Reliability-Based Service III Evaluation for Prestressed Girder Bridges Under Platoon Loads

Abstract

Platooning may benefit heavy truck transportation through fuel savings, reduced congestion, enhanced safety, and lower emissions. In the future, platoons may be able to act as mobile-WIM stations, and their permit process and allowable load limits may differ from unregulated trucks in a convoy. Previous reliability-based studies have focused on the Strength I limit state and have shown that trucks can operate at weights exceeding standard legal load limits even with short headways at operating-level reliability. However, the Service III limit state often governs prestressed concrete bridges. The AASHTO Manual for Bridge Evaluation does not specify a target reliability index (β) nor reliability-based evaluation guidance for the Service III limit state. The work presented here performed reliability analyses to investigate implicit reliability indices (β_Implicit) inferred from bridges designed according to current and past AASHTO criteria, as well as cracking probabilities. Design live loads were used to evaluate the Service III limit state for prestressed concrete NU I-girder bridges, optimally designed using LRFD and LFD/allowable stress design (ASD). Various span lengths, numbers, continuity conditions, prestress loss methods, and allowable tension stress levels were considered. Cracking probabilities ranged between 10% and 67%, which indicates that optimally designed bridges may crack during their service life. Although beyond the scope of the study, the present work suggests a reexamination of service behavior and performance is appropriate, using an alternate mechanistic approach to estimate potential cyclic damage and aid life-cycle assessment. Such assessments could provide a more rational framework for platoon operations while maintaining bridge health and safety.

Keywords

infrastructure infrastructure management and system preservation bridge and structures management structures concrete bridges innovative highway structures and appurtenances

A truck platoon consists of two or more trucks linked together in a convoy with close headways using connectivity technology and automated driving systems. Effective platooning can save fuel ( 1 , 2 ), benefit the environment ( 3 ), and improve traffic operations ( 4 ). The impact of platoons on bridge performance has been studied; however, most research has focused on strength limit states and is not based on reliability analyses ( 5 – 9 ). A few recent studies have examined the effects of platoons on bridges based on reliability analysis. Yang et al. ( 10 ) and Steelman et al. ( 11 ) determined platoon vehicle weights can potentially be increased by reducing live load coefficients of variation (CoVs) while maintaining a strength operating-level reliability index β = 2.5. Sajid et al. further parametrized bias and CoVs of platoons and determined that two-lane bridges can reliably accommodate platoons to achieve an inventory level reliability index β = 3.5 for Strength ( 12 ).

Reduced headways and increased platoon weights may affect bridge serviceability, particularly for prestressed concrete bridges where the Service III limit state typically governs. AASHTO notes that service limit states are not calibrated based on reliability but on past practice and experience. The 8th edition of AASHTO’s Standard Specifications for Highway Bridges (hereafter, Standard Specifications) first established maximum allowable tensile stress limits for prestressed concrete bridges under service loads ( 13 ). In the 11th edition of AASHTO’s Standard Specifications for Highway Bridges ( 14 ), the allowable tensile limit was revised, but additional, significant modifications have not occurred for approximately half a century. According to the current AASHTO LRFD Bridge Design Specifications (LRFD BDS) ( 15 ), the allowable tensile stress for Service III for bridges under severe corrosion conditions should be the smaller of $0.0948 \sqrt{f'_{c}}$ (kips per square inch [ksi]) or 0.3 ksi. Under no worse than moderate corrosion conditions, the allowable tensile stress is the smaller value of $0.19 \sqrt{f'_{c}}$ (ksi) or 0.6 ksi. The apparent intent of Service III is to prevent cracks at the bottom of the girder when subjected to service loads, which would typically be expected to occur for tensile stresses ranging between $0.24 \sqrt{f'_{c}}$ and $0.37 \sqrt{f'_{c}}$ (ksi) for normal weight concrete according to LRFD BDS C5.4.2.6.

A target reliability index, β, for Service III is needed to evaluate platoon effects on girder bridges in a reliability-based framework. However, reliability-based evaluation criteria for Service III and cracking probability are not specified by AASHTO. Only a few recent studies have investigated potential target β values for Service III. Wassef et al. ( 16 ) assumed that bridges designed based on the prestress loss method in the Standard Specifications ( 17 ) performed well in service. Service evaluations were performed using weigh-in-motion (WIM) data for a site with annual average daily truck traffic (AADTT) of 5,000 to characterize live load means and CoVs. Target β_Implicit were recommended for bridges according to various performance limit criteria (e.g., decompression and maximum tensile stress). LRFD BDS Table 3.4.1-4 includes an increased live load factor of 1.0 when using the refined time-dependent loss method with elastic gains, which was recommended by Wassef et al. ( 16 ). Barker et al. evaluated Service III for Wyoming bridges under heavy truck traffic resulting from extended roadway closures that would create large truck trains ( 18 ). Bridges designed and evaluated under current AASHTO LRFD criteria were assumed to perform satisfactorily. Using I-80 WIM vehicle load characteristics, roadway closures created load effects that resulted in negative reliability indices for Service III, indicating that some reliability concerns may exist for heavy-load, truck train situations.

In summary, reliability-based evaluations for Service III and their associations with mechanical performance objectives, such as limiting cracking probability, are currently not clearly established in AASHTO, thereby hampering optimal truck platoon deployment. Furthermore, the flexibility afforded to bridge designers by LRFD BDS language, such as the selection of a prestress loss calculation method, whether to use gross or transformed section properties, and whether elastic gains should be included, creates ambiguity with respect to the level of safety intended by the specifications. These options raise questions about resulting service reliabilities for bridges designed using loss calculation methods other than those incorporating refined time-dependent losses and elastic gains. The preceding discussion does not suggest that prestressed concrete bridges experience poor cracking behavior resulting from in-service loads; what it does suggest is that if platoon permits allow for increased loads, the potential for damage exists. This paper intends to address this knowledge gap via research that conducted reliability analyses for Service III and provided preliminary β_Implicit for prestressed girder bridges designed using LRFD BDS and Standard Specification requirements. This study focused on three objectives: (1) analyzing the reliability of different design options; (2) determining an β_Implicit for Service III; and (3) investigating cracking probability under current design live load conditions during bridge service lives.

Methodology

General Reliability Analysis Procedure

Structural reliability analysis can determine whether the probability of exceeding a limiting criterion is acceptable. According to Nowak and Collins, Equation 1 represents a general limit state function ( 19 ),

g (R, Q) = R - Q

(1)

where R and Q are random variables representing resistance and load effects, respectively. Monte Carlo simulation (MCS) is commonly used to conduct modern reliability analysis. MCS is performed by randomly generating N load and resistance samples, counting the number of instances in which the limit state is violated (i.e., g in Equation 1 is less than zero), and assessing the probability of failure, P_f, as a ratio of the number of failure samples, F, to the total number of samples, N. g is typically assumed to follow a normal distribution, so that β can be determined as shown in Equation 2,

β = - Φ^{- 1} (P_{f})

(2)

where −Φ^-1 is the inverse of the standard normal cumulative distribution function. The index is routinely used in structural reliability frameworks to characterize structural safety (i.e., strength limit states). Figure 1 presents typical probability density functions (PDFs) for loads, resistance, and the strength limit state function. β represents the number of standard deviations the mean value of g is from zero (Figure 1). At the Strength level, design and inventory load ratings target β = 3.5, whereas operating load ratings relax the target β to 2.5.

Figure 1.

PDFs of load, resistance, and margin of safety for strength.

The following reliability indices are defined for this paper:

Implicit reliability (index), β_Implicit, is the reliability provided by code-compliant bridge designs, that is, β is inferred from designs produced from historical practice.

Cracking reliability, β_Cracking, is the reliability against mechanical cracking limits provided by code-compliant bridge designs.

This research examined the β_Implicit for different design scenarios and evaluated cracking probability for bridges optimally designed according to service criteria in the LRFD BDS or the Standard Specifications. To evaluate bridges designed using prestress losses obtained from Standard Specifications criteria, they were designed based on appropriate live loads, load distributions, dynamic amplifications, and service load combinations from the Standard Specifications. However, all designs were evaluated using HL-93 loading for consistent comparisons. Figure 2 provides the research framework overview in which shaded cells illustrate the pathway the research team followed to complete optimal bridge designs and conduct reliability analyses.

Figure 2.

Research methodology.

Three inputs are required for reliability analyses: (1) nominal values, (2) bias factors (i.e., ratio of mean to nominal value), and (3) CoVs (i.e., ratio of standard deviation to mean). A parametric study was conducted that varied allowable tensile stresses, prestress loss methods, girder spacings, and span numbers and lengths to investigate the effects of these parameters on service β_Implicit. Optimal bridge design resistances were determined by incorporating nominal dead and nominal HL-93 ( 15 ) and HS20-44 design live loads ( 17 ). For this study, an optimal design referred to a bridge for which the capacity exactly satisfied service load requirements according to deterministic AASHTO LRFD and ASD criteria.

This study assumed that the β_Implicit based on current and past design criteria was adequate to provide satisfactory in-service performance, similar to the studies of Wassef et al. ( 16 ) and Barker et al. ( 18 ). Nominal demands, live loads, and resistances were mapped onto probabilistic distributions with characteristic means and CoVs, and the probability of failure for each parametric combination was calculated using MCS with N = 1,000,000 samples to determine β, as shown in Figure 2. The selected number of samples corresponds to a maximum perceptible reliability index of approximately β = 4.75.

This process was repeated to conduct reliability analysis for different design scenarios, identify an β_Implicit, and investigate β_Cracking for optimally designed prestressed concrete bridges for service. For this paper, β_Cracking was defined based on a bridge designed using a typical allowable stress limit ( $0.0948 \sqrt{f'_{c}}$ [ksi]) and evaluated using a range of rupture moduli between $0.24 \sqrt{f'_{c}}$ and $0.37 \sqrt{f'_{c}}$ (ksi) as specified in LRFD BDS C5.4.2.6.

Nominal Values and Design Parameters

Nominal Resistance

AASHTO Manual for Bridge Evaluation (MBE) Equation 6A.5.4.1 ( 20 ) is used for LRFD rating (LRFR) for Service III and is as follows:

RF = \frac{f_{R} - (γ_{D}) (f_{D})}{(γ_{L}) (f_{LL + IM})}

(3)

where

RF = load rating factor,

γ_D = dead load factor for rating, and

γ_L = rating live load factor.

The other load and resistance terms in Equation 3 for composite prestressed concrete girder bridges are based on gross section properties and are calculated as shown in Equations 4 to 7:

f_{R} = f_{t} + \frac{P_{e}}{A_{g}} + \frac{P_{e} e_{nc}}{S_{ncb}}

(4)

P_{e} = A_{ps} (f_{pi} - Δ f_{s})

(5)

f_{D} = \frac{(D_{gw} + D_{nc})}{S_{ncb}} + \frac{(D_{c} + D_{w})}{S_{cb}}

(6)

f_{LL + IM} = \frac{LL (1 + IM) GD F_{m}}{S_{cb}}

(7)

where

f_R = nominal flexure resistance (ksi);

f_D = nominal tensile stress from dead loads (ksi);

f_LL+IM = nominal tensile stress from live loads including impact (ksi);

f_t = nominal design allowable tensile stress (ksi);

f_pi = nominal initial prestressing (ksi);

P_e = nominal effective prestress force after losses (kip);

Δf_s = nominal prestress losses (ksi);

A_ps = nominal total area of prestressing reinforcement (in.²);

A_g = nominal gross section area (in.²);

e_nc = nominal eccentricity from the noncomposite centroid to prestress strand centroid (in.);

D_gw = nominal girder dead load moment (kip-in.);

D_nc = nominal noncomposite dead load moment (kip-in.);

D_c = nominal composite dead load moment (kip-in.);

D_w = nominal wearing dead load moment (kip-in.);

LL = nominal HL-93 design loading (LRFD bridges), or HS20-44 design loading (LFD/ASD bridges) (kip-in.);

IM = nominal impact factor; LRFD BDS Article 3.6.2 (LRFD bridges); Standard Specifications Equation 3-1 (LFD/ASD bridges);

GDF_m = nominal AASHTO moment girder distribution factor (GDF) (LRFD bridges), or nominal GDF in Standard Specifications (LFD/ASD bridges);

S_ncb = nominal noncomposite section modulus for the bottom fiber (in.³); and

S_cb = nominal composite section modulus for the bottom fiber (in.³).

The nominal optimal resistance based on LRFD BDS at Service III was determined by setting RF = 1.0. Bridges designed based on the Standard Specifications followed the same procedure using appropriate live loads, GDFs, and IMs. The nominal resistance is calculated by reforming Equation 3 as follows:

f_{t} + \frac{P_{e}}{A_{g}} + \frac{P_{e} e_{nc}}{S_{ncb}} = \frac{γ_{D} (D_{gw} + D_{nc})}{S_{ncb}} + \frac{γ_{D} (D_{c} + D_{w})}{S_{cb}} + \frac{γ_{L} LL (1 + IM) GD F_{m}}{S_{cb}}

(8)

The bottom tension stress design, f_des, was calculated as in the right-hand side of Equation 8. For a given f_t, the required effective stress, f_req, in the prestressing strands after losses was calculated using Equations 9 to 11. Then, the required effective prestress force based on f_req was determined as follows:

f_{des} = \frac{γ_{D} (D_{gw} + D_{nc})}{S_{ncb}} + \frac{γ_{D} (D_{c} + D_{w})}{S_{cb}} + \frac{γ_{L} LL (1 + IM) GD F_{m}}{S_{cb}}

(9)

f_{req} = f_{des} - f_{t}

(10)

f_{req} = \frac{P_{e}}{A_{g}} + \frac{P_{e} e_{nc}}{S_{ncb}}

(11)

P_{req} = \frac{f_{req}}{\frac{1}{A_{g}} + \frac{e_{nc}}{S_{ncb}}}

(12)

Based on the calculated f_req and corresponding P_req, the required number of strands was calculated and used to determine A_ps, e_nc, and Δf_s, which included or neglected elastic gains, depending on the design scenario under consideration, for the initial iteration. Iterative modifications were made until Equation 12 was satisfied, which meant that P_e equaled P_req in Equation 5. The resulting number of strands was not rounded to the nearest whole number to determine optimal designs and β_Implicit’s. Although the number of strands is typically specified as even integers, this study aimed to investigate theoretically optimal reliabilities.

Bridge Parameters

The LRFD BDS, Standard Specifications, and MBE address allowable tensile stress limits in positive moment regions. Accordingly, this study considered only positive moments at the middle of simple-span prestressed girder composite bridges, and at 40% of the span length from supporting abutments for two equal-span, simple, made-continuous, prestressed girder composite bridges.

The bridges were assumed to carry two traffic lanes, and interior girders were designed using NU-I girders ( 21 ). The details were as follows:

60-, 90-, 120-, and 150-ft spans;

NU 900 (60 ft), 1,100 (90 ft), 1,600 (120 ft), and 2,000 (150 ft) (based on 21 );

3.5-ft overhang length;

5 girders;

10-ft girder spacings (studies indicated insensitivity of β_Implicit with 6-, 8-, 10-, and 12-ft spacing);

Composite deck thickness 8.5 in. with 0.5 wear off and 2-in. asphalt wearing surface;

1 in. deck haunch;

1 ft 5-in. wide barrier, weight 0.124 k/ft per girder; and

Simple and continuous spans.

Four tension stress limit values (hereafter f_t) were considered for service designs. The maximum design f_t considered was the concrete modulus of rupture ( f_r), $0.24 \sqrt{f'_{c}}$ (ksi), which is larger than the 0.6 ksi upper bound for f_t in the LRFD BDS. A minimum design f_t of zero was used, commonly referred to as the decompression limit. The two most common design allowable tensile stresses were also included in the study ( $0.0948 \sqrt{f'_{c}}$ [ksi] and $0.19 \sqrt{f'_{c}}$ [ksi]). For simplicity, the coefficient before $\sqrt{f'_{c}}$ (ksi) in the design f_t is referred to here as κ. Preliminary analyses indicated that service reliability results for optimally designed bridges were not sensitive to final and initial concrete strength parameters. Initial concrete strength was used to calculate prestress loss based on the refined time-dependent loss method in the LRFD BDS, derived initially from Tadros et al. ( 22 ). Parameters considered for κ, f_t, f'_c, and f'_ci included,

f_t = $κ \sqrt{f'_{c}}$ ksi, where κ = 0, 0.0948, 0.19, and 0.24,

f'_{c_girder} = 8 ksi,

f'_{ci_girder} = 5 ksi,

f'_{c_deck} = 4 ksi, and

f'_{ci_deck} = 3.2 ksi.

Grade 270 0.6-in. low-relaxation strands with a modulus of elasticity of 28,500 ksi were used. The initial stress at transfer (f_pi) was 0.75 of the ultimate tensile strength (f_pu). Following Hanna et al., the maximum number of strands was set to 60 with those strands placed in up to 7 layers (18 strands per layer in the flange, reducing to 2 strands per layer in the web) ( 21 ). The bottom concrete cover was assumed to be 2 in., and the distance between prestressing layers was 2 in. Participation of any mild reinforcing steel in girder cross-sectional resistance was ignored.

Prestress Loss Method Parameters

Prestress loss calculations were based on gross section properties (A_g, S_ncb, and S_cb). Bridges with a 47-ft width and a travelway width of 40 ft-2 in. were subjected to two-lane HL-93 loading. The gross deck area was transformed using the modular ratio when determining S_cb. LRFD BDS Equation C5.4.2.4-2 was used to calculate the modulus of elasticity. S_cb was determined after reducing the nominal bridge deck thickness by 0.5 in. to account for a future wearing surface degradation. Three loss methods and six design scenarios were considered, as summarized in Table 1.

Table 1.

Design Scenarios for Bridges

ID	Loss computation	Design live load factor (γ_L)	Elastic gains (Y/N)	Comment
Post-1.0-Gains	Post-2005 loss method (LRFD 5.9.3.4)	1.0	Y	LRFD Table 3.4.1-4: Prestressed concrete components designed using refined estimates for time-dependent losses in Article 5.9.3.4 in conjunction with elastic gains
Post-0.8-Gains	Post-2005 loss method (LRFD 5.9.3.4)	0.8	Y	Same as scenario Post-1.0-Gains with a lower live load
Post-0.8-No-gains	Post-2005 loss method (LRFD 5.9.3.4)	0.8	N	LRFD Table 3.4.1-4: All other prestressed concrete components
Approx-0.8-Gains	Approximate loss method (LRFD 5.9.3.3)	0.8	Y	LRFD Table 3.4.1-4: All other prestressed concrete components
Approx-0.8-No-gains	Approximate loss method (LRFD 5.9.3.3)	0.8	N	LRFD Table 3.4.1-4: All other prestressed concrete components
Pre-1.0-No-gains	Pre-2005 loss method	1.0	N	Standard Specifications 9.16.2.1

Note: LRFD = Load and Resistance Factor Design.

According to LRFD BDS Tables 3.4.1-1 and 3.4.1.4 and Standard Specifications Table 3.22.1A, the design dead load factor (design γ_D) is 1.0 for service. However, the design live load factor (design γ_L) varies with prestress loss method and whether elastic gains are included in the effective prestress force. The primary considered case was Post-1.0-Gains from the table as proposed by Wassef et al. ( 16 ). Research also considered Post-0.8-Gains, which would be representative of how some bridges might have been designed between when the post-2005 losses were included in AASHTO and when the required design γ_L was increased to 1.0. Post-0.8-No-gains is technically consistent with LRFD BDS Table 3.4.1-4. Using approximate loss methods with or without elastic gain was interpreted as an instance of ambiguity or subtlety within the AASHTO LRFD BDS, as Commentary C5.9.3.3 in the LRFD BDS notes that elastic gains “should” be included unless transformed section properties are used. Because of higher prestress loss predictions and a design live load factor of 1.0, bridges designed using Pre-1.0-No-gains typically required the most strands.

Sensitivity analyses were conducted using information from bridges designed for Nebraska to examine the effects of concrete age and average relative humidity on reliability. Analysis results were negligibly influenced by a reasonable range of ages and average relative humidity values. As a result, the following parameters were used:

Prestress transfer: 1 day,

Deck placement: 30 days,

Final analysis time: 3,650 days, and

Average relative humidity: H = 65%.

Nominal Dead Loads

Realistic estimates for nominal dead loads were calculated for various bridge types and span lengths (Figure 2). As shown in Equation 8, dead load moment had four components (D_gw, D_nc, D_c, and D_w). All concrete was assumed to be normal weight (i.e., 150 pcf). Designs produced the same optimal NU sections for both simple- and two-span bridges of similar span. Preliminary analyses verified that this assumption had negligible effects on β_Implicit analysis results.

Nominal Design Live Loads

The HL-93 load model from the LRFD BDS and HS20-44 from the Standard Specifications were the nominal design live loads. The HL-93 live load consists of a truck or design tandem loading combined with a single lane load. A constant IM = 0.33 was applied to live loads in accordance with the LRFD BDS. The interior girder GDF_m for multiple lanes in Equation 8 was determined using LRFD BDS Table 4.6.2.2.2b-1. The longitudinal stiffness (K_g) term was calculated based on the designed girder cross-sectional geometry when determining GDF_m. The HS20-44 load effect was determined by the larger load effect from the truck or lane loading combined with a concentrated load as specified in the Standard Specifications. For GDF_m for multiple lanes in Equation 8, the Standard Specifications effectively specify lane GDFs of S/11, where S is the girder spacing, for prestressed concrete bridges. Standard Specifications Equation 3-1 also provides IM as a function of span length, with a maximum value of 0.30.

Statistical Parameters

Statistical parameters were implemented in reliability analyses, as shown in Figure 2. Selected statistical parameters and their sources for dead loads (D_gw, D_nc, D_c, and D_w), live loads (LL_HL-93), and selected variables affecting resistance (e_nc, e_c, A_ps, f_pi, Δf_s_{[w/o LL gain]}, f_t [κ = 0, 0.095, and 0.19], and f_r) are summarized in Table 2.

Table 2.

Statistical Parameters for Variables

Variables	Distribution	Bias	CoV	References
D_gw	Normal	1.03	0.08	Kulicki et al. ( 23 )
D_nc	Normal	1.05	0.10	Kulicki et al. ( 23 )
D_c	Normal	1.05	0.10	Kulicki et al. ( 23 )
D_w	Normal	1.00	0.25	Kulicki et al. ( 23 )
LL_HL-93	Normal	1.18	0.18	Kulicki et al. ( 23 )
e_nc, e_c	Normal	1.00	0.04	Siriaksorn and Naaman ( 24 )
A_ps	Normal	1.01176	0.0125	Siriaksorn and Naaman ( 24 )
f_p _i	Normal	0.97	0.08	Wassef et al. ( 16 ), and Gross and Burns ( 25 )
Δf_s _{(w/o LL gain)}	Normal	1.05	0.10	Wassef et al. ( 16 ), and Tadros et al. ( 22 )
f_r	Normal	0.936	0.24	Holombo and Tadros ( 26 )
f_t (κ= 0-0.19)	Deterministic	1	0	Assumption

Note: CoV = coefficient of variation; e_c = eccentricity between gross composite centroid and prestressing strand centroid.

The total HL-93 CoV represented combined uncertainty contributions from vehicle weight, dynamic amplification, and girder load distribution ( 23 ). Post-1.0-Gains from in Table 1 was used to evaluate bridge performance because this is the most recently developed method and was used to calibrate live load factors by Wassef et al. ( 16 ). The Δf_s_{(w/o LL gain)} bias and CoVs were applied to the prestress loss, considered elastic shortening loss, long-term loss, and the dead load elastic gains. The live load elastic gain is not included in these statistical parameters as its variability is included based on live load statistics and the bias was assumed to be 1.0. The statistical uncertainty for f_r used that selected by Holombo and Tadros ( 26 ), who determined that the mean and CoV of the modulus of rupture were $7.02 \sqrt{f'_{c}}$ (psi) and 0.24, respectively. The typical nominal modulus of rupture is $7.5 \sqrt{f'_{c}}$ (psi), and the bias for f_r was taken as the ratio of 7.02 to 7.5. The uncertainties for the moduli of rupture for $0.30 \sqrt{f'_{c}}$ and $0.37 \sqrt{f'_{c}}$ were assumed to be the same as for $0.24 \sqrt{f'_{c}}$ . The allowable tension limits (f_t) for κ ranged between 0 and 0.19 and were considered deterministic values since these limits were, historically, arbitrarily specified based on engineering judgment. The results were insensitive to variations of gross sectional area, section modulus, and prestressing steel and concrete moduli of elasticity. Accordingly, these parameters were also treated as deterministic having biases of one (1.0). The probabilistic mean for dynamic amplification resulting from impact, IM, was set to 10% ( 23 ).

Limit State Function for Service III

Reliability analyses were implemented by progressing through the middle, shaded portion of Figure 2 using previously defined nominal values and statistical parameters, recognizing that the effective prestress force for evaluation may have differed from the effective prestress force used in design. HL-93 loads GDF_m and IM in LRFD BDS were applied in the evaluations for single- and two-span bridges designed based on LRFD BDS and Standard Specifications. Equation 1 represents the general limit state function. The present study reformed the equation as follows:

g = R - Q = (f_{t} + \frac{P_{eval}}{A_{g}} + \frac{P_{eval} e_{nc}}{S_{ncb}}) - (\frac{(D_{gw} + D_{nc})}{S_{ncb}} + \frac{(D_{c} + D_{w})}{S_{cb}} + \frac{L L_{HL - 93} (1 + IM) GD F_{m}}{S_{cb}})

(13)

P_{eval} = A_{ps} (f_{pi} - Δ f_{S_{eval}}) = A_{ps} (f_{pi} - (Δ f_{pES} + Δ f_{pLT} - Δ {f_{Gain}}_{DL} - Δ {f_{Gain}}_{LL}))

(14)

f_{Reval} = f_{t} + \frac{P_{eval}}{A_{g}} + \frac{P_{eval} e_{nc}}{S_{ncb}}

(15)

where

P_eval = effective prestress force for the evaluation,

Δf_Seval = prestress loss for the evaluation,

Δf_pES = elastic shortening loss,

Δf_pLT = long-term prestress loss,

Δf_pGainDL = elastic gain from dead loads, and

Δf_pGainLL = elastic gain from live load.

f_Reval is available resistance to tension stress for evaluation and can be determined using Equation 15.

Results and Discussion

Overview

Optimal numbers of strands for each prestress loss method increased with span and decreased as the design tensile stress increased from κ = 0 to 0.24. Post-1.0-Gains predicted the lowest prestress losses, and Pre-1.0-No-gains gave the highest prestress losses. The approximate and post-2005 loss methods predicted similar prestress losses for cases using the same design live load factor and the same consideration of elastic gains. β_Implicit and β_Cracking results were slightly lower for simple-span bridges than for two-span bridges. Therefore, only simple-span bridge results are presented in this section. Several cases that were examined are excluded from the results for 150-ft bridges because the design required more than the maximum allowable number of prestressing strands (60 strands). For simplicity, the allowable tensile stress limit with a κ coefficient will be represented as f_t (κ) (e.g., $0.19 \sqrt{f'_{c}} = f_{t} [κ = 0.19]$ ). In the following charts and figures, “DET” indicates the use of a deterministic value for the allowable tension stress limit f_t whereas “VAR” indicates the use of a probabilistically varying value for the modulus of rupture, f_r.

Nominal Live Load Positive Moments

To provide context for this study, nominal positive live load moments are compared for platoon and HL-93 loadings for simple- and two-span bridges as a function of their span lengths and platoon headways. Nominal positive live load moments for a four-truck platoon with adjacent, routine traffic and for two lanes under HL-93 loading were determined. These two loading scenarios, and the headways between the platoons, are shown in Figure 3. Headway is the distance between the leading truck’s last axle and the first axle of the following truck (Figure 3). The platoon model was based on the notional rating load (NRL) from the study by Sivakumar ( 27 ) as shown in Figure 3. The NRL was found to govern the positive moment regions as in Steelman et al.’s study ( 11 ). Platoon headways ranged from 10 to 50 ft at increments of 10 ft and span lengths from 60 to 150 ft. According to AASHTO requirements, HL-93 was considered for one- and two-lane loading. As presented by Yang et al. ( 10 ) and Steelman et al. ( 11 ), the platoon was accompanied by 5,000 AADTT adjacent routine traffic. Adjacent routine traffic was characterized using interstate I-80 WIM data, and more details can be found in Steelman et al. ( 11 ). Multiple presence probabilities for platoons loaded with adjacent routine traffic were adopted from Ghosn et al. ( 28 ). IM was assumed to be 0.10 for all load cases, which is the same as its probabilistic mean. For platoons operating alongside routine traffic in the adjacent lane, the GDF was calculated according to LRFD BDS Equation (4.6.2.2.5-1), with additional details found in Yang et al. ( 10 ) and Steelman et al. ( 11 ).

Figure 3.

Loading scenarios for comparison of nominal live load moments.

To help identify controlling load cases, two 3D surfaces were generated for live load moments (Figure 4, a and b ). Figure 4 indicates that nominal HL-93 loading often envelopes platoon cases, except where platoons operate at small headways of 10 to 20 ft. As documented elsewhere in the literature, the primary bridges of concern are those with longer span lengths subjected to closely spaced truck platoons.

Figure 4.

Live load moments for platoons and HL-93 loadings on prestressed girder bridges: (a) simple-span bridges at the mid span and (b) two-span bridges at the 0.4L of end spans.

Live load moments generated from a platoon at a 10-ft headway with adjacent routine traffic on 150-ft simple-span and two-span bridges were about 21% greater than those induced by two lanes of HL-93 loading (Figure 4a and b). Reliability analyses, however, have indicated that four-truck platoons with lower uncertainties than typically associated with vehicular live loads can operate on such bridges and still reach target reliability indices for Strength ( 10 , 11 ). While these results indicate that four-truck platoons can safely traverse bridges for Strength based on reliability analyses, the current LRFD BDS and MBE load factors for Service III are not reliability-based.

Optimal Design and Evaluation Results

In this section, f_t (κ = 0.0948) is used to show the intermediate calculation results for design and evaluation, with those results being synthesized through reliability analyses to obtain reliability indices in succeeding sections. Figure 5a shows the tensile stresses caused by factored dead and live loads for design and evaluation. The factored f_D values were the same for both design and evaluation (Figure 5a). The f_LL+IM results for evaluation were all based on HL-93 loading with γ_L = 1.0, whereas the f_LL+IM values for design were different (Figure 5a). For bridges designed with loss methods with γ_L = 0.8 or with pre-2005 loss methods (Standard Specifications), the f_LL+IM values for design were lower than those designed with Post-1.0-Gains (Figure 5a). The f_LL+IM values did not vary with different f_t (κ) cases.

Figure 5.

Tensile stresses from factored dead and live loads, and optimal A_ps for f_t (κ = 0.0948) for various design scenarios: (a) tensile stresses resulting from factored design and evaluation loads (κ = 0–0.24) and (b) optimal A_ps for design and evaluation (κ = 0.0948).

Figure 5b presents the optimal A_ps for f_t (κ = 0.0948). For 60-ft bridges, A_ps was close for different loss methods, but A_ps varied more across design cases when span lengths increased (Figure 5b). A_ps for Approx-0.8-Gains and Post-0.8-Gains was generally close and smaller than for Post-1.0-Gains (Figure 5b). Without considering elastic gains, A_ps for 150-ft span bridges designed with approximate and post-2005 loss methods (γ_L = 0.8) was slightly larger than that for Post-1.0-Gains (Figure 5b). Generally, the Pre-1.0-No-gains produced larger A_ps than other methods in Figure 5b.

Figure 6 presents prestress losses and elastic gains for design and evaluation. Post-0.8-Gains predicted the lowest Δf_S for design, and Pre-1.0-No-gains predicted the highest Δf_S (Figure 6a). The Δf_S for Approx-0.8-Gains and Post-0.8-Gains were generally close, although the approximate loss method exhibited greater sensitivity to span length (Figure 6a). Δf_S generally increased with span length, and changes in girder section size may be responsible for slightly reduced Δf_S for 120-ft bridges (Figure 6a). Δf_S predicted by methods with elastic gains were more consistent over the studied spans than those that did not include elastic gains (Figure 6a). Since the Post-1.0-Gains was used for evaluation, the Δf_Seval and Δf_S were the same for bridges designed with Post-1.0-Gains, and Δf_Seval for evaluation reduced significantly from Δf_S for design for loss methods without elastic gains (Figure 6, a and b ). The slight difference between Δf_S and Δf_Seval for Post-0.8-Gains was because of slightly different design configurations resulting from the use of different elastic gains for the design.

Figure 6.

Prestress loss and elastic gains for design and evaluation at f_t (κ = 0.0948): (a) Δf_S for design (κ = 0.0948), (b) Δf_Seval for evaluation (κ = 0.0948), (c) Δf_Gain for design (κ = 0.0948), and (d) Δf_Gain for evaluation (κ = 0.0948).

As shown in Figure 6c, Post-1.0-Gains produced larger elastic gains than the other methods, whereas the trend for Post-0.8-Gains and Approx-0.8-Gains was similar but lower because of a lower γ_L. The elastic gains for evaluation were practically the same for different designs (Figure 6d) and contributed to differences between Δf_S and Δf_Seval in Figure 6, a and b .

Based on A_ps, f_pi, Δf_S, and Δf_Seval, the effective prestress loss force for design and evaluation can be determined using Equations 5 and 14. Flexural resistances for design and evaluation are based on Equations 4 and 15. Figure 7 presents effective prestress force and flexure resistance for design and evaluation. Figure 7a shows that Post-1.0-Gains had the largest P_e for design. The P_e for bridges for all other design scenarios were similar.

Figure 7.

Effective prestress force and resistance for design and evaluation at f_t (κ = 0.0948): (a) P_e for design (κ = 0.0948), (b) P_eval for evaluation (κ = 0.0948), (c) f_R for design (κ = 0.0948), and (d) f_Reval for evaluation (κ = 0.0948).

As expected, P_eval tracks with A_ps, was identical for P_e and P_eval for Post-1.0-Gains, and was largest for bridges designed using Pre-1.0-No-gains when that method could be proportioned according to the parameters of this study (Figure 7b). The P_eval values were close for bridges designed using loss methods including gains and using γ_L = 0.8, and were smaller than for the Post-1.0-Gains in Figure 7b owing to the lower A_ps provided from design. Similarly, bridges designed with loss methods neglecting elastic gains and using γ_L = 0.8 generally produced larger P_eval values than Post-1.0-Gains in Figure 7b. Figure 7, c and d , present flexural resistances for design and evaluation, respectively. Note that Post-0.8-No-gains, Post-0.8-Gains, Approx-0.8-No-gains, and Approx-0.8-Gains overlap in Figure 7c, exhibiting similar trends for effective prestress force and flexure resistance for design and evaluation (Figure 7). As illustrated in Figures 7d and 5b, methods requiring a larger A_ps led to a larger f_Reval. The various influences of selected loss method, live load factor, and use or neglect of elastic gains produced bridges with differing f_Reval values relative to the baseline Post-1.0-Gains case exhibiting sensitivity to span. Therefore, marginal differences in reliabilities for other methods relative to Post-1.0-Gains will vary with span.

β_Implicit and Probabilities of Exceeding Tension Limits

MCS was implemented to determine dead load, live load, and resistance according to distributions based on the nominal values and statistical parameters described previously. Figure 8a presents PDFs for Service III at f_t (κ_eva = 0.0948) for a 120-ft simple-span bridge designed using Post-1.0-Gains. Figure 8a shows resistance, load, and limit state function PDFs. The mean resistance is close to the nominal resistance for evaluation in Figure 7d, and the mean load is greater than the mean resistance. β_Implicit, representing the number of standard deviations between the mean for g and the failure threshold, was −0.61. A negative value indicates that the probability of exceeding the Service III limit state was more than 50%. Figure 8b further examines the PDF for g. Figure 8b depicts the probability that Service III f_t (κ_eva = 0.0948) will be exceeded during a bridge’s service life. As shown in Figure 8b, about 72 out of 100 of the 120-ft simple-span bridges with f_t (κ_eva = 0.0948) are expected to exceed the Service III limit state during their service life despite being optimally proportioned to satisfy Service III design criteria.

Figure 8.

Probability density functions for evaluating Service III at f_t (κ = 0.0948) for 120-ft simple-span bridges designed by using Post-1.0-Gains: (a) PDFs for R, Q, and g and (b) zoomed PDF for g.

P_eval and f_Reval values were generally close for Post-1.0-Gains, Post-0.8-No-gains, and Approx-0.8-No-gains for spans of 60 to 120 ft (Figure 7, b and d ). These three methods therefore were expected to result in similar β_Implicit. Higher nominal f_Reval values led to higher reliability indices, though f_t and P_eval uncertainties also contributed to the reliability indices. β_Implicit and corresponding probabilities of exceeding tension limits for these three loss methods are plotted in Figures 9 to 11, and key observations are presented alongside the figures.

Figure 9.

Reliability index β and corresponding probabilities of exceeding tension limits for simple-span bridges designed using Post-1.0-Gains method, similar design and evaluation f_t (κ): (a) β for Post-1.0-Gains and (b) probability of exceeding tension stress limits for Post-1.0-Gains.

Figure 10.

Reliability index β_Implicit and corresponding probabilities of exceeding tension limits for simple-span bridges designed using Post-0.8-No-gains method, similar design, and evaluation f_t (κ): (a) β for Post-0.8-No-gains and (b) probability of exceeding tension stress limits for Post-0.8-No-gains.

Figure 11.

Reliability index β_Implicit and corresponding probabilities of exceeding tension limits for simple-span bridges designed using Approx-0.8-No-gains method, similar design and evaluation f_t (κ): (a) β for Approx-0.8-No-gains and (b) probability of exceeding tension stress limits for Approx-0.8-No-gains.

P_eval and f_Reval values were generally similar for Post-0.8-Gains and Approx-0.8-Gains but lower than loss methods depicted in Figures 9 to 11 (Figure 7, b and d ). Therefore, these two methods resulted in β_Implicit that were similar, but lower than the methods presented in Figures 9 to 11. Figure 12 presents β_Implicit and corresponding probabilities of exceeding tension limits for two design methods with elastic gains and γ_L = 0.8. Figure 12a presents β for bridges designed using Post-0.8-Gains and indicates that β was approximately −1.20 and largely insensitive to f_t (κ) and span length. Lower reliabilities in Figure 12a were expected compared with those in Figure 9a because a smaller area of prestressing reinforcement resulting from a lower live load factor (Figure 5b). Figure 12b indicates that the average probability of exceeding tension limits was about 88%, higher than the 73% shown in Figure 9b. Figure 12c presents β for bridges designed using Approx-0.8-Gains, with β generally close to the results in Figure 12a, having an average of −1.10 and with slightly more sensitivity to span length. In both cases, elastic gains were considered, and the same design live load factor was used. As a result, prestress loss predictions were similar. Figure 12d indicates that the average probability of exceeding tension limits was about 86%, which is similar to the 88% in Figure 12b.

Figure 12.

Reliability index β_Implicit and corresponding probabilities of exceeding tension limits for simple-span bridges designed using Post-0.8-Gains and Approx-0.8-Gains methods, similar design, and evaluation f_t (κ): (a) β for Post-0.8-Gains, (b) probability of exceeding tension stress limits for Post-0.8-Gains, (c) β for Approx-0.8-Gains, and (d) probability of exceeding tension stress limits for Approx-0.8-Gains.

Generally, the Pre-1.0-No-gains produced higher P_eval and f_Reval values than other design loss methods, except for some 60-ft bridge span cases (Figure 7, b and d ). The β_Implicit for Pre-1.0-No-gains was therefore generally higher than that for other methods. Figure 13a presents β_Implicit for Pre-1.0-No-gains. The bridges were designed based on the Standard Specifications, but evaluated using parameters consistent with LRFR. β’s varied as a function of span length and generally increased as span increased, however, the trends for P_eval and f_Reval did not consistently parallel the values obtained using prestress loss methods in the LRFD BDS. Compared with β from other design methods, Pre-1.0-No-gains generally provided much higher reliability, except for some 60-ft simple-span Post-1.0-Gains cases (Figures 9 to 13). Figure 13b indicates that the average probability of exceeding the tension limit is about 59%, which is noticeably lower than other design methods.

Figure 13.

Reliability index β and corresponding exceeding tension limit probability for simple-span bridges designed using Pre-1.0-No-gains method, similar design and evaluation f_t (κ): (a) β for Pre-1.0-No-gains and (b) probability of exceeding tension stress limits for Pre-1.0-No-gains.

Generally, loss design methods that considered elastic gains produced a more consistent β for the considered spans (Figures 9a, 12a, and 12c). Bridges designed without considering elastic gain possessed higher β and more pronounced span length sensitivity (Figures 10a, 11a, and 13a) reflecting variation in dead load stresses and the natural result of neglecting associated elastic gain.

In summary, average β_Implicit for optimally designed Service III bridges was approximately

−0.60 for bridges designed using Post-1.0-Gains, Post-0.8-No-gains, and Approx-0.8-No-gains

−1.20 for bridges designed using Post-0.8-Gains and Approx-0.8-Gains

Designs obtained using recent calibration for Service III correspond to a β_Implicit of −0.60 ( 16 ). There is no published physical evidence that prestressed girders designed using any prestress loss methods with a live load factor of 0.8 before 2014 exhibit significant cracking in service, as stated in AASHTO LRFD C3.4.1. Consequently, this paper recommends targeting a relatively conservative β_Implicit = −0.60 to evaluate bridges for platoon loading.

β_Cracking and Cracking Probability During Service Life

Negative β_Implicit’s imply that optimally designed bridges for service are more than 50% likely to violate tensile stress limits under current design live loads at some point during their service lives. However, exceeding the service limit does not necessarily mean that they will experience flexural cracking in the precompressed tensile zone. Figure 14a presents PDFs for the likelihood of cracking at f_r (κ_eva = 0.24) for a 120-ft simple-span bridge designed using Post-1.0-Gains. This 120-ft simple-span bridge is assumed to be designed with f_t (κ = 0.0948). Figure 14a shows that the resistance mean is slightly larger than the load mean. The corresponding cracking reliability was +0.08 (Figure 14a). The shaded area in Figure 14b represents cracking cases. The shaded area here is smaller than that in Figure 8b, reflecting that cracking is less likely than violating a limit state set to a tension stress limit less than the theoretical concrete cracking strength. Figure 14b indicates that about 47 out of 100 120-ft simple-span bridges are expected to crack during their service life.

Figure 14.

Probability density functions for evaluating cracking at f_r (κ = 0.24) for 120-ft simple-span bridges designed by using Post-1.0-Gains: (a) PDFs for R, Q, and g and (b) zoomed PDF for g.

To further evaluate bridge β_Cracking, those designed using a typical f_t (κ = 0.0948) were evaluated over a range of f_r (κ). Nominal moduli of rupture were assumed equal to $0.24 \sqrt{f'_{c}}$ , $0.30 \sqrt{f'_{c}}$ , and $0.37 \sqrt{f'_{c}}$ to investigate how cracking probability varied as a function of rupture moduli.

Figure 15 presents the cracking reliability index and corresponding cracking probability for Post-1.0-Gains, Post-0.8-No-gains, and Approx-0.8-No-gains methods. Figure 15a shows β_Cracking values for simple-span bridges designed using different loss methods with f_t (κ = 0.0948) and evaluated with f_r (κ = 0.24). Consider a 60-ft simple-span bridge designed using Post-1.0-Gains as an example. The β_Cracking for this case was equal to 0.52, as shown in the leftmost purple bar of Figure 15a. As span length increased from 60 to 150 ft, β_Cracking decreased from 0.52 to −0.02 (Figure 15a) owing to increasing dead-to-live load ratios. For bridges designed using Post-0.8-No-gains and Approx-0.8-No-gains, the β_Cracking slightly increased as span length increased. Figure 15b shows that cracking probability is generally less than or equal to 50% for these three methods. A β_Cracking of 0.52 implies that 30 out of 100 optimally designed bridges for Service III are expected to crack during their service life. Increasing nominal f_r (κ = 0.24) to f_r (κ = 0.30), β_Cracking increased by about 0.30 on average (Figure 15, a and c ), with cracking probabilities being reduced by about 11% (Figure 15, b and d ). Increasing the nominal f_r (κ = 0.30) to f_r (κ = 0.37), the reliability for bridges again increased by about 0.32 on average (Figure 15, c and e ). Changing the nominal f_r (κ = 0.24) to f_r (κ = 0.37) increased the reliability index by an average of 0.62 and decreased cracking probability by an average of 20%.

Figure 15.

β_Cracking and cracking probability for simple-span bridges designed using Post-1.0-Gains, Post-0.8-No-gains, and Approx-0.8-No-gains methods, different f_r (κ = 0.24, 0.30, and 0.37): (a) β_Cracking for f_r (κ = 0.24), (b) cracking probability for f_r (κ = 0.24), (c) β_Cracking for f_r (κ = 0.30), (d) cracking probability for f_r (κ = 0.30), (e) β_Cracking for f_r (κ = 0.37), and (f) cracking probability for f_r (κ = 0.37).

Figure 16a shows β_Cracking values for simple-span bridges designed using Post-0.8-Gains and Approx-0.8-Gains with f_t (κ = 0.0948) and evaluated with f_r (κ = 0.24). The β_Cracking is always negative in Figure 16a, as expected, lower than the β_Cracking in Figure 15a. β_Cracking and probabilities were close for bridges designed using Post-0.8-Gains and Approx-0.8-Gains. Increasing nominal f_r (κ = 0.24) to f_r (κ = 0.37), reliability increased by an average of 0.68, and cracking probability decreased by an average of 26%.

Figure 16.

β_Cracking and cracking probability for simple-span bridges designed using Post-0.8-Gains and Approx-0.8-Gains methods, different f_r (κ = 0.24, 0.30, and 0.37): (a) β_Cracking for f_r (κ = 0.24), (b) cracking probability for f_r (κ = 0.24), (c) β_Cracking for f_r (κ = 0.30), (d) cracking probability for f_r (κ = 0.30), (e) β_Cracking for f_r (κ = 0.37), and (f) cracking probability for f_r (κ = 0.37).

Figure 17a shows β values for simple-span bridges designed using Pre-1.0-No-gains with f_t (κ = 0.0948) and evaluated from f_r (κ = 0.24) to f_r (κ = 0.37). β_Cracking is always positive in Figure 17a, and is generally higher than β_Cracking for the same nominal f_r in Figures 15 and 16. The maximum cracking β was 1.29 for 90-ft simple-span bridges, which implies that 9 out of 100 optimally designed Service III bridges would experience cracking during their service life (Figure 17). By increasing nominal f_r (κ = 0.24) to f_r (κ = 0.37), the reliability for bridges would increase by 0.61 on average, and cracking probability would decrease by an average of 16% (Figure 17).

Figure 17.

β_Cracking and cracking probability for simple-span bridges designed using Pre-1.0-No-gains method, different f_r (κ = 0.24, 0.30, and 0.37): (a) β_Cracking for different f_r and (b) cracking probability for different f_r..

Increasing nominal f_r (κ = 0.24) to f_r (κ = 0.30) increased β_Cracking by about 0.31, and cracking probabilities decreased by 11% on average (Figures 15 to 17). Changing the nominal f_r (κ = 0.24) to f_r (κ = 0.37) increased the reliability index by an average of 0.64 and decreased cracking probability by about 22% on average. Given that linear changes in reliabilities and cracking probabilities as a function of nominal rupture moduli were observed for all cases, the β_Cracking and cracking probability can be preliminarily estimated based on linear interpolation for nominal moduli of rupture between 0.24 and 0.37.

Current optimally designed Service III bridges with f_t (κ = 0.0948) and subjected to current design live loads have β_Cracking ranging between −0.45 and 1.29 for the considered range of nominal moduli of rupture. Therefore, the estimated cracking probability during their service lives ranges between 10% and 67% for optimally designed bridges. The highest reliabilities and lowest cracking probabilities correspond to upper-bound rupture moduli, which AASHTO notes correspond to small-depth samples that were moist-cured before testing. This means that lower reliabilities and higher cracking probabilities are likely to be more representative of in-service bridges.

Summary and Conclusions

The objective of this research was to investigate β_Implicit as a precursor to reliability-based evaluations of the effects of heavy loads from truck platoons on simple- and two-span continuous prestressed concrete bridges. Structures comprising span lengths between 60 and 150 ft and designed using the AASHTO LRFD BDS (Service III) and AASHTO Standard Specifications (allowable stress) were considered. Bridges were conservatively assumed to have an optimal service design capacity by setting the AASHTO MBE rating factor equation to 1.0. Optimal bridge design cases considered three prestress loss methods, two design live load factors, four design allowable tensile stress limits, and included or neglected elastic gains. MCS was used to determine the reliabilities of these optimally designed bridges under the current design HL-93 loading. β_Implicit values were determined for optimal designs, with values identified from this study potentially being used to guide service limit evaluations for truck platoon loads. The likelihood of exceeding service limits raised questions about the appropriateness of current AASHTO methods to evaluate potential frequent heavy loads, such as those from truck platoons, which led to additional studies investigating β_Cracking for these optimally designed bridges.

This study produced the following findings for the parameters that were considered:

Post-2005 and approximate loss methods predicted similar prestress losses for design, whereas prestress losses were predicted to be higher with the pre-2005 loss method.

β_Implicit was approximately −0.60 when averaged across all considered span lengths for bridges designed using the following loss methods:

○ Post-2005 loss method with elastic gains and using γ_L = 1.0,

○ Post-2005 loss method without elastic gains and using γ_L = 0.8, and

○ Approximate loss method without elastic gains and using γ_L = 0.8.

β_Implicit was approximately −1.20 when averaged across all considered span lengths for bridges designed using the following loss methods:

○ Post-2005 loss method with elastic gains and using γ_L = 0.8, and

○ Approximate loss method with elastic gains and using γ_L = 0.8.

A target of β_Implicit = −0.60 is presently recommended to evaluate girder bridges for platoon loading at Service III. Use of a more liberal value of −1.20 may be acceptable if supported by observations of adequate performance in service for girders designed considering elastic gains and using γ_L = 0.8.

β_Implicit’s of −0.60 and −1.20 imply a 73% to 88% probability of exceeding Service III limit during service life.

Bridge cracking probabilities range between 10% and 67% under code-specified service load, which indicates that optimally designed bridges may crack during their service life, despite the nominal expectation that cracking is avoided by using tensile stress limits less than the modulus of rupture.

A nominal change of f_r (κ = 0.24) to f_r (κ = 0.37) produced an average increase of β equal to 0.64 and a 22% average decrease in cracking probability.

β_Cracking and cracking probability change approximately linearly as a function of nominal moduli of rupture and can be preliminarily estimated using an assumed value.

Bridges designed using the Pre-2005 loss method without elastic gains and using γ_L = 1.0 possessed the highest cracking reliabilities, which indicates that heavy platoon loads pose lower risks to these bridges than other bridges designed using more recent prestress loss estimation methods.

The results presented here apply to bridges optimally designed for Service III using the AASHTO LRFD and for an allowable stress using the Standard Specifications assuming they perform well under HL-93 loadings and that the interior girder was the critical load-carrying member. Notwithstanding nominally limiting tensile stresses to less than the modulus of rupture, many bridges optimally designed for Service III have the potential to crack during their service life based on the current study findings. Considering the identified inconsistencies between implied intentions and expected outcomes for Service III designs, future research should investigate the effects of heavy loads, such as those from truck platoons, on prestressed concrete bridges accounting for potential cracking that could affect performance. Service III is primarily based on traditional/historical limits on allowable tensile stresses. Future work should reexamine service behavior and performance to investigate whether an alternate mechanistic format may be more appropriate for service criteria to account for the likelihood of cracking during a bridge’s life. Note that the assumed HL-93 loading performance basis may be conservative and improved assessments of service performance could be obtained through calibration to representative WIM data to improve β_Implicit estimates. In the future, platoons may be able to act as mobile-WIM stations, and their permitting process and allowable load limits may be different than unregulated trucks moving together in a convoy. A rational and consistent basis to evaluate anticipated performance in service will ensure long service lives and optimize the return on investment into transportation infrastructure carrying heavy truck platoons.

Footnotes

Acknowledgements

The Nebraska Department of Transportation sponsored the work described here.

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: J.S. Steelman, J.A. Puckett, D.G. Linzell, B. Yang; data collection: B. Yang; analysis and interpretation of results: B. Yang, J.A. Puckett, J.S. Steelman, D.G. Linzell; draft manuscript preparation: B. Yang, J.S. Steelman, J.A. Puckett, D.G. Linzell. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research presented in this paper was supported by the Nebraska Department of Transportation under project SPR-FY22(011): Truck Platooning Effects on Girder Bridges: Phase II-Service.

ORCID iDs

Bowen Yang

Joshua S. Steelman

Daniel G. Linzell

The opinions and conclusions expressed or implied in this paper are solely those of the authors and not necessarily those of the Nebraska Department of Transportation.

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Reliability-Based Service III Evaluation for Prestressed Girder Bridges Under Platoon Loads

Abstract

Keywords

Methodology

General Reliability Analysis Procedure

Nominal Values and Design Parameters

Nominal Resistance

Bridge Parameters

Prestress Loss Method Parameters

Nominal Dead Loads

Nominal Design Live Loads

Statistical Parameters

Limit State Function for Service III

Results and Discussion

Overview

Nominal Live Load Positive Moments

Optimal Design and Evaluation Results

βImplicit and Probabilities of Exceeding Tension Limits

βCracking and Cracking Probability During Service Life

Summary and Conclusions

Footnotes

Acknowledgements

Author Contributions

Declaration of Conflicting Interests

Funding

ORCID iDs

References

β_Implicit and Probabilities of Exceeding Tension Limits

β_Cracking and Cracking Probability During Service Life