activation energy (KJ)

The elastic modulus of GFRP reinforcement

Statistically independent, identically distributed random variables with a mean and unit variance of zero

The compressive strength of concrete (MPa)

Guaranteed tensile strength of GFRP reinforcement

The mean tensile strength of sample of test specimens

Thickness of asphalt pavement (m)

Thickness of concrete slab (m)

The ratio of neutral axis to reinforcement depth (ACI 440.1R, Eq. (7.3.2.2b))

The span length of the slab between two girders (m)

The dead load moment

The live load moment

Randomly generated from mean and standard deviation of

The standard deviations of the error terms,

The load on one middle or rear wheel of an HS20 truck

The following equation at the given time,

The universal gas constant (KJ/mol-K)

The radius of the GFRP reinforcement

The strain of service moment

Effective crack spacing

296 K (23°C)

The exposure temperatures (in K)

Normalized shear stress

Uniform weight of utility piping for slab (kN/m)

Standard deviation

Capacity of the

Time-variant reliability index,

Asphalt unit weight (kN/m^{3})

Concrete unit weight (kN/m^{3})

Concrete crushing strain in the top fiber

Longitudinal strain at mid-depth for predicted shear failures

Corresponding strain of bar failure

Corresponding strain of concrete crushing failure

Corresponding strain of shear failure

Vectors to capture the parameters,

A vector of unknown parameters introduced to fit the data

A vector of unknown parameters to fit the data of elasticity of concrete

Uncertainty factor for asphalt

Uncertainty factor for concrete

Uncertainty factor for truck

Mean tensile strength of sample of test specimens

The inverse standard normal distribution function and the probability of failure

Vectors of parameters

A vector of basic variables for bar properties

A vector of basic variables for shear

A vector of demand.

The data set is generated by Monte Carlo simulation for this study’s outcomes. Therefore, the data set can be created by any researchers when Tables

The authors declare that there is no conflict of interest regarding the publication of this paper.

The authors wish to express gratitude and sincere appreciation of the partial financial support by the University of Louisville.