Axial compression tests and numerical simulation of steel reinforced recycled concrete short columns confined by carbon fiber reinforced plastics strips

Abstract

To research the axial compression behavior of steel reinforced recycled concrete (SRRC) short columns confined by carbon fiber reinforced plastics (CFRP) strips, nine scaled specimens of SRRC short columns were fabricated and tested under axial compression loading. Subsequently, the failure process and failure modes were observed, and load-displacement curves as well as the strain of various materials were analyzed. The effects on the substitution percentage of recycled coarse aggregate (RCA), width of CFRP strips, spacing of CFRP strips and strength of recycled aggregate concrete (RAC) on the axial compression properties of columns were also analyzed in the experimental investigation. Furthermore, the finite element model of columns which can consider the adverse influence of RCA and the constraint effect of CFRP strips was founded by ABAQUS software and the nonlinear parameter analysis of columns was also implemented in this study. The results show that the first to reach the yield state was the profile steel in the columns, then the longitudinal rebars and stirrups yielded successively, and finally RAC was crushed as well as the CFRP strips was also broken. The replacement rate of RCA has little effect on the columns, and with the substitution rate of RCA from 0 to 100%, the bearing capacity of columns decreased by only 4.8%. Increasing the CFRP strips width or decreasing the CFRP strips spacing could enhance the axial bearing capacity of columns, the maximum increase was 10.5% or 11.4%, and the ductility of columns was significantly enhanced. Obviously, CFRP strips are conducive to enhance the axial bearing capacity and deformation capacity of columns. On this basis, considering the restraint effect of CFRP strips and the adverse effects of RCA, the revised formulas for calculating the axial bearing capacity of SRRC short columns confined by CFRP strips were proposed.

Graphical abstract

Keywords

steel reinforced recycled concrete / CFRP strips / short columns / axial compression behavior / recycled aggregate concrete

1 Introduction

The waste concrete is crushed to form coarse aggregate and it is added into the concrete to form recycled aggregate concrete (RAC), which can not only protect the natural environment, but also reduce the consumption of building industrial materials and save the non-renewable stones, which is in line with the development requirements of green buildings in the world [1–4]. At present, the mechanical properties of RAC material and its structure had been widely studied and preliminarily applied in engineering all over the world [5–6]. Most of the research results indicate that the physical mechanics performance of RAC is slightly lower than that of normal concrete, but it still can be used in the building structure through the reasonable design and construction [7–11]. Therefore, the application of RAC material can solve the impact of construction waste on the environment and is of great significance to the green development of construction industry.

Nevertheless, in order to develop the engineering application of RAC, improving the mechanical properties of RAC and its structure is still one of the current research focuses. Some scholars combined RAC with profile steel to form steel reinforced recycled concrete (SRRC) composite members, and studied their mechanical properties [12–14], so as to enhance the mechanical properties of RAC structures and components. The results show that profile steel used in the RAC structural components can effectively enhance the mechanical properties of RAC structure. Even so, compared with natural concrete, the use of RAC has a certain impact on the deformation capacity and axial bearing capacity of SRRC columns. Moreover, in order to enhance the vertical bearing capacity of SRRC members, some scholars had also researched the mechanical performance of SRRC filled steel tube columns [15–20]. The results show that the steel tube has good restraint capacity to the SRRC columns, which can greatly enhance the vertical bearing capacity of columns, and this kind of composite column also shows good ductility and deformation ability. However, it is not economical to adopt the steel tube to enhance the mechanical properties of SRRC columns, mainly because the large amount of steel material leads to the increase of construction cost. In addition, when exposed to the external environment, the steel tube is prone to rust and the durability as well as the service performance of structural members is reduced.

Fortunately, as a kind of reinforcing material in the field of civil engineering, carbon fiber reinforced plastics (CFRP) material has high tensile strength, good durability and appropriate cost. Therefore, it is widely used in the engineering field, mainly used for restraining and strengthening components in the engineering structures. In view of the good restraint effect of CFRP material, many scholars have applied it in reinforced concrete, so as to improve the performance of reinforced concrete components and structures. It indicates that the reinforced concrete is compressed in three directions under the strong restraint of CFRP, which can not only significantly enhance the bearing capacity of reinforced concrete, but also improve its deformation capacity [21–24]. In addition, CFRP tube can also improve the compression properties of RAC columns, and the replacement percentage of recycled coarse aggregate (RCA) has no great impact on the restraint effect of CFRP [25].

In view of this, CFRP strips were adopted to enhance the mechanical properties of SRRC columns. At present, there are few reports on the research of SRRC columns confined by CFRP strips, so it needs to be studied. In this study, nine scaled specimens of SRRC short columns confined by CFRP strips were made and experimented under axial compression loading. The effects of the replacement rate of RCA, CFRP strips spacing, CFRP strips width and RAC strength on the axial compression properties of members were investigated in detail. Furthermore, the nonlinear mechanical behavior and parametric analysis of SRRC short columns confined by CFRP strips under axial compression loading was also studied. Based on the above research, the revised calculation method of SRRC short columns confined by CFRP strips under the axial bearing capacity was proposed in this study, and its effectiveness was determined through experimental comparison, which provided a reference for engineering application.

2 Experimental investigation of SRRC short columns confined by carbon fiber reinforced plastics strips

2.1 Design and manufacture of specimens

Nine scaled (1:3) test specimens of SRRC short columns confined by CFRP strips were designed and manufactured in this experimental investigation. The test design parameters of specimens such as the replacement rate of RCA, CFRP strips spacing, CFRP strips width and RAC strength were mainly considered. See Tab.1 for the basic parameters of the specimens. The geometric sizes and section form of specimens are shown in Fig.1.

Three strength levels of RAC (i.e., C40, C50, and C60) were designed and prepared in the test. The RCA used in the specimens came from the waste concrete of disused building, with a continuous grading of 5–25 mm, and it was made by crushing, screening, cleaning, drying and other processes. The basic performance of RCA can satisfy the use stipulation of “recycled coarse aggregate for concrete” (GB/T-25177-2010 , in China) [26]. Fig.2 shows the RCA and test cubes of RAC in the test. Tab.2 and Tab.3 show the mix proportion design and basic performance indexes of RAC material, respectively.

The profile steel used in the specimens was the hot rolled profile steel with the strength grade of Q235, and the section size was 140 mm × 80 mm × 5.5 mm. HRB400 hot-rolled rebars were adopted in the test specimens, and the diameters of longitudinal rebars and stirrups were 16 and 8 mm, respectively. Tab.4 shows the basic mechanical performance of steel in the specimens. Moreover, the CFRP material adopted in this test were produced by Fischer Company in Germany, and the basic performance indexes of CFRP are listed in Tab.5. In order to learn the influence of CFRP strips width on the behavior of SRRC columns, CFRP strips width in the specimens were 35, 50, and 65 mm, respectively according to literature [27]. Fig.3 shows the construction process of CFRP strips on the specimens of columns.

When making the test specimens, the longitudinal rebars and stirrups were assembled to form the reinforcement skeletons, and they were welded together with the profile steel through the bottom plate to fix the position of profile steel and reinforcement skeletons. Then the wooden formwork of specimens was set up and the mixture of RAC material shall be evenly mixed and vibrated during pouring. Fig.4 shows the main fabrication process of SRRC short columns confined by CFRP strips.

2.2 Test loading devices and measuring points

In this research, the axial loading test of specimens was carried out through using a 500 t electro-hydraulic servo press machine and the loading devices are shown in Fig.5. To understand the failure process of columns under axial compression loading, the mixed loading procedure of force control and displacement control was used in the test. The main loading steps of experiment were described in detail as follows: 1) the specimen was preloaded before the formal loading and then unloaded to zero, so as to eliminate the clearance between the loading devices and specimen; 2) before the loads reached 80% of the estimated maximum bearing capacity, the force control was adopted, and each level of force was 1/15 of the estimated peak loads, and the loading duration of each load level was 1.0 min; 3) when the loads increased slowly, the loading procedure was changed to the displacement control, and load at the speed of 1.0 mm/min. When the axial bearing capacity of specimen decreased to 85% of the peak loads or specimen was deformed greatly and should not be loaded, which means that the specimen was seriously damaged and the axial compression test can be stopped.

Fig.6 shows the measuring points of strains and displacement meters of specimens in this study. Four transverse strain gauges were attached to the CFRP strips in the middle of the specimens to test the strain of CFRP. The vertical strain gauges were also attached to the longitudinal rebars and stirrups, respectively, so as to measure their strain development. And three longitudinal strain gauges were set in the middle of flange and web to monitor the strain law of profile steel. Moreover, four displacement meters perpendicular to the middle surface of specimens were arranged to measure the lateral displacement of specimens. One displacement meter was also arranged at the bottom of specimens to measure the relative vertical displacement of specimens.

3 Analysis of test results

3.1 Failure process and characteristics

Based on the observation of test process, the failure process and characteristics of SRRC short columns confined by CFRP strips under axial compression loading with different design parameters were basically similar. At the beginning of loading, the change of specimens was not obvious due to the small axial forces, and it was in the elastic state. When the axial force reached about 50% of peak load, there was a slight slip between CFRP strips and RAC, accompanied by a slight crisp sound. At the same time, a trivia crack appeared in RAC at the corner of specimens. As the axial load continued, some new cracks appeared one after another on the surface of specimens, and some original cracks at the corner gradually extended to the middle of specimens, and the click sound was constantly generated. When the axial force increased to about 80% of peak load, some CFRP strips on the specimens appeared the bulging deformation and their strains increased rapidly due to the axial compression loads. Meanwhile, more crevices appeared on the surface of RAC and the original crevices continued to extend, and the load-displacement curves of specimens were obviously nonlinear characteristics. It shows that the elastoplastic state of the columns had arrived. When the load of specimens was reached the peak load, some CFRP strips were gradually broken at the corner of SRRC columns. And the original cracks of RAC gradually were penetrated, leading to the spalling of partially RAC cover. With the continuous application of loads, RAC in the middle of specimens continued to crack and expanded outward, and the filamentous fracture of CFRP strips intensified. When the specimens reached the ultimate loads, CFRP strips in the middle of specimens was firstly broken with a loud sound, and large-scale spalling of partially RAC occurred. Meanwhile, the specimens had reached the ultimate failure state and the test was declared to be over. Fig.7 shows the failure characteristics and failure process of SRRC short columns confined by CFRP strips under axial compression loading.

According to the failure phenomenon of specimens, compared with the SRRC short columns fully wrapped with CFRP in Ref. [28], most of the CFRP strips are damaged in this test, and its utilization rate is high. Most of the columns fully wrapped with CFRP are damaged by fibers in the middle, which does not give full play to the performance of all CFRP. Therefore, CFRP strips wrapped SRRC columns has higher utilization rate.

3.2 Axial load-displacement curves

Fig.8 shows the axial load-displacement curves of SRRC short columns confined by CFRP strips with different design parameters. It can be obtained from Fig.8 that the columns confined by CFRP strips have the advantages of high bearing capacity and high stiffness under axial compression loading. Tab.6 shows the main load characteristic values of SRRC short columns confined by CFRP strips. The effects of different parameters on the axial compression performance of the columns are as follows.

(1) It can be obtained from Fig.8(a) and Tab.6 that the replacement ratio of RCA has little impact on the initial stiffness of columns, and the initial stiffness of fully RAC column is slightly smaller than that of normal concrete column. The load-displacement curves of specimens gradually towards the horizontal ordinate with the increase of axial loads, showing that the stiffness of columns decreases. When the maximum load is about to be reached, the bearing capacity of specimens increases slowly. After the maximum load, the falling stage of the curves of specimens has a long platform and good deformation capacity. It is mainly because these columns are strongly restrained by CFRP strips, so that they still maintain good mechanical properties in the later stage of loading. In general, the axial bearing capacity of columns decreases slightly with the increase of substitution rate of RCA, and the bearing capacity of fully RAC column only decreases about 4.8% compared with ordinary concrete column. Obviously, the results show that the substitution percentage of RCA has little effect on the axial compression properties of columns, so the replacement rate of RCA in the columns can be taken as 100%.

(2) From Fig.8(b) and Tab.6, the spacing change of CFRP strips has a great influence on the axial stiffness of columns, and the initial stiffness gradually decreases with the increase of CFRP strips spacing. When the axial load increases to the maximum load, the bearing capacity of columns decreases obviously with the increase of CFRP strips spacing. For example, the bearing capacity of column with the CFRP strips spacing of 50 mm is 10.5% lower than that of the column with the CFRP strips spacing of 25 mm. After the peak load, with the reduction of CFRP strips spacing, the longer the descending section of load-displacement curves and the slower the downward trend. That is to say, the smaller the spacing of CFRP strips or the denser the CFRP strips and the better the restraint effect of CFRP strips, which is conducive to enhance the deformation capacity and bearing capacity of columns.

(3) Fig.8(c) is the influence of CFRP strips width on the load-displacement curves of columns. The axial force of each specimen is proportional to the displacement at the beginning of loading, and the constraint of CFRP strips on the specimens is small, which means that the strengthening effect of CFRP strips has not yet been exerted. The columns entered into the elastic-plastic stage with the increase of axial loads, and the relationship between force and displacement is approximately nonlinear. The axial bearing capacity and deformation of columns increase greatly with the increase of CFRP strips width, so it can be obtained that the width of CFRP strips has a significant influence on the properties of columns. For instance, the peak force of column with the CFRP strips width of 65 mm is 11.4% higher than that of the column with the CFRP strips width of 35 mm. In addition, the slope on the descending section of curves decreases with the increasing width of CFRP strips. It indicates that the increasing CFRP strips width is good for improving the deformation of SRRC columns.

(4) According to Fig.8(d) and Tab.6, the initial stiffness of the columns is improved with the increase of RAC strength. When the strength of RAC increases from C40 to C60, the peak load of column increases by about 23%. After the maximum load, the falling stage of load-displacement curve of specimen with the RAC strength of C50 or C60 becomes steeper than that of the specimen with the strength of C40. It indicates that the higher the RAC strength is, the higher the axial bearing capacity of columns, but the more obvious the brittleness of columns is. Therefore, for the RAC strength of C60 and above, it is suggested to mix the fiber in RAC or adopt the restraint measures outside the column to reduce its brittle failure as far as possible.

In addition, compared with the peak bearing capacity of short columns in literature [28], the bearing capacity of SRRC short columns wrapped with CFRP strips is about 6% lower than that fully wrapped with CFRP, indicating that the change from fully wrapped with CFRP to wrapped with CFRP strips has little impact on the mechanical properties of SRRC short columns.

3.3 Strain characteristics of columns confined by carbon fiber reinforced plastics strips

3.3.1 Strain of profile steel and rebars

The axial load-strain relationship of profile steel and rebars in the short columns under axial compression loading is described in Fig.9, where “M, F, Y” in “M-F-1” and “M-Y-1” represent the middle position of specimens, web and flanges of profile steel, respectively. At the beginning of loading, the strain values of profile steel and rebars in the specimens are very small, and their strain changes linearly with the axial loads. When about 70% of maximum load is reached, the profile steel flange begins to yield and its strain is about 1400με. And when about 80% of maximum load is reached, the profile steel web also reaches the yield state, and its strain value is about 1900με. After maximum load, the strain growth rate of profile steel accelerates until it yields completely, and then longitudinal rebars and stirrups also reach the yield state one after another. At this moment, the strain values of longitudinal rebars and stirrups are about 2200με and 2000με, respectively. From the strain change comparison between profile steel and rebars, it can be seen that the strain of profile steel is significantly greater than that of rebars under the same axial force, and the growth rate is also faster than that of rebars. That is to say, the profile steel in the columns first reaches the yield state under the axial compression loading, then the longitudinal rebars yields. In addition, comparing Fig.9(a) and 9(b), it can be concluded that the initial strain of profile steel increases rapidly with the increase of RCA replacement rate r. However, the strain growth rate of profile steel with r = 100% slows down with the increase of load, which also causes it to yield later than the profile steel with r = 50%. Similarly, the yield of rebars slows down with the increase of RCA substitution rate. In comparison Fig.9(b)–9(d), the spacing of CFRP strips has little effect on the initial strain of profile steel and rebars. With the increase of strip spacing, the profile steel and rebars yield earlier. Comparing Fig.9(b) and 9(e), the wider the CFRP strip, the later the profile steel and rebars enter the yield time. As can be obtained from Fig.9(b) and 9(f), the higher the RAC strength level, the faster the strain growth rate of profile steel and the easier it is to yield.

3.3.2 Strain characteristics of recycled aggregate concrete

The axial load-strain curves of RAC in SRRC columns confined by CFRP strips are shown in Fig.10. It can be obtained from Fig.10 that at the beginning of loading, the trend of load-strain curves of all the columns is basically the same, and the strain of RAC varies linearly with the load. Fig.10(a) is the strain change of RAC in the columns under different replacement rate of RCA. When the load increases to 1700 kN, the strain value of RAC is about 1900με, and the strain value of RAC increases nonlinearly with the axial force. When the load increases to 80% and 100% of peak load, the strain values of RAC are about 2400με and 5000με, respectively. Under peak loads, the strain value of RAC with the replacement rate of RCA for 100% increases by 46.8% compared with ordinary concrete column. Obviously, it shows that the strain value of RAC increases obviously with the increasing substitution rate of RCA. From Fig.10(b) and 10(c), the wider the CFRP strips width or the smaller the CFRP strips spacing, the lower the strain values of RAC, which indicates that the width and spacing of CFRP strips have a significant impact on the deformation of RAC. From Fig.10(d), the strain values of RAC with the strength of C40, C50 and C60 are about 4900με, 3455με and 2094με, respectively. The strain value of RAC with the strength of C50 is 29.5% lower than that of RAC with the strength of C40, and the strain value of RAC with the strength grade of C60 is 39.4% lower than that of RAC with the strength grade of C50. It indicates that the higher the strength of RAC, the worse the deformation capacity of columns.

3.3.3 Strain of carbon fiber reinforced plastics strips

The circumferential strain of CFRP strips was measured by pasting the strain gauges on the surface of CFRP strips in the columns, so as to obtain the strain rule of CFRP strips under of axial compression loading. Fig.11 shows the axial load-strain relationship of CFRP strips in the columns. At the initial stage of loading, the strain values of CFRP strips increase slowly, and the axial force and strain of CFRP strips change almost linearly. After the cracking of RAC, the strain values of CFRP strips begin to increase rapidly. When it reaches about 80% of peak load, the strain values of CFRP strips increase nonlinearly with the force, and the strain values of CFRP strips are about 1400με. When maximum load is reached, the strain values of CFRP strips are about 2000με. At this stress stage, RAC in the columns cracks seriously and its volume expansion is also obvious, which causes to a rapid increase in the strain of CFRP strips. When the CFRP strips of columns are broken, the maximum strain of CFRP strips can reach about 13000με (i.e., CFRP- SRRC-5 specimen). Meanwhile, the strain values of the CFRP strips with the width of 65 mm and 35 mm reach 10200με and 6777με, respectively. Compared to the CFRP strips with the width of 65 mm, the strain value of CFRP strips with the width of 35 mm is reduced by about 33.5%. Besides, compared to the CFRP strips with the spacing of 50 mm, the strain value of CFRP strips with the spacing of 25 mm is increased by 60.9%. The results clearly show that the wider the CFRP strips width or the smaller the CFRP strips spacing, the stronger the restraint effect of CFRP strips on the SRRC columns.

4 Finite element analysis of SRRC columns confined by carbon fiber reinforced plastics strips

4.1 Constitutive relationship of materials in the columns

4.1.1 Constitutive model of recycled aggregate concrete

(1) Compression constitutive model of RAC

The reasonable constitutive model can well reflect the mechanical properties of materials, which is very important for the finite element calculation of structures or components. The compression constitutive model of RAC in this research is based on the compressive stress-strain curve of RAC proposed by Xiao [29]. Its equation can be expressed as follows:

(1)

where x = ε/ε₀, y = σ/f_c; ε₀ is the maximum compressive strain of RAC; f_c is the uniaxial compressive strength of RAC. The f_c values of RAC with strength of C40, C50, and C60 are 32.5, 40, and 46 MPa, respectively; the experimental research show that the replacement percentage of RCA has a certain effect on the mechanical performance of RAC, so the effect of substitution rate of RCA should be considered in the compression constitutive model of RAC. With the increasing substitution percentage of RCA, the peak strain of RAC increases gradually, so the effect of substitution rate r on the peak strain ε₀ of RAC needs to be considered, which can be calculated according to the calculation method proposed by Xiao [29], as shown in Eq. (2). In addition, the rise and fall sections on the constitutive curves of RAC are also different from the ordinary concrete, which can be adjusted by parameters a and b. The expressions of parameters a and b in Eq. (1) are shown in Eq. (3).

(2)\{\begin{array}{c}{\epsilon }_{\text{0}}=\{0.00076+[(0.626f_{\text{c}}-4.33)\times {10}^{-7}{]}^{0.5}\}\times \left[1-\frac{r}{B(r)}\right],\\ B(r)=65.715r^2-109.43r+48.989,\end{array}

(3)\{\begin{array}{c}a=2.2\times (0.748r^2-1.231r+0.975),\\ b=0.8\times (7.6483r+1.142).\end{array}

(2) Tension constitutive model of RAC

In this research, the rising stage and falling stage of tensile stress-strain curve of RAC were determined according to literatures [30–31], and the influence of substitution percentage of RCA on the tensile properties of RCA is also considered in the tension constitutive model. The peak tensile strain and tensile stress of RAC with the strength grades of C40, C50, and C60 under three substitution rates can be calculated based on Eq. (5), and then brought into Eq. (4) to obtain the tensile constitutive relationship of RAC.

(4)\{\begin{array}{c}y=1.2x-0.2x^6,x⩽1,\\ y=\frac{x}{{\alpha }_{\text{t}}{(x-1)}^{1.7}+x},x>1,\\ y=\frac{\sigma }{f_t},\\ x=\frac{\epsilon }{{\epsilon }_{\text{t}}},\end{array}

(5)\{\begin{array}{c}{f}_{\text{t}}=(cr+0.24)(f_{\text{cu}}{)}^{\frac{2}{3}},\\ {\epsilon }_{\text{t}}=(55+dr)(f_{\text{t}}{)}^{0.54}\times {10}^{-6},\end{array}

where α_t is the parameter values on the descending section of tensile stress-strain curve of RAC, which is determined according to Tab.7; f_t is the maximum tensile stress of RAC; f_cu is the cube compressive strength of RAC; ε_t is the maximum tensile strain of RAC, calculated on the basis of formula (5); In addition, the parameters c and d in Eq. (5) can be –0.06 and 14, respectively.

Based on the existing experimental research, the compression and tensile damage of RAC is the same as that of normal concrete. Because of the lack of plastic damage of RAC, in order to facilitate the calculation, so the damage model of ordinary concrete was adopted by adjusting input parameters in this paper. In ABAQUS software, the plastic damage of concrete material can not only consider the biaxial ratio, but also consider the interaction of biaxial tensile and compressive properties of concrete material, which well reflects the actual stress situation of concrete material. The plastic damage model includes two failure modes of concrete material, namely compressive failure and tensile cracking. In this research, the damage criterion of RAC is determined by controlling the equivalent stress on the yield surface of RAC. When RAC are damaged, the material will enter the softening stage of stress-strain curve. The main parameters of concrete damaged plasticity are as follows: the eccentricity ε = 0.1; the invariant stress ratio K_c = 0.6667; the dilation angle ψ = 30°; the ultimate strength rate of the biaxial compression and the uniaxial compression f_bo/f_co = 1.16 and the viscosity parameter μ = 0.0005. The yield conditions of RAC are shown in Fig.12, and Fig.13 shows a typical yield surface on the off plane of RAC.

4.1.2 Constitutive model of profile steel and rebars

The constitutive model of profile steel and rebars adopted simplified stress-strain curve in the numerical model, which is characterized by two plastic platforms, as shown in Eq. (6). The constitutive model can consider the strengthening stage of steel directly after yield, and the strength of steel remains unchanged in the plastic flow stage. In a word, the model accords with the material characteristics, considers the strengthening effect of steel, and better reflects the mechanical process of steel.

(6)

where f_y is the yield stress; ɛ_y is the yield strain; E_s is the elastic modulus; k₁, k₂, and k₃ are the control parameters of curve shape, respectively, which can be taken as k₁ = 4.5, k₂ = 45, and k₃ = 1.4.

In this study, the failure criterion of steel conforms to the Mises yield criterion, as shown in Eq. (7). It is usually considered that as follows: on the one hand, the steel is in an elastic state before yielding and the relationship of stress−strain is linear; on the other hand, under certain deformation conditions, when the elastic energy of the change of shape per unit volume of material reaches a constant, the steel material will yield and enter the plastic state.

(7){\sigma }_s=\sqrt{1/2[{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_1-{\sigma }_3)}^2]},

where σ₁, σ₂, and σ₃ are the first, second and third principal stresses of steel, respectively, and σ_s is the yield strength of steel.

4.1.3 Constitutive model of carbon fiber reinforced plastics strips

As a composite material, CFRP is considered as an elastic material, and its failure basically meets the Hashin criterion. The progressive damage theory based on the energy method is adopted in the finite element simulation. It is considered that the material will not be broken immediately after damage, but the material performance will degrade, and more loads will be borne through the surrounding undamaged elements. With the continuous decline of the bearing capacity of the surrounding material elements, the undamaged elements will gradually decrease, and the material will not be broken until the damage exceeds the specified value. Material damage is divided into: the fiber direction tensile failure, the fiber direction compression failure, the tensile failure in vertical fiber direction, the vertical fiber compression failure. Material damage is determined by the following formula.

Fiber direction tensile failure:

(8)F_{\text{ft}}={\left(\frac{{\stackrel{\wedge }{\sigma }}_{11}}{X^{\text{T}}}\right)}^2+\alpha {\left(\frac{{\stackrel{\wedge }{\sigma }}_{12}}{S^{\text{L}}}\right)}^2=1,({\stackrel{\wedge }{\sigma }}_{11}⩾0).

Fiber direction compression failure:

(9)F_{\text{tc}}={\left(\frac{{\stackrel{\wedge }{\sigma }}_{11}}{X^{\text{C}}}\right)}^2=1,({\stackrel{\wedge }{\sigma }}_{11}⩽0).

Tensile failure in vertical fiber direction:

(10)F_{\text{mt}}={\left(\frac{{\stackrel{\wedge }{\sigma }}_{22}}{Y^{\text{T}}}\right)}^2+{\left(\frac{{\stackrel{\wedge }{\sigma }}_{12}}{S^{\text{L}}}\right)}^2=1,({\stackrel{\wedge }{\sigma }}_{22}⩾0).

Vertical fiber compression failure:

(11)F_{\text{mc}}={\left(\frac{{\stackrel{\wedge }{\sigma }}_{\text{22}}}{2S^{\text{T}}}\right)}^2+\left[{\left(\frac{Y^{\text{C}}}{2S^{\text{T}}}\right)}^2-1\right]\frac{{\stackrel{\wedge }{\sigma }}_{22}}{Y^{\text{C}}}+{\left(\frac{{\stackrel{\wedge }{\sigma }}_{12}}{S^{\text{L}}}\right)}^2=1,({\stackrel{\wedge }{\sigma }}_{22}⩽0).

The above formulas are used to judge the damage initiation of the material. When the value of F reaches 1.0, it means that the material damage begins at this time, and when F is less than 1.0, it means that the material is not damaged. The six strength values in the above formulas are actually related to the material damage, it can be described as follows: the longitudinal compressive strength X^C = 1200 MPa, the longitudinal tensile strength X^T = 3565.2 MPa, the transverse compressive strength Y^C = 190 MPa, the transverse tensile strength Y^T = 62 MPa, the longitudinal shear strength S^L = 81 MPa, the transverse shear strength S^T = 81 MPa. It is incomplete to use the above four formulas for the failure analysis of CFRP material, because it can only predict the beginning of damage and lacks the stiffness degradation after damage. Therefore, Hashin [32] also puts forward the damage evolution based on energy, that is, the energy required for the complete fracture of the material along the interface. The fracture energy corresponding to the four failure modes of CFRP material input in this paper is: the longitudinal compressive fracture energy G_fcc = 75 N/mm, the longitudinal tensile fracture energy G_ftc = 50 N/mm, the transverse tensile fracture energy G_mtc = 0.25 N/mm as well as the transverse compressive fracture energy G_mcc = 0.75 N/mm. The fracture energy is taken as the specified value of failure, which is used to judge whether the material is completely broken.

4.2 Contact settings between materials

During the establishment of numerical model, profile steel and RAC were defined as the contact attributes, and the bond slip between steel and concrete is considered. The relationship between the critical tangential stress τ_crit and the normal stress p conforms to Eq. (12), and the friction coefficient μ can be taken as 0.6. In addition, CFRP and RAC were defined as the binding relationship between surfaces, and the rebars were built into RAC. The top and bottom of SRRC short column were respectively coupled at two points. The bottom of column was fixed and constrained to limit movement and rotation. The coupling point was established above the column, and the axial force was applied to the column top in the form of uniform load through the coupling point, so as to avoid local damage to the column top.

(12){\tau }_{\text{crit}}=\mu p⩾{\tau }_{\text{bond}}.

The mesh density directly affects the correctness and efficiency on the numerical simulation of column. Too large mesh will lead to very low calculation accuracy of the model, but too small mesh may lead to non-convergence and reduce the calculation speed. In order to increase the calculation accuracy, the grid at the corner of column was densified. The solid parts such as RAC and profile steel were divided into hexahedral calculation elements (C3D8R), and CFRP was divided into quadrilateral shell elements (S4R), as well as the rebars were divided into two node line elements (T3D2). The mesh division on the numerical model of columns is shown in Fig.14.

4.3 Results of finite element analysis

4.3.1 Stress nephogram characteristics

Through the calculation of numerical model, the stress nephogram and mechanical properties indexes of SRRC short columns confined by CFRP strips can be obtained, as described in Fig.15–Fig.18. At the initial stage of loading, the columns were in an elastic state and the stress of various materials was very small. The stress of specimens increased with the increase of loads, but its deformation was not obvious. After the cracking of RAC, the specimens gradually deformed obviously with the increase of force, and the middle area of columns was slightly bulging. When increasing to 80% of peak load, the stress of profile steel reached the yield strength, and the stress in the middle of steel web and steel flange exceeded 235 MPa. Meanwhile, although the stress of rebars, RAC and CFRP strips increased with the increase of force, they still had not the yield strength. When the peak load was reached, the stress of profile steel continued to increase and the yield area of profile steel gradually expanded. At this stage, the stress in the middle of longitudinal rebars exceeded 360 MPa and they had yielded, but the stirrups did not enter the yield state. The tensile stress of RAC at the corner of columns far exceeded 2.4 MPa, which shows the cracking of RAC at the corner of columns was serious. By the way, although the CFRP strips had not yet reached the yield strength, their expansion deformation was more obvious because of the cracking of RAC. When the loading ended, the stress of RAC exceeded 36 MPa, which shows that RAC was seriously damaged under the compression stress. While profile steel and rebars in the columns almost completely yielded. In addition, the stress of CFRP strips in the middle of columns reached 3216 MPa, which was close to the ultimate tensile strength, and the ultimate failure was mainly due to the fracture of CFRP strips.

As a whole, the failure characteristics of the columns indicate that the profile steel first yields, then the longitudinal rebars yields, and RAC is crushed, finally the CFRP strips are broken. Because of the good strengthening effect of CFRP strips, the columns have high bearing capacity and stiffness. In addition, when these columns are damaged, they do not show obvious brittleness and still have a certain deformation capacity.

4.3.2 Load-displacement calculated curves

Tab.8 lists the comparison results of main load characteristic values of specimens under different test design parameters. The comparisons of load-displacement curves of specimens between the test investigation and numerical simulation are shown in Fig.19. It can be obtained from Fig.19 that the test and calculation load- displacement curves of columns are basically consistent before reaching peak loads, and the slopes of both curves are almost the same. However, when peak load is reached, the simulated values on the axial bearing capacity of specimens are certain different from the test values. According to Tab.8, the maximal relative error of peak load between the experimental value and simulated value of column is 13.67% (i.e., CFRP-SRRC-8). After the peak load, the falling stage of the curves first decreases rapidly, and then tends to decrease slowly, which shows that the specimen shows good deformation ability in the later stage of loading. By the way, although there is a certain difference between the test curves and numerical curves in the descending section, the descending stage of curves has the same trend. To sum up, because the columns are restrained by CFRP strips, the bearing capacity will not drop rapidly to the end, so as to slow down the decline of bearing capacity until CFRP strips are broken. The above analysis results show that the columns have high deformation capacity and bearing capacity under the constraint of CFRP strips.

4.4 Parameter analysis of the columns confined by carbon fiber reinforced plastics strips

Through the above comparison of test and simulation of columns, there is little difference between experiment and simulation. From the above simulation results, it can be seen that this model can be suitable for the simulation of this kind of columns. Based on this numerical calculation, the parameter analysis of SRRC short columns confined by CFRP strips was put into effect in this study. The influence of extended parameters such as the profile steel strength, chamfer radius of cross-section, CFRP strips width and CFRP strips layers on the axial compression properties of columns were mainly considered, as shown in Tab.9. The geometric dimension of simulated columns was the same as that of test specimens, and the replacement rate of RCA and CFRP strips spacing were 100% and 40 mm respectively in all simulated columns. Fig.20 displays load-displacement curves of the columns under different parameters. The effect characteristics of extended parameters on the axial compression properties of columns are as follows.

(1) It can be obtained from Fig.20(a) that the profile steel strength has a great impact on the axial bearing capacity of columns. The axial bearing capacity is greatly improved with the increase of steel strength, and the bearing capacity increases by about 18.9% with the increase of steel strength from Q235 to Q390. At the beginning of loading, the columns are in the elastic state and the force is proportional to the displacement. The initial stiffness of columns is basically the same under different steel strength. When 70% of peak load is reached, the slope of curves gradually decreases, which means that the columns enter the nonlinear state. There are obvious differences in the curves of columns under different steel strength. The higher the strength of profile steel, the growth rate of bearing capacity of column is significantly greater than that of column with a low strength of profile steel. The descending stage of curves shows the characteristics of first fast and then slow in the later stage of loading, indicating that the columns have a good deformation capacity.

(2) It can be obtained from Fig.20(b) that the impact of CFRP strips width on the load-displacement curves of columns. According to Fig.20(b), the narrower the CFRP strips width, the faster the curve slope of columns changes. When the CFRP strips becomes wider, the axial bearing capacity of columns also enhances. The CFRP strips with a width less than 20 mm have little enhancement influence on the bearing capacity of columns, while the restraint effect is better when the CFRP strip width is greater than 30 mm. Compared with the column without the restraint effect of CFRP strips, the bearing capacity of it can be increased by 15% with the CFRP strip width of 65 mm.

(3) Based on Fig.20(c), it shows that the CFRP strips layers have some certain impact on the deformation performance and axial bearing capacity of columns, and the axial bearing capacity gradually increases with the increase of CFRP strips layers. If the layers number of CFRP strips is greater than 3.0, the axial bearing capacity of the column increases little, only about 3%, so it is recommended that the layers number of CFRP strips should not exceed 3.0.

(4) From Fig.20(d), the curves of columns are basically the same before the peak load is reached, which shows the change of chamfer radius of cross section has little impact on the initial stiffness of columns. The axial bearing capacity increases a little with the increase of the chamfer radius of cross section, and the declining stage of load-displacement curves shows a trend of first fast and then slows. When the section of column is close to a rectangle (that is, the section chamfer is very small), the axial bearing capacity is lower than that when the section of column is close to a circle (that is, the section chamfer is very large). The axial bearing capacity of column with chamfer radius of 30 mm is 9.9% higher than that of the rectangular column without chamfer radius.

5 Calculation on the axial bearing capacity of columns confined by carbon fiber reinforced plastics strips

Based on the above tests and numerical simulation, a bearing capacity correction calculation method is proposed for this kind of short column in this study, which can offer technical reference for the practical application of SRRC short columns confined by CFRP strips.

5.1 Mechanical mechanism of the columns under axial compression loading

It can be seen from the above analysis that under the action of vertical load, the profile steel and longitudinal rebars have reached the yield strength at the peak load, and RAC has also cracked, indicating that the three materials have given full play to their performance, so the vertical bearing capacity should be borne by these three materials, as shown in Eq. (26). In particular, RAC material is in three-dimensional compression state under the restraint of CFRP strips and transverse stirrup, so the effect of restraining and strengthening concrete should be calculated in the nominal axial bearing capacity of the columns, which is more accurate. Fig.21 describes the axial compressive stress diagram of SRRC short columns confined by CFRP strips.

As mentioned above, when the axial bearing capacity of columns is calculated, the restraint effect of RAC should be considered. At present, there are few studies on the strength calculation model of SRRC columns confined by CFRP strips. However, there are a lot of research on FRP confined reinforced concrete members. This paper modified the calculation model of FRP confined reinforced concrete based on the code ACI 440.2R-08 [33]. According to the test study, it can be concluded that the mechanical properties of RAC restrained by CFRP strips are similar to ordinary concrete, as shown in Fig.21(a). The strength model of concrete strength in FRP confined ordinary concrete is given in document [33], as follows.

(13)f_{\text{cc}}=f_{\text{co}}+3.\text{3}{\psi }_{\text{f}}{k}_{\text{a}}{f}_{\text{l}}.

where f_cc is the compressive strength of CFRP strips confined RAC; f_co is the compressive strength of unconstrained RAC; f_l is the lateral restraint force of CFRP strips, and its calculation is shown in Eqs. (22)–(23); ψ_f is the additional reduction factor of RAC strength after the restraint, and the value is 0.95; k_a is influence coefficient of section shape, and it depends on the cross-sectional area and length-width ratio h/b of the effectively restrained RAC, as shown in Eq. (18).

The above formula includes the strength of the concrete itself and the strengthening strength of FRP constraints. In addition, it also considers the adverse effect of the replacement rate of RCA in this study. Combined with the test data, the reduction factor on the replacement rate of RCA can be put forward by regression method ϕ_r [34]. Therefore, Eq. (13) is modified as follows:

(14)f_{\text{cc}}={\varphi }_{\text{r}}{f}_{\text{co}}+3.\text{3}{\psi }_{\text{f}}{k}_{\text{a}}{f}_{\text{l}},

(15){\varphi }_{\text{r}}=\text{0}\text{.88/}\left(-0.3r^2\text{+ 0}\text{.45}r+1\right),

where r is the substitution rate of RCA.

Besides, the welded transverse stirrups were adopted in SRRC short columns and it can be seen that stirrups played a strong restraint role in maximum load from the load-strain curves of stirrups, which is conducive to enhance the axial bearing capacity of SRRC columns. However, the influence of stirrups on concrete strength is not considered in ACI. The transverse binding force f_stl of stirrups on columns is considered in this study, as shown in Eq. (17). Fig.20(c) shows a simplified model diagram for the calculation of CFRP strips and stirrups constrained RAC. Therefore, Eq. (14) can be modified, as shown in Eq. (16).

(16)f_{\text{cc}}={\varphi }_{\text{r}}{f}_{\text{co}}+3.3{\psi }_{\text{f}}{k}_{\text{a}}{f}_{\text{l}}+f_{\text{stl}},

(17)f_{\text{stl}}^{}=\frac{n_{\text{st}}{f}_{\text{yt}}{A}_{\text{st}}}{s{b}_{\text{c}}},

(18)k_{\text{a}}=\frac{A_{\text{e}}}{A_{\text{c}}}{\left(\frac{b}{h}\right)}^2,

where n_st is the number of stirrup legs; A_st is the section area of stirrups; f_yt is the yield strength of stirrups; s is the spacing of stirrups; b_c is the center line distance of stirrups; A_c and A_e are the section area and effective restraint area of columns, their ratio is shown in Eqs. (19)–(20).

In addition, due to the rectangular cross-section of SRRC short columns in this research, the lateral pressure of CFRP strips on the columns is not uniform due to the stress concentration effect. When the CFRP strips wrap the columns with a circular section, the restraint stress is uniformly distributed along the periphery of columns. In view of this, in order to facilitate the calculation, the rectangular columns in this study need to be equivalent to the circular columns, as shown in Fig.21(b). The diameter D of the equivalent circular section is calculated as Eq. (21).

(19)\frac{A_{\text{e}}}{A_{\text{c}}}=\frac{1-{\displaystyle \frac{\left[\left({\displaystyle \frac{b}{h}}\right){\left(h-2r_{\text{c}}\right)}^2+\left({\displaystyle \frac{h}{b}}\right){(b-2r_{\text{c}})}^2\right]}{3A_{\text{g}}}}-{\rho }_{\text{g}}}{1-{\rho }_{\text{g}}},

(20)A_{\text{g}}=hb-\left(4-\text{π}\right){r_{\text{c}}}^{\text{2}},

(21)D=\sqrt{b^2+h^2},

where r_c is the chamfer radius of cross-section of columns; b and h are the width and height of cross section; A_g is the total cross-sectional area of columns; ρ_g is the rate of the sum of sectional area of profile steel and longitudinal rebars to the sectional area of column.

The failure process of the specimens and the load-strain curves of the CFRP strips show that the CFRP strip plays a strong restraint role at the peak load, but it has not reached the failure state of the material. The restraint of CFRP strips produces the lateral stress f_l on RAC material in the columns, it shown in Fig.18(c), which can be calculated by Eq. (22) in ACI [33]. Where the efficiency factor of strain k_ε = 0.55, ε_fu is the fracture strain of CFRP strips; t_f is the thickness of single-layer CFRP strips; ε_fe is the effective fracture strain of CFRP strips; E_f is the elastic modulus of CFRP strips.

(22)f_{\text{l}}^{}=\frac{2E_{\text{f}}n{t}_{\text{f}}{\epsilon }_{\text{fe}}}{D},

(23){\epsilon }_{\text{fe}}=k_{\epsilon }{\epsilon }_{\text{fu}}.

The test analysis shows that with the increasing number of CFRP layers, the increase degree on the axial bearing capacity of columns decreases gradually. Therefore, considering the influence of CFRP strip layers, the correction factor μ_n of confinement effect was introduced in this study, as shown in Eq. (24), where n is the layers number of CFRP strips.

(24){\mu }_{\text{n}}=\{\begin{array}{ll}1,& n=1,\\ 0.9-0.05(n-2),& n⩾2.\end{array}

Because the CFRP strips are wrapped outside SRRC short columns, the width and spacing of CFRP strips have a strong influence on the axial bearing capacity of columns, the influence coefficient k_f of CFRP strips spacing was proposed in this study. k_f is the rate of effective restraint area and the section area of RAC in the columns; s_f is the spacing of CFRP strips.

(25)k_{\text{f}}=\frac{\left(b-{\displaystyle \frac{1}{2}}{s}_{\text{f}}\right)\left(h-{\displaystyle \frac{1}{2}}{s}_{\text{f}}\right)}{bh}{k}_{\text{d}}.

Considering the effect of CFRP strips width on SRRC short columns, the influence coefficient of CFRP strips width k_d = 0.044e^23d + 0.88 was proposed, where d is the width of CFRP strips in the columns and it is suitable for d < 70 mm.

5.2 Comparisons on the nominal axial bearing capacity of columns

Based on the superposition methods, the nominal axial bearing capacity of SRRC short columns confined by CFRP strips can be calculated as the following formula:

(26)N_{\text{c}}\text{=}{\mu }_{\text{n}}{k}_{\text{f}}{f}_{\text{cc}}{A}_{\text{c}}+f_{\text{a}}{A}_{\text{a}}+f_{\text{y}}{A}_{\text{s}},

where N_c is the nominal axial bearing capacity of columns; μ_n is influence coefficient of CFRP constraint layers; k_f is influence coefficient of CFRP strips spacing; f_cc is the compressive strength of RAC confined by CFRP strips; A_c is the cross-sectional area of RAC confined by CFRP strips; f_a is the compressive strength of profile steel; A_a is the section area of profile steel; f_y is the compressive strength of longitudinal rebars; A_s is the sectional area of longitudinal rebars.

The comparison results between test values and calculated values based on the above calculation formulas are shown in Tab.10. It can be obtained from Tab.10 that the relative error between calculated values and test values is relatively small, with the average error of 0.971 and the standard deviation of 0.036. As a whole, the proposed calculation formulas can better meet the calculation accuracy requirements on the axial bearing capacity of SRRC short columns confined by CFRP strips.

6 Conclusions

(1) In this investigation, the failure modes of all SRRC short columns confined by CFRP strips under axial compression loading is almost the same. The profile steel first reached the yield state, and then the longitudinal rebars and transverse stirrups yielded one after another. Afterwards, RAC inside the columns was crushed, finally CFRP strips reached the ultimate tensile strain, which means that CFRP strips was broken and the columns lost its axial bearing capacity.

(2) Known from the test, the axial bearing capacity of columns decreases with the increasing replacement rate of RCA, but the decrease range is relatively small, and the maximum decrease is only 4.8%. Increasing the CFRP strips width or decreasing the CFRP strips spacing can increase the axial bearing capacity and deformation of columns, and the maximum increase of bearing capacity is 10.5% and 11.4%, respectively. Besides, the axial bearing capacity of columns also enhances with the increase of RAC strength, but the deformation of columns decreases gradually.

(3) The calculation results of numerical model used in the study are in good agreement with the experimental results, which shows that the numerical model established by ABAQUS software is reasonable. The numerical model considers the adverse influence on the substitution ratio of RCA and the constraint effect of CFRP strips, and well reflects the mechanical characteristics of columns.

(4) The results of parameter analysis indicate that the axial bearing capacity of columns is greatly enhanced with the increasing strength of profile steel, and the maximum increase is 18.9%. The increasing corner fillet radius of cross section is beneficial to the bearing capacity of columns. Moreover, the bearing capacity and deformation capacity of columns increases with the increasing CFRP strips layers. Nevertheless, when the layers number of CFRP strips exceeds 3, the bearing capacity of columns is enhanced slightly. Therefore, it is proposed that when the layers number of CFRP strips is 2–3, the restraint enhancement effect on the columns is the best.

(5) Considering the influence of substitution rate of RCA and the restraint effect of CFRP strips and stirrups, the modified formulas on the axial bearing capacity of SRRC short columns confined by CFRP strips was proposed. The validity of the modified formula is verified by calculation.

(6) In general, the axial bearing capacity and ductility of SRRC short columns are greatly improved by the restraint of CFRP strips. The research results provide a way for the application of RAC material.

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