Tubos de concreto reforçado com fibras de aço. Parte 2: modelo numérico para simular o ensaio de compressão diametral

Steel fibre reinforced concrete pipes. Part 2: numerical model to simulate the crushing test

Steel fibre reinforced concrete pipes. Part 2: Numerical model to simulate the crushing test

Tubos de concreto reforçado com fibras de aço. Parte 2: Modelo numérico para simular o ensaio de compressão diametral

A. de le Fuente^I; A. D. de Figueiredo^II; A. Aguado^III; C. Molins^IV; P. J. Chama Neto^V

^IDepartment of Construction Engineering, Barcelona Tech (UPC), [e-mail], C/Jordi Girona Salgado, 1-3, 08034, Barcelona (Spain)

^IIDepartment of Civil Construction Engineering, University of São Paulo (USP), [e-mail]. Caixa Postal 61548, CEP 05508-900. São Paulo (Brazil)

^IIIDepartment of Construction Engineering, Barcelona Tech (UPC), [e-mail], C/Jordi Girona Salgado, 1-3, 08034, Barcelona (Spain)

^IVDepartment of Construction Engineering, Barcelona Tech (UPC), [e-mail], C/Jordi Girona Salgado, 1-3, 08034, Barcelona (Spain)

^VCompanhia de Saneamento Básico do Estado de São Paulo (SABESP), [e-mail], São Paulo (Brasil)

ABSTRACT

This paper is part of an extensive work about the technological development, experimental analysis and numerical modeling of steel fibre reinforced concrete pipes. The first part ("Steel fibre reinforced concrete pipes. Part 1: technological analysis of the mechanical behavior") dealt with the technological development of the experimental campaign, the test procedure and the discussion of the structural behavior obtained for each of the dosages of fibre used. This second part deals with the aspects of numerical modeling. In this respect, a numerical model called MAP, which simulates the behavior of fibre reinforced concrete pipes with medium-low range diameters, is introduced. The bases of the numerical model are also mentioned. Subsequently, the experimental results are contrasted with those produced by the numerical model, obtaining excellent correlations. It was possible to conclude that the numerical model is a useful tool for the design of this type of pipes, which represents an important step forward to establish the structural fibres as reinforcement for concrete pipes. Finally, the design for the optimal amount of fibres for a pipe with a diameter of 400 mm is presented as an illustrating example with strategic interest.

Keywords: concrete pipes, fibres, crushing test, numerical model, optimal design

RESUMO

Este artigo faz parte de um extenso trabalho relacionado ao desenvolvimento tecnológico experimental e modelagem numérica de tubos de concreto reforçados com fibra de aço. Na primeira parte ("Tubos de concreto reforçado com fibras de aço. Parte 1: Análise tecnológica do comportamento mecânico"), foi apresentado o estudo experimental com enfoque tecnológico, abordando o procedimiento de ensaio e a discussão do comportamento estrutural obtido para cada consumo de fibra e tipo de reforço empregado. Nesta segunda parte, são abordados os aspectos de modelagem numérica. Neste sentido, se apresenta um modelo numérico para a simulação do comportamento de tubos de diâmetros inferiores a 1000 mm chamado MAP. São explicitadas as bases do modelo numérico e, posteriormente, seus resultados são confrontados com os obtidos experimentalmente, obtendo-se excelentes níveis de correlacão. Se conclui que a ferramenta numérica é útil para a otimização deste tipo de tubos, o que representa um avanço importante para a implantação das fibras estruturais como refuerço de tubos de concreto. Além disso, com o objetivo de proporcionar um exemplo de interesse estratégico, se apresenta a otimização do consumo de fibras para un tubo de 400 mm de diâmetro.

Palavras-chave: tubos de concreto, ensaio de compressão diametral, modelo numérico, otimização.

1. Introduction

Unreinforced concrete pipes (UCP) and steel bar reinforced concrete pipes (SBRCP) are well-known and accepted solutions for drainage and sewage pipes (Viñolas et al. [1]).

On the other hand, fibre reinforced concrete pipes (FRCP) and those reinforced with steel rebars and fibres (SBFRCP) are other underdevelopment alternatives (Haktanir et al. [2], de la Fuente et al. [3 and 4], Figuereido [5], Figueiredo et al. [6] and Lambrechts [7]). In this respect, the addition of fibres provides advantages from both the technical and the economic point of view. From the technical point of view, a substantial improvement of several mechanical properties of concrete is achieved (As'ad et al. [8]), especially with the addition of metallic fibres (Blanco [9]). Likewise, the composite solution leads to a positive structural synergy: the steel rebars perform the main strength function (Chiaia et al. [10]), whereas the fibres bridge the cracks, reducing their average spacing and width. The fibres also contribute to the strength function (Blanco et al. [11]). The use of fibres also contribute economically, because allows saving up on the assembling operations related to conventional reinforcement, reducing labor force, equipment use, and associated risks (de la Fuente et al. [12]).

FRCP and SBFRCP have already been considered as alternatives for UCP and SBRCP in several experimental campaigns both in Brazil (see Figueiredo et al. [6 and 13]) and Spain (see de la Fuente et al. [3]). However, their introduction in the market is under progress due to several factors such as: (1) the risk of damage when FRCP are manipulated; (2) the lack of calculation methods for this type of material, and (3) the difficulty to overcome the inertia towards change (Parrot [14]). Nonetheless, nowadays there are solutions for such problems: (1) polishing with emery powder in order to remove imperfections and avoid possible injuries; (2) constitutive equations to consider the tensile behavior of the steel fibre reinforced concrete (SFRC) (Hillerborg et al. [15], Vandewalle et al. [16] and Laranjeira et al. [17]), and (3) it has been verified that the incorporation of fibres improves the response of the pipe and leads to a global reduction of costs (Pedersen 1992 [18]).

Another relevant aspect related to FRCP and SBFRCP technology is the lack of recommendations and simplified calculation methods. Because of this, the design of FRCP and SBFRCP is normally carried out by trial and error: trying out several dosages and/or concrete thickness until finding an optimal amount of fibres that meet the requirements of the desired strength class in the crushing test (CT) (Figure [1]). This design procedure is hardly operative, uneconomical and inefficient due to the variety of diameters, thickness, strength classes, types of fibres and the factory limitations. For this reason, it is necessary to develop analytical and/or numerical tools that would make possible to carry out the optimal design and the verification of concrete pipes (CP), especially FRCP and SBFRCP, in order to avoid the regular procedures traditionally used.

The aim of this paper is, firstly, to introduce a model for the non-linear analysis of CP of medium-small diameter (less than 1000 mm) called Mechanical Analysis of Pipes (MAP) which is able to simulate the CT; and, secondly, to contrast the numerical and the experimental results in order to achieve the model validation.

Initially, a summarized exposition of the normalized CT procedure is presented. Then, the bases considered in the MAP model are mentioned, and the model results are contrasted with the results presented in the first part of this work (Figueiredo et al. [19]). Finally, an example of the application of MAP is presented aiming at determining the optimal amount of fibres for a pipe with 400 mm of D_i.

2. Crushing Test

The NBR 8890:2007 [20] specifies the procedures and all the details that should be observed during the execution of the CT. Both the cross and the longitudinal sections of the test configuration are schematically shown in Figure [2].

The load process and the strength requirements are function of the type of reinforcement. In the case of steel fibre reinforced concrete pipes (SFRCP) the requirements are presented below:

1. Withstand the proof load (F_c) during a minute without cracking or, in other words, without exceeding the first cracking load (F_cr). F_c is equivalent to the 67% of the minimum failure load (F_n).

2. Reach the ultimate failure load F_u, which must be higher than F_n.

3. When the load has decreased a 5% of F_u, or more, the pipe is completely downloaded and reloaded until reaching F_c. This load level must be supported for more than a minute.

4. The loading process must continue until reaching a minimum post failure load (F_min,pos) equivalent to, at least, 105% of F_c.

The purpose of this cyclic loading process is to verify if the type and amount of fibres are the suitable ones to guarantee the F_min,pos load and, indirectly, if the fibre-concrete anchorage and the post-peak strength of SFRC are appropriate (Figueiredo [5]).

3. Model for the simulation of the crushing test

The required subroutine for the simulation of the CT up to high displacement levels should take in to account paramount aspects as the cracking the post-failure response of the materials and the modeling of the SFRC behavior. In that sense, the Analysis of Evolutive Sections (AES) introduced in de la Fuente et al. [21] was used in order to deal with these aspects.

On the other hand, the MAP routine, which includes the AES model, was also developed. The bases for the structural model were already suggested by Pedersen [22] for the analysis of pipes with a small diameter. However, for this work, several changes were made as regards the behavior at the sectional level, the constitutive equations of SFRC and the possibility of considering the coexistence of steel rebars and structural fibres as reinforcement.

This section puts forward the main foundations of the AES model, highlighting the modifications introduced for this work, as well as the analytical equations of the MAP model and the calculation algorithm.

3.2 Sectional analysis model

3.2.1 Modeling the materials

The AES model discretizes the concrete in 2-D differential elements (dA_c), and the steel rebars in elements with concentrated area A_s,i in its gravity center y_s,i. Then, it assigns the suitable constitutive model to each material and integrates the stresses resulting from a given strain plane (see Figure [3a]). The total concrete strain ε_c(t,t₀), assessed at an instant of time t, is considered to be the sum of the mechanical strains ε_c^m(t₀), produced instantaneously at t_o, and the non-mechanical strains ε_c^nm(t,t₀) (see de la Fuente et al. [21] and Marí et al. [23]). In this paper, ε_c^nm(t,t₀) are not considered since the test only takes a few minutes to be executed, insufficient time to present non-mechanical strains due to the concrete creep. Likewise, according to Heger [24], shrinkage hardly influences the stress state of the pipe cross section; hence, it is also disregarded in this model.

 

For the simulation of the concrete compressive behavior the diagram suggested by Thorenfeldt et al. [25] is used, since it could be adjusted correctly to a wide range of concrete strengths and suitably simulates the post-failure response. On the other hand, the tensile behavior and concrete stiffening between cracks is described by means of the equation proposed in Collins et al. [26].

According to Bencardino et al. [27], the inclusion of metallic fibres modifies the SFRC compressive behavior depending on the volume of fibres used. In this respect, the expression suggested by Barros et al. [28] fits properly the uniaxial compressive behavior in the post-failure regime of SFRC. On the other hand, the simulation of its tensile behavior has been dealt by means of the σ_c-ε_c model (see Figure [3b]) proposed in (Vandewalle et al. [16]), because it has already been used in several numerical-experimental contrasting tests (see Pujadas [29]), guaranteeing good results.

The value of the crack width (w) is calculated considering that the crack surfaces rotate as a rigid body (see Figure [4]), forming a φ angle between the crack faces (Eq. 1). This angle is related to the sectional curvature χ by means of the length of the hinge l_bc through Eq. 2 (see Pedersen [30]). The value of l_bc varies depending on the stress level of the section; however, some authors (Pedersen [30] and Olesen [31]) establish it as a constant value of h/2, and still others (Casanova [32]) propose that it should vary depending on the crack height (s_n). For this paper, a constant value of h/2 for l_bc has been adopted, following the recommendations proposed by Pedersen [30] for the analysis of FRCP.

The steel rebars are modeled with a trilineal diagram, with the possibility of simulate the hardening response of the material (see Figure [3c]).

3.2.2 Basic hypotheses about sectional behavior

The following classical hypotheses have been adopted for the modeling the sectional behavior: (1) the sections have a symmetry axis and are subjected to straight flexo - compression; (2) perfect bond between the materials in the section; (3) sections initially planes remain planes after applying the forces; (4) shear strains are negligible and therefore are not considered; and (5) the pipe curvature does not affect on stress-strain distribution.

3.2.3 Idealization of the section

The positive signs are for: (1) the bending moments which compress the upper fibre; (2) the axial forces which compress the section; (3) compressive stresses and (4) the shortenings.

3.2.4 Equilibrium and compatibility equations

The stress-strain state (see Figure [3a]) resulting from a combination of internal efforts (normal force N and bending moment M) is defined by the stress distribution of the materials and the plane formed by the strain in the most bottom layer of the concrete section (taken as the reference layer) and the sectional curvature (ε_c,inf, χ). This state is obtained by applying the internal balance equations (Eqs. 3 and 4) and establishing the hypothesis of perfect bond between concrete and steel (Eq. 5).

Eqs. 3-5 leads to a nonlinear system of equations which is solved by using the Newton-Raphson iterative method (see Yang et al. [33]). After solving the system, the values of the unknown parameters ε_c,inf and χ, which define the strain plane, are obtained.

3.3 Structural analysis model

3.3.1 Basic hypotheses

For the simulation of the CT (Figure [1]), the following hypotheses have been considered: (1) the structure can be idealized as a medium plane piece with a curved shape and a constant radius R_m; (2) symmetry with regard to the vertical and the horizontal axes, so only a quarter of the pipe is simulated; (3) the initial curvature of the piece does not have an influence over the distribution of the stresses along the piece, nor over its deformed shape; (4) the axial and shear effects are disregarded in the assessment of the pipe displacements; (5) the continuous test is considered to be representative for the simulation of the FRCP behavior up to post-failure (Figueiredo [5]); and (6) three stages are considered:

■ Stage 1: linear elastic stage (Figure [5a]).

■ Stage 2: elastic stage with cracking at ridge (point C in Figure [5b]).

■ Stage 3: elastic stage with cracking at ridge and springs (point S in Figure [5c]).

 

3.3.2 Behavior equations

The governing equations for the structural problem implemented in MAP were deduced by Pedersen [22] for the simulation of SFRC pipes with small diameters. The strategy consists in considering that the response is represented by the three stages previously described, and that the pipe behaves elastically throughout the whole test, except the section at the ridge and at the spring line. This response pattern has also been observed in pipes tested by Figueiredo et al. [13]. The behavior of both sections is simulated with the AES model.

This paper presents the final form of the governing equations for the problem. Their analytical deduction can be found in Pedersen [22]. They are based on the energy theorems by Castigliano (see Timoshenko [34]) and on other classical considerations about the calculation of structures.

The applied force F, and the bending moments at the ridge M_c and at the spring line M_s (see Figure [5]) are the determinant parameters. They depend on the behavior regime of the pipe and, as a consequence, the analytical formulation varies depending on the stress state of the control sections.

In the elastic regime (Stage I), M_c is assessed by means the linear equation (Ec. 6). Once M_c is known, F and M_s are obtained using Eq. 7 and Eq. 8, respectively. This regime ends when a crack is formed at ridge for a rotation φ_c^crk (Eq. 9).

At Stage II, M_c (Eq. 10) is numerically obtained with the AES model since the formation of the first crack in C leads to a non linear system (Eqs. 3-5). Then, F is calculated by means of Eq. 11 and M_s is deduced by imposing external bending moment equilibrium (Eq. 12).

Stage III (Figure [5c]) starts when a crack is formed in the section S. This crack appears when a rotation φ_s^crk (Eq. 13) is reached. At this stage, F is calculated by means of Eq. 14, M_c (Eq. 10) is numerically obtained with the AES model and M_s is deduced imposing external bending moment equilibrium (Eq. 15).

The vertical displacement at the ridge (v_c) and the horizontal displacement at the springline (u_s) (see Figure [5d]) are calculated as a composition of the elastic strain of the pipe (v_c^e and u_s^e) and a plastic one (v_c^p and u_s^p) due to the rotation of the rigid body after the appearance of cracks in the critical sections.

The displacement v_c^e is calculated by means of Eq. 16 and Eq. 17.

The displacement u_s^e is deduced by means of Eqs. 18 and 19.

The displacements v_c^p and u_s^p are expressed by means of Eqs. 20-22.

Relevant results are obtained with application of the MAP model, including the curves F-v_c from the CT. These curves allow understanding the behavior of the structure at each of the stages.

3.3.3 Solution procedure

The process is initiated with zero values for the rotations at the ridge (φ_c) and spring line (φ_s). The control variable is φ_c, which increases with variable steps depending on the behavior stage. Establishing N = 0 (simple bending) for each value of φ_c at the ridge, the value of M_c is obtained by means of the AES model (Eq.6 for elastic regime, State I, and Eq. 10 for cracked regime, States II and III). After that, the values of M_s and F are calculated with the expressions previously presented:

■ At Stage I: F is obtained with Eq. 7 and M_s, by means of Eq. 8. At this stage the whole pipe, even the critical sections, works in a linear regime.

■ At Stage II: F is obtained with Eq. 11 and M_s, by means of Eq. 12. At this stage, a degree of non-linearity is introduced due to the cracking of section C (φ_c> φ_c^crk), whereas section S still works in a linear regime (φ_s< φ_s^crk). Therefore, due to the degree of hyperstatism of the system, a redistribution of moments from C to S takes place.

■ At Stage III: F is obtained with Eq. 14 and M_s, by resolving Eq. 15. At this stage, unlike at the previous ones, the balance condition (Eq. 15) is non-linear due to the fact that both section C and section S have cracked. For its solution, an iterative Newton - Raphson schema was implemented (see Yang et al. [33]). In this sense, it has to be noticed that the section S works under a bending - compression state during the whole load process (N = F and M = M_s).

The algorithm stops either when the maximum strain is reached at any of the two critical sections, or when the displacement v_c exceeds the fixed value pre-established by the user v_c,max.

Once the F and M values have been obtained, the displacements in sections C and S can be assessed with Eqs. 16-22.

This procedure guarantees good results in concrete pipes with a predominantly rigid behavior: pipes with a small-medium diameter (300 - 1000 mm) and with moderate reinforcement densities. With these hypotheses, it can be guaranteed that, in most cases, the cracks are concentrated in sections C and S, while the rest of the pipe works with its entire section (de la Fuente et al. [3]). In the opposite case, cracks appear intermediately and the model deviates from the experimental results and it is necessary to resort to other models capable of considering the distributed cracking, such as the one presented in de la Fuente et al. [12].

4. Contrasting experimental and numerical results

With the aim of verifying the suitability of the MAP model for the simulation of the mechanical response of FRCP subjected to CT, the experimental results presented in the first part of this work (see Figueiredo et al. [19]) were contrasted. So, a comparison of the curves F-v_c captured during the test for the pipes with D_i of 600 mm from series 2 (displacement measured in the spigot) is done.

All the pipes in these series of tests were manufactured with the same concrete composition, although there were some modifications in the water consumption in order to improve the workability due to the use of fibres (DRAMIX^® RC-80/60-BN). The pipes were manufactured and tested at the same age with the aim of reaching, at least, the resistance class EA2 established in NBR 8890:2007 [20].

4.1 Modeling the materials

The modeling of the compressive behavior of SFRC was performed with the equation suggested by Barros et al. [28], considering a characteristic compressive strength (f_ck) of 50 MPa at 28 days according to the tests performed during the regular quality control in the factory.

On the other hand, for the simulation of its tensile response, the trilinear diagram proposed by Vandewalle et al. [16] was used (see Figure [3b]). However, due to the lack of flexural tests (see Vandewalle et al. [35]), in order to determine the values of concrete tension stress σ_i, the expressions (Eqs. 23-24) calibrated in Barros et al. [36] have been used to determine the values of the residual flexural strength f_Ri as a function of C_f. In this regard, the type of fibres used both in this campaign and in the one carried out by Barros et al. [36] are the same: (DRAMIX^® RC-80/60-BN). Table [1] shows the values established for σ_i, ε_i and E_cm in order to model the tensile behavior of SFRC.

4.2 Results obtained

Figures 6a, 6b and 6c show the curves F-v_c obtained both experimentally (individual and average values) and numerically for pipes with C_f of 10, 20 and 40 kg/m³, respectively.

 

With reference to what has been previously explained in section 3 and to the requirements of the EA2 class of the NBR 8890:2007, the load F_cr must be equal or higher than the load F_c (90 kN), and F_u must be equal or higher than 135 kN for this type of pipes. Finally, the maximum post-failure load F_max,pos measured in the curve F-v_c must reach, at least, the 105% of F_c (94.5 kN). In this sense, since the test was carried out in a continuous manner, it was established that the value of F_max,pos is associated to a v_c of 3 mm (see Figueiredo et al. [19]), and is called F_3mm.

Based on the results presented in Fig. 6a, it is deduced that the MAP model fits properly to the experimental results for the amount of 10 kg/m³, particularly in the linear elastic regime and in the post-failure regime. For the latter, the numerical results tend toward the experimental maximum values for displacements higher than 3 mm. This might indicate that the values f_R,i and/or σ_i of the constitutive equation of tensioned SFRC (see Fig. 3b) are slightly higher than the real ones. The values for F_cr, F_u and F_3mm obtained numerically are 98 kN, 114 kN and 88 kN, respectively. Therefore, neither F_u nor F_3mm reach the minimum values stipulated by NBR 8890:2007 for the EA2 class. Thus, it can be stated that, according to the model, the amount of 10 kg/m³ is not enough to guarantee that level of requirements.

The results gathered in Fig. 6b, concerning the dosage of 20 kg/m³, highlight that the simulation by means of the numerical model guarantees values close to the experimental ones. Still, the model exceeds the experimental results with displacements higher than 5.5 mm, which can be due to the considered excessive values of f_R,i and/or σ_i, as in the case of pipes with 10 kg/m³ of fibres for this range of displacements. The values for F_cr, F_u and F_3mm obtained numerically are: 98 kN, 123 kN and 108 kN, respectively. Consequently, according to the model, with 20 kg/m³ the EA2 class would not be reached, since the load F_u (123 kN) is lower than the required 135 kN.

Finally, for the pipes with 40 kg/m³ of fibres (Fig. 6c), it can be noticed that the numerical model adjusts suitably to the average experimental results at all the stages. In this case, F_cr is 98 kN and F_u is 156 kN, values higher than those specified for a pipe of 600 mm from the EA2 class (NBR 8890:2007). It should be noted that, due to the hardening behavior, there is no way to assess the load F_min,pos for a displacement v_c of 3 mm as in the previous cases. However, it can be asserted that the load F_min,pos from these pipes exceeds the value of 94.5 kN, since the pipes with 20 kg/m³ of fibres showed values 14.3% higher than this (108 kN) according to the numerical model.

Table [2] gathers the average experimental and numerical values for F_c, F_u and F_3mm. The parameter ξ is the relative error of the numerical value with regard to the experimental data. Positive values of ξ indicate that the experimental data exceeds the numerical one, and vice versa.

With regard to the results illustrated in Table [2], it is deduced that:

■ The load F_cr obtained numerically is independent from the C_f, since tension σ₁ is related exclusively to the concrete matrix (see Figure [3b] and Table [1]). F_cr depends exclusively on σ₁, h and D_i (see de la Fuente et al. [3]). The model tends to overestimate F_cr with regard to the average experimental results between a 4.3% (pipe with 10 kg/m³) and a 6.5% (pipe with 40 kg/m³). This can be due to the incorporation, during the mixing, of additional water (the bigger the amount of fibres, the bigger the quantity added) into the concrete used for the contrasted pipes (see Figueiredo et al. [19]).

■ The model underestimates the maximum load F_u with regard to the experimental values between a 3.7% (pipe with 40 kg/m³) and a 13.6% (pipe with 10 kg/m³). For this load level, the matrix of concrete has already cracked and the fibres work with the maximum efficiency; then, the reason for this difference in the results can be due to the fact that the values of σ_i used in the constitutive equation might be too conservative for low levels of displacement. In this regard, de la Fuente et al. [38] proves that the fibres in concrete pipes manufactured with traditional systems (for example, turbo-compression) work oriented towards the stress flow, practically guaranteeing their maximum efficiency. One of the ways to consider this fact is by using an amplification coefficient of the parameters σ_i. So, the use constitutive equations incorporating this effect, in order to take into account the preferential orientation of the fibres could be considered as a good alternative, as suggested by Laranjeira [39].

■ As regards the load F_3mm, the numerical values obtained are 6.4% (10 kg/m³) and a 6.9% (20 kg/m³) lower in comparison with the experimental data. This could also be due to the underestimation of the parameters σ_i adopted to model the tensile response of SFRC in this regime of v_c.

To sum up, the MAP model adjusts satisfactorily to the experimental results, even considering that the input parameters of the constitutive equation used to model the tension response of SFRC are calibrated from concretes with a f_ck ranging between 25 and 30 MPa (see Barros et al. [36]), as opposed to the reported 50 MPa. Similarly, the equation used does not take into account the effect of the preferential orientation of the fibres within the wall of the pipe. Therefore, the MAP model tends to underestimate the experimental results in most of the cases, yet these differences do not exceed the 13.6%. Nevertheless, the results can be considered a success, taking into account the multitude of variables involved in the problem, their uncertainty and the difficulties for the direct experimental determination of some of them.

5. Example of the optimal design of the amount of fibres

The following example of MAP model application to a pipe with D_i of 400 mm, h of 67 mm and a total length of 2500 mm is purposed to illustrate the methodology of the design of the optimal C_f in SFRCP.

This diameter was chosen for two reasons: (1) because it is within the range of diameters for which the hypotheses from the MAP model are appropriate, and (2) because it is a commercial diameter which lately is losing its market share in favor of the plastic pipes. Therefore, new alternatives and improvements are necessary for the concrete pipe to be competitive once again (Viñolas et al. [1]).

For the analysis, a range of C_f between 0 kg/m³ and 30 kg/m³ was established. The first amount would correspond to an unreinforced concrete pipe (UCP), and the second one has been established as a maximum value because of economic criteria.

Likewise, the requirement of a minimum early compressive strength is necessary for demolding and to manipulate the pipe. In order to satisfy this requirement an f_ck of 45 MPa at age of 28 days has been assumed in order to carry out the analysis.

Table 3 presents the values for the mechanical parameters used to simulate the tension behavior of SFRC.

The values of F_c and F_u fixed in NBR 8890:2007 for the strength classes EA2, EA3 and EA4 for pipes with D_i = 400 mm are gathered in the Table 4.

Fig. 7 shows the curves F-v_c obtained with the MAP model. From those curves, it is deduced that:

■ The load F_cr obtained in the four cases (96 kN) is enough to reach the load F_c stipulated in classes EA2 (60 kN) and EA3 (90 kN). However, the value of F_c for class EA4 (120 kN) cannot be reached by means of the addition of metallic fibres only (stress σ₁ is independent from C_f).

■ The 90 kN established for F_u in class EA2 are exceeded if a fibre dosage of 10 kg/m³ (93 kN) is used. But, in order to reach the 135 kN stipulated for class EA3, according to the numerical model, at least 30 kg/m³ of fibres (136 kN) are required. On the other hand, the C_f necessary to achieve the 180 kN established for class EA4 is not under the economically competitive values.

■ In order to reach the 63 kN specified for the F_min,pos of class EA2 and the 94.5 kN of class EA3, 10 kg/m³ (64 kN) and 20 kg/m³ (99 kN) are required, respectively.

Therefore, keeping in mind that the three strength requirements must be fulfilled simultaneously for a fixed C_f, it is possible to conclude that 10 kg/m³ of fibres would be required to achieve the EA2 class requirements, and 30 kg/m³ would be required for class EA3, according to the numerical model.

Alternatively, in order to achieve class EA4 with an economically attractive reinforcement configuration, a composite solution could be proposed (fibres + bars). On one hand, the load F_c (F_w=0.25mm in case of SBFRCP) would be reached thanks to the use of fibres. On the other hand, the steel bars would guarantee higher failure strength due to their higher efficiency and strategic position within the section.

Fig. 7 also shows a simulation considering a reinforcement consisting of 10 kg/m³ of fibres and 7Ф5/ml of CA60 steel bars. This strategy leads to an F_w=0.25mm = 125 kN (F_c = 120 kN) and also to an ultimate failure load F_u strictly equal to 180 kN. Consequently, class EA4 would be reached, with F_u being in this case the critical parameter in the design process.

When comparing the curve F-v_c obtained SFRCP with 30 kg/m³ of fibres and the one obtained for the SBFRCP (10 kg/m³ + 7Ф5/m) for values of v_c up to 1.2 mm (service range), it can be observed that the behavior of the former is better. This highlights the fact that the normative prescription which refers to the load F_c is much more restrictive for FRCP, if compared with the conventional SBRCP. While FRCP are not allowed any cracking symptoms (F_cr>F_c), the SBRCP are allowed to reach cracking with a width of up to 0.25 mm (F_w=0.25mm>F_c). In the opinion of the authors, this criterion restricts the range of application of fibres to this product. Besides, it is not in accordance with the experimental and numerical results, especially when it is known that the inclusion of the suitable type and volume of fibres improves considerably the cracking behavior of the concrete structures.

6. Conclusions

This paper introduces the MAP model for the analysis of concrete pipes with mid-low diameters (lower than 1000 mm) and reinforced with traditional steel rebars and/or steel fibres. The bases for this model were already introduced by Pedersen [22], but this paper uses the most recent constitutive equations to simulate the behavior of SFRC. The obtained degree of correlation between experimental and model results can be considered excellent, having obtained numerical results with an average relative error of 7.0% in the safe of safety. To improve this aspect, the constitutive equation of tensioned SFRC could be adjusted taking into account that the fibres are oriented towards a preferential direction within the wall of the pipe. In short, the MAP model can be considered to be a suitable tool for designing the optimal configuration of the reinforcement for this type of pipes. It is intended to be used in precast plants where the required technology to manufacture and test pipes is available. It leads to savings as regards both time and economical resources, since it avoids the extensive test programs required in order to find the optimal amount of reinforcement. This model is especially interesting when a geometrical condition of the pipe, the type of fibres or the strength class is modified, or simply if the factory wishes to make the design tables of the most commercial pipes.

With the aim of illustrating the model capability, an example of optimal design for a pipe with a diameter of 400 mm has been included. The conclusions established were that, according to the model, class EA2 would be obtained with 10 kg/m³ of fibres; class EA3, with 30 kg/m³; and with 10 kg/m³ + 7Ф5/m of CA60 steel, class EA4 from the NBR 8890:2007 would be also reached.

Nowadays, several experimental campaigns are being carried out with the purpose of expanding the data bank used to contrast the model and adjust, if necessary, the convenient bases or parameters.

7. Acknowledgements

The authors of this document wish to express their appreciation for the financial support received through the Research Project BIA2010-17478: Procesos constructivos mediante hormigones reforzados con fibras.

Likewise, Professor Antonio D. de Figueiredo wishes to thank the support provided by CAPES -Coordenação de Aperfeiçoamento de Pessoal de Nível Superior- for having awarded him the postdoctoral grant that allowed him to participate in this work.

8. References

Received: 11 Feb 2011

Accepted: 20 Oct 2011

Available Online: 01 Feb 2012

9. Nomenclature

A:

Pipe spigot. A_c: Concrete area. A_s,i: Area of the i-th steel bar. B: Pipe socket.

dA_c:

Differential concrete area. C_f: Fibre dosage.

D_i:

Internal diameter of the pipe. D_o: Outside diameter of the pipe. E_cm: Average Young modulus of the concrete.

E_s:

Young modulus of the steel. F: Applied load on the pipe. F_c: Proof load of the pipe. F_cr: First cracking load of the pipe. F_n: Minimum failure load (established) of the pipe. F_max,pos: Maximum post-failure load (simulated) of the pipe. F_min,pos: Minimum post-failure load (established) of the pipe.

F_u:

Ultimate failure load of the pipe. F_3mm: Post-failure load for a 3.0 mm vertical displacement of the key (2^st series of the tested pipes). f_ck: Characteristic compressive concrete strength. f_cm: Average compressive concrete strength. f_ctm,fl: Average flexural tensile concrete strength.

f_Ri:

Residual flexural concrete strength. f_s,max: Maximum steel strength. f_y: Yielding strength of the steel. h: Thickness of the concrete wall.

I:

Moment of inertia around the horizontal axis (I = h³/12). k_h: Size factor. l: Length of the pipe. l_bc: Length of the hinge. M: Applied bending moment. M_c: Bending moment at ridge. M_s: Bending moment at springline.

M_θ:

Bending moment at the angular coordinate θ. N: Applied normal force. N_θ: Normal force at the angular coordinate θ. n_s: Number of steel bars. R_m: Average radius of the pipe. s_n: Height of the crack. t: Instant of calculus. t_o: Age of loading. u_S: Horizontal displacement of the springline. u_s^e: Elastic horizontal displacement of the springline. u_s^p: Plastic horizontal displacement of the springline. v_c: Vertical displacement of the ridge.

v_c^e:

Elastic vertical displacement of the ridge.

v_c^p:

Plastic vertical displacement of the ridge. v_c,max: Maximum vertical displacement of the ridge. w: Crack width. x_n: Neutral axis depth. y: Height of the analyzed element. y_c: Height of the analyzed concrete layer. y_G: Height of the gravity center of the section. y_s,i: Height of the analyzed i-th steel bar. β: Angle between the supports and the centre of the pipe. ε: Strain. ε_c: Concrete strain. ε_i: Tensile strain of the SFRC. ε_c^m: Mechanical concrete strain. ε_c^nm: Non-mechanical concrete strain. ε_s: Steel strain. ε_c,sup: Strain of the concrete section top layer. ε_c,inf: Strain of the concrete section bottom layer. ε_s,i: Strain of the i-th steel bar. ε_y: Yielding steel strain. θ: Angular coordinate. ξ: Relative error of the numerical value. σ_c: Concrete stress. σ_i: Steel fibre concrete tension stress. σ_s: Steel stress. σ_s,i: I-th steel bar stress. σ_u: Ultimate concrete compressive stress. φ: Angle between the crack faces. φ_c: Rotation of the ridge. φ_c^crk:

Cracking rotation of the ridge.

φ_s: Rotation of the springline. φ_s^crk: Cracking rotation of the springline. χ: Sectional curvature. χ_c: Curvature of the section of the ridge.

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Datas de Publicação

• Publicação nesta coleção
  04 Jul 2013
• Data do Fascículo
  Fev 2012

Histórico

• Recebido
  11 Fev 2011
• Aceito
  20 Out 2011