Comparative study of steel-concrete composite beams for railway bridges

Santos, Ruan Richelly; Carvalho, Hermes; Santos, Rodrigo Felipe; Bittencourt, Túlio Nogueira; Caldas, Rodrigo Barreto

doi:10.1590/S1983-41952024000300002

Ruan Richelly Santos

data curation

formal analysis

investigation

methodology

writing

Universidade de São Paulo – USP, Escola Politécnica, Departamento de Engenharia de Estruturas e Geotécnica, São Paulo, SP, Brasil

https://orcid.org/0000-0002-3799-2248

Hermes Carvalho

conceptualization

formal analysis

methodology

resources

supervision

revision

Universidade Federal de Minas Gerais – UFMG, Departamento de Engenharia de Estruturas, Belo Horizonte, MG, Brasil

https://orcid.org/0000-0002-4652-8068

Rodrigo Felipe Santos

formal analysis

investigation

writing

Universidade Federal de Viçosa – UFV, Departamento de Engenharia Civil, Viçosa, MG, Brasil

https://orcid.org/0000-0002-1986-4779

Túlio Nogueira Bittencourt

validation

visualization

revision

Universidade de São Paulo – USP, Escola Politécnica, Departamento de Engenharia de Estruturas e Geotécnica, São Paulo, SP, Brasil

https://orcid.org/0000-0001-6523-2687

Rodrigo Barreto Caldas

validation

visualization

revision

Universidade Federal de Minas Gerais – UFMG, Departamento de Engenharia de Estruturas, Belo Horizonte, MG, Brasil

https://orcid.org/0000-0002-0851-7632

Corresponding author: Ruan Richelly Santos. E-mail: ruanrichelly@usp.br

Conflict of interest: Nothing to declare.

Author contributions: RRS: data curation, formal analysis, investigation, methodology, writing; HC: conceptualization, formal analysis, methodology, resources, supervision, revision; RFS: formal analysis, investigation, writing; TNB and RBC: validation, visualization, revision.

Editors: Rebecca Gravina, Guilherme Aris Parsekian.

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Figure 1
Composite beams cross-section examples: (a) traditional solution, (b) beam with composite dowels, (c) pre-cambered composite beam.

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Figure 2
Steel-concrete composite beams with composite dowels (Adapted from Feldmann et al. [⁴4 M. Feldmann, M. Kopp, and D. Pak, “Composite dowels as shear connectors for composite beams – background to the german technical approval,” Steel Constr., vol. 9, no. 2, pp. 80–88, 2016, http://dx.doi.org/10.1002/stco.201610020.
http://dx.doi.org/10.1002/stco.201610020... ]).

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Figure 3
Pre-cambered composite beams, Preflex [¹⁰10 Christmann and Pfeifer. “The Preflex girder for bridge building.” https://www.cpbau.de/en/products/preflexgirder/ (accessed Sept. 01, 2022).
https://www.cpbau.de/en/products/preflex... ].

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Figure 4
Typical section of a composite bridge with steel I-girders [¹⁸18 D. C. Iles, Design Guide for Steel Railway Bridges. Berkshire, UK: Steel Construction Institute, 2004.].

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Figure 5
Development timeline of steel-plated connectors with drilled holes and composite dowel connectors (Adapted from Cardoso et al. [²⁸28 H. S. Cardoso, O. P. Aguiar, R. B. Caldas, and R. H. Fakury, “Composite dowels as load introduction devices in concrete-filled steel tubular columns,” Eng. Struct., vol. 219, Sep 2020, http://dx.doi.org/10.1016/j.engstruct.2020.110805.
http://dx.doi.org/10.1016/j.engstruct.20... ]).

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Figure 6
Composite dowels geometry: (a) puzzle, (b) clothoidal.

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Figure 7
Notation for the section along a composite dowel [⁴4 M. Feldmann, M. Kopp, and D. Pak, “Composite dowels as shear connectors for composite beams – background to the german technical approval,” Steel Constr., vol. 9, no. 2, pp. 80–88, 2016, http://dx.doi.org/10.1002/stco.201610020.
http://dx.doi.org/10.1002/stco.201610020... ].

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Figure 8
Geometry notation for the double composite beam with composite dowels.

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Figure 9
Composite dowels failure modes [²2 M. Kopp et al., “Composite dowels as shear connectors for composite beams – Background to the design concept for static loading,” J. Construct. Steel Res., vol. 147, pp. 488–503, Aug 2018, http://dx.doi.org/10.1016/j.jcsr.2018.04.013.
http://dx.doi.org/10.1016/j.jcsr.2018.04... ], [⁴4 M. Feldmann, M. Kopp, and D. Pak, “Composite dowels as shear connectors for composite beams – background to the german technical approval,” Steel Constr., vol. 9, no. 2, pp. 80–88, 2016, http://dx.doi.org/10.1002/stco.201610020.
http://dx.doi.org/10.1002/stco.201610020... ].

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Figure 10
Reinforcement details in composite girder with reinforced concrete web and composite dowels [¹⁴14 Deutsches Institut für Bautechnik, Allgemeine Bauartgenehmigung der Stahl-verbundträger mit Verbunddübelleisten in Klothoiden- und Puzzleform, Z-26.4-56, 2018.].

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Figure 6
Composite dowels geometry: (a) puzzle, (b) clothoidal.

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Figure 7
Notation for the section along a composite dowel [⁴4 M. Feldmann, M. Kopp, and D. Pak, “Composite dowels as shear connectors for composite beams – background to the german technical approval,” Steel Constr., vol. 9, no. 2, pp. 80–88, 2016, http://dx.doi.org/10.1002/stco.201610020.
http://dx.doi.org/10.1002/stco.201610020... ].

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Figure 8
Geometry notation for the double composite beam with composite dowels.

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Figure 9
Composite dowels failure modes [²2 M. Kopp et al., “Composite dowels as shear connectors for composite beams – Background to the design concept for static loading,” J. Construct. Steel Res., vol. 147, pp. 488–503, Aug 2018, http://dx.doi.org/10.1016/j.jcsr.2018.04.013.
http://dx.doi.org/10.1016/j.jcsr.2018.04... ], [⁴4 M. Feldmann, M. Kopp, and D. Pak, “Composite dowels as shear connectors for composite beams – background to the german technical approval,” Steel Constr., vol. 9, no. 2, pp. 80–88, 2016, http://dx.doi.org/10.1002/stco.201610020.
http://dx.doi.org/10.1002/stco.201610020... ].

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Figure 10
Reinforcement details in composite girder with reinforced concrete web and composite dowels [¹⁴14 Deutsches Institut für Bautechnik, Allgemeine Bauartgenehmigung der Stahl-verbundträger mit Verbunddübelleisten in Klothoiden- und Puzzleform, Z-26.4-56, 2018.].

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Figure 9
Composite dowels failure modes [²2 M. Kopp et al., “Composite dowels as shear connectors for composite beams – Background to the design concept for static loading,” J. Construct. Steel Res., vol. 147, pp. 488–503, Aug 2018, http://dx.doi.org/10.1016/j.jcsr.2018.04.013.
http://dx.doi.org/10.1016/j.jcsr.2018.04... ], [⁴4 M. Feldmann, M. Kopp, and D. Pak, “Composite dowels as shear connectors for composite beams – background to the german technical approval,” Steel Constr., vol. 9, no. 2, pp. 80–88, 2016, http://dx.doi.org/10.1002/stco.201610020.
http://dx.doi.org/10.1002/stco.201610020... ].

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Figure 10
Reinforcement details in composite girder with reinforced concrete web and composite dowels [¹⁴14 Deutsches Institut für Bautechnik, Allgemeine Bauartgenehmigung der Stahl-verbundträger mit Verbunddübelleisten in Klothoiden- und Puzzleform, Z-26.4-56, 2018.].

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Figure 11
Sign convention for the cross-section’s internal and external effects [⁷7 S. G. Morano and C. Mannini, “Preflex beams: a method of calculation of creep and shrinkage effects,” J. Bridge Eng., vol. 11, no. 1, pp. 48–58, 2006, http://dx.doi.org/10.1061/ASCE1084-0702200611:148.
http://dx.doi.org/10.1061/ASCE1084-07022... ].

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Figure 12
Reference deck cross-section.

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Figure 13
Courbon’s method: (a) girder bridge deck eccentrically loaded, (b) transverse section with equivalent loading, (c) transverse deflection profile and girder reaction (Adapted from Binjola [³⁴34 A. K. Binjola, “Load distribution in girders bridges by different methods,” M.S. thesis, University of Roorkee, Roorkee, 1988.]).

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Figure 14
Flowchart for the composite beams design.

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Figure 13
Courbon’s method: (a) girder bridge deck eccentrically loaded, (b) transverse section with equivalent loading, (c) transverse deflection profile and girder reaction (Adapted from Binjola [³⁴34 A. K. Binjola, “Load distribution in girders bridges by different methods,” M.S. thesis, University of Roorkee, Roorkee, 1988.]).

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Figure 14
Flowchart for the composite beams design.

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Figure 15
Cross-section geometry notation: (a) traditional composite beam; (b) composite beam with composite dowels; (c) pre-cambered composite beam.

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Figure 16
Equivalent steel weight per unit deck surface to slenderness ratio.

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Figure 17
TRA-S20-LH15-W deck cross-section (dimensions in mm).

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Figure 18
Construction details of TRA-S20-LH15-W solution (dimensions in mm).

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Figure 19
Equivalent steel weight per unit deck surface to slenderness ratio for beams with composite dowels.

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Figure 20
PCB-S20-LH20-W deck cross-section (dimensions in mm).

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Figure 21
Construction details of PCB-S20-LH20-W solution (dimensions in mm).

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Figure 22
Equivalent steel weight per unit deck surface to slenderness ratio for pre-cambered composite beams.

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Figure 23
PFX-S20-LH15-W deck cross-section (dimensions in mm).

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Figure 24
Construction details of PFX-S20-LH15-W solution (dimensions in mm).

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Figure 25
Cost analysis: (a) 20-meter span, (b) 25-meter span; (c) 30-meter span.

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Figure 26
Cost analysis accounting for a beam height restriction: (a) 20-meter span, (b) 25-meter span; (c) 30-meter span.

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Figure 16
Equivalent steel weight per unit deck surface to slenderness ratio.

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Figure 17
TRA-S20-LH15-W deck cross-section (dimensions in mm).

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Figure 18
Construction details of TRA-S20-LH15-W solution (dimensions in mm).

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Figure 19
Equivalent steel weight per unit deck surface to slenderness ratio for beams with composite dowels.

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Figure 20
PCB-S20-LH20-W deck cross-section (dimensions in mm).

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Figure 21
Construction details of PCB-S20-LH20-W solution (dimensions in mm).

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Figure 22
Equivalent steel weight per unit deck surface to slenderness ratio for pre-cambered composite beams.

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Figure 23
PFX-S20-LH15-W deck cross-section (dimensions in mm).

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Figure 24
Construction details of PFX-S20-LH15-W solution (dimensions in mm).

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Figure 25
Cost analysis: (a) 20-meter span, (b) 25-meter span; (c) 30-meter span.

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Figure 26
Cost analysis accounting for a beam height restriction: (a) 20-meter span, (b) 25-meter span; (c) 30-meter span.

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Table 1
Equivalent cost for materials.

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Table 2
Geometric information of the sections obtained: traditional solution.

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Table 3
Geometric information of the sections obtained: composite beams with composite dowels.

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Table 4
Geometric information of the sections obtained: pre-cambered composite beam.

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Case Short-term Long-term ULS SLS FLS ULS SLS FLS Mpl,Rd/Med or σel/σEd VRd/ VL,Rd VL,Rd/ VL,Rd Displacement (mm) VL,Rd/ VL,Rd ΔτC/ λφΔτ Δσc,lf./ λφΔσE,lf Mpl,Rd/Med ou σel/σEd VRd/ VL,Rd VL,Rd/ VL,Rd Displacement (mm) VL,Rd/ VL,Rd ΔτC/ λφΔτ Δσc,lf./ λφΔσE,lf TRA-V20-LH15-S 1.63 1.52 1.40 25.46 1.48 1.49 2.93 1.50 1.38 1.30 32.50 1.47 1.47 2.69 TRA-V20-LH20-S 1.53 1.09 1.10 27.07 1.17 1.15 4.60 1.50 1.09 1.03 38.92 1.17 1.15 4.30 TRA-V20-LH25-S 1.24 1.95 1.18 27.57 1.25 1.24 8.76 1.12 1.73 1.12 47.27 1.27 1.26 8.78 TRA-V25-LH15-S 1.65 1.13 1.39 31.42 1.48 1.63 3.11 1.47 1.12 1.35 39.76 1.53 1.69 2.92 TRA-V25-LH20-S 1.54 1.44 1.31 32.16 1.39 1.48 4.92 1.46 1.35 1.27 46.28 1.44 1.54 4.71 TRA-V25-LH25-S 1.24 1.53 1.03 32.02 1.09 1.19 9.50 1.12 1.44 1.03 54.43 1.16 1.28 9.57 TRA-V30-LH15-S 1.61 1.30 1.01 38.39 1.07 1.27 3.18 1.41 1.22 1.01 48.87 1.15 1.36 3.00 TRA-V30-LH20-S 1.48 1.66 1.33 38.46 1.41 1.63 5.14 1.41 1.55 1.37 55.15 1.55 1.78 4.91 TRA-V30-LH25-S 1.19 1.61 1.42 36.05 1.50 1.79 10.95 1.06 1.51 1.45 62.51 1.63 1.95 11.13

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Case Short-term Long-term ULS SLS FLS ULS SLS FLS Mpl,Rd/Med or σel/σEd VRd/ VL,Rd VL,Rd/ VL,Rd Displacement (mm) VL,Rd/ VL,Rd ΔτC/ λφΔτ Δσc,lf./ λφΔσE,lf Mpl,Rd/Med or σel/σEd VRd/ VL,Rd VL,Rd/ VL,Rd Displacement (mm) VL,Rd/ VL,Rd ΔτC/ λφΔτ Δσc,lf./ λφΔσE,lf PCB-V20-LH15-S 2.17 4.93 1.02 16.61 1.26 1.55 5.50 2.17 5.00 3.79 23.51 4.68 1.14 4.25 PCB-V20-LH20-L 1.42 3.53 1.01 20.65 1.25 1.49 7.14 1.42 3.56 1.57 30.95 1.93 1.06 5.17 PCB-V20-LH20-S 1.63 3.29 1.00 26.54 1.24 1.53 5.06 1.63 3.32 2.10 39.73 2.59 1.05 3.56 PCB-V20-LH25-S 1.10 2.17 1.25 27.43 1.54 1.88 8.54 1.48 2.17 1.49 45.36 1.84 1.19 5.47 PCB-V25-LH15-S 1.94 4.70 1.56 21.45 1.92 1.70 5.19 1.94 4.74 2.74 28.21 3.39 1.09 3.67 PCB-V25-LH20-S 1.60 2.84 1.10 31.31 1.36 1.80 5.67 1.60 2.87 3.38 31.79 4.16 1.21 3.89 PCB-V25-LH25-S 1.00 3.07 1.50 31.43 1.84 1.86 9.53 1.84 3.07 8.45 34.79 10.37 1.05 5.41 PCB-V30-LH15-S 1.59 4.63 1.16 29.69 1.43 1.57 4.69 1.59 4.71 2.67 38.62 3.29 1.07 3.60 PCB-V30-LH20-S 1.46 3.56 1.50 35.39 1.85 1.66 6.35 1.46 3.59 1.44 42.96 1.77 1.06 4.40 PCB-V30-LH25-S 1.45 3.34 1.55 36.14 1.89 1.99 10.08 1.09 3.34 1.24 39.70 1.52 1.18 6.20

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Case ULS SLS FLS Mpl,Rd/Med or σel/σEd VRd/ VL,Rd VL,Rd/VL,Rd Deck VL,Rd/VL,Rd C1 Displacement (mm) VL,Rd/VL,Rd Deck VL,Rd/VL,Rd C1 ΔτC/λφΔτ Deck ΔτC/λφΔτ C1 Δσc,lf./ λφΔσE,lf PFX-V20-LH15-S 1.31 1.31 1.33 1.05 17.83 1.41 1.11 1.43 1.13 5.09 PFX-V20-LH20-S 2.35 1.05 1.18 1.07 27.22 1.26 1.14 1.26 1.14 5.33 PFX-V20-LH25-S 2.72 1.06 1.11 1.93 29.67 1.18 2.04 1.18 2.04 9.81 PFX-V25-LH15-S 1.39 1.62 1.27 1.07 31.27 1.34 1.13 1.19 1.00 3.91 PFX-V25-LH20-S 1.31 1.66 1.03 1.13 31.67 1.09 1.20 1.20 1.32 6.08 PFX-V25-LH25-S 2.40 1.01 1.19 1.10 35.21 1.26 1.16 1.43 1.32 12.69 PFX-V30-LH15-S 1.35 1.26 1.35 1.25 24.55 1.44 1.32 1.74 1.60 6.22 PFX-V30-LH20-S 1.28 2.24 1.24 1.17 35.34 1.31 1.24 1.55 1.47 6.55 PFX-V30-LH25-S 2.19 1.14 1.20 1.17 38.76 1.26 1.24 1.61 1.58 16.00

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Material	Equivalent S460 steel
1 $k g$ of structural steel S460, $f_{y} = 460 M P a$	1.00 kg
1 $k g$ of reinforcing steel, $f_{y} = 500 M P a$	0.64 kg
1 $m^{3}$ of concrete, $f_{c k} = 50 M P a$	50.00 kg
1 $m^{3}$ of concrete, $f_{c k} = 90 M P a$	68.00 kg

Case	Steel section						Stud bolts		Composite section
Case	h (mm)	^bf,sup ^(mm)	t_f,sup (mm)	b_f,inf (mm)	^tf,inf ^(mm)	t_w (mm)	Studs/ section	Type	Number of beams	H (mm)	b (mm)	h_c (mm)	Steel weight/deck area (kg/m²)	Concrete volume/deck area (m³/m²)
TRA-S20-LH15-W	893.00	450.00	22.40	550.00	44.50	19.00	4	Ø22x150	4	1333.00	3167.50	440.00	124.24	0.44
TRA-S20-LH20-W	560.00	450.00	25.00	580.00	76.00	16.00	4	Ø22x150	6	1000.00	2111.67	440.00	232.44	0.44
TRA-S20-LH25-W	360.00	450.00	22.40	850.00	89.00	37.50	4	Ø22x150	9	800.00	1407.78	400.00	528.78	0.44
TRA-S25-LH15-W	1227.00	450.00	22.40	550.00	52.50	19.00	4	Ø22x150	4	1667.00	3167.50	440.00	150.44	0.44
TRA-S25-LH20-W	810.00	450.00	16.00	600.00	76.00	16.00	4	Ø22x150	7	1250.00	1810.00	440.00	278.16	0.44
TRA-S25-LH25-W	560.00	450.00	22.40	960.00	89.00	19.00	4	Ø22x150	10	1000.00	1267.00	440.00	643.10	0.44
TRA-S30-LH15-W	1560.00	400.00	25.00	550.00	54.00	22.40	4	Ø22x150	4	2000.00	3167.50	440.00	180.18	0.44
TRA-S30-LH20-W	1060.00	450.00	25.00	615.00	82.00	19.00	4	Ø22x150	7	1500.00	1810.00	440.00	345.22	0.44
TRA-S30-LH25-W	760.00	450.00	19.00	1200.00	89.00	16.00	4	Ø22x150	10	1200.00	1267.00	440.00	777.47	0.44

Case	Steel section					Connectors			Composite section
Case	Section	h_mr (mm)	b_f (mm)	t_f (mm)	t_w (mm)	h_d (mm)	e_x (mm)	e_y (mm)	Number of beams	H (mm)	b (mm)	h_c (mm)	c_o (mm)	c_u (mm)	b_c (mm)	Steel weight/deck area (kg/m²)	Concrete volume/deck area (m³/m²)
PCB-S20-LH15-W	-	713.00	300.00	37.50	31.50	180.00	450.00	650.00	3	1333.00	4223.33	440.00	440.00	220.00	568.50	120.64	0.58
PCB-S20-LH20-L	W 920x420x656	403.50	431.00	62.00	34.50	180.00	450.00	912.00	4	1024.00	3167.50	440.00	440.00	270.00	827.50	190.78	0.64
PCB-S20-LH20-W	-	380.00	400.00	37.50	31.50	180.00	450.00	850.00	4	1000.00	3167.50	440.00	440.00	130.00	768.50	127.52	0.60
PCB-S20-LH25-W	-	180.00	720.00	44.50	37.50	180.00	450.00	1490.00	6	800.00	2111.67	440.00	440.00	135.50	1402.50	275.34	0.76
PCB-S25-LH15-W	-	1047.00	500.00	22.40	38.00	180.00	450.00	550.00	4	1667.00	3167.50	440.00	440.00	200.00	481.00	124.91	0.65
PCB-S25-LH20-W	-	425.00	400.00	44.50	31.50	180.00	450.00	850.00	4	1250.00	3167.50	440.00	440.00	200.00	768.50	147.29	0.70
PCB-S25-LH25-W	-	380.00	785.00	37.50	25.00	180.00	450.00	1620.00	7	1000.00	1810.00	440.00	440.00	335.50	1545.00	328.83	1.11
PCB-S30-LH15-W	-	1380.00	250.00	25.00	22.40	180.00	450.00	550.00	3	2000.00	4223.33	440.00	440.00	220.00	477.60	135.74	0.64
PCB-S30-LH20-W	-	880.00	405.00	44.50	22.40	180.00	450.00	860.00	4	1500.00	3167.50	440.00	440.00	835.50	787.60	181.66	0.82
PCB-S30-LH25-W	-	580.00	825.00	37.50	22.40	180.00	450.00	1700.00	7	1200.00	1810.00	440.00	440.00	542.50	1627.60	372.88	1.42

Case	Steel section				Stud bolts				Bottom flange concrete				Composite section
Case	h (mm)	b_f,sup (mm)	t_f,sup (mm)	b_f,inf (mm)	t_f,inf (mm)	t_w (mm)	Studs/section	Type	Concrete volume/deck area (m³/m²)	Reinforcement weight/deck area (kg/m²)	Number of beams	H (mm)	b (mm)	h_c (mm)	Steel weight/deck area (kg/m²)	Concrete volume/deck area (m³/m²)
PFX-S20-LH15-W	813.00	450.00	22.40	650.00	31.50	19.00	4	Ø22x150	0.04	2.98	4	1333.00	3167.50	440.00	111.21	0.44
PFX-S20-LH20-W	500.00	600.00	31.50	800.00	52.50	22.40	4	Ø22x150	0.03	4.97	5	1000.00	2534.00	440.00	217.01	0.44
PFX-S20-LH25-W	320.00	450.00	25.00	800.00	89.00	25.00	4	Ø22x150	0.04	6.71	9	800.00	1407.78	400.00	487.32	0.44
PFX-S25-LH15-W	1127.00	450.00	22.40	650.00	44.50	22.40	4	Ø22x150	0.04	2.24	3	1667.00	4223.33	440.00	116.36	0.44
PFX-S25-LH20-W	710.00	450.00	31.50	650.00	50.85	25.00	4	Ø22x150	0.08	4.47	6	1250.00	2111.67	440.00	233.34	0.44
PFX-S25-LH25-W	460.00	450.00	25.00	900.00	89.00	16.00	4	Ø22x150	0.13	9.94	10	1000.00	1267.00	440.00	598.86	0.44
PFX-S30-LH15-W	1430.00	450.00	22.40	637.00	44.50	22.40	4	Ø22x150	0.07	2.98	4	2000.00	3167.50	440.00	170.50	0.44
PFX-S30-LH20-W	960.00	450.00	31.50	650.00	44.50	25.00	4	Ø22x150	0.11	5.22	7	1500.00	1810.00	440.00	282.11	0.44
PFX-S30-LH25-W	630.00	450.00	25.00	1100.00	89.00	16.00	4	Ø22x150	0.23	12.43	10	1200.00	1267.00	440.00	725.70	0.44

P_{R k (C L; P Z)} = m i n \{\begin{matrix} P_{s h, k} = η_{D} e_{x}^{2} \sqrt{f_{c k}} (1 + ρ_{D}) \\ P_{p o, k} = χ_{x} χ_{y} 90 h_{p o}^{1.5} \sqrt{f_{c}} (1 + ρ_{D, i}) \\ P_{p l, k} = 0.25 e_{x} t_{w} f_{y} \end{matrix}

D s = |k_{f, L} \frac{D V S_{y}}{I_{y} t_{w}}| + |k_{f, G} (\frac{D N}{A} + \frac{D M}{I_{y}} z_{D})| \leq 1.3 f_{y}

η_{N} (t) = \frac{ζ (t_{0}, t) ∙ (\frac{A_{c}}{A_{s}} + \frac{A_{c} d^{2}}{I_{s}}) + [(1 - α) ∙ ζ (t_{0}, t) + α] ∙ n_{i}}{n_{i} + \frac{A_{c}}{A_{s}} + \frac{A_{c} d^{2}}{I_{s}} - n_{28} α ∙ [ζ (t_{0}, t) - 1]} n_{28}

η_{M} (t) = \frac{ζ (t_{0}, t) - α_{M} [ζ (t_{0}, t) - 1] ∙ γ}{1 - n_{28} α_{M} [ζ (t_{0}, t) - 1] ∙ β (t)} n_{28}

β (t) = \frac{n_{i} + \frac{A_{c}}{A_{s}}}{n_{N} (t) + \frac{A_{c}}{A_{s}}} ∙ \frac{n_{N} (t) + \frac{A_{c}}{A_{s}} + \frac{A_{c} d^{2}}{I_{s}}}{\frac{I_{c}}{I_{s}} (n_{i} + \frac{A_{c}}{A_{s}}) + n_{i} (n_{i} + \frac{A_{c}}{A_{s}} + \frac{A_{c} d^{2}}{I_{s}})}

γ = \frac{n_{i} (n_{i} + \frac{A_{c}}{A_{s}} + \frac{A_{c} d^{2}}{I_{s}})}{\frac{I_{c}}{I_{s}} (n_{i} + \frac{A_{c}}{A_{s}}) + n_{i} (n_{i} + \frac{A_{c}}{A_{s}} + \frac{A_{c} d^{2}}{I_{s}})}

[1] Corresponding author: Ruan Richelly Santos. E-mail: ruanrichelly@usp.br

Brasil

Brasil

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Abstract