On axial compressive behavior of steel fiber reinforced concrete confined by FRP

Abstract

In this study, the co-effects of steel fibers and FRP confinement on the concrete behavior under the axial compression load are investigated. Thus, the experimental tests were conducted on 18 steel fiber-reinforced concrete (SFRC) specimens confined by FRP. Moreover, 24 existing experimental test results of FRP-conﬁned specimens tested under axial compression are gathered to compile a reliable database for developing a mathematical model. In the conducted experimental tests, the concrete strength was varied as 26 MPa and 32.5 MPa and the steel fiber content was varied as 0.0%, 1.5%, and 3%. The specimens were confined with one and two layers of glass fiber reinforced polymer (GFRP) sheet. The experimental test results show that simultaneously using the steel fibers and FRP confinement in concrete not only significantly increases the peak strength and ultimate strain of concrete but also solves the issue of sudden failure in the FRP-confined concrete. The simulations confirm that the results of the proposed model are in good agreement with those of experimental tests.

Keywords

compressive strength fiber-reinforced concrete fiber-reinforced polymers experimental test mathematical model

Introduction

Concrete is considered as brittle material that makes its application limited in constructions undertaken in seismically active zones (Caballero-Morrison et al., 2013; Zhang et al., 2019b; Zheng and Ozbakkaloglu, 2017). Thus, enormous research has been done to increase the strength and deformability of concrete either by adding the internal fibers in the concrete or by external lateral conﬁnement. The performed studies confirm that the brittle behavior of conventional concrete can be improved by the internal steel fibers (Afroughsabet and Ozbakkaloglu, 2015; Afroughsabet et al., 2016; Bencardino et al., 2008; Madandoust et al., 2015; Xie and Ozbakkaloglu, 2015; Zohrevand and Mirmiran, 2012). Adding the internal steel fibers in the concrete transmits loads across cracks which leads to decreasing isolated major crack and controlling the crack propagation (Xie and Ozbakkaloglu, 2015). Thus, the ductility and strength of concrete can be improved. The curiosity to understand the mechanical behavior of steel fiber-reinforced concrete (SFRC) has led to a large number of experimental work over the years (Ayan et al., 2011; Düğenci et al., 2015; Gesoglu et al., 2016; Hannawi et al., 2016; Hassan et al., 2012; Kaïkea et al., 2014; Li et al., 2017; Oliver et al., 2012; Ou et al., 2011; Tokgoz and Dundar, 2012; Wu et al., 2016). Various types of steel fiber with different tensile strength, shape, and size have been used in SFRC. Existing research shows that at a given steel fiber content ( $V_{f}$ ), the peak axial stress ( $f_{c_{0}}'$ ), and corresponding axial strain ( $ε_{c_{0}}$ ) of concrete decrease with increasing the fiber aspect ratio ( $A_{R}$ ). It was also shown that the uniform fiber distribution in the concrete can significantly improve the mechanical properties of concrete (Li et al., 2017; Wu et al., 2016). Soroushian and Bayasi (1991) studied the impact of different types of steel fiber including straight-round, crimped-rectangular, crimped-round, hooked-collated fibers, and hooked-single in normal-strength concrete. They reported that hooked-end fiber has the best performance as compared to the others. These results are in consonance with the results reported by Katzer (2006). Bayasi and Gebman (2002) investigated the effects of steel fiber on seismic beam-column joints. They reported that the confining effect of steel fibers can reduce the demand for lateral reinforcement in seismic beam-column joints. Mechanical responses of steel-and-synthetic-fiber-reinforced pipes (FRCPs) were examined by Lee at al. (2019). They asserted that steel fibers absorb energy more significantly than synthetic fibers while used as reinforcement. Farhan et al. (2018) investigated the behavior of concrete column specimens reinforced by steel fibers under different loading conditions. Three types of steel fibers including straight micro steel fiber, deformed macro steel fiber, and hybrid steel fiber were used in reinforcing the concrete column specimens. The test results showed that the addition of different types of steel fibers resulted in significant improvements in the ductility of the specimens. Hung and Hsieh (2020) performed a comparative study on shear failure behavior of squat high-strength steel reinforced concrete. The results showed that the addition of steel fibers transformed the shear critical damage pattern from concrete spalling to localized diagonal cracks. An experimental investigation on steel fiber RC hollow columns under eccentric loading was performed by Azzawi and Abolmaail (2020). They asserted that the addition of steel fiber increased the strength capacity and ductility of columns.

It is well known that the ductility and compressive strength of concrete can be significantly increased by the lateral confinement (Deng and Qu, 2015; Dundar et al., 2015; Ilki et al., 2008; Kusumawardaningsih and Hadi, 2010; Mansouri et al., 2018; Ozbakkaloglu and Vincent, 2013; Rousakis et al., 2008; Saberi et al., 2020; Smith et al., 2010; Wu and Jiang, 2013). In general, it is reported that the ductility and compressive strength of confined concrete are increased with increasing FRP conﬁnement ratio defined as the ratio of the maximum confining pressure to the unconfined concrete strength. As the confined concrete is under compression load, the damage in concrete is distributed symmetrically and gradually increased before the FRP rupture as mentioned by Ceccato et al. (2017). With the sudden rupture of the FRP jacket, the strain and damage localization occur in the specimen.

Since high-strength concrete specimens confined by FRP exhibit highly brittle compressive behavior, Xie and Ozbakkaloglu (2015) experimentally studied the effects of steel fiber on increasing the ductility of these specimens. Regardless of FRP layers as a parameter, it is found that fiber volume fraction significantly affects the compressive behavior rather than the fiber shape and fiber aspect ratio. Gholampour and Ozbakkaloglu (2018) presented the ﬁnite element (FE) model for the compressive behavior of confined SFRC specimens. To build the FE model, results of previous studies together with additional axial compression tests conducted on actively conﬁned SFRC specimens were used. By considering the effect of the steel ﬁber parameters on the mechanical behavior of conﬁned SFRC specimens, they reported that application of the models of conﬁned plain concrete to predict the behavior of conﬁned SFRC specimens are not accurate. However, although considerable research has been experimentally and numerically conducted in the past to understand or to model the behavior of FRP-conﬁned concrete, little attention has been paid to improve the sudden failure in concrete confined by FRP. Moreover, there is not any analytical model to predict the compressive behavior of FRP-confined SFRC specimen and the application of models of conﬁned plain concrete are not accurate. Thus, this work experimentally investigates the co-effects of steel ﬁbers and FRP conﬁnement on axial compressive strength of concrete. Furthermore, a mathematical model is developed to model behavior of SFRC specimens confined by FRP. In this method, each stress-strain curve is approximated by Jacobi polynomials and the unknown coefficients in linear combination of the polynomials are related to the specimen features through the nonlinear equations optimized by least square method.

Experimental program and test setup

Specimen preparation and material properties tests

Totally, 18 concrete cylinders in which their diameter and height are $D = 100$ mm and $h = 200$ mm are managed and prepared as test setup of this research, respectively. This program contains two sets of 9 specimens. Each set consists of a control cylinder which is not confined by composite jackets or reinforced by steel fiber prepared from corresponding concrete batch to provide control data. Concrete mix designs for two sets are given in Table 1. The cylinders are prepared in a controlled condition at room temperature. Different mixing designs for both sets result in two different concrete strengths 26 MPa and 32.5 MPa at 28 days. High-strength sulfur was added to the ends of the concrete cylinders. This material prevents the composite jacket from directly bearing the axial compression force.

Table 1.

Concrete mix design for two sets.

Test set	${f'}_{c}$ (MPa)	W/C	Water (kg/m³)	Cement (kg/m³)	Fine aggregates (kg/m³)	Coarse aggregates (kg/m³)
Set 1	26	0.56	220	330	938	786
Set 2	32.5	0.46	225	417	891	746

In this work, the hook-end steel fibers are used according to their positive effects on crack bridging and arresting. If the fiber length is less than the size of the coarse aggregates, it can affect the bridging effect of fiber for avoiding crack growth. However, in this study the maximum size of the coarse aggregates is 9.5 mm where fiber length is 25 mm. The steel fiber properties are in Table 2. The volume fractions of steel fibers are selected as 0.0 %, 1.5%, and 3.0% with the aspect ratios of 50.

Table 2.

Fiber properties.

Fiber species	Aspect ratio	Fiber length (mm)	Fiber type	Density (g/cm³)	Tensile strength (MPa)
Steel fiber	50	25	Hook end	7.8	>600

Table 3 shows the material properties of the Glass fiber unidirectional sheet used to confine the specimen. Information on the properties of the composite is provided by the factory. Two parts of FRP epoxy adhesive are epoxy resin binder and thixotropic epoxy adhesive mixed in the ratio of 3:1. We used a thin layer of epoxy resin on the concrete surface before wrapping the Glass fiber sheet. Since the fibers placed in the hoop direction provide maximum confining stress to the core and resist lateral expansion in the concrete due to compressive stresses, we aligned all fibers along the hoop direction with a 100 mm overlap.

Table 3.

FRP properties.

FRP species	Thickness $t_{FRP}$ (mm)	Young’s modulus $E_{FRP}$ (MPa)	Tensile strength (MPa)	Ultimate strain
Glass	0.50	25,000	275	0.011

Six specimens of each set are reinforced by different volume fraction of steel fiber and six specimens confined by different thicknesses of FRP to investigate their effects on mechanical properties of concrete. The details of tested specimens are tabulated in Table 4.

Table 4.

Details of specimen.

Label	Confinement	$V_{f} %$	${f'}_{c}$	Label	Confinement	$V_{f} %$	${f'}_{c}$
UR-UC-26	–	0	26	UR-UC-32.5	–	0	32.5
R-1.5-UC-26	–	1.5%	26	R-1.5-UC-32.5	–	1.5%	32.5
R-3-UC-26	–	3%	26	R-3-UC-32.5	–	3%	32.5
UR-C-1-26	1 Layer	0	26	UR-C-1-32.5	1 Layer	0	32.5
R-1.5-C-1-26	1 Layer	1.5%	26	R-1.5-C-1-32.5	1 Layer	1.5%	32.5
R-3-C-1-26	1 Layer	3%	26	R-3-C-1-32.5	1 Layer	3%	32.5
UR-C-2-26	2 Layer	0	26	UR-C-2-32.5	2 Layer	0	32.5
R-1.5-C-2-26	2 Layer	1.5%	26	R-1.5-C-2-32.5	2 Layer	1.5%	32.5
R-3-C-2-26	2 Layer	3%	26	R-3-C-2-32.5	2 Layer	3%	32.5

Label starts with “R” or “UR” for reinforced and unreinforced specimens, respectively. For reinforced specimens, the hyphen is followed by percentage of steel ﬁber. Then, the hyphen is followed by “C” or “UC” for confined and unconfined specimens, respectively. For confined specimens, the hyphen is followed by the number of FRP layers. The last number is related to the compressive strength of plain concrete. As an example, UR-UC-26 denotes the unreinforced and unconfined specimen with concrete strength 26 MPa or R-3-C-2-32.5 denotes the specimen with plain concrete strength 32.5 MPa reinforced by 3% steel ﬁber content and confined by two layers of FRP.

Test setup and instrumentation

The uniaxial compressive experiments were conducted by a universal electro-hydraulic servo testing machine-Schenck with the load capacity of 1000 kN. The testing machine composed of a vertical Hydraulic jack that was instrumented with a control system to measure the acting forces and displacements. The control system uses internal displacement transducer and the linear position sensors that allow measuring the relative displacement in the hydraulic jack. From these measurements, the resultant axial forces and displacements are calculated and stored in the data acquisition system.

A displacement control loading with the speed 0.005 mm/s was considered. At the beginning of each loading, a pre-loading was applied, approximately 10% of the compressive strength to stabilize the system. In order to calibrate the testing machine measurement system and verify the measured data, we conducted an additional experimental test with four linear variable displacement transducers (LVDT) with a maximal range of 5 mm according to Figure 1.

Figure 1.

Test setup and instrumentation.

Figure 2 compares the mean value of the displacement data recorded by the four assembled LVDTs and the displacements recorded by the internal displacement transducer of the testing system. As it can be seen, the mean value of the four LVDTs is close to the transducer result. Thus, after calibration of internal transducer mounted in servo testing machine-Schenck, the recorded data of the machine was used during the experiments to measure the axial force-deformation.

Figure 2.

Comparison of mean value of four LVDT results and internal displacement transducer result.

Results and discussions

Failure modes

Figure 3 illustrates the failure modes of the tested specimens. Moreover, a detailed illustration of the damage in the concrete of each specimen is shown in Figure 4. Cracking and spalling occurred suddenly in plain specimens at the peak load. In contrast to the plain specimens, the SFRC specimens without confinement failed due to the occurrence of a shear crack throughout the height of the specimen. The steel ﬁbers around the shear crack were pulled out or ruptured. The integration of concrete remains and there is no concrete loss. Thus, the steel fibers can eﬀectively control the crack propagations and spalling of concrete. Consequently, the ductility and strength of concrete are increased.

Figure 3.

Failure patterns observed in specimens.

Figure 4.

Illustration of damage in concrete.

The diﬀerent groups of confined specimens have diﬀerent failure modes, although all specimens failed due to the hoop rupture of FRP. The plain specimens failed in rather explosive manners immediately after the FRP rupture and then, the specimens split into many concrete segments. Increasing FRP layers provide higher lateral pressure on specimens. Thus, as the rapture happens suddenly in high FRP confinement, the explosive behavior becomes more severe. In SFRC specimens, the explosion of specimens was eﬀectively restricted. It is noticeable that increasing the fiber content causes the FRP rupture to occur in larger axial displacement. This can be attributed to the effect of fibers on restraining the crack propagations that decreases the lateral strain at a given axial strain.

Stress-strain curves of specimens

Figure 5 shows the stress-strain curves of FRP-confined SFRC specimens. In specimens without confinement the curves don’t feature the second ascending part in contrast to the moderately and heavily confined plain specimens (Jiang and Teng, 2007). In highly and moderately confined FRP concrete under high compression, crack growth causes increase in lateral deformation, and as a result increase in lateral pressure in concrete. With further increase in loading, concrete crushes, and thereafter the load carrying capacity depends on pressure and lateral stiffness of FRP. With increasing axial loading, the rapture suddenly happens in FRP and crashed concrete cannot carried out any more loading and the stress strain curve fells down suddenly. The experiments show that contrary to two-layer specimens the plain specimens with one-layer FRP (weakly confined concrete) feature a descending branch. This result is in agreement with the results of Jiang and Teng (2007). In reinforced specimens, increasing the steel ﬁber content results in increasing the peak stress and strain as compared to the control specimens. After the peak stress, with increasing loading, the rate of losing the concrete strength is reduced with increasing the steel fiber content. For specimens with one FRP layer as shown in Figure 5(b) and (e), the stress-strain curves of specimens with both 1.5% and 3% fiber content are increased approximately until the strain of 0.007 and 0.005 for the concrete strength of 26 MPa and 32.5 MPa, respectively. Then, the curves of specimens with 1.5% fiber continue with a descending branch, whereas specimens reinforced by 3% fiber, feature the second ascending part with a very little slope until the strain 0.0014 and 0.0012 for the concrete strength 26 MPa and 32.5 MPa, respectively. Finally, the specimens with 3% fiber continue with a larger descending part as compared with specimens with 1.5% fiber. This can be attributed to the high pre-peak energy absorbed by specimens with 3% fiber in comparison to the specimens with 1.5% fiber. In Figure 5(b) and (e), the effect of fiber content on the peak strain of specimen with one-layer FRP can be observed. It is shown that increasing the fiber content from 0% to 1.5% has a negligible effect on peak strain for specimens with one FRP layer.

Figure 5.

Stress-strain curve of FRP-confined SFRC specimens.

In the case of specimens confined with two FRP layers, increasing fiber content generally has a little effect on the peak strain due to the fact that the higher pressure applied on specimen restricts the width of cracks in concrete before the rapture happens in FRP sheet. In general, increasing the steel ﬁber content in specimen results in diminishing the abrupt failure property of specimens and increasing the ultimate strain because of the property of ﬁbers in bridging the cracks.

Peak strain

Figure 6 depicts the effects of different FRP layers and different fiber contents on the peak strain values of stress-strain curves. The most increase in peak strain due to the addition of steel ﬁbers is related to the confined specimens. This point corresponds to the strain that the failure occurs in FRP. It is noticeable that for confined specimens, the peak axial strain in plain specimens is less than reinforced ones. This is because of the property of ﬁbers in bridging the cracks that result in decreasing the lateral displacement of the specimen at a given axial strain. Thus, the peak strain corresponding to FRP failure occurs in higher axial strain. Generally, the specimens with higher compressive strength show more brittle behavior and less peak strain. Moreover, the peak strain is increased with increasing the confinement and reinforcement.

Figure 6.

The effects of FRP layers and steel fiber content on the peak strain of stress-strain curve: (a) specimen with $f_{c}'$ = 26 (MPa) and (b) specimen with $f_{c}'$ = 32.5 (MPa).

Peak stress

Figure 7 depicts the peak stress of stress-strain curves. As is expected, the peak stress increases with increasing the compressive strength of the plain specimen. By adding 1.5% steel fiber to the plain specimens, the peak stress of specimens without FRP, and with one and two FRP layers is increased averagely 28%, 21%, and 18%, respectively. These values for the specimens with 3% steel fiber are 40%, 35%, and 28%, respectively. Thus, the effect of steel fiber on peak stress is decreased by increasing confinement. The peak stress of specimens confined by one FRP layer is increased averagely 19%, 13%, and 15% for those without fiber, with 1.5% and 3% fiber in comparison to the unconfined specimens, respectively. These values for the two FRP layers are 36%, 26%, and 24%, respectively. Thus, FRP confinement increases the peak stress of plain specimens more than SFRC specimens and with increasing the steel fibers content this effect is reduced.

Figure 7.

The effects of FRP layer and steel fiber content on the peak stress of stress-strain curve: (a) specimen with $f_{c}'$ = 26 (MPa) and (b) specimen with $f_{c}'$ = 32.5 (MPa).

Pre-peak energy

The pre-peak energy of each specimen is defined as the area under the stress-strain curve before its peak point. Pre-peak energy includes the elastic energy and the energy absorbed from the yield to the peak stress point. This energy shows the capacity of specimen to absorb energy without losing the strength. Variations of pre-peak energy with the steel fiber content and FRP layers are shown in Figure 8. As can be seen, adding steel fiber and increasing the FRP layers increase the pre-peak energy of specimens, because reinforcing and confining the specimen increase the strain and stress of peak point as discussed previously. A visible increase in pre-peak energy is observed in specimens confined by two FRP layers in comparison to the unconfined concrete. Moreover, reinforcing the specimens without confinement has a relatively negligible effect on pre-peak energy. In terms of quantity, the pre-peak energy of specimens with one FRP layer is increased averagely 108%, 67%, and 166% for the specimens without fiber, with 1.5% and 3% fiber as compared to the unconfined specimens, respectively. These values for the specimens with two FRP layers are 458%, 270%, and 197%, respectively. Thus, high pre-peak energy values can be achieved by high FRP confinement or by the combination of low FRP confinement and high steel fiber content.

Figure 8.

The effects of FRP layer and steel fiber content on the pre-peak energy: (a) specimens with $f_{c}'$ = 26 (MPa) and (b) specimens with $f_{c}'$ = 32.5 (MPa).

Post-peak energy

The post-peak energy of each specimen is defined as the area under the stress-strain curve between its peak point and the ultimate point. It can be observed in Figure 9, the plain specimens show very low post-peak energy values. This is due to the post-peak behavior of unreinforced concrete with or without confinement. However, the low fiber content in unconfined specimens cannot significantly increase the post-peak energy. As it was observed in stress-strain curves, the specimens with low fiber content and low confinement absorb the lower pre-peak energy than the specimens with high reinforcement-content and high FRP confinement. Thus, the rate of stress reduction is decreased after the peak point for low confined and SFRC specimens. Consequently, the post-peak energy is increased for these specimens in comparison to others. For confined specimens, adding 1.5% and 3% steel fiber averagely increases the post-peak energy 295% and 370% as compared to the unconfined specimens, respectively.

Figure 9.

The effects of FRP layer and steel fiber content on the post-peak energy of specimens: (a) specimens with $f_{c}'$ = 26 (MPa) and (b) specimens with $f_{c}'$ = 32.5 (MPa).

Energy dissipation capacity

Figure 10 shows a comparison between the energy dissipation capacity of specimens. This parameter is defined as the total area under the stress-strain curve of specimens before the ultimate failure. As can be seen, by increasing the steel fiber content and FRP layers, the energy dissipation capacity of specimens is increased. This is due to the fact that the steel fibers in the concrete transmit loads across the cracks which lead to decreasing major crack and controlling the propagation of the crack in specimen. Also, the FRP layers restrict transverse deformations of specimens and increase their bearing load capacity. In comparison to the unconfined specimens, the energy dissipation capacity of specimens without fiber, with 1.5% and 3% fiber increases 150%, 188%, and 79% by adding two FRP layers, respectively.

Figure 10.

The effects of FRP layer and steel fiber content on energy dissipation capacity of specimens: (a) specimens with $f_{c}'$ = 26 (MPa) and (b) specimens with $f_{c}'$ = 32.5 (MPa).

Mathematical model

In design-oriented models, the expressions of the axial stress–strain curve are predefined, and the parameters are directly calibrated with test data. In analysis oriented models, an actively confined concrete model is applied to predict the axial stress at a given axial strain and confining pressure. The confining pressure is calculated explicitly with the properties of the confining materials and the lateral-to-axial strain relationship of the concrete (Yang and Feng, 2020). In the proposed model, Jacobi polynomials considered as basis functions. Choosing the Jacobi polynomial is closely related to its advantages used extensively in mathematical analysis and practical applications. As mentioned by Bahşı et al. (2016), Jacobi polynomials comprise all of the Legendre, Chebyshev, and Gegenbauer polynomials and, therefore, is the comprehensive polynomial solution technique. The coefficients in linear combination of basis functions are unknown and govern the output curves. To calibrate the model parameters and find an optimal relation between input features including $f_{co}^{'}$ , $V_{f}$ , and $E_{FRP} \times t_{FRP} / t_{FRP} D D$ and the coefficients, we define an optimization problem. The objective function is the difference between stress-strain curves of the model and experimental test. Finally, the problem is solved by least square method. The details of proposed model is as following.

Let $J_{n}^{(α, β)} (z), n = 0, 1, \dots$ denotes a group of basis functions applied in this study for the approximation of stress-strain curves. The orthogonality of the Jacobi polynomial is expressed as (Zhang et al., 2019a)

\int_{- 1}^{1} p (z) J_{m}^{(α, β)} (z) J_{n}^{(α, β)} (z) d z = \frac{2^{α + β + 1}}{2 n + α + β + 1} \frac{Γ (n + α + 1) Γ (n + β + 1)}{Γ (n + α + β + 1) n!} δ_{m n} (α, β > - 1)

(1)

where $p (z) = {(1 - z)}^{α} {(1 + z)}^{β}$ $\forall z \in [- 1, 1]$ is a function of the parameters $α$ and $β$ . The Jacobi polynomial can be defined by following three recursive relation

\begin{array}{l} J_{0}^{(a, ß)} (z) = 1 \\ J_{1}^{(a, ß)} (z) = \frac{1}{2} (a - ß) + \frac{1}{2} (a + ß + 2) z \\ 2 n (n + a + ß) (2 n + a + ß - 2) J_{n}^{(a, ß)} (z) = (2 n + a + ß - 1) \times \\ [(2 n + a + ß) (2 n + a + ß - 2) z + a^{2} - ß^{2}] J_{n - 1}^{(a, ß)} (z) - 2 (n + a - 1) (n + ß - 1) (2 n + a + ß) J_{n - 2}^{(a, ß)} (z) \end{array}

(2)

The Jacobi polynomial boundary values at $z = - 1$ and $z = 1$ are

{\begin{matrix} J_{n}^{(α, β)} (z = 1) = (\begin{matrix} n + α \\ n \end{matrix}) \\ J_{n}^{(α, β)} (z = - 1) = {(- 1)}^{n} (\begin{matrix} n + β \\ n \end{matrix}) \end{matrix}

(3)

To satisfy the condition of the stress-strain curves at $z = 0$ , Jacobi polynomials $J_{m}^{(a, b)}$ are multiplied by $z$ .

Φ_{mj} = z_{j} P_{m}^{(a, b)} (z_{j})

(4)

In the second step, we propose an optimized strategy to find a nonlinear relation between the specimen features and the coefficients for the linear combination of Jacobi polynomials. Consider $n$ input curve that digitized to $t_{λ}$ data points ${({\vec{z}}^{1}_{1 \times t_{1}}, {\vec{y}}^{1}_{1 \times t_{1}}), . . . ., ({\vec{z}}_{1 \times t_{n}}^{n}, {\vec{y}}_{1 \times t_{n}}^{n})}$ . Each curve is correspondingly a function of $r$ input features as $X = {{\vec{x}}_{1 \times r}^{1}, . . ., {\vec{x}}_{1 \times r}^{n}}$ . We define $m^{th}$ Jacobi polynomial coefficient in $j^{th}$ curve $F_{mj}$ as following

F_{mj} = \sum_{i = 1}^{c} τ_{ij} \sum_{q = 0}^{r} p_{miq} x_{qj}

(5)

where the membership degree of $i^{th}$ input feature data in $j^{th}$ cluster $τ_{ij}$ is expressed as

τ_{ij} = \frac{{‖ \vec{x}_{i} - {\vec{μ}}_{j} ‖}_{2}}{\sum_{k = 1}^{c} {‖ \vec{x}_{i} - {\vec{μ}}_{j} ‖}_{2}} .

(6)

in which $μ = {{\vec{μ}}_{1}, \dots, {\vec{μ}}_{c}}$ are the cluster centers calculated by an iterative procedure as follows

{\vec{μ}}_{i} = \frac{\sum_{j = 1}^{N} u_{ij}^{h} {\vec{x}}_{j}}{\sum_{j = 1}^{N} u_{ij}^{h}},

(7)

u_{ij} = {[\sum_{k = 1}^{c} {(\frac{‖ {\vec{x}}_{j} - {\vec{μ}}_{i} ‖_{A}^{2}}{‖ {\vec{x}}_{j} - {\vec{μ}}_{k} ‖_{A}^{2}})}^{\frac{1}{h - 1}}]}^{- 1} .

(8)

An optimization problem is defined to calculate the unknown coefficients $p_{miq}$ . The considered objective function is the difference between the forecast data and the target data as follows:

Obj = \sum_{j = 1}^{N} {(Outpu t_{j} - y_{j})}^{2} = \sum_{j = 1}^{N} {(\sum_{m = 0}^{w} F_{mj} Φ_{mj} - y_{j})}^{2}

(9)

where $w$ is the number of Jacobi polynomials.

By setting the derivative of equation (9) equals to zero, we obtain $N$ equations to calculate the unknown coefficients $p_{miq}$ as follows:

\sum_{m = 0}^{w} Φ_{mj} \sum_{i = 1}^{c} τ_{ij} \sum_{q = 0}^{r} p_{miq} x_{qj} - y_{j} = 0 \forall j \in [1, \sum_{λ = 1}^{n} t_{λ}],

(10)

The matrix form of equation (10) is as following.

\begin{matrix} [\begin{matrix} τ_{11} Φ_{01} & . . . & τ_{11} Φ_{k 1} & . . . & τ_{c 1} Φ_{01} & . . . & τ_{c 1} Φ_{k 1} & τ_{11} Φ_{01} x_{11} & . . . & τ_{c 1} Φ_{w 1} x_{11} & . . . & τ_{11} Φ_{01} x_{r 1} & . . . & τ_{c 1} Φ_{w 1} x_{r 1} \\ τ_{12} Φ_{12} & . . . & τ_{12} Φ_{k 2} & . . . & τ_{c 2} Φ_{12} & . . . & τ_{c 2} Φ_{k 2} & τ_{12} Φ_{12} x_{12} & . . . & τ_{c 2} Φ_{w 2} x_{12} & . . . & τ_{12} Φ_{12} x_{r 1} & . . . & τ_{c 2} Φ_{w 2} x_{r 2} \\ ⋮ & . . . & ⋮ & . . . & ⋮ & . . . & ⋮ & ⋮ & . . . & ⋮ & . . . & ⋮ & . . . & ⋮ \\ τ_{1 j} Φ_{1 j} & . . . & τ_{1 j} Φ_{kj} & . . . & τ_{cj} Φ_{1 j} & . . . & τ_{cj} Φ_{kj} & τ_{1 j} Φ_{1 j} x_{1 j} & . . . & τ_{cj} Φ_{wj} x_{1 j} & . . . & τ_{1 j} Φ_{1 j} x_{r 1} & . . . & τ_{cj} Φ_{wj} x_{rj} \\ ⋮ & . . . & ⋮ & . . . & ⋮ & . . . & ⋮ & ⋮ & . . . & ⋮ & . . . & ⋮ & . . . & ⋮ \\ τ_{1 N} Φ_{1 N} & . . . & τ_{1 N} Φ_{kN} & . . . & τ_{cN} Φ_{1 N} & . . . & τ_{cN} Φ_{kN} & τ_{1 N} Φ_{1 N} x_{1 N} & . . . & τ_{cN} Φ_{wN} x_{1 N} & . . . & τ_{1 N} Φ_{1 N} x_{rN} & . . . & τ_{cN} Φ_{wN} x_{rN} \end{matrix}] \end{matrix}

(11)

[\begin{matrix} p_{010} \\ ⋮ \\ p_{k 10} \\ ⋮ \\ p_{0 c 0} \\ ⋮ \\ p_{k c 0} \\ p_{011} \\ ⋮ \\ p_{w c 1} \\ ⋮ \\ p_{10 r} \\ ⋮ \\ p_{w c r} \end{matrix}] = [\begin{matrix} y_{1} \\ y_{2} \\ ⋮ \\ y_{j} \\ ⋮ \\ y_{N} \end{matrix}] .

Equation (11) is solved by the least square method to calculate the unknown coefficients. Figure 11 shows the flowchart of the proposed model.

Figure 11.

Flowchart of proposed model.

Since the parameters $a$ and $b$ make the shape of basis functions used for approximation of the stress-strain curves, the optimization of $a$ and $b$ by least square method is not possible and these two parameters must be specified before least square optimization process. However, we use try and error process for finding these two parameters.

To calibrate the model parameters for prediction of stress strain curves of FRP-confined SFRC specimens under monotonic axial compression loading, totally 42 experimental test results including 18 experimental tests conducted in this study and 24 experimental tests gathered from literatures are used. Table 5 shows the FRP-conﬁned SFRC database.

Table 5.

FRP-conﬁned SFRC database.

Source	Label	$D$ (mm)	$t_{f}$ (mm)	$E_{f}$ (MPa)	Fiber content %	$f_{co}^{'}$ (MPa)
Present Study	S1	100	0	25,000	0	26
	S2		0.5		0
	S3		1		0
	S4		0		1.5
	S5		0.5		1.5
	S6		1		1.5
	S7		0		3
	S8		0.5		3
	S9		1		3
	S10		0		0	32.5
	S11		0.5		0
	S12		1		0
	S13		0		1.5
	S14		0.5		1.5
	S15		1		1.5
	S16		0		3
	S17		0.5		3
	S18		1		3
Jang and Yun (2018)	S19	100	0	–	1	60
	S20				1.5
	S21				2
Xie and Ozbakkaloglu (2015)	S22	152	1.2	118,200	1.2	125
Xie and Ozbakkaloglu (2015)	S23
Jiang and Teng (2007)	S24	152	0.165	250,500	0	35.9
	S25		0.33
	S26		0.495			34.3
	S27		1.27	21,800		38.5
	S28		2.54
	S29		0.165	250,000		41.1
	S30		0.33	247,000		38.9
	S31		0.17	80,100		39.6
	S32		0.51	80,100		39.6
	S33		0.17	80,100		45.9
	S34		0.34
	S35		0.51
	S36		0.68	240,700		38
	S37		1.02
	S38		1.36
	S39		0.11	260,000		44.2
	S40		0.22
	S41		0.11			37.7
	S42		0.33	250,500		47.6

In the proposed model, four first sentences of Jacobi polynomials with five clusters are considered. Figure 12 shows the basis functions defined in equation (4).

Figure 12.

Basis functions.

In order to calculate the membership degree $τ_{ij}$ , the cluster centers are evaluated by using equations (6)–(8). The cluster centers are as following.

[\begin{matrix} C 1 \\ C 2 \\ C 3 \\ C 4 \\ C 5 \end{matrix}] = [\begin{matrix} 0.075 & 0.306 & 0.020 \\ 0.750 & 0.304 & 0.000 \\ 0.246 & 0.319 & 0.000 \\ 0.050 & 0.269 & 0.726 \\ 0.433 & 1.000 & 0.667 \end{matrix}]

Notice that the membership degree $τ_{ij}$ and coefficients in linear combination of basis function $F_{mj}$ are defined as function of specimen features. The input features are concrete compressive strength, steel fiber content, and jacket modulus ( $E_{FRP} \times t_{FRP} / t_{FRP} D D$ ). Moreover, the parameters $a = 1, b = 0.5$ are set. Table 6 shows the details of specimen features, the calculated membership degree of each specimen, and finally coefficients of basis functions for 42 experimental test.

Table 6.

Details of specimen features, membership degree, and coefficients of basis functions.

SN	Specimen features			Membership degree $τ_{ij}$					Coefficients of basis functions $F_{mj}$
S1	0.0	26.0	0.0	0.524	0.086	0.242	0.089	0.058	1.04E+08	−3.47E+08	9.74E+07	−2.21E+08
S2	125.0	26.0	0.0	0.544	0.079	0.252	0.076	0.050	6.71E+07	−2.22E+08	6.32E+07	−1.42E+08
S3	250.0	26.0	0.0	0.489	0.082	0.308	0.072	0.049	3.93E+07	−1.28E+08	3.72E+07	−8.14E+07
S4	0.0	26.0	1.5	0.199	0.109	0.173	0.412	0.107	2.28E+07	−7.08E+07	2.20E+07	−4.46E+07
S5	125.0	26.0	1.5	0.196	0.112	0.176	0.409	0.108	1.14E+07	−3.51E+07	1.11E+07	−2.21E+07
S6	250.0	26.0	1.5	0.195	0.118	0.182	0.394	0.110	−1.42E+06	6.20E+06	−1.14E+06	4.15E+06
S7	0.0	26.0	3.0	0.138	0.109	0.132	0.479	0.142	6.02E+06	−1.62E+07	6.18E+06	−9.90E+06
S8	125.0	26.0	3.0	0.136	0.110	0.131	0.479	0.143	3.16E+06	−8.47E+06	3.24E+06	−5.14E+06
S9	250.0	26.0	3.0	0.137	0.114	0.133	0.469	0.147	−2.25E+06	8.93E+06	−1.92E+06	5.88E+06
S10	0.0	32.5	0.0	0.594	0.071	0.211	0.074	0.049	1.11E+08	−3.69E+08	1.03E+08	−2.36E+08
S11	125.0	32.5	0.0	0.683	0.052	0.182	0.049	0.034	6.44E+07	−2.13E+08	6.06E+07	−1.36E+08
S12	250.0	32.5	0.0	0.587	0.060	0.265	0.052	0.036	3.27E+07	−1.06E+08	3.11E+07	−6.71E+07
S13	0.0	32.5	1.5	0.197	0.107	0.172	0.415	0.110	2.87E+07	−8.98E+07	2.77E+07	−5.66E+07
S14	125.0	32.5	1.5	0.194	0.109	0.174	0.413	0.110	1.75E+07	−5.47E+07	1.69E+07	−3.45E+07
S15	250.0	32.5	1.5	0.193	0.116	0.180	0.397	0.114	4.43E+06	−1.25E+07	4.45E+06	−7.72E+06
S16	0.0	32.5	3.0	0.136	0.107	0.130	0.481	0.146	1.16E+07	−3.39E+07	1.15E+07	−2.11E+07
S17	125.0	32.5	3.0	0.134	0.108	0.129	0.481	0.147	9.05E+06	−2.73E+07	8.86E+06	−1.71E+07
S18	250.0	32.5	3.0	0.135	0.112	0.131	0.470	0.152	3.72E+06	−1.02E+07	3.78E+06	−6.19E+06
S19	0.0	60.0	1.0	0.281	0.123	0.231	0.229	0.136	4.09E+07	−1.19E+08	4.07E+07	−7.41E+07
S20	0.0	60.0	1.5	0.206	0.116	0.184	0.341	0.153	3.68E+07	−1.08E+08	3.66E+07	−6.74E+07
S21	0.0	60.0	2.0	0.152	0.101	0.141	0.456	0.151	5.24E+07	−1.67E+08	5.02E+07	−1.05E+08
S22	933.2	125.0	2.5	0.090	0.089	0.093	0.122	0.607	5.12E+07	−1.65E+08	4.88E+07	−1.04E+08
S23	933.2	125.0	1.5	0.104	0.104	0.110	0.111	0.571	8.24E+07	−2.69E+08	7.79E+07	−1.71E+08
S24	271.9	35.9	0.0	0.590	0.055	0.275	0.047	0.033	3.05E+07	−9.89E+07	2.90E+07	−6.27E+07
S25	543.8	35.9	0.0	0.137	0.050	0.756	0.033	0.025	3.78E+07	−1.25E+08	3.56E+07	−7.93E+07
S26	815.8	34.3	0.0	0.213	0.175	0.465	0.082	0.066	2.16E+07	−7.02E+07	2.06E+07	−4.46E+07
S27	182.1	38.5	0.0	0.820	0.027	0.111	0.025	0.018	4.47E+07	−1.48E+08	4.21E+07	−9.40E+07
S28	364.3	38.5	0.0	0.380	0.063	0.470	0.050	0.037	3.45E+07	−1.13E+08	3.26E+07	−7.20E+07
S29	271.4	41.1	0.0	0.579	0.055	0.285	0.047	0.034	4.13E+07	−1.36E+08	3.89E+07	−8.66E+07
S30	536.3	38.9	0.0	0.045	0.016	0.921	0.010	0.008	4.72E+07	−1.57E+08	4.42E+07	−1.00E+08
S31	89.6	39.6	0.0	0.742	0.042	0.146	0.041	0.029	8.86E+07	−2.97E+08	8.28E+07	−1.89E+08
S32	268.8	39.6	0.0	0.600	0.052	0.270	0.045	0.033	3.69E+07	−1.21E+08	3.49E+07	−7.70E+07
S33	89.6	45.9	0.0	0.617	0.063	0.213	0.061	0.045	1.06E+08	−3.57E+08	9.92E+07	−2.28E+08
S34	179.2	45.9	0.0	0.612	0.059	0.234	0.054	0.040	7.79E+07	−2.61E+08	7.28E+07	−1.66E+08
S35	268.8	45.9	0.0	0.513	0.067	0.320	0.057	0.043	5.81E+07	−1.93E+08	5.45E+07	−1.23E+08
S36	1076.8	38.0	0.0	0.188	0.320	0.315	0.094	0.083	1.84E+07	−5.91E+07	1.75E+07	−3.75E+07
S37	1615.2	38.0	0.0	0.000	1.000	0.000	0.000	0.000	9.55E+06	−2.99E+07	9.21E+06	−1.89E+07
S38	2153.6	38.0	0.0	0.133	0.492	0.163	0.103	0.110	6.26E+06	−1.91E+07	6.10E+06	−1.20E+07
S39	188.2	44.2	0.0	0.652	0.052	0.213	0.047	0.035	6.87E+07	−2.30E+08	6.43E+07	−1.47E+08
S40	376.3	44.2	0.0	0.347	0.068	0.492	0.053	0.040	4.47E+07	−1.48E+08	4.20E+07	−9.41E+07
S41	188.2	37.7	0.0	0.808	0.029	0.119	0.026	0.019	4.13E+07	−1.36E+08	3.90E+07	−8.64E+07
S42	543.8	47.6	0.0	0.201	0.077	0.629	0.051	0.042	4.48E+07	−1.48E+08	4.22E+07	−9.39E+07

In Figure 13, the experimental stress-strain curves of 42 specimens are compared with those of the proposed model. The experimental and numerical results are specified by the solid line and dashed line, respectively. As can be seen, the proposed model successfully predicts the experimental test results.

Figure 13.

Comparison of the results of experimental tests and proposed model.

Conclusion

This paper concerns investigating the behavior of steel fiber reinforced concrete confined by FRP layers under axial compression load. In this aim, we experimentally studied the effects of steel fiber content varied as 0.0%, 1.5%, and 3%, FRP layers varied as 0, 1, 2, and concrete strength varied as 26 and 32.5 MPa. Moreover, we proposed a mathematical model for SFRC confined by FRP. The performed analysis on 18 stress-strain curves made the following conclusions:

Confinement is the most effective parameter on peak strain so that two FRP layers can increase the peak strain averagely 213% for plain specimens. Reinforcement has no significant effect on peak stain of specimens confined by two FRP layers. However, 3% reinforcement caused increasing the peak strain of specimens confined by one FRP layer averagely 116% in comparison to the plain specimens.

One and two FRP layers increased averagely the peak stresses of plain specimens 19% and 36%. By adding 1.5% and %3 fiber reinforcement, the peak stresses of specimens with one FRP layer were increased averagely 21% and 35% as compared to the plain specimens, respectively. These values for the two FRP layers were 18% and 28 %, respectively.

Pre-peak energy is a function of peak stress and peak strain. Thus, the effect of FRP-confinement is prominent in the pre-peak energy of specimens. However, the experimental results showed that specimens with 3% fiber and one FRP layer can provide the same pre-peak energy as specimens with two FRP layers.

By adding 1.5% steel fiber and confining with one FRP layer, the peak strain was not increased significantly. Thus, the stress reduction rate after peak point was decreased because of absorbing lower pre-peak energy as compared with specimens with two FRP layers and 3% steel fiber. Consequently, the post-peak energy was maximum in these specimens and increased significantly in comparison to the control specimens.

The Energy dissipation capacity of specimens can be considerably increased by the combination of steel fiber reinforcement and FRP confinement. This energy value was averagely increased 352% for the specimen with 1.5% steel fiber and one FRP layer in comparison to the plain specimen.

To model the behavior of FRP-confined SFRC specimens, a novel model was proposed. The proposed model was based on the linear combination of Jacobi polynomials so that the polynomial coefficients were optimally related to the sample properties. The simulation results performed on 42 stress-strain curves showed that the proposed model can be used successfully to predict the behavior of FRP-confined SFRC specimens under axial compression load.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hossein Saberi

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