https://doi.org/10.14311/APP.2022.33.0431 Acta Polytechnica CTU Proceedings 33:431–436, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague STRUCTURAL FLEXURAL STRENGTHENING THROUGH MATERIAL BONDED TO THE CONCRETE SUBSTRATE Victor Hugo Dalosto de Oliveiraa, Rodolfo Alves Carvalhoa, ∗, Rodolfo Rabelo Nevesb, Luana Ferreira Borgesa, Paulo Chaves de Rezende Martinsa, Marcos Honorato de Oliveiraa, João Paulo de Almeida Siqueiraa, Rafael de Almeida Sobrala, Sidney Luan Teixeira de Nazaréc, Carlos Henrique de Moura Cunhac a Department of Civil and Environmental Engineering, University of Brasília, SG-12, Campus Darcy Ribeiro, Asa Norte, Brasília, DF, 70910-990, Brazil b Department of Materials Engineering and Construction, Federal University of Minas Gerais, Av. Antônio 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 ∗ corresponding author: rodolfoalvescarvalho@yahoo.com.br Abstract. This article proposes an alternative method for the structural design of reinforced concrete elements strengthened in bending by metallic plates or fiber-reinforced polymer (FRP) bonded to the concrete substrate. It is proposed a new calculation procedure for the strengthening using thin adhered material bonded to the element surface that dispenses the iterative process generally used in the design. The proposed routine is validated by comparison with other methods. A practical example is also presented, applying the procedure to an element of a building where a load change was foreseen. As result, it was verified that the proposed procedure provides values similar to the trial-and-error method used in the FRP strengthening design. Results are also coherent with other methods available in the literature for metallic plates. Therefore, since this routine obtains similar values without using an iterative method, its applicability in the design becomes advantageous. Keywords: Attached materials, calculation guide, structural strengthening. 1. Introduction When constructions do not fit the required capac- ity for correct use, rehabilitation is necessary. The reasons for rehabilitation may be pathological mani- festations, accidents, natural disasters or changes in the building use. To restore the necessary resistance of the building, actions such as recovery or structural strengthening may be necessary [1–3]. The primary function of a structural retrieval is to improve the resistant capacity of any structure. The strengthening can be performed by associating new materials to the original structure, forming a compos- ite system that supports the applied loads. Examples of these processes are the addition of steel bars to the original section, reinforced concrete jacketing, exter- nal prestressing tendons and elements adhered to the substrate, such as steel sheets and composite materi- als [4]. This study proposes a new calculation methodol- ogy for the flexural design of reinforced concrete ele- ments associated with small thickness bonded mate- rials to the surface, as external steel sheets or fiber- reinforced polymer (FRP). The main goal herein is to establish a calculation guide for structural strength- ening design, which fits the requirements of the Brazilian standard NBR 6118: 2014 [5]. It should be noted that only the flexural capacity is concerned in this paper. Considerations related to anchor capacity or the possibility of debonding of the strengthening are not made. 2. Literature review and numerical simulation This section presents some methods commonly used for the flexural strengthening design of reinforced concrete beams and slabs present in the literature. These methods consider the insertion of steel sheets and carbon fiber reinforced polymer sheets bonded to the surface. The calculation guide proposed in this paper follows the literature review. 2.1. Calculation models for metal sheet strengthening There are several calculation methods in the liter- ature for the flexural strengthening using bonded metal sheets and steel profiles. Some of the most relevant are described below. 2.1.1. J. Bresson Model Metal sheet dimensioning with Bresson’s method is carried out by balancing the bending moments in re- lation to the most compressed fiber. Materials are 431 https://doi.org/10.14311/APP.2022.33.0431 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en V. H. D. D. Oliveira, R. A. Carvalho, R. R. Neves et al. Acta Polytechnica CTU Proceedings Figure 1. Stress diagram of a strengthened beam. Source: Adapted from Silveira [6]. assumed to have a linear elastic behavior; concrete tensile strength is neglected and stresses are limited to their respective elastic admissible values [6]. Thus, the strengthening plate area is obtained by equation 1: As = ! MDead + MLive " + (fc1 + fc2 ) b·y2 6 − (fs1 + fs2 ) As d fR · h (1) As shown in Figure 1, design takes into account dead and live loads, referred, respectively, to the forces already acting in the section when the strength- ening is carried out and after the rehabilitation. 2.1.2. Campagnolo Model In this model, strengthening design is performed us- ing classical formula for bending, considering com- posite beams theory [1]. Thus, the tensile stress cal- culation for the steel plate is obtained as follows in equation 2. fR = ER Ec · Mu (h − xII ) Ix (2) 2.1.3. Cánovas model The design is performed on the basis that materi- als behave under their maximum allowable elastic stresses. Stress distribution before and after the strengthening execution are considered. Balance and design equations are obtained by superimposing the stress diagrams [1], as illustrated in Figure 1. Thus, the necessary area of additional strengthen- ing, AR, is obtained by balancing external and inter- nal moments as in equation 3: Mu = (As fs + AR fR) (d − x′) (3) 2.1.4. Ziraba and Hussein model The authors proposed a method in which the strengthening design is done at the ultimate limit state, following the specifications of ACI 318 [1, 7]. Stress resultants and the strengthening area are ob- tained from the balancing of moments in the section taken at the point of the compressive stress resultant in concrete, as reproduced in Figure 2, resulting in an iterative process, as shown in equation 4. Msd = Rs # d − y 2 $ + RR # hR − y 2 $ (4) 2.2. Calculation models for carbon fiber reinforced polymer Design of strengthening systems composed of car- bon fibers, according to Machado [2] and the Amer- ican code ACI 440 [8], may be done through a trial- and-error procedure, that gives the required area of strengthening at the ultimate limit state. According to these methods, a position for the neu- tral axis (x) is initially arbitrated. Then, deforma- tions (ε) and stresses (f ) of materials are calculated using compatibility and equilibrium equations. It is then verified if the resulting forces are balanced (equation 5), and if the resistant bending moment gotten is greater than the external moment applied (equation 6). If any of the conditions is not satisfied, a new neutral axis depth is chosen, and the procedure is repeated. Nsd = Rs + R′s + RR + RC (5) Msd = Rs # d − yx 2 $ + R′s # yx 2 − d′ $ + RR # hR − yx 2 $ (6) 2.3. Proposed calculation model This section contains the method proposed by the authors. It is a calculation guide used for structural strengthening design, addressing the association of thin-thickness materials adhered to the surface. 2.3.1. Calculation premise The following hypotheses are adopted: • The design is carried out at the ultimate limit state, disregarding the concrete tensile strength. • Plane sections remain plane after loading, and the deformations are proportional to their distance to the neutral axis (Bernoulli criterion). 432 vol. 33/2022 Structural strengthening through bonded material Figure 2. Stress and strain diagram for a section in the strengthened condition. • Perfect adhesion exists between the concrete, the steel reinforcement, and the thin adhered material (Navier criterion). • The cross-sectional failure is characterized when the specific deformation of any material reaches its limit (ultimate) value. Figure 2 illustrates the behavior of a rectangu- lar cross-section of a strengthened concrete element, submitted to simple normal flexure. The rectangu- lar stress diagram presented in section 17.2.2 of the Brazilian standard NBR 6118: 2014 [5] is adopted. The compression stresses in concrete are distributed in a rectangle of depth y, with peak stress equal to α · fcd, according to Figure 2. The value of y is defined as (equation 7): y = λ · x (7) In which, λ is given by equation (8) for fck ≤ 50 MPa, and λ is given by equation 9 for fck > 50 MPa: y = 0.8 (8) y = 0.8 − (fck − 50) % 400 (9) The α parameter is defined according to the con- crete class as follows: For concrete classes up to C50: α = 0.85 (10) For concrete classes from C50 up to C90: α = 0.85 & 1 − (fck − 50) % 200 ' (11) 2.3.2. Longitudinal strains Considering Bernoulli’s hypothesis, the deformation of any point within the section is proportional to its distance to the neutral axis, given as a function only of its location (y) and the curvature radius (ρ). Knowing that the strengthening material bonded to the element has a minimal thickness, about a few millimeters, it can be considered that its specific lon- gitudinal deformation (εr ) is equal to that of the con- crete fiber substrate to which it is glued. Thus, the deformation of the material, taken as constant along its thickness, can be obtained by equation 12: 1 ρ = εc x = εs (d − x) = εRF (h − x) = ε ′s (d − x) (12) When the applied loads, including the dead load, are not eliminated before the strengthening applica- tion, the material added to the section is not stressed by all these loads. In this way, the original structure is subjected to pre-existing stresses and has already suf- fered deformations before strengthening. Under these conditions, it is necessary to know these stresses to determine the actual stress acting on the strength- ening. Thus, strengthening deformations can be ob- tained using equation 13: εR = (εRF − εRI ) ≤ εU (13) in which εRF is the specific deformation of the con- crete substrate when all loads are applied, and εRI is the specific deformation in the section at the moment of strengthening application. 2.3.3. Failure criteria Analyzing the failure criteria, the ultimate concrete deformations are defined in section 8.2.10.1 of the Brazilian standard [5], by εc2 and εcu, as shown in equations 14 to 17: For concretes classes up to C50: εc2 = 0.2 % (14) εcu = 0.35 % (15) For concretes classes from C50 up to C90: εc2 = 0.2 % + 0.0085 % (fck − 50) 0.53 (16) εcu = 0.35 % + 3.5 % & (90 − fck) % 100 '4 (17) Considering Hooke’s law valid up to yield stress (fy ) of any steel element, it is assumed that the 433 V. H. D. D. Oliveira, R. A. Carvalho, R. R. Neves et al. Acta Polytechnica CTU Proceedings Figure 3. Strengthened beam scheme. Source: Adapted from Machado [2]. stresses are proportional to the deformations. In this way, the tensile resultant in the pre-existent steel re- inforcement (Rs) and in the strengthening added to the element (RR) can be done by equations equa- tion 18 and 19. Rs = As · Es · εs ≤ As · fSy (18) RR = AR · Es · εs ≤ AR · fRy (19) It is important to note that, as the carbon fiber has high tensile strength, linear elastic behavior is expected for this type of material until failure of the critical section. Finally, to prevent excessive plastic deformation in critical sections, the maximum allowable elongations are limited to = 1%. 2.3.4. Ductility criteria In reinforced concrete structures, proper measures must be taken to ensure adequate ductility of all structural elements up to the ultimate limit state. According to ACI 318-14 [7], if the tensile stress in the reinforcement is greater than εs = 0,5%, the sec- tion is characterized as ductile, presenting warnings about imminent rupture, such as the presence of large deformations and cracks. If the steel deformation is less than this value, a fragile rupture condition is ex- pected, with few warnings of an imminent collapse. Section 14.6.4.3 of the Brazilian standard NBR 6118: 2014 [5], in order to guarantee a ductile be- havior for the slabs and beams, establishes that the depth of the neutral axis at the ultimate limit state must be limited to the following values: equation 20 for ≤ 50 MPa, and equation 21 for 50 MPa < fck ≤ 90 MPa. x ≤ 0.45 · d (20) x ≤ 0.35 · d (21) 2.3.5. Strengthening equation The equilibriums equation of strengthening supposes that dimensions of the cross-section, the reinforce- ment position, and the material properties are known. The procedure described below must consider the bal- ance of the section, the compatibility of deformations, and its failure mode. It was considered that the internal resisting bend- ing moment must be at least equal to the external moment applied to the section. In this way, the equi- librium equation of moments can be carried out at the contact surface between concrete and the newly added strengthening. One of the variables is elimi- nated with this procedure. Therefore, the equilibrium of moments in a strengthened section is: ( M = Msd Msd = Rc (h − x′) + Rs (h − d) + R ′s (h − d ′) Msd = α fcd bw y # h − y 2 $ + As fs (h − d) + A ′s f ′ s (h − d ′) (22) Knowing that the compressed concrete block is given as a function of equation 7, the position of the neutral axis can be obtained by as the positive root of equation 23, derived from equation 22 by algebraic manipulation: ! −0.5 α fcd bw λ2 " x2 + (α fcd bw λ h) x + [A ′s f ′ s (h − d ′) − As fs (h − d) − Msd] = 0 (23) With the neutral axis position, equations 12 and 13 are used to identify the domain of ultimate limit state of the behavior of the section, and to determine the specific deformations along its height. The sec- tional stresses distribution from existing load action during the strengthening can be considered, taking 434 vol. 33/2022 Structural strengthening through bonded material Variable Iterative process [2] Suggested model(Authors, 2021) εRI 0.228 ! 0.272 ! εRF 7.000 ! 7.324 ! εR 6.772 ! 7.052 ! L · N 0.228 cm 22.31 cm fR 1544.0 MPa 1607.9 MPa Strengthening area 0.588 cm2 0.563 cm2 Table 1. Calculated carbon fiber strengthened area. Figure 4. Representation of beam characteristics of the building. Calculation model Strengthening area Difference Suggested model 2.054 cm2 - J. Bresson 4.091 cm2 99.17 % Campagnolo 3.684 cm2 79.36 % Ziraba e Hussein 2.905 cm2 41.43 % Cánovas 2.286 cm2 11.30 % Table 2. Comparison of the calculated strengthening sheet area. into account the cracked section properties, if neces- sary. Considering a cross-section submitted only to flexure forces, the resultant strengthening force is de- termined as presented in equation 24. ( F = Nsd Nsd = Rc + Rs + R ′s + RR RR = Nsd − α fcd bw λ x − As fs − A ′s f ′ s (24) Once the resulting force and specific deformation have been determined, the strengthening area neces- sary to resist to the applied moment can be calculated using equation 19. Finally, it is still necessary to assess whether the section mode of failure satisfies the ductility criteria set out in section 2.3.4. It is worth to mention that the proposed equation 24 also allows the direct ap- plication of the environmental reduction factors, as exposed in ACI 440 [8], to the strengthening resul- tant. 3. Results and discussions A comparison between the design methods mentioned and the calculation model proposed in this paper was made. For this, some examples available in the lit- erature and a study case were taken for. Analyzing example 4.1 of the Manual by Machado [2], a concrete beam was supposed to be strengthened through the association of carbon fiber sheets. All necessary data about the beam, the applied loads, and the material properties are shown in Figure 3. The proposed calculation routine was compared with the iterative procedure used in Machado [2]. The design was carried out using the criteria of the Brazil- ian standard NBR 6118: 2014 [5], adopting the same parameters for the materials. The comparative re- sults are shown in Table 1. The actual strengthening provided had an effective area of 0.594 cm2, composed of two carbon fiber lay- ers, 0.165 mm thick and 18 cm width. It appears that the recommended procedure provides a value very close to the trial-and-error process for necessary car- bon fiber strengthening area (less than 4.25%). This difference may be due to the tolerance criteria applied in the iterative process [2]. 435 V. H. D. D. Oliveira, R. A. Carvalho, R. R. Neves et al. Acta Polytechnica CTU Proceedings The second example is the case of a building, where the live load was foreseen to be increased. The addi- tional strengthening was conceived through the asso- ciation of metal sheets. Figure 4 shows the character- istics of one of the beams that needed to be strength- ened. Table 2 shows the section sizing of the steel plate obtained from the different methods. It is essential to highlight that, although the pro- posed method presents similarities with that devel- oped by Ziraba and Hussein - both consider the strengthening design at the ultimate limit state, their results are quite different. This fact is due, not only, but mainly because of criteria that govern ACI 318 [7] is different from those of NBR 6118 [5] and because of the consideration of pre-existing deformations con- sidered in the proposed calculation. 4. Conclusions This paper shows that it is possible to design fiber- reinforced polymer strengthening using a different ap- proach from the trial and error method, frequently used in technical design. The proposed method leads to encouraging results with reasonable accuracy if compared with the classical ones. No iterative cal- culations are needed. The method provides the depth of the neutral axis without evaluating the stress resultant of the internal reinforcement. Consequently, it is possible to con- sider its application for other thin materials bonded to the substrate, such as metal sheets. There was a distinct difference in the results ob- tained with other literature models for sheet metal design. The authors attribute these divergences mainly to the difference between some calculation hy- potheses adopted, such as the design based on ulti- mate limit state instead of the Allowable Stress De- sign methods, where material resistances are limited to its limit of elasticity. The proposed method also allows the usage of the Load and Resistance Factor Design, the consideration of pre-existing stresses, and the application of environmental reduction factors di- rectly to the strengthening, which are not possible in some other methods. Thus, the numerical analysis shown that the pro- posed method presents good efficiency. More exhaus- tive laboratory tests, experimental and/or numerical, may be foreseen to improve and calibrate the param- eters of the method proposed herein. Acknowledgements This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES). References [1] V. C. M. D. Souza, T. Ripper. Patologia, recuperação e reforço de estruturas de concreto, São Paulo: Pini, 1998. [2] A. P. Machado. Manual de reforço das estruturas de concreto armado com fibras de carbono, São Paulo: Viapol, 2015. [3] P. K. Mehta, P. J. M. Monteiro. Concreto: Microestrutura, propriedades e materiais 2, São Paulo: Ibracon, 2014. http://www.viapol.com.br/media/ 97576/manual-fibra-de-carbono.pdf. [4] A. P. A. Reis. Reforço de vigas de concreto armado por meio de barras de aço adicionais ou chapas de aço e argamassa de alto desempenho Dissertação de mestrado Universidade de São Paulo, São Carlos, 1998. [5] Associação Brasileira de Normas Técnicas. ABNT NBR 6118: projeto de estruturas de concreto - procedimento, Rio de Janeiro: ABNT, 2004. [6] S. S. Silveira. Dimensionamento de vigas de concreto armado reforçadas com chapas coladas com resina epóxi Dissertação de mestrado Universidade Federal Fluminense, Niterói, 1997. [7] American Concrete Institute 2014 ACI 318-14: Building Code Requirements for Structural Concrete, Farmington Hills: ACI, 2014. [8] American Concrete Institute 2007 ACI 440.2R-08: Guide for the design and construction of externally bonded FRP systems for strengthening concrete structures, Farmington Hills: ACI, 2007. 436 http://www.viapol.com.br/media/97576/manual-fibra-de-carbono.pdf