Abstracts
Abstract
The evaluation of the deflection in beams is an indispensable step of the structural design. Currently, many standard codes adopt the Branson’s model. However, the Branson’s model underestimates the deflection of beams with a reinforcement rate of less than 1%. Therefore, this study proposes a new alternative to quantify the deflections in reinforced concrete beams, based on the Lumped Damage Mechanics (LDM). LDM is a nonlinear theory, which uses concepts from Fracture and Damage Mechanics combined with plastic hinges. The viability of the proposed model was verified through comparisons with results from experimental works developed by other authors and the application of the Branson’s model. The obtained results showed that the proposed calculation model had a good approximation of the experimental data with satisfactory accuracy and equivalent values to the Branson’s model in the investigated scenarios.
Keywords:
reinforced concrete beams; deflection; lumped damage mechanics; plastic hinge; structural design
Resumo
A avaliação de flechas em vigas é uma etapa indispensável no projeto estrutural. Atualmente, muitas normas adotam o modelo de Branson. Entretanto, o modelo de Branson subestima a flecha de vigas com taxa de armadura menor do que 1%. Desta forma, este estudo propõe uma alternativa para quantificar flechas em vigas de concreto armado com base na Mecânica do Dano Concentrado (MDC). A MDC é uma teoria não linear que utiliza conceitos das Mecânicas da Fratura e do Dano combinados com rótulas plásticas. A viabilidade do modelo proposto foi verificada por meio da comparação com resultados experimentais obtidos por outros autores bem como a aplicação do modelo de Branson. Os resultados obtidos mostram que o modelo proposto tem boa aproximação aos dados experimentais com acurácia satisfatória e valores equivalentes ao modelo de Branson nos cenários investigados.
Palavraschave:
vigas de concreto armado; flecha; mecânica do dano concentrado; rótula plástica; projeto estrutural
1 INTRODUCTION
Structural response in service is an important issue in civil engineering. Design codes around the world estimate the immediate deflection of reinforced concrete (RC) beams based on the equation proposed by Branson [^{1}1 D. E. Branson, Instantaneous and Timedependent Deflections of Simple and Continuous Reinforced Concrete Beams, (HPR Report 7, pp. 178). College Park, MD, USA: Alabama Highway Department, Bureau of Public Roads, 1963.], [^{2}2 D. E. Branson, Deformation of Concrete Structures, New York, NY, USA: McGrawHill, 1977.] for estimating the equivalent inertia moments of cracked RC members (e.g., [^{3}3 ACI Committee 318, Building code requirements for structural concrete, ACI 31819, 2019.][^{6}6 Standards Australia, Australian Standard for Concrete Structures, AS3600, 2018.]). Such design codes present small variations for the formulation proposed by Branson [^{1}1 D. E. Branson, Instantaneous and Timedependent Deflections of Simple and Continuous Reinforced Concrete Beams, (HPR Report 7, pp. 178). College Park, MD, USA: Alabama Highway Department, Bureau of Public Roads, 1963.], [^{2}2 D. E. Branson, Deformation of Concrete Structures, New York, NY, USA: McGrawHill, 1977.].
Recently, other researchers have initiated improvements to Branson’s model, especially for concrete reinforced with other materials, such as fibrereinforced polymers [^{7}7 E. G. Nawy and G. E. Neuworth, "Fiberglass reinforced concrete slabs and beams," J. Struct. Div., vol. 103, no. 2, pp. 421440, 1977, http://dx.doi.org/10.1061/JSDEAG.0004559.
http://dx.doi.org/10.1061/JSDEAG.0004559...
][^{20}20 T. Lou and T. L. Karavasilis, "Timedependent assessment and deflection prediction of prestressed concrete beams with unbonded CFRP tendons," Compos. Struct., vol. 194, pp. 263276, 2018, http://dx.doi.org/10.1016/j.compstruct.2018.04.013.
http://dx.doi.org/10.1016/j.compstruct.2...
]. Gribniak et al. [^{21}21 V. Gribniak, V. Cervenka, and G. Kaklauskas, "Deflection prediction of reinforced concrete beams by design codes and computer simulation," Eng. Struct., vol. 56, pp. 21752186, 2013, http://dx.doi.org/10.1016/j.engstruct.2013.08.045.
http://dx.doi.org/10.1016/j.engstruct.20...
] presented a statistical study of the immediate deflections of RC beams evaluated by different design codes [^{3}3 ACI Committee 318, Building code requirements for structural concrete, ACI 31819, 2019.], [^{22}22 Comité Européen de Normalisation, Design of Concrete Structures  Part 1: General Rules and Rules for Buildings, Eurocode 2, 2004.], [^{23}23 Concrete and Reinforced Concrete Research and Technology Institute, Concrete and Reinforced Concrete Structures Without Prestressing, SP 521012003, 2006.] and smeared crack numerical analysis with several finite elements. Despite the accuracy of the finite element analysis (FEA) presented in [^{21}21 V. Gribniak, V. Cervenka, and G. Kaklauskas, "Deflection prediction of reinforced concrete beams by design codes and computer simulation," Eng. Struct., vol. 56, pp. 21752186, 2013, http://dx.doi.org/10.1016/j.engstruct.2013.08.045.
http://dx.doi.org/10.1016/j.engstruct.20...
], its application to design engineering practice is unfeasible.
Lumped damage mechanics (LDM) appears as an interesting alternative to FEA because of its use of few finite elements resulting from its combination of key concepts from classic fracture [^{24}24 D. Broek, Elementary Engineering Fracture Mechanics, Dordrecht, Netherlands: Martinus Nijhoff Publishers, 1984.] and damage mechanics [^{25}25 J. Lemaitre and J. L. Chaboche Mécaniques des matériaux solides, Paris, France: Dunod, 1985.] with plastic hinges. For a review on LDM, see [^{26}26 J. FlórezLópez, M. E. Marante, and R. Picón Fracture and damage mechanics for structural engineering frames: state of the art and industrial applications, Hershey, Pennsylvania, USA: IGI Global, 2015, https://doi.org/10.4018/9781466663794
https://doi.org/10.4018/978146666379...
].
Hence, this study mainly aims to propose a simplified formulation for estimating the immediate deflections of RC beams based on the LDM framework. Different from the work in [^{1}1 D. E. Branson, Instantaneous and Timedependent Deflections of Simple and Continuous Reinforced Concrete Beams, (HPR Report 7, pp. 178). College Park, MD, USA: Alabama Highway Department, Bureau of Public Roads, 1963.], [^{2}2 D. E. Branson, Deformation of Concrete Structures, New York, NY, USA: McGrawHill, 1977.], which is solely based on experimental observations, the model proposed in the current study is supported by the popular and widely accepted concepts of fracture and damage mechanics, such as effective stress, strain equivalence hypothesis and the Griffith energy criterion.
2 DEFLECTION OF REINFORCED CONCRETE BEAMS
To describe this nonlinear behaviour of RC beams, Branson [^{1}1 D. E. Branson, Instantaneous and Timedependent Deflections of Simple and Continuous Reinforced Concrete Beams, (HPR Report 7, pp. 178). College Park, MD, USA: Alabama Highway Department, Bureau of Public Roads, 1963.] performed an experimental study on rectangular and ‘T’ beams applied with uniformly distributed shortterm loads.
A formulation was subsequently proposed to calculate the immediate deflection based on an effective moment of inertia. This formula establishes a proportional relationship between the moment of inertia of the gross concrete section about the centroidal axis I_{g}; and the moment of inertia of the cracked section transformed into concrete I_{cr}. Based on a multiplier factor, the ratio between the first cracking moment M_{cr} and the maximum moment in the beam due to service loads at the deflection stage M_{a} is calculated. Branson’s model [^{1}1 D. E. Branson, Instantaneous and Timedependent Deflections of Simple and Continuous Reinforced Concrete Beams, (HPR Report 7, pp. 178). College Park, MD, USA: Alabama Highway Department, Bureau of Public Roads, 1963.] is expressed by Equation 1.
Exponent m equal to 3 is adopted for the calculation of a reference section for the entire span. The calculation considers the sum of the effects of the loss of stiffness and the contribution of the concrete in the traction area between cracks, in the cracked region of the span, and also the region without visible cracks. For the calculation of an individual section, exponent m is assigned with a value of 4 [^{27}27 Committee 435 Chairman, et al., "Deflections of continuous concrete beams", J. Aci., vol. 70, no. 12, pp. 781787, Dec. 1973.].
According to Bischoff [^{15}15 P. H. Bischoff, "Deflection calculation of FRP reinforced concrete beams based on modifications to the existing Branson equation," J. Compos. Constr., vol. 11, no. 1, pp. 414, 2007, http://dx.doi.org/10.1061/(ASCE)10900268(2007)11:1(4).
http://dx.doi.org/10.1061/(ASCE)1090026...
], Branson’s Model [^{1}1 D. E. Branson, Instantaneous and Timedependent Deflections of Simple and Continuous Reinforced Concrete Beams, (HPR Report 7, pp. 178). College Park, MD, USA: Alabama Highway Department, Bureau of Public Roads, 1963.] works well for RC beams with a reinforcement rate between 1% and 2%, which was the standard reinforcement rate in the past. However, the equation underestimates the deflection of RC beams with a reinforcement rate below 1%; corroborating the results obtained in this study.
3 LUMPED DAMAGE MECHANICS
Consider the beam element depicted in Figure 1, where L denotes the span. The deformed shape of such beam can be described by two relative rotations at edges i and j i.e. ϕ_{i} and ϕ_{j}, respectively (Figure 1a). These relative rotations, now called generalised deformations [^{28}28 H. G. Powell, "Theory for nonlinear elastic structures," J. Struct. Div., vol. 95, no. 12, pp. 26872701, 1969.], are conjugated to two bending moments (m_{i} and m_{j}) named generalised stresses [^{28}28 H. G. Powell, "Theory for nonlinear elastic structures," J. Struct. Div., vol. 95, no. 12, pp. 26872701, 1969.] (Figure 1a).
Deformed shape of a beam element: (a) generalised deformations and stresses; (b) transverse displacement field.
The transverse displacement along the beam element is represented by a cubic polynomial function w(x) (Figure1b). Then, the boundary conditions are:
Therefore, the transverse displacement field is described as follows:
Now, considering that the generalised deformations are elastic, i.e. ϕ_{i}^{e} and ϕ_{j}^{e}, the bending moment distribution along the beam element can be written as:
As the bending moments at the edges of the beam element are m_{i} and m_{j} (Figure 1a), then:
Equations 5, 6 can be rewritten in terms of generalised deformations, i.e.
Equations 7, 8 can be expressed in matrix form, as:
where {Φ^{e}} = {ϕ_{i}^{e} ϕ_{j}^{e}}^{T} is the matrix of elastic generalised deformations, {M} = {m_{i} m_{j}}^{T} is the matrix of generalised stresses, [F_{0}] is the elastic flexibility matrix, described by:
and the superscript $T$ means ‘transpose of’.
LDM states that a beam element is understood as a composition of an elastic beam with two inelastic hinges at its edges (Figure 2a). Therefore, such hinges are responsible for inelastic effects.
Lumped damage mechanics for RC beams: (a) elastic beam with inelastic hinges, (b) reinforcement yielding and (c) concrete cracking.
Under the deformation equivalence hypothesis [^{26}26 J. FlórezLópez, M. E. Marante, and R. Picón Fracture and damage mechanics for structural engineering frames: state of the art and industrial applications, Hershey, Pennsylvania, USA: IGI Global, 2015, https://doi.org/10.4018/9781466663794
https://doi.org/10.4018/978146666379...
], the matrix of generalised deformations {Φ} can be expressed as a sum of three parts:
being {Φ^{e}} the elastic part, {Φ^{p}} = {ϕ_{i}^{p} ϕ_{j}^{p}}^{T} the plastic part, accounting for reinforcement yielding at the hinges (Figure 3b), and {Φ^{d}} the damaged one, expressed by [^{29}29 A. Cipollina, A. LópezInojosa, and J. FlórezLópez, "A simplified damage mechanics approach to nonlinear analysis of frames," Comput. Struc., vol. 54, pp. 11131126, 1995, http://dx.doi.org/10.1016/00457949(94)00394I.
http://dx.doi.org/10.1016/00457949(94)0...
]:
where, [C(D)] is the matrix of additional flexibility due to damage variables at the hinges (${d}_{i}$ and ${d}_{j}$); it represents concrete cracking (Figure 2c).
Graphical representation of damage measurement (adapted from [^{29}29 A. Cipollina, A. LópezInojosa, and J. FlórezLópez, "A simplified damage mechanics approach to nonlinear analysis of frames," Comput. Struc., vol. 54, pp. 11131126, 1995, http://dx.doi.org/10.1016/00457949(94)00394I.
http://dx.doi.org/10.1016/00457949(94)0... ]).
Finally, the following expression is obtained by substituting Equations 9 and 12 in (11):
where [F(D)] is the flexibility matrix of a damaged beam element; it is described as:
Note that both terms of the main diagonal of [F(D)] present the inertia moment penalised by a damage variable i.e. I_{g}(1 d_{i}) and I_{g}(1 d_{j}). Henceforth, this study focuses on only one of the hinges. The damage variable of such hinge is described herein without an index (d).
The concept of effective inertia moment (I_{eff}) is then introduced as a function of d [^{30}30 D. L. N. F. Amorim, S. P. B. Proença, and J. FlórezLópez, "Simplified modeling of cracking in concrete: application in tunnel linings," Eng. Struct., vol. 70, pp. 2335, 2014, http://dx.doi.org/10.1016/j.engstruct.2014.03.031.
http://dx.doi.org/10.1016/j.engstruct.20...
]:
Cippolina et al. [^{29}29 A. Cipollina, A. LópezInojosa, and J. FlórezLópez, "A simplified damage mechanics approach to nonlinear analysis of frames," Comput. Struc., vol. 54, pp. 11131126, 1995, http://dx.doi.org/10.1016/00457949(94)00394I.
http://dx.doi.org/10.1016/00457949(94)0...
] experimentally presented a simple way to quantify the damage variable. Such experiment consisted of a simply supported beam, such as that depicted in Figure 3a.
During the test, unloadingreloading cycles were performed to quantify the beam stiffness (Figure 3b). For the first unloadingreloading cycle, the applied load was lower than the threshold for concrete cracking as an alternative to measure elastic stiffness (S_{0}). Thereafter, the stiffness values for concrete cracking were obtained (Figure 3b), i.e. S(d). The damage variable $d$ for any cycle is then calculated [^{29}29 A. Cipollina, A. LópezInojosa, and J. FlórezLópez, "A simplified damage mechanics approach to nonlinear analysis of frames," Comput. Struc., vol. 54, pp. 11131126, 1995, http://dx.doi.org/10.1016/00457949(94)00394I.
http://dx.doi.org/10.1016/00457949(94)0...
]:
Experimental observations (e.g. [^{26}26 J. FlórezLópez, M. E. Marante, and R. Picón Fracture and damage mechanics for structural engineering frames: state of the art and industrial applications, Hershey, Pennsylvania, USA: IGI Global, 2015, https://doi.org/10.4018/9781466663794
https://doi.org/10.4018/978146666379...
], [^{29}29 A. Cipollina, A. LópezInojosa, and J. FlórezLópez, "A simplified damage mechanics approach to nonlinear analysis of frames," Comput. Struc., vol. 54, pp. 11131126, 1995, http://dx.doi.org/10.1016/00457949(94)00394I.
http://dx.doi.org/10.1016/00457949(94)0...
], [^{30}30 D. L. N. F. Amorim, S. P. B. Proença, and J. FlórezLópez, "Simplified modeling of cracking in concrete: application in tunnel linings," Eng. Struct., vol. 70, pp. 2335, 2014, http://dx.doi.org/10.1016/j.engstruct.2014.03.031.
http://dx.doi.org/10.1016/j.engstruct.20...
]) show that the damage variable can be easily associated with the plastic bending moment (M_{p}) and ultimate bending moment (M_{u}), both of which are known quantities of classic RC theory.
Despite its accuracy at the load bearing condition of structural elements, the necessity of a lumped damage approach to analyse deflection in beams was observed by the application of the classic LDM for reinforced concrete structures [^{26}26 J. FlórezLópez, M. E. Marante, and R. Picón Fracture and damage mechanics for structural engineering frames: state of the art and industrial applications, Hershey, Pennsylvania, USA: IGI Global, 2015, https://doi.org/10.4018/9781466663794
https://doi.org/10.4018/978146666379...
] to a deflection test. In order to illustrate this issue, note that the classic LDM cracking evolution criterion is based on the generalised Griffith criterion, where the energy release rate (G) is equal to a crack resistance function (R), both defined as follows [^{26}26 J. FlórezLópez, M. E. Marante, and R. Picón Fracture and damage mechanics for structural engineering frames: state of the art and industrial applications, Hershey, Pennsylvania, USA: IGI Global, 2015, https://doi.org/10.4018/9781466663794
https://doi.org/10.4018/978146666379...
]:
being R_{0} the initial crack resistance and q a parameter associated to the longitudinal reinforcement.
Then, three conditions are known by the bending moment vs. damage obtained by Equation 17 i.e. by the classic LDM (see Figure 4):
where d_{u} is the ultimate damage i.e. the damage value at the ultimate condition.
Experimental bending moment vs. damage results from Álvares [^{31}31 M. S. Álvares, "Study of a damage model for concrete: formulation, parametric identification and application using the finite element method," M.S. thesis, Dept. Struct. Eng, Univ. São Paulo, São Carlos, SP, 1993.] compared with the classic LDM [^{26}26 J. FlórezLópez, M. E. Marante, and R. Picón Fracture and damage mechanics for structural engineering frames: state of the art and industrial applications, Hershey, Pennsylvania, USA: IGI Global, 2015, https://doi.org/10.4018/9781466663794
https://doi.org/10.4018/978146666379... ].
For an experimental analysis carried out by Álvares [^{31}31 M. S. Álvares, "Study of a damage model for concrete: formulation, parametric identification and application using the finite element method," M.S. thesis, Dept. Struct. Eng, Univ. São Paulo, São Carlos, SP, 1993.] (Figure 4), the conditions in (1820) result in: R_{0}= 0.32kNmm, q=  106.37kNmm and d_{u}= 0.63. The experimental damage, depicted in Figure 4, is obtained by the following relation:
being w the deflection.
4 PROPOSED MODEL
The proposed formulation for calculating deflection is derived by inserting an effective moment of inertia I_{eff}, calculated according to Equation 2.
However, differently from classic LDM, the proposed approach must present a bending moment vs. damage relation closer to experimental analysis in order to evaluate deflected beams in service.
Therefore, an exponential expression for the acting moment (M_{a}) is proposed i.e.
In classic LDM, d_{u} is numerically obtained by solving the system composed by Equations 19, 20. However, for practical applications, the following equation is a satisfactory approximation for d_{u}:
where d_{p} is the plastic damage i.e. the damage when the reinforcement is about to yield.
Again, for practical applications, a reasonable approximation for d_{p} is:
Therefore, by substituting Equations 24, 25 in Equation 23, $d$ is the acting damage on the structural element is calculated according to Equation 26:
where M_{a} is the acting moment, M_{cr} is the first cracking moment, M_{u} is the ultimate moment.
5 RESULTS
To analyse the proposed model, literature data including beams with different dimensions, strengths, elasticity modules and reinforcement rates were used. The selected works provided the results of the increase in displacements with the applied loads, in addition to the necessary information for the application in both Proposed and Branson’s models.
As an illustration, the experimental results of Álvares [^{31}31 M. S. Álvares, "Study of a damage model for concrete: formulation, parametric identification and application using the finite element method," M.S. thesis, Dept. Struct. Eng, Univ. São Paulo, São Carlos, SP, 1993.] and Fernandes [^{32}32 S. A. Fernandes, "Concrete deformations and behavior of beams analysis subjected to simple bending," M.S. thesis, Dept. Struct. Eng, Univ. São Paulo, São Carlos, SP, 1996.] are shown in Figures 4 and 5, respectively. The application of the Proposed Model to such experiments provided a satisfactory behaviour (Figures 4 and 5).
Experimental bending moment vs. damage results from Álvares [^{31}31 M. S. Álvares, "Study of a damage model for concrete: formulation, parametric identification and application using the finite element method," M.S. thesis, Dept. Struct. Eng, Univ. São Paulo, São Carlos, SP, 1993.] compared with the classic LDM [^{26}26 J. FlórezLópez, M. E. Marante, and R. Picón Fracture and damage mechanics for structural engineering frames: state of the art and industrial applications, Hershey, Pennsylvania, USA: IGI Global, 2015, https://doi.org/10.4018/9781466663794
https://doi.org/10.4018/978146666379... ] and the proposed model.
Subsequently, the Proposed Model was compared with 72 experiments by several authors [^{33}33 S. A. Ashour, F. F. Wafa, and M. I. Kamal, "Effect of the concrete compressive strength and tensile reinforcement ratio on the flexural behavior of fibrous concrete beams," Eng. Struct., vol. 22, no. 9, pp. 11451158, 2000., http://dx.doi.org/10.1016/S01410296(99)000528.
http://dx.doi.org/10.1016/S01410296(99)...
][^{41}41 M. I. Mousa, "Flexural behaviour and ductility of high strength concrete (HSC) beams with tension lap splice," Alex. Eng. J., vol. 54, no. 3, pp. 551563, 2015, http://dx.doi.org/10.1016/j.aej.2015.03.032.
http://dx.doi.org/10.1016/j.aej.2015.03....
] that were brought together in the work of Melo [^{42}42 L. M. A. Melo, "Evaluation of displacements in concrete beams with steel reinforcement and fiber reinforced polymers," M.S. dissertation, Fed. Univ. Alagoas, Maceió, AL, 2019.]. These works provided the necessary properties for the calculation of the deflection by the methods studied in this work, in addition to the values of force versus displacement for service situation. The properties of beams are depicted in ^{Appendix A}
APPENDIX A
Table A1
Beams data with fc from 20MPa to 50MPa and reinforcement rate from 0 a 1%.
Authors
Beam
b (mm)
h (mm)
Service load (kN)
fc (MPa)
ρ (%)
Experimental deflection (mm)
Proposed Model (mm)
Branson's Model (mm)
Sharifi [39]
SCCB1
200
300
13.27
31.60
0.51
4.38
3.52
1.59
Gribniak [37]
S1
280
300
65.68
47.30
0.37
8.25
10.10
5.03
S1R
281
299
65.68
47.30
0.37
7.53
9.95
5.02
S2
280
300
67.20
48.70
0.37
7.96
9.83
5.05
S2R
282
300
66.46
48.20
0.37
8.02
9.86
4.93
S3
277
300
61.74
41.10
0.36
9.35
9.52
5.03
S3R
281
299
62.69
41.20
0.36
8.41
9.21
4.98
Silva [40]
V25A Sub
150
150
26.30
27.40
0.70
5.40
4.17
3.68
Table A2
Beams data with fc from 20MPa to 50MPa and reinforcement rate from 1% to 2%.
Author
Beam
b (mm)
h (mm)
Service load (kN)
fc (MPa)
ρ (%)
Experimental deflection (mm)
Proposed Model (mm)
Branson's Model (mm)
Sharifi [39]
SCCB2
200
300
33.19
32.84
1.05
6.90
5.54
4.86
SCCB3
200
300
43.72
28.84
1.44
7.35
5.62
5.15
Ashour et al. [33]
BN2
200
250
53.18
48.61
1.02
10.70
16.15
13.56
BN3
200
250
76.19
48.61
1.53
12.47
17.21
15.00
Rashid and Mansur [35]
A211
250
400
324.39
42.80
1.96
9.64
9.73
8.55
Silva [40]
V25A Super
150
150
37.00
35.77
1.40
5.40
4.41
4.08
Table A3
Beams data with fc from 50MPa to 90MPa and reinforcement rate from 0 to 1%.
Authors
Beam
b (mm)
h (mm)
Service load (kN)
fc (MPa)
ρ (%)
Experimental deflection (mm)
Proposed Model (mm)
Branson's Model (mm)
Gribniak [37]
S4
277
300
62.21
54.20
0.36
8.90
9.41
4.01
S4R
283
301
63.62
54.20
0.36
9.50
9.04
3.89
Elrakib [38]
B503
250
400
61.50
52.00
0.30
3.59
7.66
1.72
B753
250
400
77.51
73.00
0.38
4.67
7.73
1.83
Mousa [41]
A1
150
200
37.84
55.00
0.75
3.26
7.64
5.22
A2
150
200
37.84
55.00
0.75
3.11
7.64
5.22
A3
150
200
37.84
55.00
0.75
3.64
7.64
5.22
A4
150
200
37.84
55.00
0.75
2.82
7.64
5.22
B1
150
200
38.13
65.00
0.75
4.29
7.59
4.83
B2
150
200
38.13
65.00
0.75
4.09
7.59
4.83
B3
150
200
38.13
65.00
0.75
3.25
7.59
4.83
B4
150
200
38.13
65.00
0.75
3.53
7.59
4.83
B5
150
200
38.13
65.00
0.75
3.97
7.59
4.83
B6
150
200
38.13
65.00
0.75
3.96
7.59
4.83
B7
150
200
38.13
65.00
0.75
3.26
7.59
4.83
B8
150
200
38.13
65.00
0.75
3.09
7.59
4.83
B14
150
200
34.59
65.00
0.75
3.61
8.50
4.68
Table A4
Beams data with fc from 50MPa to 90MPa and reinforcement rate from 1% to 2%.
Authors
Beam
b (mm)
h (mm)
Service load (kN)
fc (MPa)
ρ (%)
Experimental deflection (mm)
Proposed Model (mm)
Branson's Model (mm)
Ashour et al. [33]
BM2
200
250
55.00
78.50
1.02
10.73
15.42
11.89
BM3
200
250
80.29
78.50
1.53
13.14
16.46
13.92
Bernardo and Lopes [34]
B1
120
270
75.10
79.20
1.40
12.67
11.04
9.24
B2
120
270
111.05
78.90
1.94
14.13
12.25
10.34
B3
120
270
110.99
78.50
1.94
14.55
12.25
10.34
C1
120
270
111.60
82.90
1.94
16.34
12.25
10.32
C2
120
270
111.73
83.90
1.94
17.35
12.25
10.32
D1
120
270
75.79
88.00
1.24
12.79
11.08
9.19
A1
120
270
74.13
62.90
1.40
10.16
11.02
9.35
A2
120
270
106.07
64.90
1.94
12.15
12.54
10.62
A3
120
270
105.90
64.10
1.94
10.95
12.54
10.62
Maghsoudi and Bengar [36]
B2
200
300
187.73
70.50
1.05
5.75
3.15
2.29
B3
200
300
277.85
70.80
1.70
5.33
3.13
2.63
BC2
200
300
186.81
63.48
1.05
4.98
3.16
2.37
BC3
200
300
275.61
63.21
1.70
4.90
3.09
2.61
Rashid and Mansur [35]
B211a
250
400
346.29
73.60
1.96
10.83
9.78
8.48
Mousa [41]
B9
150
200
64.73
65.00
1.34
5.86
8.24
6.50
B10
150
200
64.73
65.00
1.34
4.59
8.24
6.50
B11
150
200
64.73
65.00
1.34
4.37
8.24
6.50
B12
150
200
64.73
65.00
1.34
4.80
8.24
6.50
Silva [40]
V50A Super
150
150
48.99
53.90
1.40
2.65
5.78
4.90
Table A5
Beams data with fc from 50MPa to 90MPa and reinforcement rate from 2% to 3%.
Authors
Beam
b (mm)
h (mm)
Service load (kN)
fc (MPa)
ρ (%)
Experimental deflection (mm)
Proposed Model (mm)
Branson's Model (mm)
Ashour et al. [33]
BM4
200
250
104.09
78.50
2.04
13.94
17.28
14.98
Bernardo and Lopes [34]
C3
120
270
137.18
83.60
2.48
18.25
12.49
10.91
C4
120
270
140.37
83.40
2.48
16.99
12.17
10.65
A4
120
270
128.27
63.20
2.48
15.07
12.69
11.14
A5
120
270
128.95
65.10
2.48
15.35
12.72
11.15
Maghsoudi and Bengar [36]
BC4
200
300
359.41
71.45
2.09
7.26
3.42
2.94
B4
200
300
360.01
72.80
2.09
7.18
3.48
2.99
Rashid and Mansur [35]
B311
250
400
495.86
72.80
2.95
13.67
10.35
9.13
B312
250
400
495.86
72.80
2.95
13.92
10.35
9.13
B313
250
400
495.86
72.80
2.95
13.40
10.35
9.13
B321
250
400
499.63
77.00
2.95
14.39
10.25
9.03
B331
250
400
495.86
72.80
2.95
14.73
10.12
8.93
C211
250
400
415.89
85.60
2.37
12.98
10.01
8.73
C311
250
400
480.51
88.10
2.77
14.70
10.24
8.99
Table A6
Beams data with fc from 50MPa to 90MPa and reinforcement rate greater than 3%.
Authors
Beam
b (mm)
h (mm)
Service load (kN)
fc (MPa)
ρ (%)
Experimental deflection (mm)
Proposed Model (mm)
Branson's Model (mm)
Bernardo and Lopes [34]
D2
120
270
169.34
85.80
3.18
19.35
12.95
11.39
D3
120
270
169.41
86.00
3.18
15.11
12.95
11.39
Maghsoudi and Bengar [36]
BC5
200
300
668.36
72.98
4.11
6.62
3.72
3.30
B5
200
300
664.90
71.00
4.11
6.38
3.79
3.37
BC6
200
300
669.11
73.42
4.11
6.90
3.65
3.24
BC7
200
300
668.36
72.98
4.11
5.63
3.54
3.14
Rashid and Mansur [35]
C411
250
400
598.96
85.60
3.57
15.30
10.67
9.44
C511
250
400
707.01
88.10
4.33
18.04
11.01
9.77
(Tables A1 to A6).
In order to analyse these results, it was necessary to group them according to the compressive strength (f_{c}) and the reinforcement rate. Figures 6 and 7 show the results of the deflections of reinforced concrete beams with f_{c} between 20 and 50 MPa. Figures 8, 9, 10, 11, 12 and 13, show the results of the deflections of the reinforced concrete beams with f_{c} between 50 and 90 MPa.
Experimental results from Álvares [^{31}31 M. S. Álvares, "Study of a damage model for concrete: formulation, parametric identification and application using the finite element method," M.S. thesis, Dept. Struct. Eng, Univ. São Paulo, São Carlos, SP, 1993.] compared with Branson’s model and the proposed model.
Experimental results from Fernandes [^{32}32 S. A. Fernandes, "Concrete deformations and behavior of beams analysis subjected to simple bending," M.S. thesis, Dept. Struct. Eng, Univ. São Paulo, São Carlos, SP, 1996.] compared with Branson’s model and the proposed model.
To statistically compare the Proposed Model results with the responses provided by the experiments considered and by Branson’s Model, the most appropriate multiple comparison tests were used for each situation, according to the normality and homoscedasticity of the data considered. The normality test employed was the ShapiroWilk one, considering normality when the pvalue was greater than the 5% significance level. In relation to the homoscedasticity test, both Bartlett and Levene were used. The Bartlett test was applied when the sample had a normal distribution; otherwise, Levene test was adopted. The studied samples met the conditions of homoscedasticity when the pvalue was above the 5% significance level.
Regarding the equivalence test between groups, ANOVA test was applied when the conditions of normality and homoscedasticity were satisfied, with the identification of difference between groups by the FisherBonferroni test, considering that for pvalue greater than 0.05 no difference was found. When the hypotheses related to the ANOVA test were not verified, the nonparametrical KruskalWallis test was used, adopting a significance level of 5%.
To easier the analysis, the results were divided according to the concrete resistance f_{c} and the reinforcement rate, as specified in Tables 1, 2 to 3.
Beams analysis with f_{c} from 20 to 50 MPa with reinforcement rate from 0 to 1% and reinforcement rate from 1% to 2%.
Beams analysis with f_{c} from 50 to 90 MPa with reinforcement rate from 0 to 1% and reinforcement rate from 1% to 2%.
Beams analysis with f_{c} from 50 to 90 MPa with reinforcement rate from 2% to 3% and reinforcement rate greater than 3%.
According to the results obtained by the statistical analysis presented in Tables 1 to 3, it is possible to verify that:

For f_{c} from 20 to 50 MPa and rates from the studied reinforcement, the statistical tests applied to the samples

showed not only equivalent variance but also equal mean values between deflection obtained by either the experiments or the proposed model, derived from TDC. For reinforcement rates from 1% to 2%, it was also found that the normal distribution is associated with the results of the experimental deflections and the proposed formulation forecast.

For f_{c} from 20 to 50 MPa and rates from the studied reinforcement, the statistical tests applied to the samples showed that the average deflection provided by the proposed model and Branson’s model are equivalent, with adherence to the respective experimental results averages.

For f_{c} from 50 to 90 MPa and reinforcement rates from 1% to 2% and greater than 3%, statistical tests applied to the samples showed not only equivalent variance values but also equal mean values between deflection obtained by either the experiments or the proposed model. At these reinforcement rates, it was also found that the average deflection values provided by both proposed and Branson models are equivalent, with adherence to the respective experimental results means.

For the f_{c} in the range of 50 to 90 MPa, for reinforcement rates from 0 to 1%, the statistical tests applied to the samples showed the equality of the medians of the deflection values obtained by the experiments and the Branson’s Model, both being different medians that obtained through the proposed formulation. However, it should be noted that, for these rates, the nonnormality of the data for all groups was observed and the difference in the variance of the experimental results in relation to the variance of the values from the Branson’s Model and the proposed formulation.
6 CONCLUSIONS
The main objective of this work was to propose an effective moment of inertia for the calculation of deflection in reinforced concrete beams, using the formulations of the Lumped Damage Mechanics as a basis, and to evaluate the proposed calculation model, comparing it with experimental results and with the Branson’s Model.
The Proposed Model presented a satisfactory behaviour when the evolution of the deflection was verified and compared with the experimental works of Álvares [^{31}31 M. S. Álvares, "Study of a damage model for concrete: formulation, parametric identification and application using the finite element method," M.S. thesis, Dept. Struct. Eng, Univ. São Paulo, São Carlos, SP, 1993.] and Fernandes [^{32}32 S. A. Fernandes, "Concrete deformations and behavior of beams analysis subjected to simple bending," M.S. thesis, Dept. Struct. Eng, Univ. São Paulo, São Carlos, SP, 1996.].
The statistical tests used showed that the reinforced concrete beams with the f_{c} in the range of 20 to 50 MPa, there is an equivalence of variance and equality of means between the Proposed Model and the experimental response, both in the reinforcement rates between 0 and 1% as well as the reinforcement rate of 1% to 2%. In this range of f_{c}, the Proposed Model also proved to be equivalent to the Branson’s Model.
In the f_{c} range between 50 and 90 MPa and reinforcement rate from 0 to 1%, the statistical tests showed the equality of the medians between the deflection values of the experiment and the Branson’s Model, but both are different from the median of the Proposed Model. In this group of beams the nonnormality of the data and difference in variance between the experiment and the calculation methods studied were also verified.
Through the study, it was possible to verify that the application of the results of the Proposed Model provides a good approximation of the experimental response, equivalent to that provided by the Branson’s Model in most of the investigated scenarios.
ACKNOWLEDGEMENTS
The authors acknowledge Professor Severino Pereira Cavalcanti Marques and Engineer Laila Monteiro Alves for providing the experimental data of the beams used in the development of her MSc dissertation [^{42}42 L. M. A. Melo, "Evaluation of displacements in concrete beams with steel reinforcement and fiber reinforced polymers," M.S. dissertation, Fed. Univ. Alagoas, Maceió, AL, 2019.]. The first author acknowledges CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) for the financial support during the development of this research.

Financial support: None.

Data Availability: The data that support the findings of this study are openly available in the appendix of this paper.

How to cite: A. A. F. Souza, W. S. Assis, and D. L. N. F. Amorim, “Deflection estimate of reinforced concrete beams by the lumped damage mechanics,“ Rev. IBRACON Estrut. Mater., vol. 16, no. 5, e16505, 2023, https://doi.org/10.1590/S198341952023000500005
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Publication Dates

Publication in this collection
16 Jan 2023 
Date of issue
2023
History

Received
10 Aug 2022 
Accepted
08 Dec 2022