https://doi.org/10.14311/APP.2022.33.0424 Acta Polytechnica CTU Proceedings 33:424–430, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague COMPARISON BETWEEN CONCRETE STRESS DIAGRAMS ACCORDING TO THE BRAZILIAN STANDARD ABNT NBR 6118: 2014 Victor Hugo Dalosto de Oliveiraa, ∗, Rodolfo Alves Carvalhoa, Luana Ferreira Borgesa, Paulo Chaves de Rezende Martinsa, Rodolfo Rabelo Nevesb, João Paulo de Almeida Siqueiraa, Gabriel Yves da Silva Oliveirac, Cintia Adriana Azevedo de Liz Anhaiad a Department of Civil and Environmental Engineering, University of Brasília, SG-12, Campus Darcy Ribeiro, Asa Norte, Brasília, DF, 70910-900, Brazil b Department of Materials Engineering and Construction, Federal University of Minas Gerais, Av. Antø̂nio Carlos, 6627, Pampulha, Belo Horizonte, MG, 31270-901, Brazil c School of Exact, Architecture and Environment, Catholic University of Brasília, QS 07 - Lote 01, Taguatinga, Brasília, DF, 71966-700, Brazil d Department of Civil Engineering, Faculty of Pinhais, Av. Camilo di Lellis, 1065, Centro, Pinhais, PR, 83323-000, Brazil ∗ corresponding author: victordalosto@gmail.com Abstract. This study aims to compare methods for the determination of concrete properties by means of the stress diagrams present in the Brazilian standard ABNT NBR 6118: 2014. The area under the stress diagram, the internal reactions, and the application point of the resulting reactions for the parabola-rectangle and rectangular block diagrams are present in order to compare them. Deductions and numerical examples were used, and different results were obtained for each formulation. This is due to non-consideration of the relationship between stress and strain in the simplified rectangular block. The rectangular block is applicable only for cases in which the concrete reaches the ultimate strain. These cases are those that concrete crushing determines the section failure in compression with steel yielding in tension (domain 3) or without steel yielding (domain 4). Keywords: Concrete behavior, parabola-rectangle diagram, rectangular block. 1. Introduction Exhaustion of the stress capacity of a reinforced concrete element subjected to normal stress can be achieved by crushing the concrete under compres- sion or by an excessive plastic strain of the steel. However, concrete failure is difficult to identify. It is conventional to assume that this failure occurs when the material reaches a maximum compressive strain determined by experimental results [1]. Ac- cording to American standard ACI 318-14 [2], the maximum compressive strain of concrete is around 0.003 to 0.004 under normal conditions, and it may reach 0.008 under special conditions. The Brazilian standard ABNT NBR 6118: 2014 [3] states a max- imum specific strain of 0.0035 for concretes up to class C50 (characteristic compressive strength up to 50 MPa). This Brazilian standard adopts an idealized stress- strain diagram to represent concrete behavior, in which the stress distribution takes place according to a parabola-rectangular diagram [3]. This stan- dard allows to replace the parabola-rectangle diagram with a rectangle of equivalent depth, as a simplifica- tion of calculation [3]. ABNT NBR 6118: 2014 [3] states that the difference in results with both formu- lations is small, so there is no need to make any ad- justment. The rectangular block does not represent the actual stress distribution within the compressed concrete zone, but provides reasonably the same com- pressive force [4]. However, Mendes Neto [5] noticed that the design of structures with the rectangular block provides a different and unsafe result compared to the parabola- rectangle diagram result. Formulations for stress, in- ternal reaction, and the point of application of the re- sulting reaction are presented in this paper by means of the parabola-rectangle diagram and the rectangu- lar block. Solutions are compared for some cases to verify if the use of the rectangular block leads to rea- sonable and safe results. 424 https://doi.org/10.14311/APP.2022.33.0424 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 33/2022 Concrete Diagrams (ABNT NBR 6118: 2014) Figure 1. Idealized stress-strain diagram. Source: Adapted from the ABNT NBR 6118: 2014 [3]. 2. Materials and numerical methods This section presents the concepts and formulations considered for the comparison between parabola- rectangle and rectangular block diagrams. ABNT NBR 6118: 2014 [3] uses an idealized stress-strain diagram to represent concrete compression at the ul- timate limit state, as shown in Figure 1, in which εc is the concrete strain; εc2 is the assumed specific com- pression strain of concrete at the beginning of the plastic level; εcu is the assumed maximum useable compression strain in the concrete; fck is the char- acteristic strength of the concrete; fcd is the design strength of the concrete; and σc is the concrete stress [3, 6]. This diagram is similar to that presented in section 7.2.3.1.5 of the fib Model Code 2010 [7] and section 3.1.7 of the Eurocode 2 [8]. Equation (1) defines this diagram (Figure 1): σc = 0.85fcd ! 1 − " 1 − εc εc2 #n$ (1) in which n = 2 for fck ≤ 50 MPa; and n is given by Equation 2 for fck > 50 MPa. n = 1.4 + 23.4 % (90 − fck) & 100 '4 (2) For concretes with fck ≤ 50 MPa: εc2 = 0.002; and εcu = 0.0035. Equations 3 and 4 give the values of εc2 and εcu for concretes with fck > 50 MPa. εc2 = 0.002 + 0.000085 (fck − 50) 0.53 (3) εcu = 0.0026 + 0.035 % (90 − fck) & 100 '4 (4) The Brazilian standard [3] states that it is possi- ble to replace this diagram with a rectangle of height y = λx (in which x is the effective depth of the neu- tral axis), and constant stress αc fcd, without loss of quality in results. For concretes with fck ≤ 50 MPa: λ = 0.8, and αc = 0.85. Equations 5 and 6 give values of λ and αc for concretes with fck > 50 MPa. Thus, the strains and stresses in a rectangular cross-section at the ultimate limit state can be represented as in Figure 2. λ = 0.8 − (fck − 50) & 400 (5) αc = 0.85 % 1 − (fck − 50) & 200 ' (6) The area under the stress diagram can be obtained by integrating stresses in the domain of the specific strains. Thus, Equation 7 is used when strain values at the most compressed fiber are less than εc2, and Equation 8 is used if strain values are larger than εc2. A(ε) = 0.85 fcd ( εc 0 ! 1 − " 1 − ε εc2 #n$ dε (7) A(ε) = 0.85 fcd ( εc2 0 ! 1 − " 1 − ε εc2 #n$ dε + 0.85 fcd ( εc εc2 dε (8) Considering the Bernoulli hypothesis (plane sec- tions remain plane), the longitudinal strain varies proportionally with the distance to the neutral axis [2]. Therefore, if the specific strain is known, the neutral axis is determined. Thus, as the resultant of internal reactions equals the volume under the stress diagram, its value becomes known after determining the stress and cross-sectional compressed area. The center of gravity of the compression region is obtained by dividing the stress-strain diagram into infinitesimal areas stress-strain diagram into infinites- imal areas σ dε. The point of application of the re- sultant of normal concrete stresses can be determined by Equation 9, in which the integral of the stress area in the strain domain is determined by Equation 7 and 8. ε = ( A xdA ( A dA = ( ε εσdε ( ε σdε (9) 2.1. Strain Domains Conventionally, failure of a reinforced concrete sec- tion occurs when the specific strain of concrete or steel (or both) reaches its ultimate value. The stan- dard ABNT NBR 6118: 2014 [3] defines that the ul- timate limit state is characterized when the distri- bution of strains in the section is within one of the specified strain domains. Figure 3 illustrates these strain domains, in which εyd is the strain in the ten- sion reinforcement at failure; h is the overall height of a cross-section; d is the distance from the most com- pressed fiber to the centroid of the longitudinal rein- forcement on the tension side of the member; and d′ is the distance from the most compressed fiber to the centroid of the longitudinal compression steel [3, 6]. Thus, section failure can occur under excessive plastic strains of the steel characterized by the straight-line 425 V. H. D. D. Oliveira, R. A. Carvalho, L. F. Borges et al. Acta Polytechnica CTU Proceedings Figure 2. Strain and stress diagrams of concrete. Figure 3. Strain domains in the ultimate limit state. Source: Adapted from the ABNT NBR 6118: 2014 [3]. a, within domains 1 and 2, or by crushing of the com- pressed concrete defined by domains 3, 4, 4a, 5, and the straight-line b. The ultimate limit state of domain 2 is character- ized by the strain εyd = 1%, and the failure of the cross-section occurs without the concrete reaching its maximum strength. Fusco (1981) [1] divides domain 2 into two subdomains: 2a (εc from zero up to εc2) and 2b (from εc2 to εcu). This subdivision aims to identify the value at which the use of compression re- inforcement becomes efficient [1]. Such consideration becomes important here because it marks the point at which pseudo-plastification of concrete begins. While the section is in subdomain 2a, concrete stresses are less than 0.85 fcd. When concrete strain exceeds εc2, the concrete pseudo-plastification and the constant compression region arise. When the section deforma- tion approaches domain 3, the specific strains become higher, and the plasticized zone expands. Therefore, two different equations are required to represent the behavior of compressed concrete, Equations 7 and 8. The strain εcu characterizes the ultimate limit state of domains 3, 4 and 4a. In these domains, the failure of the section occurs by crushing the concrete in com- pression. Although the neutral axis goes down in the section and the compressed area increases, the stress diagram proportions remain constant. In these do- mains, it is possible to find the stresses and reactions in concrete only with Equation 8. The strain εcu also characterizes the ultimate limit state of domain 5, with concrete failure in compres- sion. In domain 5, the entire section is under non- uniform compression. The strains converge to a con- stant value εc2 as the section approaches the line b. Thus, the concrete properties in domain 5 are de- fined by Equation 8, changing the lower limit of the first integral. 2.2. Calculations with a computer program The MATLAB software was used for the preparation of scripts in order to make a comparison between the diagrams. The stresses, internal reactions, and cen- ter of gravity of the parabola-rectangle diagram were obtained through these scripts, that solve equations 7, 8, and 9. The results were compared with those obtained by the rectangular block. This analysis is called "Approach 1". Another comparison between the diagrams, with a different approach (Approach 2), was performed. Values were assigned to the position of the neutral axis, and the internal reactions were obtained with Equation 10 for the parabola-rectangle diagram, and with Equation 11 for the rectangular block. Rc = ( A σ dA (10) Rc = αc fcd bw λ x (11) 426 vol. 33/2022 Concrete Diagrams (ABNT NBR 6118: 2014) Figure 4. Concrete strains and stresses in the ultimate limit state. Figure 5. Representation of calculation solution. 3. Results and discussions Initially, it should be emphasized that the use of stress diagrams does not apply to line a and domain 1, since there are no compressive stresses in concrete, and its tensile stresses are neglected at the ultimate limit state. 3.1. Approach 1 Based on the parameters analyzed, the rectangular diagram presents only one way to describe the behav- ior of the concrete. However, the parabola-rectangle diagram has five different behaviors: 1. While the structure is in domain 2a, there is only a parabolic excerpt (σc < 0.85 fcd); 2. When the pseudo-plastification begins in domain 2b, the excerpt of constant stress 0.85 fcd appears, and there is an expansion of this plastic zone until the section reaches domain 3; 3. For domains 3, 4 and 4a, there is an expansion of the compressed region, in which the stress dis- tribution increases proportionally in the parabolic excerpt and in the constant stress excerpt; 4. For sections in domain 5, there is an overlap of the plastic zone, where the specific strains across the section converge to εc2 and the stresses converge to 0.85 fcd; 5. For line b, the entire section is under uniform com- pression (σc = 0.85 fcd). Figure 4 illustrates the stress distribution and be- haviors in the strain domains. Results of the parabola-rectangle diagram can be obtained by equations 7 and 8, calculating the area as a function of the neutral axis position (x). For com- parison, the solution is represented as proportions of equivalent rectangles of constant stress of 0.85 fcd and nominal height y1 and y2, referring to the por- tions of constant and parabolic stresses distribution, respectively. Figure 5 illustrates the solution, and the results are listed in Table 1. Domain 4a represents a small region of the do- mains, and it ends when the section is fully com- pressed, so its relationships are similar to those of domains 3 and 4. For domain 5 and line b, the neu- tral axis relation is no longer represented as a function of x/d and becomes a function of x/h, as shown in Table 2. Thus, the nominal compression of the parabola- rectangle stress diagram is similar to the rectangular diagram in domains 3, 4 and 4a, where the height y = 0.8095x is equivalent to the height adopted by simplification (y = 0.80x). However, the point of ap- plication of the resulting reaction is different in each type of diagram. In addition, a considerable differ- 427 V. H. D. D. Oliveira, R. A. Carvalho, L. F. Borges et al. Acta Polytechnica CTU Proceedings (x/d) y/x y1/x y2/x C.G./x Domain 2a 0.01 1460 / 29403 ∼= 0.050 - 1460 / 29403 391 / 1168 ∼= 0.335 0.02 710 / 7203 ∼= 0.099 - 710 / 7203 191 / 568 ∼= 0.336 0.03 1380 / 9409 ∼= 0.147 - 1380 / 9409 373 / 1104 ∼= 0.338 0.04 335 / 1728 ∼= 0.194 - 335 / 1728 91 / 268 ∼= 0.340 0.05 260 / 1083 ∼= 0.240 - 260 / 1083 71 / 208 ∼= 0.341 0.06 630 / 2209 ∼= 0.285 - 630 / 2209 173 / 504 ∼= 0.343 0.07 8540 / 25947 ∼= 0.329 - 8540 / 25947 337 / 976 ∼= 0.345 0.08 590 / 1587 ∼= 0.372 - 590 / 1587 41 / 118 ∼= 0.347 0.09 3420 / 8281 ∼= 0.413 - 3420 / 8281 319 / 912 ∼= 0.350 0.10 110 / 243 ∼= 0.453 - 110 / 243 31 / 88 ∼= 0.352 0.11 11660 / 23763 ∼= 0.491 - 11660 / 23763 301 / 848 ∼= 0.355 0.12 255 / 484 ∼= 0.527 - 255 / 484 73 / 204 ∼= 0.358 0.13 12740 / 22707 ∼= 0.561 - 12740 / 22707 283 / 784 ∼= 0.361 0.14 3290 / 5547 ∼= 0.593 - 3290 / 5547 137 / 376 ∼= 0.364 0.15 180 / 289 ∼= 0.623 - 180 / 289 53 / 144 ∼= 0.368 0.16 860 / 1323 ∼= 0.650 - 860 / 1323 16 / 43 ∼= 0.372 0.166667 2 / 3 ∼= 0.667 - 2 / 3 3 / 8 ∼= 0.375 Domain 2b 0.17 172 / 255 ∼= 0.675 2 / 85 166 / 255 22019 / 58480 ∼= 0.377 0.18 94 / 135 ∼= 0.696 4 / 45 82 / 135 6451 / 16920 ∼= 0.381 0.19 68 / 95 ∼= 0.716 14 / 95 54 / 95 9977 / 25840 ∼= 0.386 0.20 11 / 15 ∼= 0.733 1 / 5 8 / 15 43 / 110 ∼= 0.391 0.21 236 / 315 ∼= 0.749 26 / 105 158 / 315 39211 / 99120 ∼= 0.396 0.22 42 / 55 ∼= 0.764 16 / 55 26 / 55 3697 / 9240 ∼= 0.400 0.23 268 / 345 ∼= 0.777 38 / 115 154 / 345 49859 / 123280 ∼= 0.404 0.24 71 / 90 ∼= 0.789 11 / 30 19 / 45 3481 / 8520 ∼= 0.409 0.25 4 / 5 ∼= 0.800 2 / 5 2 / 5 33 / 80 ∼= 0.412 Domains 3 and 4 0.259259 17 / 21 ∼= 0.810 3 / 7 8 / 21 99 / 238 ∼= 0.416∼ 1 Table 1. Section properties in the ultimate limit state (Domains 2, 3 and 4). ence is noted for the nominal compression and the point of application of the resultant reaction within sections in domains 2 and 5, and in line b. 3.2. Approach 2 A rectangular section with bw = 1, d = 1, h = 1.2 and fck = 20 MPa was analyzed, in which bw is the section width. Regardless of the values adopted for the cross-section, the difference obtained by the dia- grams remains proportional. For this reason, dimen- sionless values were used for bw, h and d. The results for the ultimate limit state are shown in Figure 6. As a reference, the limits of the strain domains were marked, where CA-50 steel (steel with 500 MPa yield strength) was adopted. The same fact can be verified for concrete strength group II (C55 to C90): there is a considerable dif- ference between results with the parabola-rectangle diagram and the rectangular block simplification. A rectangular section with bw = 1, d = 1, h = 1.2 and fck = 80 MPa was analyzed. The relationship be- tween internal reaction and neutral axis position is shown in Figure 7. 4. Conclusions Although the rectangular block simply represents the behavior of the concrete at the imminence of failure, it does not reflect the relation between stresses and strains where concrete is under limit strain of failure. This simplification considers that the material works at its full capacity of strength all over the compressive zone. The nominal compression and point of application of the concrete reaction are almost equal with both diagrams (parabola-rectangle or rectangular block) in the design of linear elements sections submitted pre- dominantly to bending load (domains 3 and 4). How- ever, it can result in unsafe design in sections in do- mains 2 or 5, typically slabs or columns, respectively. The difference between the diagrams is amplified for high strength concrete elements. Since high resis- tance concretes have lower ductility and an explosive and brittle failure, they have a smaller plastic region and, consequently, values of εc2 close to εcu, with a smaller excerpt of constant stresses. Thus, the calcu- lated difference between the stress diagrams becomes larger. 428 vol. 33/2022 Concrete Diagrams (ABNT NBR 6118: 2014) (x/h) y/h y1/h y2/h C.G./h Domain 5 1.00 17 / 21 ∼= 0.810 3 / 7 8 / 21 99 / 238 ∼= 0.416 1.05 133349 / 158949 ∼= 0.839 3 / 7 65228 / 158949 805443 / 1866886 ∼= 0.431 1.10 39989 / 46389 ∼= 0.862 3 / 7 20108 / 46389 247923 / 559846 ∼= 0.443 1.15 188621 / 214221 ∼= 0.881 3 / 7 96812 / 214221 1192347 / 2640694 ∼= 0.452 1.20 13709 / 15309 ∼= 0.895 3 / 7 7148 / 15309 87963 / 191926 ∼= 0.458 1.25 10085 / 11109 ∼= 0.908 3 / 7 5324 / 11109 13095 / 28238 ∼= 0.464 1.30 71741 / 78141 ∼= 0.918 3 / 7 38252 / 78141 470187 / 1004374 ∼= 0.468 1.35 323861 / 349461 ∼= 0.927 3 / 7 174092 / 349461 2139027 / 4534054 ∼= 0.472 1.40 5669 / 6069 ∼= 0.934 3 / 7 3068 / 6069 37683 / 79366 ∼= 0.475 1.45 403829 / 429429 ∼= 0.940 3 / 7 219788 / 429429 2698803 / 5653606 ∼= 0.477 1.50 4469 / 4725 ∼= 0.946 3 / 7 2444 / 4725 30003 / 62566 ∼= 0.480 1.75 27725 / 28749 ∼= 0.964 3 / 7 15404 / 28749 37791 / 77630 ∼= 0.487 2.00 2477 / 2541 ∼= 0.975 3 / 7 1388 / 2541 17019 / 34678 ∼= 0.491 3.00 1685 / 1701 ∼= 0.991 3 / 7 956 / 1701 2343 / 4718 ∼= 0.497 4.00 13061 / 13125 ∼= 0.995 3 / 7 7436 / 13125 91107 / 182854 ∼= 0.498 5.00 335 / 336 ∼= 0.997 3 / 7 191 / 336 234 / 469 ∼= 0.499 10.0 94205 / 94269 ∼= 0.999 3 / 7 53804 / 94269 131823 / 263774 ∼= 0.500 50.0 ∼= 1.000 3 / 7 0.571 ∼= 0.500 Line b ∞ 1 3 / 7 4 / 7 0.5 Table 2. Section properties in the ultimate limit state (Domain 5 and line b). Figure 6. Internal reactions for concrete class C20 (fck = 20 MPa). The percentage difference between the two approaches is marked at some representative points, referred to results obtained with the parabola-rectangle diagram. Figure 7. Internal reaction for concrete class C80 (fck = 80 MPa). The percentage difference between the two approaches is marked at some representative points, referred to results obtained with the parabola-rectangle dia- gram. 429 V. H. D. D. Oliveira, R. A. Carvalho, L. F. Borges et al. Acta Polytechnica CTU Proceedings Acknowledgements This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. References [1] P. B. Fusco. Estruturas de concreto: Solicitações normais, Rio de Janeiro: Guanabara Dois, 1981. [2] American Concrete Institute ACI 318-14. Building code requirements for structural concrete (Farmington Hills: ACI), 2014. [3] Associação Brasileira de Normas Técnicas. ABNT NBR 6118: 2014: Projeto de estruturas de concreto - procedimento, Rio de Janeiro: ABNT, 2014. [4] A. H. Mattock, L. B. Kriz, E. Hognestad. Rectangular concrete stress distribution in ultimate strength design. Journal Proceedings, 1961. [5] F. N. Mendes. Bloco retangular versus parábola-retângulo: Comparação na flexão composta Anais do 4◦ Simpósio EPUSP sobre estruturas de concreto, São Paulo: EPUSP, 2006. [6] J. K. Wight. Reinforced concrete: mechanics and design, Hoboken: Pearson, 7th edition, 2016. [7] Fib. Fédération Internationale du Béton. Model Code for concrete structures 2010, Lausanne: fib, 2013. https://doi.org/10.1002/9783433604090. [8] European Committee for Standardization 2004 EN 1992-1-1. Eurocode2: Design of concrete structures - Part 1-1: General rules and rules for buildings, Brussels: CEN, 2004. 430 https://doi.org/10.1002/9783433604090