Fatigue safety level provided by Brazilian design standards for a prestressed girder highway bridge

Author contributions: ALC: conceptualization, methodology, formal analysis, writing, proofreading; ELP: data curation, writing review, proofreading; TNB: data curation, funding acquisition, supervision, proofreading; ATB: formal analysis, writing review, proofreading.

Editors: Mauro Real, Guilherme Aris Parsekian.

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Figure 1
Brazilian live load model, dimension in meters (adapted from NBR 7188 [¹1 Associação Brasileira de Normas Técnicas, Carga Móvel Rodoviária e de Pedestres em Pontes, Viadutos, Passarelas e Outras Estruturas, NBR 7188, 2013.]).

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Figure 2
S-N curves for reinforcement (adapted from fib [¹⁴14 Fédération Internationale du Béton, vol. 2, Model Code 2010. Lausanne, Suiça: FIB, 2010.] and NBR 6118 [¹⁵15 Associação Brasileira de Normas Técnicas, Projeto de Estruturas de Concreto – Procedimento, NBR 6118, 2014.]]).

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Figure 3
HS-WIM system: a) layout and sensors position [⁴4 A. L. Carneiro, E. L. Portela, and T. N. Bittencourt, "Development of Brazilian highway live load model for unlimited fatigue life," Rev. IBRACON Estrut. Mater., vol. 13, no. 4, pp. e13407, 2020, http://dx.doi.org/10.1590/S1983-41952020000400007.
http://dx.doi.org/10.1590/S1983-41952020... ], [⁵5 E. L. Portela, "Analysis and development of a live load model for Brazilian concrete bridges based on WIM data," Ph.D. dissertation, Polytech. Sch., Univ. São Paulo, São Paulo, Brasil, 2018. Accessed: Apr. 15, 2020. [Online]. Available: https://www.teses.usp.br/teses/disponiveis/3/3144/tde-11122018-151658/pt-br.php
https://www.teses.usp.br/teses/disponive... ]; b) weighting sensor before grout in BR-381 [⁴4 A. L. Carneiro, E. L. Portela, and T. N. Bittencourt, "Development of Brazilian highway live load model for unlimited fatigue life," Rev. IBRACON Estrut. Mater., vol. 13, no. 4, pp. e13407, 2020, http://dx.doi.org/10.1590/S1983-41952020000400007.
http://dx.doi.org/10.1590/S1983-41952020... ], [⁵5 E. L. Portela, "Analysis and development of a live load model for Brazilian concrete bridges based on WIM data," Ph.D. dissertation, Polytech. Sch., Univ. São Paulo, São Paulo, Brasil, 2018. Accessed: Apr. 15, 2020. [Online]. Available: https://www.teses.usp.br/teses/disponiveis/3/3144/tde-11122018-151658/pt-br.php
https://www.teses.usp.br/teses/disponive... ], ^[33]33 Agência Nacional de Transportes Terrestres. Desenvolvimento do Modelo de Deterioração de Pavimentos Asfálticos com Uso de Instrumentação e Sistema Weigh in Motion (Relatório final). Brasília, D.F.: ANTT, 2016.; c) final appearance of the pavement in BR-381 [⁵5 E. L. Portela, "Analysis and development of a live load model for Brazilian concrete bridges based on WIM data," Ph.D. dissertation, Polytech. Sch., Univ. São Paulo, São Paulo, Brasil, 2018. Accessed: Apr. 15, 2020. [Online]. Available: https://www.teses.usp.br/teses/disponiveis/3/3144/tde-11122018-151658/pt-br.php
https://www.teses.usp.br/teses/disponive... ], [³³33 Agência Nacional de Transportes Terrestres. Desenvolvimento do Modelo de Deterioração de Pavimentos Asfálticos com Uso de Instrumentação e Sistema Weigh in Motion (Relatório final). Brasília, D.F.: ANTT, 2016.] (position sensors, which detect when vehicles change lanes, was not installed).

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Figure 4
Histogram for GVW for class 3M6 (the horizontal axis shows the average values for each interval).

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Figure 5
Histogram for weight for first axle for class 3M6 (the horizontal axis shows the average values for each interval).

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Figure 6
Histogram for weight for double tandem for class 3M6 (the horizontal axis shows the average values for each interval).

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Figure 7
Histogram for weight for triple tandem for class 3M6, which the horizontal axis shows the average values for each interval (the histogram for both triple tandem are similar).

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Figure 8
Histogram for weight for single axle for class 3I3, which the horizontal axis shows the average values for each interval (the histogram for single axles for 3I3 are similar).

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Figure 9
Bridge section.

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Figure 10
Bridge girder details, dimensions in meters (EG: exterior girder; IG: interior girder).

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Figure 11
Values of dependent variable ∑(1/N_i) for stirrups in Lognormal probability paper considering limited prestressing and ADTT = 5000.

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Table 1
Parameters of characteristic S-N curves from fib [¹⁴14 Fédération Internationale du Béton, vol. 2, Model Code 2010. Lausanne, Suiça: FIB, 2010.].

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Table 6
Random variables (indicated in and Section 4).

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Table 2
Parameters of S-N curves for a 50% confidence level (adapted from Crespo-Minguillón and Casas [⁶6 C. Crespo-Minguillón and J. R. Casas, "Fatigue reliability analysis of prestressed concrete bridges," J. Struct. Eng., vol. 124, no. 12, pp. 1458–1466, 1998.]).

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Table 3
Parameters of variable

D M_{i}

(adapted from Crespo-Minguillón and Casas [⁶6 C. Crespo-Minguillón and J. R. Casas, "Fatigue reliability analysis of prestressed concrete bridges," J. Struct. Eng., vol. 124, no. 12, pp. 1458–1466, 1998.]).

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Table 4
Statistics for twelve most frequent classes of BR-381 from September 2016 to May 2017.

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Table 5
Reinforcement areas according to prestress levels.

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Table 9
Moments for dependent variables and reliability indexes (β) for stirrups (interior girder) for design service life of 50 years.

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Table 7
Moments for dependent variables and reliability indexes (β) for prestressing steel (exterior girder) for design service life of 50 years.

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Table 8
Moments for dependent variables and reliability indexes (β) for reinforcing steel (exterior girder) for design service life of 50 years.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

Case	m₁	m₂	$Δ σ_{R s k}$ (MPa) at N* = 10⁶ cycles	$Δ σ_{R s k}$ (MPa) at 10⁸ cycles	K₁ = N x $Δ σ$ ^m1	K₂ = N x $Δ σ$ ^m2
Reinforcing steel (straight bars ϕ_s ≤ 16 mm)	5	9	210	125	4.08 x 10¹⁷	7.45 x 10²⁶
Prestressing steel (postensioning curved tendons)	3	7	120	65	1.73 x 10¹²	4.90 x 10²⁰

Category	Variable	Distribution	Mean / Nominal	CV¹	Remarks
Material characteristics	f_c (f_ck = 40 MPa)	Normal	1.16	0.11	Santiago [²⁵25 W. C. Santiago, "Calibração baseada em confiabilidade dos coeficientes parciais de segurança das principais normas brasileiras de projeto estrutural", Doctoral dissertation, Univ. S. Paulo, São Carlos, SP, Brasil, 2019.]
	E_cs	Normal	1.04	0.04	Santiago [²⁶26 W. C. Santiago, Calibração baseada em confiabilidade dos coeficientes parciais de segurança de normas brasileiras de projeto estrutural (Texto do exame de qualificação). São Carlos, SP: Universidade de São Paulo, 2017.]
	E_p	Normal	1.03	0.02	Santiago [²⁶26 W. C. Santiago, Calibração baseada em confiabilidade dos coeficientes parciais de segurança de normas brasileiras de projeto estrutural (Texto do exame de qualificação). São Carlos, SP: Universidade de São Paulo, 2017.]
	E_s	Normal	1.03	0.02	E_p
	A_p, A_p,b	Lognormal	1.03	0.01	Santiago [²⁵25 W. C. Santiago, "Calibração baseada em confiabilidade dos coeficientes parciais de segurança das principais normas brasileiras de projeto estrutural", Doctoral dissertation, Univ. S. Paulo, São Carlos, SP, Brasil, 2019.]
	A_s, A_sw	Lognormal	1.03	0.01	A_p
	σ_0p	Normal	0.97	0.08	Used in Wassef et al. [⁸8 W. G. Wassef et al., Calibration of AASHTO LRFD Concrete Bridge Design Specifications for Serviceability, NCHRP Project 12-83. Washington, DC: National Academy of Sciences, 2014. Accessed: Apr. 15, 2020. [Online]. Available: : https://www.nap.edu/catalog/22407/calibration-of-aashto-lrfd-concrete-bridge-design-specifications-for-serviceability https://www.nap.edu/catalog/22407/calibr... ]²
	Δσ_p	Normal	1.05	0.10	Used in Wassef et al. [⁸8 W. G. Wassef et al., Calibration of AASHTO LRFD Concrete Bridge Design Specifications for Serviceability, NCHRP Project 12-83. Washington, DC: National Academy of Sciences, 2014. Accessed: Apr. 15, 2020. [Online]. Available: : https://www.nap.edu/catalog/22407/calibration-of-aashto-lrfd-concrete-bridge-design-specifications-for-serviceability https://www.nap.edu/catalog/22407/calibr... ]³
Girder geometric data	b_s, b_w, b_i, b_f	Normal	1.00	0.04	Used in Wassef et al. [⁸8 W. G. Wassef et al., Calibration of AASHTO LRFD Concrete Bridge Design Specifications for Serviceability, NCHRP Project 12-83. Washington, DC: National Academy of Sciences, 2014. Accessed: Apr. 15, 2020. [Online]. Available: : https://www.nap.edu/catalog/22407/calibration-of-aashto-lrfd-concrete-bridge-design-specifications-for-serviceability https://www.nap.edu/catalog/22407/calibr... ]⁴
	h_v, h_f, h_s, h_i	Normal	1.00	0.025	Used in Wassef et al. [⁸8 W. G. Wassef et al., Calibration of AASHTO LRFD Concrete Bridge Design Specifications for Serviceability, NCHRP Project 12-83. Washington, DC: National Academy of Sciences, 2014. Accessed: Apr. 15, 2020. [Online]. Available: : https://www.nap.edu/catalog/22407/calibration-of-aashto-lrfd-concrete-bridge-design-specifications-for-serviceability https://www.nap.edu/catalog/22407/calibr... ]⁴
	t_s, t_i	Normal	1.00	0.025	h_s, h_i
	d_p, d_s	Normal	1.00	0.04	Used in Wassef et al. [⁸8 W. G. Wassef et al., Calibration of AASHTO LRFD Concrete Bridge Design Specifications for Serviceability, NCHRP Project 12-83. Washington, DC: National Academy of Sciences, 2014. Accessed: Apr. 15, 2020. [Online]. Available: : https://www.nap.edu/catalog/22407/calibration-of-aashto-lrfd-concrete-bridge-design-specifications-for-serviceability https://www.nap.edu/catalog/22407/calibr... ]⁴
Load effect for dead loads	M_g1, V_g1, T_g1	Normal	1.03	0.08	Nowak [²⁷27 A. S. Nowak, Calibration of the LRFD Bridge Design Code (NCHRP Report, 368). Washington, D.C.: Transportation Research Board, National Research Council, 1999.]
	M_g2, V_g2, T_g2	Normal	1.05	0.10	Nowak [²⁷27 A. S. Nowak, Calibration of the LRFD Bridge Design Code (NCHRP Report, 368). Washington, D.C.: Transportation Research Board, National Research Council, 1999.]
	M_g3, V_g3, T_g3	Normal	1.10	0.25	Nowak [²⁷27 A. S. Nowak, Calibration of the LRFD Bridge Design Code (NCHRP Report, 368). Washington, D.C.: Transportation Research Board, National Research Council, 1999.]

Case	Δσ (MPa)	N	m	K = N x $Δ σ$ ^m
Reinforcing steel (straight bars)	≥ 245	≤ 2 x 10⁶	6	4.33 x 10²⁰
	< 245	> 2 x 10⁶	9	6.39 x 10²⁷
	> 205	< 10⁷	9	6.39 x 10²⁷
	≤ 205	> 10⁷	11	2.69 x 10³²
Prestressing steel (postensioning curved tendons)	≥ 165	≤ 10⁶	3	165³ x 10⁶
Prestressing steel (postensioning curved tendons)	< 165	> 10⁶	7	165⁷ x 10⁶

Case	Δσ (MPa)	Mean	Standard deviation	$α$	$u$
Reinforcing steel (straight bars)	≥ 245	1.104	0.463	2.57	1.24
	< 245	1.154	0.556	2.19	1.30
	> 205	1.154	0.556	2.19	1.30
	≤ 205	1.169	0.618	1.97	1.32
Prestressing steel (postensioning curved tendons)	≥ 165	1.041	0.274	4.28	1.14
Prestressing steel (postensioning curved tendons)	< 165	1.072	0.367	3.21	1.20

Class (DNIT)	Frequency (%)	GVW (kN)	Legal GVW (kN)
2C	14.96	62.0 (min.)	168.0
		272.5 (max.)
		100.8 (aver.)
2S1	1.89	69.9 (min.)	273.0
		359.8 (max.)
		154.2 (aver.)
3C	23.33	66.7 (min.)	242.0
		417.1 (max.)
		166.7 (aver.)
2S2	13.03	99.3 (min.)	347.0
		497.4 (max.)
		190.9 (aver.)
2I2	2.05	100.1 (min.)	378.0
		529.6 (max.)
		196.8 (aver.)
4CD	2.68	96.0 (min.)	305.0
		504.1 (max.)
		242.8 (aver.)
2S3	9.45	124.2 (min.)	436.0
		704.5 (max.)
		325.1 (aver.)
3S2	1.98	128.2 (min.)	420.0
		630.1 (max.)
		260.0 (aver.)
3S3	14.77	139.8 (min.)	509.3
		869.2 (max.)
		414.6 (aver.)
3I3	3.13	143.1 (min.)	556.5
		760.2 (max.)
		448.0 (aver.)
3D4	4.5	183.3 (min.)	599.0
		1026.1 (max.)
		502.1 (aver.)
3M6	2.68	240.8 (min.)	777.0
		1192.7 (max.)
		660.2 (aver.)

Prestress level	Prestressing steel (A_p)	Longitudinal reinforcing steel (A_s)	Two-legged stirrup (A_sw/s)
Limited	40 cm² (40 strands; ϕ_p = 12.7mm)	24.13 cm² (12 bars; ϕ_s = 16mm)	14.02 cm²/m (ϕ_s = 12.5mm; s = 17.5cm)
Complete	48 cm² (48 strands; ϕ_p = 12.7mm)	12.06 cm² (6 bars; ϕ_s = 16mm)	8.98 cm²/m (ϕ_s = 10mm; s = 17.5cm)

ADTT (average daily truck traffic)	Prestress level	∑(1/N_i) - Lognormal		DM - Weibull		β
ADTT (average daily truck traffic)	Prestress level	Mean	Standard deviation	Mean	Standard deviation	β
2500	Limited	6.108 x 10^-5	6.874 x 10^-3	1.169	0.618	4.6
2500	Complete	5.512 x 10^-5	3.472 x 10^-4	1.169	0.618	5.3
5000	Limited	1.224 x 10^-4	1.379 x 10^-2	1.169	0.618	4.4
5000	Complete	1.102 x 10^-4	6.945 x 10^-4	1.169	0.618	5.0
7500	Limited	1.832 x 10^-4	2.062 x 10^-2	1.169	0.618	4.3
7500	Complete	1.654 x 10^-4	1.042 x 10^-3	1.169	0.618	4.9

ADTT (average daily truck traffic)	Prestress level	∑(1/N_i) - Lognormal		DM - Weibull		β
ADTT (average daily truck traffic)	Prestress level	Mean	Standard deviation	Mean	Standard deviation	β
2500	Limited	6.302 x 10^-3	1.516 x 10^-2	1.072	0.367	4.1
2500	Complete	1.233 x 10^-3	1.319 x 10^-3	1.072	0.367	5.8
5000	Limited	1.260 x 10^-2	3.031 x 10^-2	1.072	0.367	3.7
5000	Complete	2.464 x 10^-3	2.636 x 10^-3	1.072	0.367	5.4
7500	Limited	1.891 x 10^-2	4.550 x 10^-2	1.072	0.367	3.4
7500	Complete	3.697 x 10^-3	3.954 x 10^-3	1.072	0.367	5.2

ADTT (average daily truck traffic)	Prestress level	∑(1/N_i) - Lognormal		DM - Weibull		β
ADTT (average daily truck traffic)	Prestress level	Mean	Standard deviation	Mean	Standard deviation	β
2500	Limited	1.247 x 10^-3	1.271 x 10^-2	1.169	0.618	4.0
2500	Complete	7.825 x 10^-4	1.803 x 10^-3	1.169	0.618	4.6
5000	Limited	2.494 x 10^-3	2.544 x 10^-2	1.169	0.618	3.7
5000	Complete	1.565 x 10^-3	3.608 x 10^-3	1.169	0.618	4.3
7500	Limited	3.740 x 10^-3	3.812 x 10^-2	1.169	0.618	3.5
7500	Complete	2.348 x 10^-3	5.412 x 10^-3	1.169	0.618	4.1

[1] Corresponding author: Anselmo Leal Carneiro. E-mail: anselmo.lc@hotmail.com

Brasil

Brasil

Fatigue safety level provided by Brazilian design standards for a prestressed girder highway bridge

Nível de segurança à fadiga proporcionado pelas normas brasileiras de projeto em relação à uma ponte rodoviária com longarinas protendidas

Abstract