Experimental study on local bond-slip constitutive model between steel and steel fiber reinforced concrete structure of circular cross-section

Abstract

The interface slip between shape steel and concrete is one of the biggest problems in numerical simulation of composite structures, this paper aims to investigate interfacial bond-slip behavior between steel and steel fiber reinforced concrete. For this purpose, a series of push-out tests on 15 circular section specimens which were designed and fabricated with different steel fiber ratio, interface embedded length and concrete cover thickness have been done. Experimental results show that average bond strength versus free end slip curves of each specimen is similar, and then a basic average bond strength-free end slip constitutive model was proposed in this paper. Meanwhile, considering the distribution regularities of steel strains, local bond strength and relative slip along the interface embedded length at different load levels were obtained through the mechanical derivation, the results show that the maximum local bond strength was located near the free end, and the local bond strength away from free end tended to be steady. Based on the basic average bond strength-free end slip constitutive model, two position functions were proposed to describe the local bond-slip constitutive model considering the variation of positions. Two position functions are of great significance for further research on SSFRC non-linear properties by FEM.

Keywords

Steel and steel fiber reinforced concrete push-out test bond strength position functions local bond-slip constitutive model

Introduction

Steel reinforced concrete (SRC), which provides the advantages of high strength, high stiffness and pronounced seismic resistance (Tong et al., 2016, 2017, 2018), is in the family of composite structures. It has been widely applied in modern civil engineering (Elmy and Nakamura, 2017; Tsai et al., 2018), especially in high-rise buildings and areas of high earthquake requirement (Boyd et al., 1995; Hsu et al., 2004; Paultre et al., 2010; Shi et al., 2012; Wu et al., 2018). Choi et al. (2016) pointed out that more than half of high-rise buildings over 200 m were completed in 2014, which employed SRC composite structures. The paper of Xue et al. (2018) showed that SRC frame with irregular section columns is suitable for the construction in the high seismic intensity zone.

Choi et al. (2018) pointed out that the lower angles of inclination for the diagonal reinforcement, when accorded to the detailing requirements specified in ACI 318 (2014), result in reinforcement congestion and construction difficulties. The existence of both rebar cages and reinforcing steel causes the conflict between rebar cages and reinforcing steel in space (Ye et al., 2013). The space conflict leads to two construction problems: (1) inconvenience of forming rebar cages, which results in the discontinuity of rebars and stirrups and (2) difficulty of casting concrete, as shown in Figure 1(a), which leads to poor quality of concrete casting. Presently, the common methods used to solve these problems include opening at the flange or web for stirrups to close up (Figure 1(b)) (Yang et al., 2016) or using special connectors for fixing rebars to the shape steel (Figure 1(c)) (Nzabonimpa and Hong, 2018; Seo and Varma, 2017). However, such solutions not only increase construction difficulty and project investment, but also lower the integral bearing capacity of SRC composite structures.

Figure 1.

Construction difficulties of steel reinforced concrete. (a) Poor quality of concrete, (b) Openings at steel, and (c) Couplers.

Steel fiber reinforced concrete (SFRC) has been widely applied in buildings and bridges, which is made by mixing a certain ratio of steel fiber into plain concrete (Lim and Nawy, 2005; Yoo and Moon, 2018). Harajli (2006) studied the effect of confinement using steel fibers and indicated that confining the concrete with steel fiber reinforcement increased the bond strength and reduced the bond degradation. The experimental results of Chao et al. (2009) indicated that the bridging effect can effectively enhance post-cracking tensile capacity and limit crack width, which was provided by these irregular steel fibers in SFRC after cracking. Li et al. (2018) carried out an experimental investigation on the stress-strain behavior of SFRC subjected to uniaxial monotonic and cyclic tension, which showed that the determinant of concrete failure mode changes from tensile cracks to shear cracks with increasing fiber volume fraction. In order to solve the construction problems of SRC composite structures, achieving the goal of avoiding the inconvenient procedure of forming rebar cages and improving the pouring quality of concrete, the steel and steel fiber reinforced concrete (SSFRC) structures are adapted by using steel fibers instead of rebar cages. In the existing literature, many scholars have confirmed the feasibility of using steel fibers to replace shear reinforcement bars or even rebar cages in reinforced concrete structures (Boita et al., 2017; Dinh et al., 2010; Ding et al., 2011; Ghazizadeh and Cruz-Noguez, 2018; Hung and Hu, 2018; Valipour et al., 2016; Yoo et al., 2017)

The composite action of SRC is provided by the force transfer between steel and the concrete, which is obtained by relying on bonding properties (Tao and Yu, 2012). Roeder et al. (1999) evaluated bond stress in SRC, and came to the conclusion that bond stress is vital to the response of SRC structures, and shear connectors are required if the bond stress demand exceeds that capacity. Zheng et al. (2016) obtained local bond strength through the mechanical derivation. The paper of Wang et al. (2017) presented the results of bond strength in SRC columns after fire exposure, and the outcome showed that the reduction in bond strength was as high as 54.2% due to fire exposure. The experimental test carried out by Liu et al. (2018) established calculation formula of bond strength for steel reinforced recycled concrete. Further, the bridging effect provided by steel fibers enhances the post-crack tensile properties of concrete, and improves the toughness of concrete and the bonding properties (Tóth et al., 2019).

Similar to steel reinforced concrete, the basis to ensure the joint work of the two materials is the bonding properties between steel and SFRC (Hamad and Abou Haidar, 2016; Morishita and Tomii, 1982). The transfer of force across the interface between steel and SFRC plays a major role in controlling the response of SSFRC structures (Chen and Teng, 2001). The constitutive relation analysis is of great significance to numerical simulation of mechanical properties. The paper of Meng et al. (2016) pointed out that the constitutive model can reflect the bonding mechanism and provides references for theoretical analysis and engineering applications. Yuan et al. (2018) investigated the interfacial bond behavior between basalt fiber reinforced polymer sheet (BFRP) and SFRC. In addition, the analytical bond strength models and interfacial bond-slip models were proposed. Song et al. (2021) studied the interfacial bond-slip behavior between the corrugated steel plate and concrete, and proposed a bond-slip constitutive model of the corrugated steel and reinforced concrete. Constitutive model could provide a certain basis for establishing the relevant theoretical calculation methods. Yu et al. (2021) analyzed bond-slip behaviors self-stressing steel slag concrete filled steel tubular (SSSCFST) columns, a bond-slip constitutive model of SSSCFST columns was established, and the distribution law of the longitudinal compressive stress, inter-facial bond stress, and relative slip were obtained. The understanding of the bond properties between steel and SFRC is the basis for the capacity analysis of SSFRC structure.

Currently, design methods for reinforced concrete members are mainly based on the rectangular section. Mohammed et al. (2019) put forward a simplified new approach for designing RC specimens with circular section which converts circular section specimens to square section specimens with equivalent area, the test results showed that the RC specimens with circular section and its equivalent square specimens have similar nonlinear performance. The paper of Mousa et al. (2019) evaluated the applicability of crack-control models, which were based on rectangular section, to RC bridge members with a circular section, and it was considered that the formula can be applied to circular section after redefining parameters. Same as RC structure, there are few design methods of SRC based on circular section. Furthermore, buildings with traditional style become more and more popular, which mostly use circular section specimens. Therefore, 15 SSFRC specimens were fabricated in this paper to propose formulas based on circular section instead of rectangular section.

The experiment research conducted herein focused on evaluating the average bond strength and local bond strength along the interface at different positions of steel embedded in SFRC through the standard push-out tests. Based on the basic constitutive model of average bond strength versus free end slip, two position functions were proposed to establish the constitutive model of the local bond strength versus relative slip along the interface. Thus, it gives a complete description of the local bond strength at any given free end slip about different positions along the interface, which is of great significance for further research on SSFRC non-linear properties by FEM.

Experimental program

Materials

Steel and steel fibers

No.10 H-shape steel with Chinese standard GB/T 706-2008 (2008) and shear rippled steel fibers were selected in this test. The mechanical properties of steel flange were tested according to Chinese standard GB/T 228.1-2010 (2010). The yield strength and ultimate tensile strength of steel flange are 298.64 MPa and 409.07 MPa respectively, the elasticity modulus of steel flange is 2.074 × 105 MPa. The information of steel fibers, shown in Table 1, was acquired according to the test report provided by Jiangsu Dongda Engineering Testing Technology Co., Ltd.

Table 1.

Information of steel fibers.

Common name	Properties	Magnitude and units
Shear rippled steel fiber	Unit weight	7850 kg/m³
	Ultimate strength	670 MPa
	Elastic modulus	2.01 × 10⁵ MPa
	Poisson’s ratio	0.26
	Nominal diameter (D_f)	0.6 mm
	Average length (L_f)	30 mm
	Aspect ratio (L_f/D_f)	50

Concrete

The mix proportion of SFRC in this test was designed according to the CECS 38: 92 (1992). The mix proportion of SFRC is shown in Table 2. The materials used for concrete casting are as follows: Normal Portland cement, CEM I-42.5R, was used, with 28-day compressive strength of 42.5 MPa. Nature coarse aggregate with continuous gradation was used, and their maximum and minimum grain size was 5 mm and 15 mm, respectively. Fine aggregate was medium sand. Urban tap water was used in this test. Standard cubic compressive strength

(f_{c u, k})

of SFRC was designed to be 40 MPa. The compressive strength of different steel fiber ratios was obtained by three cubic blocks (150 mm × 150 mm × 150 mm), respectively. The average value of three cubic blocks was calculated, which is listed in Table 2.

Table 2.

Mix proportion of steel fiber reinforced concrete and compressive strength of concrete blocks.

Concrete type	ρ_sf, %	W/C	Mix proportion of concrete (kg/m³)				Steel fiber (kg/m³)	f_cu (MPa)
Concrete type	ρ_sf, %	W/C	Water	Cement	FA	CA	Steel fiber (kg/m³)	f_cu (MPa)
C-1	1	0.48	185.0	385.4	738.2	1125.9	78.5	44.7
C-2	2	0.48	185.0	385.4	757.0	1078.1	157.0	47.6
C-3	3	0.48	185.0	385.4	775.8	1030.2	235.5	58.1

*Note: FA-Fine aggregate; CA-Coarse aggregate.

Fabrication of specimens

A total of 15 circular section SSFRC specimens were fabricated. The final pouring result shows that without the main reinforcement and stirrups, the pouring of steel fiber reinforced concrete is not disturbed and the pouring quality can be guaranteed. As Table 3 shown, the steel fiber ratio (ρ_sf: 1%, 2%, 3%), interface embedded length (l_e: 250–960 mm) and concrete cover thickness (C_ss: 19.5 39.5, 64.5 mm) were taken into consideration as the design parameters. C_ss is defined as the minimum distance from any part of shape steel to the outer surface of SFRC, as shown in Figure 2.

Table 3.

Design parameters of specimens.

No	Specimens	ρ_sf, %	l_e (mm)	D (mm)	l _e /D	C_ss (mm)	L (mm)
1	C1-2-20	1	320	160	2	19.5	390
2	C1-4-20	1	640	160	4	19.5	710
3	C1-6-20	1	960	160	6	19.5	1030
4	C2-2-20	2	320	160	2	19.5	390
5	C2-4-20	2	640	160	4	19.5	710
6	C2-6-20	2	960	160	6	19.5	1030
7	C3-2-20	3	320	160	2	19.5	390
8	C3-4-20	3	640	160	4	19.5	710
9	C3-6-20	3	960	160	6	19.5	1030
10	C2-2-40	2	400	200	2	39.5	470
11	C2-3-40	2	600	200	3	39.5	670
12	C2-4-40	2	800	200	4	39.5	870
13	C2-1-65	2	250	250	1	64.5	320
14	C2-2-65	2	500	250	2	64.5	570
15	C2-3-65	2	750	250	3	64.5	820

*Note: L is the length of shape steel.

Figure 2.

Test setup and sketch of specimens. (a) Test setup and (b) Failure mode of specimens.

All the specimens were cast vertically. During the process of casting, the upper part and lower part of the shape steel extended the SFRC 50 and 20 mm, respectively. All specimens were demolded after 3 days, and then were moved to the curing room under natural conditions for 28 days.

Test setup and measurement scheme

A series of push-out tests of SSFRC were carried out by using the test setup as Figure 2 shown. The lower part of concrete supported on special loading plate, which bore the load applied by the TYA-2000 type electric hydraulic test machine, is called concrete loading end. During the test, the supporting load on the upper part was applied to the shape steel. The upper part of the concrete is guaranteed to stay in an unconstrained condition, which is called concrete free end. An H-typed hole, with slightly larger size compared with the inner shape steel, was dug in the middle of the special loading plate. Two displacement transducers were applied to measure the free end slip (S). The free end slip is defined as the mean value of the two displacement transducers. Due to the uneven distribution of steel strain along the interface embedded length, strain gauges were attached to the steel flange along the interface embedded length before casting. The layout of strain gauges is shown in Figure 3.

Figure 3.

Lay out of strain gauges.

Prior to the loading, the upper and lower surfaces of all specimens were pure smooth to provide a level surface and to ensure even distribution of the applied loads. In the process of loading, the load control was adopted until the ultimate load with the rate of 1 kN/s firstly, and then displacement control method was adopted with the rate of 0.01 mm/s (Wu et al., 2019). At the final stage of loading, end loading when load stayed constant or slip value reached 20 mm.

Experimental results and discussion

Average bond strength-free end slip curve

Figure 4 demonstrates the P-S curves for 15 specimens, from which we can see that the P-S curves are very similar. The entire P-S curves can be summarized into four stages, which are made of the elastic stage, yield stage, steep descending stage and gentle descending stage. During the test, the whole process of interface failure is very complex. Defining the yield point and yield load is helpful to judge the initial damage and initial relative slip of the interface. Geometric drawing method, equal energy method, and R. Park method were used to calculate the yield point on the load-slip curves at the free end. The yield load (P_y) at the free end is defined as the average value of two or three methods which have similar results.

Figure 4.

Load-free end slip curves. (a) No. 1–8 specimens and (b) No. 9–15 specimens.

The average bond strength at the interface between steel and SFRC from test results is defined by equation below:

\bar{τ} = \frac{P}{C_{a} l_{e}}

(1)

where P is the applied load, C_a is the perimeter of shape steel and l_e is the interface embedded length.

According to equation (1), the average bond strength $(\bar{τ})$ versus free end slip (S) curves can be obtained, which is shown in Figure 5

Figure 5.

Average bond strength-free end slip curves. (a) No. 1–8 specimens and (b) No. 9–15 specimens.

Average bond strength-free end slip $(\bar{τ} - S)$ constitutive relationship

Obviously, the $\bar{τ} - S$ curves have the similar shape to the P-S curves. Based on this, we proposed a typical $\bar{τ} - S$ constitutive relationship model, as shown in Figure 6.

Figure 6.

A typical $\bar{τ} - S$ constitutive model.

In the model, the elastic stage (OA section), yield stage (AB section), steep descending stage (BC section), and gentle descending stage (CD section) were defined. Point A, B, C and D are the characteristic points of the $\bar{τ} - S$ constitutive model. Point A is the limit point of elasticity increase of $\bar{τ} - S$ curve, corresponding characteristic average bond strength and characteristic free end slip are $\bar{τ} y$ and S_y, respectively. Point B is the peak point of $\bar{τ} - S$ curve, corresponding characteristic average bond strength and characteristic free end slip are $\bar{τ} u$ and S_u. The tangent slope of point C is the same with the slope of line BD. Point C is defined as the critical state in which the descending degree of average bond strength turned from steep to gentle, corresponding characteristic average bond strength and characteristic free end slip are ${\bar{τ}}_{r}$ and S_r. Point D is the limit point of the descent of average bond strength, average bond strength maintains constant after point D, corresponding characteristic average bond strength and characteristic free end slip are ${\bar{τ}}_{f}$ and S_f.

In the OA section, average bond strength increased elastically to ${\bar{τ}}_{y}$ , and the corresponding applied load is P_y. With the average bond strength increasing to ${\bar{τ}}_{y}$ , the $\bar{τ} - S$ curve starts to yield.

In the AB section, a quadratic relationship between average bond strength and free end slip was defined. In this stage, the hardening of average bond strength lasts until ultimate average bond strength $({\bar{τ}}_{u})$ , and the corresponding applied load is P_u.

In the BC section, a hyperbolic relationship between average bond strength and free end slip was defined. In this stage, average bond strength descends steeply until ${\bar{τ}}_{r}$ , and the corresponding applied load is P_r.

In the CD section, an exponential relationship between average bond strength and free end slip was defined. In this stage, descending degree of average bond strength turned from steep to gentle until

{\bar{τ}}_{f}

, and the corresponding applied load is P_f. Table 4 shows the values of characteristic points.

Table 4.

Values of characteristic points.

No	P_y (kN)	$\bar{τ} y$ (MPa)	S_y (mm)	P_u (kN)	$\bar{τ} u$ (MPa)	S_u (mm)	P_r (kN)	$\bar{τ} r$ (MPa)	S_r (mm)	P_f (kN)	$\bar{τ} f$ (MPa)	S_f (mm)
C1-2-20	134.9	0.916	1.020	138.6	0.942	1.080	51.9	0.353	3.030	30.0	0.204	20.0
C1-4-20	228.2	0.775	1.769	235.5	0.800	2.433	86.5	0.294	3.289	44.0	0.149	20.0
C1-6-20	296.2	0.671	2.058	322.0	0.729	2.440	112.0	0.254	6.234	73.3	0.166	20.0
C2-2-20	149.1	1.013	1.111	152.9	1.039	1.395	52.9	0.359	4.684	36.0	0.245	20.0
C2-4-20	208.7	0.709	0.903	219.0	0.744	1.085	65.6	0.223	2.961	0.0	0.000	20.0
C2-6-20	333.2	0.755	1.451	344.5	0.780	1.563	107.0	0.242	6.663	76.0	0.172	20.0
C3-2-20	173.6	1.179	1.441	178.6	1.213	1.617	96.5	0.656	5.385	79.0	0.537	20.0
C3-4-20	281.6	0.957	1.451	297.6	1.011	2.203	176.8	0.601	5.149	120.0	0.408	20.0
C3-6-20	319.0	0.722	2.434	372.0	0.842	4.959	191.0	0.433	9.531	158.0	0.358	20.0
C2-2-40	282.3	1.534	1.522	287.4	1.562	2.050	154.3	0.839	2.926	114.3	0.621	20.0
C2-3-40	301.3	1.092	1.403	334.0	1.210	1.761	181.3	0.657	6.049	135.5	0.491	20.0
C2-4-40	354.3	0.963	1.996	376.5	1.023	2.312	167.0	0.454	6.802	99.6	0.271	20.0
C2-1-65	216.8	1.885	0.912	220.0	1.913	1.056	97.6	0.849	4.270	58.6	0.510	20.0
C2-2-65	332.3	1.445	1.470	360.0	1.565	1.887	152.9	0.665	4.767	112.7	0.490	20.0
C2-3-65	358.5	1.039	1.729	422.0	1.223	2.747	194.3	0.563	5.491	144.0	0.417	20.0

The average bond strength versus free end slip curve for SSFRC can be expressed as the equations below:

\bar{τ} (S) = {\begin{matrix} a S, \\ b S^{2} + c S + d \\ S / (e S - f), \\ \begin{array}{l} g k^{S}, \\ {\bar{τ}}_{f}, \end{array} \end{matrix} \begin{matrix} (0 \leq S < S_{y}) \\ , (S_{y} \leq S < S_{u}) \\ (S_{u} \leq S < S_{r}) \\ \begin{array}{l} (S_{r} \leq S < S_{f}) \\ (S_{f} \leq S) \end{array} \end{matrix}

(2)

where

a = \frac{{\bar{τ}}_{y}}{S_{y}},

b = \frac{({\bar{τ}}_{y} - {\bar{τ}}_{u})}{{(S_{y} - S_{u})}^{2}},

c = \frac{- 2 S_{u} ({\bar{τ}}_{y} - {\bar{τ}}_{u})}{{(S_{y} - S_{u})}^{2}},

d = {\bar{τ}}_{u} + \frac{S_{u}^{2} ({\bar{τ}}_{y} - {\bar{τ}}_{u})}{{(S_{y} - S_{u})}^{2}} e = \frac{(S_{u} - {\bar{τ}}_{u} S_{r} / {\bar{τ}}_{r})}{({\bar{τ}}_{u} S_{u} - {\bar{τ}}_{u} S_{r})}

f = \frac{{\bar{τ}}_{r} S_{u} S_{r} - {\bar{τ}}_{u} S_{u} S_{r}}{{\bar{τ}}_{u} {\bar{τ}}_{r} S_{u} - {\bar{τ}}_{u} {\bar{τ}}_{r} S_{r}},

g = \frac{{\bar{τ}}_{r}}{{({\bar{τ}}_{f} / {\bar{τ}}_{r})}^{\frac{S_{r}}{(S_{f} - S_{r})}}},

k = {(\frac{{\bar{τ}}_{f}}{{\bar{τ}}_{r}})}^{\frac{S_{r}}{(S_{f} - S_{r})}}

Statistical regression of characteristic points

Considering the steel fiber ratio, concrete cover thickness, interface embedded length, the concrete strength, and the statistical regression equations for characteristic average bond strength ${\bar{τ}}_{y}, {\bar{τ}}_{u}, {\bar{τ}}_{r}, and {\bar{τ}}_{f}$ are proposed below:

{\bar{τ}}_{y} = (0.0219 + 0.0639 C_{ss} / D - 0.0030 l_{e} / D + 0.2066 ρ_{sf}) f_{cu, k}

(3)

{\bar{τ}}_{u} = (0.0186 + 0.0806 C_{ss} / D - 0.0026 l_{e} / D + 0.2483 ρ_{sf}) f_{cu, k}

(4)

{\bar{τ}}_{r} = (0.0044 + 0.0387 C_{ss} / D - 0.0014 l_{e} / D + 0.3286 ρ_{sf}) f_{cu, k}

(5)

{\bar{τ}}_{f} = (0.0011 + 0.0263 C_{ss} / D - 0.0009 l_{e} / D + 0.3262 ρ_{sf}) f_{cu, k}

(6)

The comparison of the test and fitting values is shown in Figure 7. It can be seen in Figure 7 that the fitting values are in good agreement with the test values. It needs to be mentioned that the specimen C2-4-20 was subjected to accidental eccentricity during the loading, which led to abnormal performance.

Figure 7.

Comparison of test values and fitting values of characteristic average bond strength. (a) Comparison of ${\bar{τ}}_{y}$ (b) Comparison of ${\bar{τ}}_{u}$ , (c) Comparison of ${\bar{τ}}_{r}$ , and (d) Comparison of ${\bar{τ}}_{f}$ .

The characteristic free end slip is mainly affected by interface embedded length. As shown in Figure 8, through fitting the relationship between characteristic free end slip and interface embedded length, equations for characteristic free end slip S_y, S_u, S_r, and S_f are proposed below:

S_{y} = - 2.1581 \times 10^{- 6} l_{e}^{2} + 0.0041 l_{e}

(7)

S_{u} = - 1.7653 \times 10^{- 6} l_{e}^{2} + 0.0047 l_{e}

(8)

S_{r} = - 5.0460 \times 10^{- 6} l_{e}^{2} + 0.0122 l_{e}

(9)

S_{f} = 20 mm

(10)

Figure 8.

Test values and fitting curves of characteristic free end slip. (a) Fitting curve of S_y, (b) Fitting curve of S_u, and (c) Fitting curve of S_r.

Note that according to the P-S curve and loading system, S_f is defined as 20 mm. The comparison of the test and fitting values is shown in Figure 9. It can be seen in Figure 9 that the fitting values are in good agreement with the test values.

Figure 9.

Comparison of test values and fitting values of characteristic free end slip. (a) Comparison of S_y, (b) Comparison of S_u, and (c) Comparison of S_r.

Comparison of test curves and fitting curves

As the coordinates of characteristic points are determined, then according to equations (2)–(10), the fitting curves of $\bar{τ} - S$ can be obtained. The comparison of test curves and fitting curves for 15 SSFRC specimens is presented in Figure 10. According to Figure 10, the fitting results of $\bar{τ} - S$ curves based on the proposed typical $\bar{τ} - S$ constitutive model are satisfactory, and the model can reproduce the slip process appropriately.

Figure 10.

Comparison of test curves and fitting curves of $\bar{τ} - S$ (a) C1-2-20, (b) C1-4-20, (c) C1-6-20, (d) C2-2-20, (e) C2-4-20, (f) C2-6-20, (g) C3-2-20, (h) C3-4-20, (j) C3-6-20, (k) C2-2-40, (l) C2-3-40, (m) C2-4-40, (n) C2-1-65, (p) C2-2-65, and (q) C2-3-65.

Local bond strength and relative slip

Distribution of steel strain along the interface embedded length

Figure 11 presents the steel strain distribution along interface embedded length at different load levels, which were obtained by strain gauges. Distance away from the free end is defined as x. According to Figure 11, it can be evidently seen that the steel strain distribution along the interface embedded length is close to the exponential relationship in the whole process of loading. Therefore, the steel strain distribution can be expressed by the exponential equation:

ε (x) = l e^{m x}

(11)

where ε(x) is the strain of shape steel at location x, x is the distance away from the free end, both l and m are shown in Figure 11, which can be obtained by the exponential fitting. Note that the steel strain distribution obtained by strain gauges of specimen C1-4-20 was eliminated from Figure 11 due to its abnormal performance.

Figure 11.

Steel strain distribution along interface embedded length at different load levels. (a) C1-2-20, (b) C1-6-20, (c) C2-2-20, (d) C2-4-20, (e) C2-6-20, (f) C3-2-20, (g) C3-4-20, (h) C3-6-20, (j) C2-2-40, (k) C2-3-40, (l) C2-4-40, (m) C2-1-65, (n) C2-2-65, and (p) C2-3-65.

Distribution of local bond strength along the interface embedded length

In this study, the distribution of local bond strength was obtained through mechanical derivation. The applied load P satisfies the equilibrium equation below:

P = A_{s} E_{s} ε (x) + \int_{0}^{x} τ (x) C_{a} d x

(12)

where A_s is the cross-section area of shape steel, E_s is the elasticity modulus of steel flange, C_a is the perimeter of shape steel, and τ(x) is the local bond strength at location x. Then, the local bond strength τ(x) can be expressed as follows:

τ (x) = - \frac{A_{s} E_{s}}{C_{a}} \frac{d ε (x)}{d x} = - \frac{A_{s} E_{s}}{C_{a}} m l e^{m x}

(13)

The distribution of local bond strength was calculated and shown in Figure 12. It is obvious that the local bond strength increases with the growing of applied load, and the maximum local bond strength is located near the free end.

Figure 12.

Local bond strength distribution along interface embedded length at different load levels. (a) C1-2-20, (b) C1-6-20, (c) C2-2-20, (d) C2-4-20, (e) C2-6-20, (f) C3-2-20, (g) C3-4-20, (h) C3-6-20, (j) C2-2-40, (k) C2-3-40, (l) C2-4-40, (m) C2-1-65, (n) C2-2-65, and (p) C2-3-65.

Distribution of relative slip along the interface embedded length

The relative slip ΔS between steel and SFRC can be expressed as follows:

Δ S (x) = S + (\frac{A_{s} E_{s}}{A_{c} E_{c}} - 1) \int_{0}^{x} ε (x) d x

(14)

where S is the free end slip, A_c is the cross-section area of SFRC, and E_c is the elasticity modulus of SFRC.

Substituting equation (11) into equation (14) gives:

Δ S (x) = S + (\frac{A_{s} E_{s}}{A_{c} E_{c}} - 1) (\frac{l e^{m x}}{m} - \frac{l}{m})

(15)

which indicates that the distribution of relative slip along interface embedded length is exponential and the maximum relative slip is located at the free end. Based on equation (15), Figure 13 demonstrates the distribution of relative slip along the interface embedded length at different load levels. In order to demonstrate the exponential relationship, the distribution of relative slip at ultimate load, including specimen C2-2-20, C2-4-20 and C2-6-20, are listed in Figure 14.

Figure 13.

Distribution of relative slip along interface embedded length at different load levels. (a) C1-2-20, (b) C1-6-20, (c) C2-2-20, (d) C2-4-20, (e) C2-6-20, (f) C3-2-20, (g) C3-4-20, (h) C3-6-20, (j) C2-2-40, (k) C2-3-40, (l) C2-4-40, (m) C2-1-65, (n) C2-2-65, and (p) C2-3-65.

Figure 14.

Distribution of relative slip along interface embedded length at ultimate load (a) C2-2-20, (b) C2-4-20, and (c) C2-6-20.

Local bond-slip relationship

Local bond-slip relationship of different positions τ= τ(ΔS, x)

The typical $\bar{τ} - S$ constitutive model can only express the variation of average bond strength according to the free end slip. However, it cannot reflect the variation of local bond strength according to the relative slip at different positions. In order to express the variation of local bond strength between steel and SFRC accurately, the values of local bond strength (τ) and relative slip (ΔS) at different positions should be calculated. On the basis of Figure 12 and Figure 13, which demonstrate the values of τ and ΔS under different load levels at the same position x, a local bond-slip relationship at position x before ultimate load can be obtained. As the position x varied, the different curves of local bond strength-relative slip at different positions for one specimen were able to be acquired and classified into a set of curves τ= τ(ΔS, x), which are shown in Figure 15.

Figure 15.

Relationship of local bond strength versus relative slip at different positions. (a) C1-2-20, (b) C1-6-20, (c) C2-2-20, (d) C2-4-20, (e) C2-6-20, (f) C3-2-20, (g) C3-4-20, (h) C3-6-20, (j) C2-2-40, (k) C2-3-40, (l) C2-4-40, (m) C2-1-65, (n) C2-2-65, and (p) C2-3-65.

Comparing the features of the set of curves τ= τ(ΔS, x) and $\bar{τ} - S$ curves in Figure 15, it is obvious that the τ= τ(ΔS, x) curves have the similar shape to the $\bar{τ} - S$ curves. Based on this, we proposed two position functions to describe their relationship, which can be expressed as follows:

G (x) = \frac{τ (x)}{\bar{τ}}

(16)

F (x) = \frac{Δ S (x)}{S}

(17)

The determination of G(x) and F(x)

The determination of G(x)

Through the statistical regression analysis about the average value of τ_u(x)/ $\bar{τ} u$ in the position x for 14 specimens (except specimen C1-4-20), position function G(x) can be acquired. Figure 16(a) shows the procedure of statistical regression analysis, which indicated that a cubic polynomial has the highest degree of fitting. The relevant fitting results are shown in Figure 16(b).

Figure 16.

Determination of G(x). (a) Statistical regression of G(x) and (b) Fitting result of G(x).

The determination of F(x)

By the same procedure as above, position function F(x) can be acquired through the statistical regression analysis about the average value of ΔS_u(x)/S_u in the position x for 14 specimens. Figure 17(a) shows the procedure of statistical regression analysis, which indicated that a cubic polynomial has the highest degree of fitting. The relevant fitting results are shown in Figure 17(b).

Figure 17.

Determination of F(x). (a) Statistical regression of G(x) and (b) Fitting result of G(x).

The local bond-slip constitutive model considering the variation of positions

On the basis of the typical $\bar{τ} - S$ constitutive model and two position functions G(x) and F(x), the local bond strength versus relative slip constitutive model along the interface embedded length can be obtained and is demonstrated in Figure 18. According to Figure 18, the local bond strength versus relative slip curve along the interface embedded length τ= τ(ΔS, x) for SSFRC can be expressed as the equations below:

τ (Δ S, x) = {\begin{matrix} a Δ S (x), \\ b Δ S {(x)}^{2} + c Δ S (x) + d \\ Δ S (x) / (e Δ S (x) - f), \\ \begin{array}{l} g k^{Δ S (x)}, \\ τ_{f} (x), \end{array} \end{matrix} \begin{matrix} (0 \leq Δ S (x) < Δ S_{y} (x)) \\ , (Δ S_{y} (x) \leq Δ S (x) < Δ S_{u} (x)) \\ (Δ S_{u} (x) \leq Δ S (x) < Δ S_{r} (x)) \\ \begin{array}{l} (Δ S_{r} (x) \leq Δ S (x) < Δ S_{f} (x)) \\ (Δ S_{f} (x) \leq Δ S (x)) \end{array} \end{matrix}

(18)

in which,

G (x) = - 1.3225 {(x / l_{e})}^{3} + 5.6629 {(x / l_{e})}^{2} - 7.6205 x / l_{e} + 3.7193

;

F (x) = - 0.0461 {(x / l_{e})}^{3} + 0.1205 {(x / l_{e})}^{2} - 0.1530 x / l_{e} + 0.9974;

τ_{y} (x) = G (x) {\bar{τ}}_{y}, τ_{u} (x) = G (x) {\bar{τ}}_{u}, τ_{r} (x) = G (x) {\bar{τ}}_{r}, τ_{f} (x) = G (x) {\bar{τ}}_{f};

Δ S_{y} (x) = F (x) S_{y}, Δ S_{u} (x) = F (x) S_{u}, Δ S_{r} (x) = F (x) S_{r}, Δ S_{f} (x) = F (x) S_{f};

a = \frac{τ_{y} (x)}{Δ S_{y} (x)}, b = \frac{(τ_{y} (x) - τ_{u} (x))}{{(Δ S_{y} (x) - Δ S_{u} (x))}^{2}}, c = \frac{- 2 Δ S_{u} (x) (τ_{y} (x) - τ_{u} (x))}{{(Δ S_{y} (x) - Δ S_{u} (x))}^{2}}, d = τ_{u} (x) + \frac{Δ S_{u} {(x)}^{2} (τ_{y} (x) - τ_{u} (x))}{{(Δ S_{y} (x) - Δ S_{u} (x))}^{2}}, e = \frac{(Δ S_{u} (x) - τ_{u} (x) Δ S_{r} (x) / τ_{r} (x))}{(τ_{u} (x) Δ S_{u} (x) - τ_{u} (x) Δ S_{r} (x))}, f = \frac{τ_{r} (x) Δ S_{u} (x) Δ S_{r} (x) - τ_{u} (x) Δ S_{u} (x) Δ S_{r} (x)}{τ_{u} (x) τ_{r} (x) Δ S_{u} (x) - τ_{u} (x) τ_{r} {(x)}_{r} Δ S_{r} (x)}, g = \frac{τ_{r} (x)}{{(τ_{f} (x) / τ_{r} (x))}^{Δ S_{r} (x) / (Δ S_{f} (x) - Δ S_{r} (x))}}, k = {(\frac{τ_{f} (x)}{τ_{r} (x)})}^{Δ S_{r} (x) / (Δ S_{f} (x) - Δ S_{r} (x))}

Figure 18.

The local bond-slip constitutive model.

The characteristic average bond strength ${\bar{τ}}_{y}, {\bar{τ}}_{u}, {\bar{τ}}_{r}, {\bar{τ}}_{f}$ and characteristic free end slip S_y, S_u, S_r, and S_f are able to acquire according to equations (3)–(10).

Conclusions

The interfacial local bond strength between steel and SFRC has been investigated in this paper. A total of 15 circular section SSFRC were tested under monotonic load. The following conclusions can be drawn from this paper:

(i) There are four characteristic points of the $\bar{τ} - S$ curve, and the regression equations of corresponding characteristic average bond strength ${\bar{τ}}_{y}, {\bar{τ}}_{u}, {\bar{τ}}_{r}, {\bar{τ}}_{f}$ and characteristic free end slip S_y, S_u, S_r, S_f are proposed. Based on the characteristic values, typical $\bar{τ} - S$ constitutive relationship model is proposed.

(ii) The fitting results of $\bar{τ} - S$ curves, which were based on the proposed typical $\bar{τ} - S$ constitutive relationship model, are satisfactory, and the model can reproduce the slip process appropriately.

(iii) It can be seen that the distribution regularities of steel strain along the interface embedded length is close to the exponential relationship in the whole process of loading. According to this and the mechanical derivation, the distributions of steel strain, local bond strength and relative slip along the interface embedded length at different positions were acquired.

(iv) The distribution of local bond strength is uneven along the interface embedded length, the maximum local bond strength is located near the free end, and the local bond strength away from free end tends to be steady.

(v) Comparing the features of the set of curves τ= τ(ΔS, x) and $\bar{τ} - S$ curves found that the τ= τ (ΔS, x) curves have the similar shape to the $\bar{τ} - S$ curves. Based on this, two position functions G(x), F(x) are proposed to describe their relationship. Then the constitutive model of τ= τ(ΔS, x) is obtained on the basis of basic $\bar{τ} - S$ constitutive model and two position functions, which is of great significance for further research on SSFRC non-linear properties by FEM.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (Grant No.51208175) and Fundamental Research Funds for the Central Universities (Grant No. B200202067 and Grant No. B210203021).

ORCID iD

Kai Wu

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Experimental study on local bond-slip constitutive model between steel and steel fiber reinforced concrete structure of circular cross-section

Abstract

Keywords

Introduction

Experimental program

Materials

Steel and steel fibers

Concrete

Fabrication of specimens

Test setup and measurement scheme

Experimental results and discussion

Average bond strength-free end slip curve

Average bond strength-free end slip ( τ ¯ − S ) constitutive relationship

Statistical regression of characteristic points

Comparison of test curves and fitting curves

Local bond strength and relative slip

Distribution of steel strain along the interface embedded length

Distribution of local bond strength along the interface embedded length

Distribution of relative slip along the interface embedded length

Local bond-slip relationship

Local bond-slip relationship of different positions τ= τ(ΔS, x)

The determination of G(x) and F(x)

The determination of G(x)

The determination of F(x)

The local bond-slip constitutive model considering the variation of positions

Conclusions

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iD

References

Average bond strength-free end slip $(\bar{τ} - S)$ constitutive relationship