# Analysis of Bearing Capacity and Seismic Performance of Circular RC Columns Strengthened with Externally Wrapped Steel Plates

Authors: Chunli Zhou; Xiaowei Li; Dongbo Wang; Shunxiang Xia
Journal: Advances in Civil Engineering
 (2019)

---

## Abstract

Based on a practical project, the construction method of circular reinforced concrete (RC) columns strengthened with externally wrapped steel plates was introduced in this paper. In order to study the bearing capacity and seismic performance of circular RC columns after strengthening, a formula for calculating the bearing capacity of circular RC columns strengthened with externally wrapped steel plates was derived, with reference to theCode for Design of Strengthening Concrete Structure. Through ABAQUS software, the vertical monotone static loading analysis and the horizontal low-cyclic reversed loading analysis were carried out, respectively, before and after the strengthening of the full-size RC columns in this project. The results showed that the bearing capacity of the normal section of the RC column after strengthening was about 80% higher than that before strengthening, and the results of FEM software were in accord with the calculation results of theoretical formula to some degree. Under horizontal low-cyclic reversed loading, the columns after strengthening had both plumper hysteresis curves and higher ductility factors and equivalent viscous damping coefficient than those before strengthening, indicating the energy dissipation capacity, plastic deformation capacity, and seismic performance of the RC columns after strengthening were all obviously improved.


---

## Body

## 1. Introduction

So far, the type of China’s housing structure is still dominated by RC structure [1], a large proportion of which has a history of over 30–40 years. These houses need to be tested and strengthened for various reasons, such as poor construction quality, improper use, changing the usage of houses, or adding layers. In addition, more and more homes will need to be tested and strengthened over time [2]. RC structures are generally strengthened for bearing members including RC columns, RC beams and foundations, etc. This paper only discusses the strengthening of RC columns. Many strengthening methods exist for RC columns, mainly focusing on using different materials to wrap the RC columns to increase the bearing capacity and ductility. The commonly used materials for wrapping are RC, shape steel, fiber-reinforced polymer/plastic (FRP), etc. [3, 4].Strengthening with RC for wrapping is a traditional method, also known as structure member strengthening with increasing section area [5, 6]. In this method, according to the stress situation of RC column, RC is wrapped externally on one side, two sides, or all sides of the column to increase the column section area and reinforcement steel so as to improve the bearing capacity, deformation capacity, and bending stiffness. In all the strengthening methods of RC column, increasing section area method is especially suitable for the case that the bearing capacity is insufficient due to the small section of the column [7]. The advantages of this kind of strengthening method are simple construction technology, strong adaptability, and existing long-term using experience [8]. The disadvantages are the long time cost for wet construction, strictly controlled load limits during concrete curing, and the reduced usable area due to the increase of column section area [9].The method of structure member strengthening with FRP is widely used and is one of the hotspots in the field of strengthening at present [10, 11]. According to different types of fibers, it can be divided into carbon fiber (CFRP), glass fiber (GFRP) [12–15], aramid fiber (AFRP) [16], basalt fiber (BFRP) [17], hybrid fiber (HFRP) [18], and other strengthening methods. Among them, CFRP and GFRP strengthening methods are the most widely studied. The FRP strengthening method is suitable for RC columns with axial compression, small eccentric compression, or insufficient ductility [19]. When the bearing capacity of large eccentric columns is insufficient, it is necessary to wrap multiple layers of fiber cloth in most cases to obtain more distinct effect of strengthening, which is obviously uneconomic [20]. The advantages of FRP strengthening method are light weight, high strength, corrosion resistance, moisture resistance, convenient construction, generally no need for overlap, and adapting to the paste requirements of curved concrete [21], while the disadvantages are the limitation for the environmental temperature and the need to make special protection treatment. If the protection is not appropriate, it is easy to suffer fire and man-made damage [22]. Although FRP has the advantage of being fast and efficient in strengthening damaged circular RC columns, generally speaking, the research on the compressive performance of damaged circular RC columns strengthened by FRP is still very limited in various countries [23].In the method of structure member strengthening with externally wrapped steel, angle steel is wrapped externally around two or four corners of RC column and welded on flat-steel hoop plate to enhance the rigidity and mechanical performance of column [24]. It is widely used in RC column strengthening and is also a traditional strengthening method, which is divided into two kinds: dry strengthening and wet strengthening [25]. In the dry strengthening method, there is no connection between the angle steel and the original RC column; even though the cement mortar is filled, the shear force on the bonding surface cannot be effectively transferred, and the deformation between them is difficult to coordinate [26]. In the wet strengthening method, the latex cement, polymer mortar, or epoxy resin is poured between the angle steel and the original RC column so that the angle steel and column can work cooperatively and bear load together. The dry strengthening method is relatively simple in construction, while the effect of the wet strengthening method in increasing carrying capacity is better than that of the dry strengthening method [27]. Therefore, the wet strengthening method is more widely used in practice. The advantage of structure member strengthening with external steel is that the section size of the members increases little, but the bearing capacity increases greatly. For square column or rectangular column, the method of wrapping angle steel is mostly adopted, and angle steel is welded crosswise into a whole by hoop plate. For circular RC column, the method of wrapping flat steel and hoop is commonly adopted [28]. During the strengthening, the angle steels are bonded to the original RC column so that the transverse deformation of original RC column is restrained by angle steel, which improves not only the bearing capacity but also the ductility of column greatly. However, the transverse deformation of concrete produces lateral extrusion on the angle steel, which makes the angle steel in the compression-flexure state, resulting in the reduction of compressive bearing capacity of angle steel [29]. In addition, the stress hysteresis phenomenon often exists in angle steel, which affects the strength of angle steel to give full play [30]. Therefore, the design strength of angle steel shall be reduced in the design of the wet strengthening method with externally wrapped steel [31]. After the wet strengthening with externally wrapped steel, due to the thin postpouring concrete layer and the existence of angle steel and hoop plate, the bond between the postpouring concrete layer and original RC column is weakened. In the limit state, the postpouring concrete layer is likely to peel off first [32]. Therefore, the effect of the concrete in the postpouring layer is omitted in the calculation of strengthening design.Based on engineering projects, this paper introduces the strengthening method of circular RC columns with externally wrapped cold-formed steel plates. In this strengthening method, two semicylindrical steel plates are installed with chemical anchors after the surface of original RC columns is roughened, then two plates are welded into steel pipes on the spot, and finally grouting material or fine stone concrete is poured in the gap to form a new composite structure. Compared with the previous strengthening methods, the externally wrapped steel pipe strengthening method has its own advantages: (1) Convenient construction. The section shapes can be changed [33]. The external steel pipe itself can be used as a formwork for newly cast concrete, which is not only convenient for construction but also can meet the requirements of changing section shape from square to round. (2) Little effect on the external dimension of the members. Compared with the method of structure member strengthening with increasing section area, the section size and structural weight of the strengthened members are not increased greatly in the method of structure member strengthening with externally wrapped steel pipes, which will not affect the use space of building. (3) Good overall performance. Due to the constraints of the steel pipes, the overall performance of the strengthened members is better, and the bond between the new and old concrete joint surfaces is stronger. Meanwhile, the lateral deformation capacity of the strengthened columns is also enhanced [34], making the original RC columns in a good three-way stress state [35]. (4) High compressive bearing capacity and good seismic performance [36]. Due to the effective constraint of the round steel pipes on the original RC, the compressive bearing capacity, ductility, and energy dissipation capacity of the members are significantly improved, and the brittle failure of RC columns under high axial compression ratio is effectively improved. (5) High shear and flexural bearing capacity [37]. The strengthened columns have high shear and flexural bearing capacity. Thus, the frame structure has strong interlayer deformation capacity, which can effectively prevent the brittle shear failure of short columns. (6) High temperature resistance and good fire resistance [38]. The existence of external steel pipes improves the fire resistance limit of RC column. The external surfaces of steel pipes need not fireproof treatment because after the strengthened members get fire, even if the external steel pipes fail at high temperature, the concrete will not flake off and the RC inside steel pipes can continue to bear the load. (7) High local buckling capacity of steel pipes [39]. From the beginning of the loading on RC, the steel pipes have a transverse constraint effect on it. The longitudinal stress in the steel pipes mainly comes from the bonding friction between steel pipes and concrete. Because the longitudinal force generated by bonding friction is much less than the longitudinal stress generated by the direct bearing load of steel pipes, local buckling of thin-walled steel pipes due to compression can be avoided or postponed. Therefore, the constraint effect on RC and good tensile properties of steel pipes can be fully utilized, to achieve a better combination of steel pipes and RC.The research on the method of structure member strengthening with externally wrapped steel pipes began in the early 1980s, but compared with other strengthening methods, there are not many engineering cases and research literatures available. Park et al. [40] made 6 specimens to study the seismic performance of RC bridge piers with externally wrapped steel pipes, of which 4 were continuous casing models and 2 were discontinuous casing models. The low-cyclic and repeated load test showed that the RC bridge piers with externally wrapped steel pipes had good seismic performance, predictable strength, and ductility capacity that exceeded the requirements of current codes on New Zealand bridges and buildings. Meanwhile, it also showed that the seismic performance of discontinuous casing model was better, while the seismic performance of continuous casing model was lower than that of discontinuous casing model due to local buckling. Priestley et al. [41, 42] used circular and elliptic steel pipes to strengthen the circular and square piers of RC bridge, respectively, and conducted experimental research and theoretical analysis on their shear strength. The study showed that the strengthening with externally wrapped steel pipes could significantly improve the shear and flexural capacity of the strengthened columns. Tomii et al. [43] proposed to disconnect the steel pipe at the ends of the column so that the steel pipe would mainly play a restraining role and did not directly bear the axial pressure, indicating that the strengthened circular column had better seismic performance, but the strengthened square column was not good in seismic performance. Cai Jian et al. [33, 44] studied the mechanical properties of axial and eccentric compression on square columns strengthened by externally wrapped circular steel pipes. However, the gap was filled with fine stone concrete, which was difficult to construct, and the seismic performance was not studied. Wang Mei-hua et al. took the strengthening of RC columns in the reconstruction process of China Minsheng Bank Building as background, carried out uniaxial compression test of columns strengthened by externally wrapped circular steel pipes, and studied the construction technology of strengthening columns with initial stress, which provided reference for the design and construction of similar projects [45]. Lu Yi-yan and Xue Ji-feng et al. conducted axial compression tests on short columns and medium-long columns with circular cross section strengthened with externally wrapped steel pipes and self-compacting concrete. After analyzing the influence of steel ratio, strength of later-pouring self-compacting concrete on the bearing capacity, deformation regularity, and failure characteristics of the specimens, they proposed a simplified formula for calculating the bearing capacity of RC circular short columns and medium-long columns strengthened with externally wrapped steel pipes and self-compacting concrete [46–48]. Hu Xiao et al. carried out axial compression tests on RC short columns strengthened with externally wrapped steel pipes and grouting material. Based on the test results, the formula for calculating the axial compression bearing capacity of the strengthened members was established [49, 50]. The above literatures showed that the externally wrapped steel pipes can effectively improve the compressive bearing capacity and ductility of the strengthened members due to the increase of the constraint effect on the core RC and enhance the seismic performance. The seismic performance of the members strengthened with discontinuous steel pipes was better, and the seismic performance of the members strengthened with continuous steel pipes was less than ideal due to the local buckling under compression. However, the above literatures only studied the axial compressive performance or horizontal seismic performance, and mostly, the gaps in RC columns strengthened with wrapped steel pipes were filled with cement mortar, resulting in low overall workability. Though the strengthening method with discontinuous steel pipes was of better seismic performance, the ultimate bearing capacity of RC columns was not improved greatly. The identification results of the project in this paper determined that the strengthened columns required not only a significant increase in axial compressive bearing capacity but also a good ductility and seismic performance. If using the discontinuous steel pipes for strengthening, the axial compressive capacity could not meet the requirements. Meanwhile, the construction period given by party A was very short, only 33 days (during summer vacation). Therefore, this paper introduced and studied the strengthening method with continuous externally wrapped steel pipes, providing references for engineering applications and theoretical development of such strengthening technology.


## 2. Engineering Project of Strengthening

### 2.1. Engineering Background

The hall-type public student canteen of a school is a two-story RC frame structure, built in 2003, with a construction area of 1200 square meters and a height of 4.2 meters on both the first and second floors. Due to the increasing number of students, the usable area of the canteen was far from meeting the dining requirements of students. Therefore, the canteen needed add-layer expansion. The plan was to add two-floor multifunctional restaurants. In May 2015, the school entrusted an evaluation institution to appraise the canteen. After consulting the project acceptance documents, investigating the current situation of application, and conducting on-site nondestructive test, the strength and aging degree of concrete, material quality of steel, and other material properties of the frame beam and column or other load-bearing members were in line with the requirements of the current national codes and standards. There was no deformation or crack in the whole structure. Therefore, the key content to identify was checking and verifying the bearing capacity of the foundation, as well as the RC columns of the first and second floors and the RC beams of the second floor after adding two layers. According to the design drawings of the third and fourth floors to be added, combined with the currentCode for Design of Concrete Structures (GB 50010-2010) [51] and Load Code for the Design of Building Structures (GB 50009-2012) [52], the bearing capacity of the foundation, the RC columns of the first and second floors, and the RC beams of the second floor could not meet the requirements. In order to meet the requirements proposed by party A that try not to change the original use space and the strengthening construction period, the strengthening scheme given by the evaluation institution was as follows: the foundation would be strengthened with post-installed rebar technology to increase the foundation section area and the RC columns of the first and second floors and the RC beams of the second floor would be strengthened with integrated externally wrapped steel plates. The site of strengthening RC beams and columns on the second floor with externally wrapped steel plates is shown in Figure 1. The original RC columns were circular section, diameter d = 450 mm, with 6B22 longitudinal compression steel bar and A10@100 spiral stirrup. The reinforcement is shown in Figure 2. RC beams were rectangular section, with size b = 300 mm and h = 600 mm. The concrete strength grades were all C30. This paper only focuses on the introduction and further analysis of the strengthening of RC columns, which were strengthened by steel plates with the thickness of t = 8 mm.Figure 1
Photos of circular RC columns strengthened by steel plates: (a) before strengthening; (b) during strengthening; (c) after strengthening.
(a)
(b)
(c)Figure 2
Reinforcement diagram of the original RC columns.


### 2.2. Construction Method of Strengthening

The schematic diagram of the externally wrapped steel plates strengthening method for circular RC columns in the engineering project is shown in Figure3. The construction process is as follows:(1)
After shovelling away the paint layer of the RC columns, clean the concrete surface, remove loose mortar and floating dust, trim the protruding parts, polish them with a grinder, and then blow off the dust with a fan (Figure1(a)).
(2)
Roughen the inside of the steel plate by a grinding machine. Machine the steel plate into a semicylindrical shape, according to the size of RC column.
(3)
Drill the steel plate and then drill the RC column through the former hole. Install the two semicylindrical steel plates with chemical anchors and complete the welding. The edge distance of chemical anchors is 200 mm and the spacing is 300 mm [53], as shown in Figure 3.
(4)
Pour epoxy resin into the gap between the steel plate and RC column with the grouting pipe. During the pouring process, knock the steel plate gently on the external surface to judge the compaction effect of the pouring according to the sound.
(5)
After the fully curing of epoxy resin, treat the steel surface with rust removal, corrosion prevention, and fire prevention.
(6)
Spot weld steel wire mesh on the external surface of steel plate, with the mesh size less than 5 cm × 5 cm (Figure1(b)). Then, trowel it with 1 : 3 cement mortar and spray paint for restoration (Figure 1(c)) (this process is a special requirement by party A of this project; the first five processes are usually enough).Figure 3
Profile diagram of RC column after strengthening. (1) Epoxy resin; (2) steel plate for strengthening; (3) chemical anchor bolt; (4) welded steel wire mesh on the steel plate's surface; (5) troweled with 1 : 3 cement mortar; (6) the surface sprayed with paint for recovery.


## 2.1. Engineering Background

The hall-type public student canteen of a school is a two-story RC frame structure, built in 2003, with a construction area of 1200 square meters and a height of 4.2 meters on both the first and second floors. Due to the increasing number of students, the usable area of the canteen was far from meeting the dining requirements of students. Therefore, the canteen needed add-layer expansion. The plan was to add two-floor multifunctional restaurants. In May 2015, the school entrusted an evaluation institution to appraise the canteen. After consulting the project acceptance documents, investigating the current situation of application, and conducting on-site nondestructive test, the strength and aging degree of concrete, material quality of steel, and other material properties of the frame beam and column or other load-bearing members were in line with the requirements of the current national codes and standards. There was no deformation or crack in the whole structure. Therefore, the key content to identify was checking and verifying the bearing capacity of the foundation, as well as the RC columns of the first and second floors and the RC beams of the second floor after adding two layers. According to the design drawings of the third and fourth floors to be added, combined with the currentCode for Design of Concrete Structures (GB 50010-2010) [51] and Load Code for the Design of Building Structures (GB 50009-2012) [52], the bearing capacity of the foundation, the RC columns of the first and second floors, and the RC beams of the second floor could not meet the requirements. In order to meet the requirements proposed by party A that try not to change the original use space and the strengthening construction period, the strengthening scheme given by the evaluation institution was as follows: the foundation would be strengthened with post-installed rebar technology to increase the foundation section area and the RC columns of the first and second floors and the RC beams of the second floor would be strengthened with integrated externally wrapped steel plates. The site of strengthening RC beams and columns on the second floor with externally wrapped steel plates is shown in Figure 1. The original RC columns were circular section, diameter d = 450 mm, with 6B22 longitudinal compression steel bar and A10@100 spiral stirrup. The reinforcement is shown in Figure 2. RC beams were rectangular section, with size b = 300 mm and h = 600 mm. The concrete strength grades were all C30. This paper only focuses on the introduction and further analysis of the strengthening of RC columns, which were strengthened by steel plates with the thickness of t = 8 mm.Figure 1
Photos of circular RC columns strengthened by steel plates: (a) before strengthening; (b) during strengthening; (c) after strengthening.
(a)
(b)
(c)Figure 2
Reinforcement diagram of the original RC columns.


## 2.2. Construction Method of Strengthening

The schematic diagram of the externally wrapped steel plates strengthening method for circular RC columns in the engineering project is shown in Figure3. The construction process is as follows:(1)
After shovelling away the paint layer of the RC columns, clean the concrete surface, remove loose mortar and floating dust, trim the protruding parts, polish them with a grinder, and then blow off the dust with a fan (Figure1(a)).
(2)
Roughen the inside of the steel plate by a grinding machine. Machine the steel plate into a semicylindrical shape, according to the size of RC column.
(3)
Drill the steel plate and then drill the RC column through the former hole. Install the two semicylindrical steel plates with chemical anchors and complete the welding. The edge distance of chemical anchors is 200 mm and the spacing is 300 mm [53], as shown in Figure 3.
(4)
Pour epoxy resin into the gap between the steel plate and RC column with the grouting pipe. During the pouring process, knock the steel plate gently on the external surface to judge the compaction effect of the pouring according to the sound.
(5)
After the fully curing of epoxy resin, treat the steel surface with rust removal, corrosion prevention, and fire prevention.
(6)
Spot weld steel wire mesh on the external surface of steel plate, with the mesh size less than 5 cm × 5 cm (Figure1(b)). Then, trowel it with 1 : 3 cement mortar and spray paint for restoration (Figure 1(c)) (this process is a special requirement by party A of this project; the first five processes are usually enough).Figure 3
Profile diagram of RC column after strengthening. (1) Epoxy resin; (2) steel plate for strengthening; (3) chemical anchor bolt; (4) welded steel wire mesh on the steel plate's surface; (5) troweled with 1 : 3 cement mortar; (6) the surface sprayed with paint for recovery.


## 3. Calculation of Bearing Capacity

### 3.1. Formula of Bearing Capacity before Strengthening

According to theCode for Design of Concrete Structures, the bearing capacity of the normal section of a circular RC column with spiral stirrup before strengthening can be written as(1)Nu0=0.9fc0Acor0+fy0′As0′+2αfyv0Ass0,where fc0 is the design value of axial compressive strength of concrete in original member; Acor0 represents the area of core section of original member, taking the area of concrete section within the scope of the inner surface of the indirect reinforcement; fy0′ is the design value of compressive strength of steel reinforcement in original member; As0′ is the cross-sectional area of compressive reinforcement in original member; α represents the reduction coefficient of the constraining on concrete by indirect reinforcement, taking 1.0 when the concrete strength grade does not exceed C50, taking 0.85 when the concrete strength grade is C80, and the intermediate value can be determined by linear interpolation; fyv0 is the design value of tensile strength of indirect reinforcement in original member; Ass0 represents the conversion sectional area of spiral indirect reinforcement in original member, Ass0=πdcor0Ass1/s0; dcor0 is the diameter of the core section in the original member, taking the distance between the inner surface of the indirect reinforcement; Ass1 represents the cross-sectional area of the single spiral indirect reinforcement in the original member; and s0 is the spacing of the indirect reinforcement along the axis of the member.


### 3.2. Formula of Bearing Capacity after Strengthening

The normal section bearing capacity of the column strengthened by the wet-type method with externally wrapped steel plates can be calculated by the full section. However, the design value of strength of external steel plate shall be multiplied by the reduction factor of 0.9. The improvement of bearing capacity after strengthening consists of two parts: one is the direct contribution of strengthening steel plate to the bearing capacity of RC column and the other is the improvement of bearing capacity due to the increasing constraint degree on RC column by the strengthening steel plate [54]. Therefore, according to formula (1) and in combination with Code for Design of Strengthening Concrete Structure (GB50367-2013) [55] and Code for Design of Concrete Structures, formula (2) for the bearing capacity of the normal section of a circular RC column strengthened with externally wrapped steel plates can be directly derived:(2)Nu=0.9ψscfc0Acor0+fy0′As0′+2αfyv0Ass0+αafa′Aa′,where Nu is the ultimate axial compression bearing capacity of the normal section of the RC column after strengthening; ψsc is the improvement coefficient of bearing capacity to consider the constraint effect of strengthening steel plate on RC column, taking 1.15 here for circular section columns; αa is the strength utilization coefficient of the steel plate for strengthening, taking 0.9; fa′ is the design value of compression strength of the strengthening steel plate, which is determined according to the Standard for Classification of Steel Structures (GB 50017-2017) [56]; and Aa′ is the cross-sectional area of the strengthening steel plate, taking πdt in the conservative approximate calculation formula. The meanings of other symbols are consistent with formula (1).


## 3.1. Formula of Bearing Capacity before Strengthening

According to theCode for Design of Concrete Structures, the bearing capacity of the normal section of a circular RC column with spiral stirrup before strengthening can be written as(1)Nu0=0.9fc0Acor0+fy0′As0′+2αfyv0Ass0,where fc0 is the design value of axial compressive strength of concrete in original member; Acor0 represents the area of core section of original member, taking the area of concrete section within the scope of the inner surface of the indirect reinforcement; fy0′ is the design value of compressive strength of steel reinforcement in original member; As0′ is the cross-sectional area of compressive reinforcement in original member; α represents the reduction coefficient of the constraining on concrete by indirect reinforcement, taking 1.0 when the concrete strength grade does not exceed C50, taking 0.85 when the concrete strength grade is C80, and the intermediate value can be determined by linear interpolation; fyv0 is the design value of tensile strength of indirect reinforcement in original member; Ass0 represents the conversion sectional area of spiral indirect reinforcement in original member, Ass0=πdcor0Ass1/s0; dcor0 is the diameter of the core section in the original member, taking the distance between the inner surface of the indirect reinforcement; Ass1 represents the cross-sectional area of the single spiral indirect reinforcement in the original member; and s0 is the spacing of the indirect reinforcement along the axis of the member.


## 3.2. Formula of Bearing Capacity after Strengthening

The normal section bearing capacity of the column strengthened by the wet-type method with externally wrapped steel plates can be calculated by the full section. However, the design value of strength of external steel plate shall be multiplied by the reduction factor of 0.9. The improvement of bearing capacity after strengthening consists of two parts: one is the direct contribution of strengthening steel plate to the bearing capacity of RC column and the other is the improvement of bearing capacity due to the increasing constraint degree on RC column by the strengthening steel plate [54]. Therefore, according to formula (1) and in combination with Code for Design of Strengthening Concrete Structure (GB50367-2013) [55] and Code for Design of Concrete Structures, formula (2) for the bearing capacity of the normal section of a circular RC column strengthened with externally wrapped steel plates can be directly derived:(2)Nu=0.9ψscfc0Acor0+fy0′As0′+2αfyv0Ass0+αafa′Aa′,where Nu is the ultimate axial compression bearing capacity of the normal section of the RC column after strengthening; ψsc is the improvement coefficient of bearing capacity to consider the constraint effect of strengthening steel plate on RC column, taking 1.15 here for circular section columns; αa is the strength utilization coefficient of the steel plate for strengthening, taking 0.9; fa′ is the design value of compression strength of the strengthening steel plate, which is determined according to the Standard for Classification of Steel Structures (GB 50017-2017) [56]; and Aa′ is the cross-sectional area of the strengthening steel plate, taking πdt in the conservative approximate calculation formula. The meanings of other symbols are consistent with formula (1).


## 4. FEM Analysis

FEM software ABAQUS is widely used in the strengthening field. Simulation with this software before strengthening is a direct and economic method, which plays a vital role in the calculation of strengthening design and the prediction of strengthening effect [57]. In this paper, FEM simulation calculation and comparative analysis are carried out for the RC columns before and after strengthening in the engineering case.


### 4.1. Constitutive Relation of Materials

#### 4.1.1. Constitutive Relation of Concrete

The concrete damaged plasticity (CDP) model is adopted in the constitutive relationship of concrete material [58, 59]. This model can be applied to unidirectional, cyclic, and dynamic loading and has good convergence. The whole-process constitutive relation curve of concrete tension and compression is shown in Figure 4.Figure 4
Constitutive relation of concrete.The stress-strain curve of concrete under uniaxial tension can be determined by the following formula:(3)σc=1−dtEcεc,dt=1−ρt1.2−0.2x5,x≤1,1−ρtαtx−11.7+x,x>1,x=εcεt,r,ρt=ft,rEcεt,r,where σc represents the concrete stress; εc represents the concrete strain; Ec is the elastic modulus of concrete; αt is the parameter value of the descending section in the uniaxial tensile stress-strain curve of concrete; ft,r is the representative value of uniaxial tensile strength of concrete, which can be taken ft, ftk, or according to the actual needs of structural analysis; εt,r represents the peak tensile strain of concrete corresponding to the representative value of uniaxial tensile strength ft,r; dt is the damage parameter of concrete in uniaxial tensile; ft is the design value of the axial tensile strength of concrete; ftk is the standard value of tensile strength of concrete; ftm is the average tensile strength of concrete, ftm=ftk/1−1.645δc; and δc represents the variation coefficient of concrete strength.The stress-strain curve of concrete under uniaxial compression can be determined by the following formula:(4)σc=1−dcEcεc,dc=1−ρcnn−1+xn,x≤1,1−ρcαcx−12+x,x>1,x=εcεc,r,ρc=fc,rEcεc,r,n=Ecεc,rEcεc,r−fc,r,where αc is the parameter value of the descending section in the uniaxial compression stress-strain curve of concrete; fc,r is the representative value of uniaxial compression strength of concrete, which can be taken fc, fck, or fcm; εc,r represents the peak compression strain of concrete corresponding to the representative value of uniaxial compression strength ft,r; dc is the damage parameter of concrete in uniaxial compression; fc is the design value of the axial compression strength of concrete; fck is the standard value of compression strength of concrete; and fcm is the average compression strength of concrete, fcm=fck/1−1.645δc.The material behavior parameters of concrete required for ABAQUS modeling are shown in Table1.Table 1
The material behavior parameters of concrete.
ρ (t/mm3)
E
c (MPa)
v
c
θ (°)
e
f
b0/fc0
k
μ
2.4 × 10−9
3.0 × 104
0.2
30
0.1
1.16
0.667
0.005
Note. ρ represents the concrete density; vc is Poisson's ratio of concrete; θ is the expansion angle; e is the eccentricity ratio; fb0 represents the strength of concrete under biaxial constant pressure; fc0 is the uniaxial strength of concrete; k is the ratio of the second stress invariant on tension and compression meridian; μ is the coefficient of viscosity.


#### 4.1.2. Constitutive Relation of Steel

The material constitutive relationship of compression reinforcement, stirrup, and steel plate all adopts the ideal bilinear elastoplastic model. The constitutive relation curve is shown in Figure5 [60], which can be expressed by the following formula:(5)σs=Esεs,εs≤εy,σy,εs>εy,where σs is the stress of steel; εs is the strain of steel; and Es represents the elastic modulus of steel.Figure 5
Constitutive relation of steel.In this paper, the compressive reinforcement, the stirrups, and the steel plates of the RC columns analyzed adopt HRB335, HPB235, and Q235, respectively, the measured yield strength was 361 MPa, 275 MPa, and 287 MPa, respectively, and the measured tensile strength was 464 MPa, 398 MPa, and 431 MPa respectively. According to GB 50010-2010 and GB 50017-2017, the elastic modulusEs of the three kinds of steel is 2.0 × 105 MPa, 2.1 × 105 MPa, and 2.06 × 105 MPa, respectively. Poisson’s ratio all takes 0.3, and the density all takes ρs = 7.85 × 10−9 tonne/mm3.


### 4.2. FEM Modeling

The part between the two reverse bending points of the frame columns on the first and second floors was taken as the study object, and the modeling analysis was carried out with ABAQUS software. The model was composed of three parts: steel bar skeleton, concrete, and steel plate for strengthening. The bar skeleton was simulated by linear element of truss (T3D2), which was subjected only to axial force but not shear force and bending moment. Three-dimensional linear reduced integrated element (C3D8R) was selected for concrete and strengthening steel plate [61], which was one integral point less than the complete integral element in each direction with higher calculation efficiency.In the simulation of interaction, the slip between the reinforcement skeleton and concrete was not taken into account. The steel bar skeleton was taken as a built-in unit, which was built into the concrete unit by Embedded command. The interaction between concrete and strengthening steel plate could be simulated by Tie command, since epoxy resin was poured between the two and chemical anchors were used as the fixed connection to ensure the costressing and coordinated deformation of the two.When simulating the modeling boundary conditions of the frame columns, in order to ensure that no out-of-plane instability occurred during loading, the boundary conditions on the bottom of the columns were set as U1 = U2 = U3 = 0 and those of the top on the columns were set as U1 = U2 = 0 and UR2 = UR3 = 0 [62]. The FEM models of steel bar skeleton, concrete, strengthening steel plate, and strengthened frame column are shown in Figure 6, where Figure 6(d) is the loading diagram with boundary conditions.Figure 6
FEM models: (a) FEM model of steel bar skeleton; (b) FEM model of concrete; (c) FEM model of steel plates for strengthening; (d) FEM model of frame columns after reinforcement.
(a)
(b)
(c)
(d)


### 4.3. FEM Calculation Results and Analysis

#### 4.3.1. Analysis of Axial Compression Action

In order to avoid local failure caused by concentrated load, two disc-shaped discrete rigid sheets (Figure6(d)) are set at the bottom and the top of the column, respectively, to apply concentrated load, and the displacement-controlled monotone static loading method is adopted to apply axial pressure [63].(1) Relationship between Axial Compression Load and Axial Strain. The vertical monotonic static loading simulation analysis was carried out on RC columns before and after strengthening, respectively, with ABAQUS software, and the relationship curve between axial compression load and axial strain is shown in Figure 7. Combined with Figure 7 and ABAQUS postprocessing data, the ultimate axial compression bearing capacities of the normal section of the RC frame column before and after strengthening are 2765 kN and 4896 kN, respectively. The strengthening brings a 77% improvement degree of the bearing capacity. The curve before strengthening is divided into the elastic stage, elastic-plastic stage, and instability decline stage. The ultimate instability occurs in the elastic-plastic stage, that is, the bending failure, and the strain in the instability is 0.0021. The curve after strengthening is divided into the elastic stage, elastic-plastic stage, plastic stage, and instability decline stage. The ultimate instability occurs in the plastic stage, that is, the bending failure, and the strain in the instability is 0.0058. The ultimate strains at the time of failure before and after strengthening are 0.0053 and 0.0094, respectively. The result indicates that the axial compression deformation capacity of the strengthened frame column is improved, and the material properties are given full play.Figure 7
Relationship between axial compression and strains.(2) Comparison between FEM Calculation and Theoretical Formula Calculation. According to the design drawing of the proposed additional layers and the Load Code for the Design of Building Structures, the design value of the maximum bearing capacity of the normal section of the frame column on the first floor is calculated to be 3746 kN. According to formula (1) and the Code for Design of Concrete Structures, the ultimate bearing capacity of the normal section of the frame column before strengthening is calculated to be 2809 kN. By using formula (2) in combination with the Code for Design of Strengthening Concrete Structure, the ultimate bearing capacity of the normal section of the strengthened frame column is calculated to be 5100 kN. The ultimate bearing capacity of normal section of frame column after strengthening calculated by the theoretical formula is 82% higher than that before strengthening. The data analyzed by ABAQUS software and the data calculated by theoretical formulae are summarized as shown in Table 2. It can be seen that the bearing capacities before strengthening calculated by both the FEM method and theoretical formulae are less than the designed value of bearing capacity, which does not meet the requirements, while the calculated bearing capacities after strengthening are both greater than the designed value, meeting the requirements. In addition, the ratios before and after strengthening between the FEM analysis results and the calculation results by the theoretical formulae are 0.98 and 0.96, respectively, indicating that the FEM calculation is in good agreement with the theoretical formula calculation.Table 2
Review and comparison of bearing capacity.
Design value (kN)
Before reinforcement
After reinforcement
FEM calculated value (kN)
Formula (1) (kN)
Ratio
FEM calculated value (kN)
Formula (2) (kN)
Ratio
3746
2765
2809
0.98
4896
5100
0.96
Note. The ratio is the ratio between the FEM analysis results and the formula calculation results.


#### 4.3.2. Analysis of Horizontal Low-Cyclic Loading

In order to avoid local damage at the action position of concentrated horizontal load and tensile failure at the beam end, a discrete rigid frame loading-frame (Figure6(d)) is set in the horizontal direction to apply horizontal low-cyclic reversed loading. Referring to Specification for Seismic Test of Buildings (JGJ101-2015) [64], the vertical constant load N is firstly applied according to the axial compression ratio, and then the horizontal low cycle load is applied. Force-displacement mixed control loading mode is adopted. Force-control loading mode is adopted before yield, in which case each grade of load is reversed once. Displacement-control loading mode is adopted after yield, in which case the deformation value takes the horizontal displacement value Δy of the members at yield. The multiple of Δy is used as the range for the displacement-control loading mode, with each grade of load repeated three times [28], loading until the component failure or load dropped to about 85% of the maximum load. The loading system [65] is shown in Figure 8.Figure 8
Loading system.In this paper, the loading system shown in Figure8 is adopted by ABAQUS software, and seismic analysis is carried out for 6 working conditions when the control axial pressure ratios (u) of frame column before and after strengthening are 0.3, 0.6, and 0.9, respectively. The analyzed stress field is shown in Figure 9. When u equals to 0.9 of the strengthened frame column. The material strength of both reinforcement and concrete of the frame column after strengthening is fully developed.Figure 9
Stress field: (a) stress field of reinforcement skeleton; (b) stress field of concrete; (c) stress field of steel plate.
(a)
(b)
(c)(1) P-Δ Hysteretic Curve. The P-Δ hysteretic curves of frame columns before and after strengthening in 6 working conditions are shown in Figure 10. Before strengthening, the shapes of the frame columns’ P-Δ hysteresis loops under three axial compression ratios are all bow with “rheostriction,” indicating that the circular RC frame column with spiral stirrups has a strong energy dissipation capacity. After strengthening, the shapes of the frame column’s hysteresis loop under three axial compression ratios are fusiform, which is fuller than the bow before strengthening, and the ultimate load and ultimate displacement are significantly improved from that before strengthening, indicating that the strengthened frame column has a strong energy dissipation ability and good seismic performance. In all hysteresis curves, the hysteresis loops of each grade of load repeated three times basically coincide. The hysteresis curves are stable, indicating no obvious stiffness degradation phenomenon [66].Figure 10
Hysteretic curves: (a) before reinforcement (u = 0.3); (b) before reinforcement (u = 0.6); (c) before reinforcement (u = 0.9); (d) after reinforcement (u = 0.3); (e) after reinforcement (u = 0.6); (f) after reinforcement (u = 0.9).
(a)
(b)
(c)
(d)
(e)
(f)(2) Skeleton Curves and Ductility Coefficients. The skeleton curves corresponding to all hysteresis curves before and after strengthening are shown in Figure 11. With the increase of axial compression ratio, the peak loads corresponding to all skeleton curves increase and the ultimate displacements decrease, which indicates that increasing axial pressure ratio can improve the horizontal resistance of the column but reduce the ductility of the column. Compared with that before strengthening, both the peak load and limit displacement increase to different extent after strengthening, indicating that the energy dissipation capacity is obviously enhanced. Skeleton curves can be divided into two categories: one with a descending stage and the other without. The skeleton curves before strengthening have obvious descending stage. A more obvious descending is occurred with increasing the axial compression ratio. After strengthening, the skeleton curves have no descending stage when u equals to 0.3 or 0.6; when u = 0.9, it is the skeleton curve with descending stage. Whether the skeleton curve has a descending stage or not can reflect the change of ductility coefficient and the following is a brief discussion on the change of ductility coefficient.Figure 11
Skeleton curves: (a) before reinforcement; (b) after reinforcement.
(a)
(b)Ductility coefficient (μ) is the ratio of displacement when the horizontal load is reduced to 85% of the maximum load to yield displacement; it is the index of ductility deformation ability of structure and can reflect the seismic performance of structure. The influence of axial compression ratio on ductility coefficient before and after strengthening of frame column is shown in Figure 12. Before strengthening, the ductility coefficient decreases with increasing the axial compression ratio. After strengthening, the ductility coefficient is infinite when the axial compression ratio is less than 0.6. When the axial compression ratio is greater than 0.6, the ductility coefficient also decreases with the increase of the axial compression ratio. Moreover, the ductility coefficient curve after strengthening is much higher than that before strengthening, proving that the strengthened frame column has good plastic deformation capacity and good seismic performance.Figure 12
The influence of axial compression ratio on ductility factor.(3) Analysis of Energy Consumption. There is no unified standard to evaluate the energy dissipation capacity of components till now. The abovementioned hysteretic curve, skeleton curve, and ductility coefficient can be used to evaluate the energy dissipation capacity qualitatively instead of quantitatively. Generally, equivalent viscous damping coefficient and energy dissipation coefficient (the energy dissipation coefficient is 2π of the equivalent viscous damping coefficient) are used to quantitatively determine the energy dissipation capacity of components [67]. The formula of equivalent viscous damping coefficient is as follows:(6)he=12π⋅ShSOAB+SOCD,and the symbols in the formula are shown in Figure 13: Sh represents the area of hysteretic loop; SOAB is the area of triangle OAB; and SOCD is the area of triangle OCD.Figure 13
Hysteretic loop.The relation curve between the equivalent viscous damping coefficienthe and the recurrence number n of horizontal load is the energy dissipation curve, as shown in Figure 14. As can be seen from Figure 14, when n is less than 12, the greater the axial pressure ratio u is, the higher he is, indicating that increasing the axial pressure ratio is beneficial to the energy consumption of members at the early loading stage. When n is greater than 12, the larger u is, the smaller he is, indicating that increasing axial compression ratio is unfavorable to energy dissipation of members at the later loading stage. Therefore, the axial compression ratio of columns shall be limited, regarding the earthquake resistance. Before strengthening, with the increase of n, he presents an upward trend; when n is less than 6, with the members in the elastic stage, he is small and increases slowly, and energy dissipation capacity is low; when n is within the range of 6∼18, with the members in the elastoplastic stage, he increases fast, and the energy dissipation capacity increases rapidly; when n is greater than 18, with the members in the plastic stage, he slowly increases to the maximum value, and the energy dissipation capacity reaches a high degree. After strengthening, when n is less than 12, he increases fast and energy dissipation capacity increases rapidly; when n is greater than 12, he reaches the maximum value, and the energy dissipation capacity of members no longer increases, and the energy dissipation capacity of members with u = 0.9 slightly decreases. Compared with that before strengthening, he after strengthening has been obviously improved, which proves that the frame column after strengthening has good energy dissipation capacity and seismic performance.Figure 14
Energy-absorbed curve: (a) before reinforcement; (b) after reinforcement.
(a)
(b)


### 4.4. Validation of FEM Model

In order to fully verify the accuracy and effectiveness of the proposed model, the FEM modeling method introduced in Section4.2 is used to conduct modeling analysis on the samples in similar literature [41, 42, 45, 68, 69], and the analysis results are compared with the test data. The following is a brief introduction to the experimental overview of literature [41, 42, 45, 68, 69]:In literature [41, 42], Priestley et al. conducted a systematic study on bridge piers strengthened with externally wrapped steel and introduced a set of design methods for strengthening, which guided a large number of bridge strengthening projects in California and other regions. In order to study the brittle shear failure mode of the built circular bridge piers, four models of the piers strengthened with external steel plates were made, with the model scale of 1 : 2.5. They carried out shear tests in the horizontal direction with the control axial compression ratios of 0.06 and 0.18, respectively. The research showed that the shear and flexural capacity of the strengthened column could be greatly improved by the externally wrapped steel strengthening method. Wang [45] made four RC columns strengthened by circular steel tubes, of which two were strengthened by steel pipes without initial load, and the other two were strengthened by steel pipes with initial axial load of 500 kN and 1000 kN, respectively. The gaps between steel pipes and RC column were filled with cement mortar material of the same grade and high-strength nonshrinkage grouting material, respectively. After the axial compression tests on 4 specimens, the results showed that the RC column strengthened by circular steel pipes could greatly improve the bearing capacity and ductility. There was no slippage between the outer steel pipes and the core RC columns. The two deformed with coordination and worked together.After searching, there are relatively few studies on the experimental research of circular RC columns strengthened with steel plates. In a broad sense, the circular RC columns strengthened by externally wrapped steel plates belong to the category of circular tube confined RC (hereinafter referred to as CTRC) columns [68, 69], and the research on CTRC columns is relatively extensive. The test in literature [68] took the frame columns as the research object and assumed that the columns were fixed at both ends. First, the columns were subjected to a constant vertical axial pressure load controlled by the axial pressure ratio, and then repeated horizontal load was applied at the top of the columns to simulate the seismic load. A total of 4 CTRC column specimens were made, one filled with normal strength concrete (C30) and three filled with high-strength concrete (C60). The test results showed that the CTRC columns had good flexural capacity, ductility, and energy dissipation capacity, which effectively improved the brittle failure of high-strength RC columns under high axial compression ratio. Zhou et al. [69] made 16 CTRC column specimens and conducted eccentric compression test on them with slenderness ratio, eccentricity, and steel content ratio as parameters. According to the mechanical characteristics, the specimens were divided into two modes, A and B: a model steel pipe wall connected up and down (only slotted at both ends of the specimens); in mode B, the middle part of the steel pipe wall was also slotted, which divided the steel pipe wall into upper and lower parts. The test results showed that the long CTRC columns had good ductility and still maintained a high bearing capacity after the peak; with the increase of slenderness ratio and eccentricity, the stiffness of specimens decreased, but ductility increased; mode A and B had no obvious influence on the bearing capacity and ductility of the specimens. The reinforcement diagram of specimens in literature [41, 42, 45, 68, 69] is shown in Figure 15, and other test parameters are shown in Table 3.Figure 15
Reinforcement diagram of the specimens in literature: (a) literature [41, 42]; (b) literature [45]; (c) literature [68]; (d) literature [69].
(a)
(b)
(c)
(d)Table 3
Specimen parameters and comparison of ultimate bearing capacity.
Specimens
e (mm)
u
L (mm)
D (mm)
t (mm)
f
y (MPa)
f
yz (MPa)
f
yh (MPa)
f
c (MPa)
P
ut (kN)
P
uf (kN)
P
uf/Put
Loading methods
Data sources
C2R
—
0.06
1220
610
4.8
348
324
359
34
734
639
0.87
Horizontal shear
Literature [41, 42]
C4R
—
0.06
1220
610
4.8
348
324
324
35
957
813
0.85
C6R
—
0.18
1220
610
3.2
286
469
324
40
1024
850
0.83
C8R
—
0.18
915
610
3.2
286
469
324
31
1228
1105
0.90
ZY2-1
—
—
1200
280
4.0
433
400
380
36
4358
3879
0.89
Axial compression
Literature [45]
ZY2-2
—
—
1200
280
4.0
433
400
380
36
4594
4181
0.91
ZY3-1
—
—
1200
280
4.0
433
400
380
36
4700
4277
0.91
ZY3-2
—
—
1200
280
4.0
433
400
380
36
4581
4260
0.93
CTRC-60-8
—
0.8
1200
226
2.7
436
506
305
53
232
218
0.94
Horizontal circulation
Literature [68]
CTRC-60-6
—
0.6
1200
226
2.7
436
506
305
53
199
186
0.93
CTRC-60-3
—
0.3
1200
226
2.7
436
506
305
53
139
133
0.96
CTRC-30-8
—
0.8
1200
226
2.7
436
506
305
39
164
150
0.92
C200-6-25-a
25
—
1200
200
1.5
364
477
286
56
1926
1887
0.98
Eccentric compression
Literature [69]
C200-6-25-b
25
—
1200
200
1.5
364
477
286
56
1758
1688
0.96
C200-6-50-a
50
—
1200
200
1.5
364
477
286
56
1317
1225
0.93
C200-6-50-b
50
—
1200
200
1.5
364
477
286
56
1238
1201
0.97
C200-10-25-a
25
—
2000
200
1.5
364
477
286
56
1602
1522
0.95
C200-10-25-b
25
—
2000
200
1.5
364
477
286
56
1520
1490
0.98
C200-10-50-a
50
—
2000
200
1.5
364
477
286
56
1121
1054
0.94
C200-10-50-b
50
—
2000
200
1.5
364
477
286
56
1136
1102
0.97
C240-6-25-a
25
—
1440
240
1.5
364
477
286
56
2518
2468
0.98
C240-6-25-b
25
—
1440
240
1.5
364
477
286
56
2464
2292
0.93
C240-6-50-a
50
—
1440
240
1.5
364
477
286
56
1957
1879
0.96
C240-6-50-b
50
—
1440
240
1.5
364
477
286
56
1872
1685
0.90
C240-10-25-a
25
—
2400
240
1.5
364
477
286
56
2219
2175
0.98
C240-10-25-b
25
—
2400
240
1.5
364
477
286
56
2212
2124
0.96
C240-10-50-a
50
—
2400
240
1.5
364
477
286
56
1909
1756
0.92
C240-10-50-b
50
—
2400
240
1.5
364
477
286
56
1687
1653
0.98
Note. e is the eccentricity; u is the axial compression ratio; L, D, and t represent the steel pipe length, outer diameter, and wall thickness, respectively; fy, fyz, and fyh are the measured yield strength of steel pipe, longitudinal bar, and stirrup, respectively; fc is the compressive strength of concrete; Put represents the test value of bearing capacity; Puf is the FEM simulation value of bearing capacity.The analysis results are compared with the test data, as shown in Table3 and Figure 16. As can be seen from Table 3 and Figure 16, the results of FEM simulation are all smaller than the experimental data in the literatures. The biggest error is in the comparison with literature [41, 42], with Puf/Put of 0.83. The probable reasons may be the following: differences between Chinese and American codes on the value of material strength, where American codes have higher reliability and safety; differences on the structure types, where literature [41, 42] studied piers of bridges, while the FEM simulation in this paper was RC columns of buildings; and differences in time, where experiment in literature [41, 42] was carried out in the early 1990s, with a history of nearly 30 years. Compared with literature [45, 68, 69], Puf/Put is between 0.89 and 0.98. The FEM simulation results are consistent with the experimental data, and the simulation results tend to be safer. The above analysis can fully prove the accuracy and applicability of the FEM model adopted in this paper.Figure 16
Comparison of ultimate bearing capacity between FEM simulation value and experimental value [41, 42, 45, 68, 69].


## 4.1. Constitutive Relation of Materials

### 4.1.1. Constitutive Relation of Concrete

The concrete damaged plasticity (CDP) model is adopted in the constitutive relationship of concrete material [58, 59]. This model can be applied to unidirectional, cyclic, and dynamic loading and has good convergence. The whole-process constitutive relation curve of concrete tension and compression is shown in Figure 4.Figure 4
Constitutive relation of concrete.The stress-strain curve of concrete under uniaxial tension can be determined by the following formula:(3)σc=1−dtEcεc,dt=1−ρt1.2−0.2x5,x≤1,1−ρtαtx−11.7+x,x>1,x=εcεt,r,ρt=ft,rEcεt,r,where σc represents the concrete stress; εc represents the concrete strain; Ec is the elastic modulus of concrete; αt is the parameter value of the descending section in the uniaxial tensile stress-strain curve of concrete; ft,r is the representative value of uniaxial tensile strength of concrete, which can be taken ft, ftk, or according to the actual needs of structural analysis; εt,r represents the peak tensile strain of concrete corresponding to the representative value of uniaxial tensile strength ft,r; dt is the damage parameter of concrete in uniaxial tensile; ft is the design value of the axial tensile strength of concrete; ftk is the standard value of tensile strength of concrete; ftm is the average tensile strength of concrete, ftm=ftk/1−1.645δc; and δc represents the variation coefficient of concrete strength.The stress-strain curve of concrete under uniaxial compression can be determined by the following formula:(4)σc=1−dcEcεc,dc=1−ρcnn−1+xn,x≤1,1−ρcαcx−12+x,x>1,x=εcεc,r,ρc=fc,rEcεc,r,n=Ecεc,rEcεc,r−fc,r,where αc is the parameter value of the descending section in the uniaxial compression stress-strain curve of concrete; fc,r is the representative value of uniaxial compression strength of concrete, which can be taken fc, fck, or fcm; εc,r represents the peak compression strain of concrete corresponding to the representative value of uniaxial compression strength ft,r; dc is the damage parameter of concrete in uniaxial compression; fc is the design value of the axial compression strength of concrete; fck is the standard value of compression strength of concrete; and fcm is the average compression strength of concrete, fcm=fck/1−1.645δc.The material behavior parameters of concrete required for ABAQUS modeling are shown in Table1.Table 1
The material behavior parameters of concrete.
ρ (t/mm3)
E
c (MPa)
v
c
θ (°)
e
f
b0/fc0
k
μ
2.4 × 10−9
3.0 × 104
0.2
30
0.1
1.16
0.667
0.005
Note. ρ represents the concrete density; vc is Poisson's ratio of concrete; θ is the expansion angle; e is the eccentricity ratio; fb0 represents the strength of concrete under biaxial constant pressure; fc0 is the uniaxial strength of concrete; k is the ratio of the second stress invariant on tension and compression meridian; μ is the coefficient of viscosity.


### 4.1.2. Constitutive Relation of Steel

The material constitutive relationship of compression reinforcement, stirrup, and steel plate all adopts the ideal bilinear elastoplastic model. The constitutive relation curve is shown in Figure5 [60], which can be expressed by the following formula:(5)σs=Esεs,εs≤εy,σy,εs>εy,where σs is the stress of steel; εs is the strain of steel; and Es represents the elastic modulus of steel.Figure 5
Constitutive relation of steel.In this paper, the compressive reinforcement, the stirrups, and the steel plates of the RC columns analyzed adopt HRB335, HPB235, and Q235, respectively, the measured yield strength was 361 MPa, 275 MPa, and 287 MPa, respectively, and the measured tensile strength was 464 MPa, 398 MPa, and 431 MPa respectively. According to GB 50010-2010 and GB 50017-2017, the elastic modulusEs of the three kinds of steel is 2.0 × 105 MPa, 2.1 × 105 MPa, and 2.06 × 105 MPa, respectively. Poisson’s ratio all takes 0.3, and the density all takes ρs = 7.85 × 10−9 tonne/mm3.


## 4.1.1. Constitutive Relation of Concrete

The concrete damaged plasticity (CDP) model is adopted in the constitutive relationship of concrete material [58, 59]. This model can be applied to unidirectional, cyclic, and dynamic loading and has good convergence. The whole-process constitutive relation curve of concrete tension and compression is shown in Figure 4.Figure 4
Constitutive relation of concrete.The stress-strain curve of concrete under uniaxial tension can be determined by the following formula:(3)σc=1−dtEcεc,dt=1−ρt1.2−0.2x5,x≤1,1−ρtαtx−11.7+x,x>1,x=εcεt,r,ρt=ft,rEcεt,r,where σc represents the concrete stress; εc represents the concrete strain; Ec is the elastic modulus of concrete; αt is the parameter value of the descending section in the uniaxial tensile stress-strain curve of concrete; ft,r is the representative value of uniaxial tensile strength of concrete, which can be taken ft, ftk, or according to the actual needs of structural analysis; εt,r represents the peak tensile strain of concrete corresponding to the representative value of uniaxial tensile strength ft,r; dt is the damage parameter of concrete in uniaxial tensile; ft is the design value of the axial tensile strength of concrete; ftk is the standard value of tensile strength of concrete; ftm is the average tensile strength of concrete, ftm=ftk/1−1.645δc; and δc represents the variation coefficient of concrete strength.The stress-strain curve of concrete under uniaxial compression can be determined by the following formula:(4)σc=1−dcEcεc,dc=1−ρcnn−1+xn,x≤1,1−ρcαcx−12+x,x>1,x=εcεc,r,ρc=fc,rEcεc,r,n=Ecεc,rEcεc,r−fc,r,where αc is the parameter value of the descending section in the uniaxial compression stress-strain curve of concrete; fc,r is the representative value of uniaxial compression strength of concrete, which can be taken fc, fck, or fcm; εc,r represents the peak compression strain of concrete corresponding to the representative value of uniaxial compression strength ft,r; dc is the damage parameter of concrete in uniaxial compression; fc is the design value of the axial compression strength of concrete; fck is the standard value of compression strength of concrete; and fcm is the average compression strength of concrete, fcm=fck/1−1.645δc.The material behavior parameters of concrete required for ABAQUS modeling are shown in Table1.Table 1
The material behavior parameters of concrete.
ρ (t/mm3)
E
c (MPa)
v
c
θ (°)
e
f
b0/fc0
k
μ
2.4 × 10−9
3.0 × 104
0.2
30
0.1
1.16
0.667
0.005
Note. ρ represents the concrete density; vc is Poisson's ratio of concrete; θ is the expansion angle; e is the eccentricity ratio; fb0 represents the strength of concrete under biaxial constant pressure; fc0 is the uniaxial strength of concrete; k is the ratio of the second stress invariant on tension and compression meridian; μ is the coefficient of viscosity.


## 4.1.2. Constitutive Relation of Steel

The material constitutive relationship of compression reinforcement, stirrup, and steel plate all adopts the ideal bilinear elastoplastic model. The constitutive relation curve is shown in Figure5 [60], which can be expressed by the following formula:(5)σs=Esεs,εs≤εy,σy,εs>εy,where σs is the stress of steel; εs is the strain of steel; and Es represents the elastic modulus of steel.Figure 5
Constitutive relation of steel.In this paper, the compressive reinforcement, the stirrups, and the steel plates of the RC columns analyzed adopt HRB335, HPB235, and Q235, respectively, the measured yield strength was 361 MPa, 275 MPa, and 287 MPa, respectively, and the measured tensile strength was 464 MPa, 398 MPa, and 431 MPa respectively. According to GB 50010-2010 and GB 50017-2017, the elastic modulusEs of the three kinds of steel is 2.0 × 105 MPa, 2.1 × 105 MPa, and 2.06 × 105 MPa, respectively. Poisson’s ratio all takes 0.3, and the density all takes ρs = 7.85 × 10−9 tonne/mm3.


## 4.2. FEM Modeling

The part between the two reverse bending points of the frame columns on the first and second floors was taken as the study object, and the modeling analysis was carried out with ABAQUS software. The model was composed of three parts: steel bar skeleton, concrete, and steel plate for strengthening. The bar skeleton was simulated by linear element of truss (T3D2), which was subjected only to axial force but not shear force and bending moment. Three-dimensional linear reduced integrated element (C3D8R) was selected for concrete and strengthening steel plate [61], which was one integral point less than the complete integral element in each direction with higher calculation efficiency.In the simulation of interaction, the slip between the reinforcement skeleton and concrete was not taken into account. The steel bar skeleton was taken as a built-in unit, which was built into the concrete unit by Embedded command. The interaction between concrete and strengthening steel plate could be simulated by Tie command, since epoxy resin was poured between the two and chemical anchors were used as the fixed connection to ensure the costressing and coordinated deformation of the two.When simulating the modeling boundary conditions of the frame columns, in order to ensure that no out-of-plane instability occurred during loading, the boundary conditions on the bottom of the columns were set as U1 = U2 = U3 = 0 and those of the top on the columns were set as U1 = U2 = 0 and UR2 = UR3 = 0 [62]. The FEM models of steel bar skeleton, concrete, strengthening steel plate, and strengthened frame column are shown in Figure 6, where Figure 6(d) is the loading diagram with boundary conditions.Figure 6
FEM models: (a) FEM model of steel bar skeleton; (b) FEM model of concrete; (c) FEM model of steel plates for strengthening; (d) FEM model of frame columns after reinforcement.
(a)
(b)
(c)
(d)


## 4.3. FEM Calculation Results and Analysis

### 4.3.1. Analysis of Axial Compression Action

In order to avoid local failure caused by concentrated load, two disc-shaped discrete rigid sheets (Figure6(d)) are set at the bottom and the top of the column, respectively, to apply concentrated load, and the displacement-controlled monotone static loading method is adopted to apply axial pressure [63].(1) Relationship between Axial Compression Load and Axial Strain. The vertical monotonic static loading simulation analysis was carried out on RC columns before and after strengthening, respectively, with ABAQUS software, and the relationship curve between axial compression load and axial strain is shown in Figure 7. Combined with Figure 7 and ABAQUS postprocessing data, the ultimate axial compression bearing capacities of the normal section of the RC frame column before and after strengthening are 2765 kN and 4896 kN, respectively. The strengthening brings a 77% improvement degree of the bearing capacity. The curve before strengthening is divided into the elastic stage, elastic-plastic stage, and instability decline stage. The ultimate instability occurs in the elastic-plastic stage, that is, the bending failure, and the strain in the instability is 0.0021. The curve after strengthening is divided into the elastic stage, elastic-plastic stage, plastic stage, and instability decline stage. The ultimate instability occurs in the plastic stage, that is, the bending failure, and the strain in the instability is 0.0058. The ultimate strains at the time of failure before and after strengthening are 0.0053 and 0.0094, respectively. The result indicates that the axial compression deformation capacity of the strengthened frame column is improved, and the material properties are given full play.Figure 7
Relationship between axial compression and strains.(2) Comparison between FEM Calculation and Theoretical Formula Calculation. According to the design drawing of the proposed additional layers and the Load Code for the Design of Building Structures, the design value of the maximum bearing capacity of the normal section of the frame column on the first floor is calculated to be 3746 kN. According to formula (1) and the Code for Design of Concrete Structures, the ultimate bearing capacity of the normal section of the frame column before strengthening is calculated to be 2809 kN. By using formula (2) in combination with the Code for Design of Strengthening Concrete Structure, the ultimate bearing capacity of the normal section of the strengthened frame column is calculated to be 5100 kN. The ultimate bearing capacity of normal section of frame column after strengthening calculated by the theoretical formula is 82% higher than that before strengthening. The data analyzed by ABAQUS software and the data calculated by theoretical formulae are summarized as shown in Table 2. It can be seen that the bearing capacities before strengthening calculated by both the FEM method and theoretical formulae are less than the designed value of bearing capacity, which does not meet the requirements, while the calculated bearing capacities after strengthening are both greater than the designed value, meeting the requirements. In addition, the ratios before and after strengthening between the FEM analysis results and the calculation results by the theoretical formulae are 0.98 and 0.96, respectively, indicating that the FEM calculation is in good agreement with the theoretical formula calculation.Table 2
Review and comparison of bearing capacity.
Design value (kN)
Before reinforcement
After reinforcement
FEM calculated value (kN)
Formula (1) (kN)
Ratio
FEM calculated value (kN)
Formula (2) (kN)
Ratio
3746
2765
2809
0.98
4896
5100
0.96
Note. The ratio is the ratio between the FEM analysis results and the formula calculation results.


### 4.3.2. Analysis of Horizontal Low-Cyclic Loading

In order to avoid local damage at the action position of concentrated horizontal load and tensile failure at the beam end, a discrete rigid frame loading-frame (Figure6(d)) is set in the horizontal direction to apply horizontal low-cyclic reversed loading. Referring to Specification for Seismic Test of Buildings (JGJ101-2015) [64], the vertical constant load N is firstly applied according to the axial compression ratio, and then the horizontal low cycle load is applied. Force-displacement mixed control loading mode is adopted. Force-control loading mode is adopted before yield, in which case each grade of load is reversed once. Displacement-control loading mode is adopted after yield, in which case the deformation value takes the horizontal displacement value Δy of the members at yield. The multiple of Δy is used as the range for the displacement-control loading mode, with each grade of load repeated three times [28], loading until the component failure or load dropped to about 85% of the maximum load. The loading system [65] is shown in Figure 8.Figure 8
Loading system.In this paper, the loading system shown in Figure8 is adopted by ABAQUS software, and seismic analysis is carried out for 6 working conditions when the control axial pressure ratios (u) of frame column before and after strengthening are 0.3, 0.6, and 0.9, respectively. The analyzed stress field is shown in Figure 9. When u equals to 0.9 of the strengthened frame column. The material strength of both reinforcement and concrete of the frame column after strengthening is fully developed.Figure 9
Stress field: (a) stress field of reinforcement skeleton; (b) stress field of concrete; (c) stress field of steel plate.
(a)
(b)
(c)(1) P-Δ Hysteretic Curve. The P-Δ hysteretic curves of frame columns before and after strengthening in 6 working conditions are shown in Figure 10. Before strengthening, the shapes of the frame columns’ P-Δ hysteresis loops under three axial compression ratios are all bow with “rheostriction,” indicating that the circular RC frame column with spiral stirrups has a strong energy dissipation capacity. After strengthening, the shapes of the frame column’s hysteresis loop under three axial compression ratios are fusiform, which is fuller than the bow before strengthening, and the ultimate load and ultimate displacement are significantly improved from that before strengthening, indicating that the strengthened frame column has a strong energy dissipation ability and good seismic performance. In all hysteresis curves, the hysteresis loops of each grade of load repeated three times basically coincide. The hysteresis curves are stable, indicating no obvious stiffness degradation phenomenon [66].Figure 10
Hysteretic curves: (a) before reinforcement (u = 0.3); (b) before reinforcement (u = 0.6); (c) before reinforcement (u = 0.9); (d) after reinforcement (u = 0.3); (e) after reinforcement (u = 0.6); (f) after reinforcement (u = 0.9).
(a)
(b)
(c)
(d)
(e)
(f)(2) Skeleton Curves and Ductility Coefficients. The skeleton curves corresponding to all hysteresis curves before and after strengthening are shown in Figure 11. With the increase of axial compression ratio, the peak loads corresponding to all skeleton curves increase and the ultimate displacements decrease, which indicates that increasing axial pressure ratio can improve the horizontal resistance of the column but reduce the ductility of the column. Compared with that before strengthening, both the peak load and limit displacement increase to different extent after strengthening, indicating that the energy dissipation capacity is obviously enhanced. Skeleton curves can be divided into two categories: one with a descending stage and the other without. The skeleton curves before strengthening have obvious descending stage. A more obvious descending is occurred with increasing the axial compression ratio. After strengthening, the skeleton curves have no descending stage when u equals to 0.3 or 0.6; when u = 0.9, it is the skeleton curve with descending stage. Whether the skeleton curve has a descending stage or not can reflect the change of ductility coefficient and the following is a brief discussion on the change of ductility coefficient.Figure 11
Skeleton curves: (a) before reinforcement; (b) after reinforcement.
(a)
(b)Ductility coefficient (μ) is the ratio of displacement when the horizontal load is reduced to 85% of the maximum load to yield displacement; it is the index of ductility deformation ability of structure and can reflect the seismic performance of structure. The influence of axial compression ratio on ductility coefficient before and after strengthening of frame column is shown in Figure 12. Before strengthening, the ductility coefficient decreases with increasing the axial compression ratio. After strengthening, the ductility coefficient is infinite when the axial compression ratio is less than 0.6. When the axial compression ratio is greater than 0.6, the ductility coefficient also decreases with the increase of the axial compression ratio. Moreover, the ductility coefficient curve after strengthening is much higher than that before strengthening, proving that the strengthened frame column has good plastic deformation capacity and good seismic performance.Figure 12
The influence of axial compression ratio on ductility factor.(3) Analysis of Energy Consumption. There is no unified standard to evaluate the energy dissipation capacity of components till now. The abovementioned hysteretic curve, skeleton curve, and ductility coefficient can be used to evaluate the energy dissipation capacity qualitatively instead of quantitatively. Generally, equivalent viscous damping coefficient and energy dissipation coefficient (the energy dissipation coefficient is 2π of the equivalent viscous damping coefficient) are used to quantitatively determine the energy dissipation capacity of components [67]. The formula of equivalent viscous damping coefficient is as follows:(6)he=12π⋅ShSOAB+SOCD,and the symbols in the formula are shown in Figure 13: Sh represents the area of hysteretic loop; SOAB is the area of triangle OAB; and SOCD is the area of triangle OCD.Figure 13
Hysteretic loop.The relation curve between the equivalent viscous damping coefficienthe and the recurrence number n of horizontal load is the energy dissipation curve, as shown in Figure 14. As can be seen from Figure 14, when n is less than 12, the greater the axial pressure ratio u is, the higher he is, indicating that increasing the axial pressure ratio is beneficial to the energy consumption of members at the early loading stage. When n is greater than 12, the larger u is, the smaller he is, indicating that increasing axial compression ratio is unfavorable to energy dissipation of members at the later loading stage. Therefore, the axial compression ratio of columns shall be limited, regarding the earthquake resistance. Before strengthening, with the increase of n, he presents an upward trend; when n is less than 6, with the members in the elastic stage, he is small and increases slowly, and energy dissipation capacity is low; when n is within the range of 6∼18, with the members in the elastoplastic stage, he increases fast, and the energy dissipation capacity increases rapidly; when n is greater than 18, with the members in the plastic stage, he slowly increases to the maximum value, and the energy dissipation capacity reaches a high degree. After strengthening, when n is less than 12, he increases fast and energy dissipation capacity increases rapidly; when n is greater than 12, he reaches the maximum value, and the energy dissipation capacity of members no longer increases, and the energy dissipation capacity of members with u = 0.9 slightly decreases. Compared with that before strengthening, he after strengthening has been obviously improved, which proves that the frame column after strengthening has good energy dissipation capacity and seismic performance.Figure 14
Energy-absorbed curve: (a) before reinforcement; (b) after reinforcement.
(a)
(b)


## 4.3.1. Analysis of Axial Compression Action

In order to avoid local failure caused by concentrated load, two disc-shaped discrete rigid sheets (Figure6(d)) are set at the bottom and the top of the column, respectively, to apply concentrated load, and the displacement-controlled monotone static loading method is adopted to apply axial pressure [63].(1) Relationship between Axial Compression Load and Axial Strain. The vertical monotonic static loading simulation analysis was carried out on RC columns before and after strengthening, respectively, with ABAQUS software, and the relationship curve between axial compression load and axial strain is shown in Figure 7. Combined with Figure 7 and ABAQUS postprocessing data, the ultimate axial compression bearing capacities of the normal section of the RC frame column before and after strengthening are 2765 kN and 4896 kN, respectively. The strengthening brings a 77% improvement degree of the bearing capacity. The curve before strengthening is divided into the elastic stage, elastic-plastic stage, and instability decline stage. The ultimate instability occurs in the elastic-plastic stage, that is, the bending failure, and the strain in the instability is 0.0021. The curve after strengthening is divided into the elastic stage, elastic-plastic stage, plastic stage, and instability decline stage. The ultimate instability occurs in the plastic stage, that is, the bending failure, and the strain in the instability is 0.0058. The ultimate strains at the time of failure before and after strengthening are 0.0053 and 0.0094, respectively. The result indicates that the axial compression deformation capacity of the strengthened frame column is improved, and the material properties are given full play.Figure 7
Relationship between axial compression and strains.(2) Comparison between FEM Calculation and Theoretical Formula Calculation. According to the design drawing of the proposed additional layers and the Load Code for the Design of Building Structures, the design value of the maximum bearing capacity of the normal section of the frame column on the first floor is calculated to be 3746 kN. According to formula (1) and the Code for Design of Concrete Structures, the ultimate bearing capacity of the normal section of the frame column before strengthening is calculated to be 2809 kN. By using formula (2) in combination with the Code for Design of Strengthening Concrete Structure, the ultimate bearing capacity of the normal section of the strengthened frame column is calculated to be 5100 kN. The ultimate bearing capacity of normal section of frame column after strengthening calculated by the theoretical formula is 82% higher than that before strengthening. The data analyzed by ABAQUS software and the data calculated by theoretical formulae are summarized as shown in Table 2. It can be seen that the bearing capacities before strengthening calculated by both the FEM method and theoretical formulae are less than the designed value of bearing capacity, which does not meet the requirements, while the calculated bearing capacities after strengthening are both greater than the designed value, meeting the requirements. In addition, the ratios before and after strengthening between the FEM analysis results and the calculation results by the theoretical formulae are 0.98 and 0.96, respectively, indicating that the FEM calculation is in good agreement with the theoretical formula calculation.Table 2
Review and comparison of bearing capacity.
Design value (kN)
Before reinforcement
After reinforcement
FEM calculated value (kN)
Formula (1) (kN)
Ratio
FEM calculated value (kN)
Formula (2) (kN)
Ratio
3746
2765
2809
0.98
4896
5100
0.96
Note. The ratio is the ratio between the FEM analysis results and the formula calculation results.


## 4.3.2. Analysis of Horizontal Low-Cyclic Loading

In order to avoid local damage at the action position of concentrated horizontal load and tensile failure at the beam end, a discrete rigid frame loading-frame (Figure6(d)) is set in the horizontal direction to apply horizontal low-cyclic reversed loading. Referring to Specification for Seismic Test of Buildings (JGJ101-2015) [64], the vertical constant load N is firstly applied according to the axial compression ratio, and then the horizontal low cycle load is applied. Force-displacement mixed control loading mode is adopted. Force-control loading mode is adopted before yield, in which case each grade of load is reversed once. Displacement-control loading mode is adopted after yield, in which case the deformation value takes the horizontal displacement value Δy of the members at yield. The multiple of Δy is used as the range for the displacement-control loading mode, with each grade of load repeated three times [28], loading until the component failure or load dropped to about 85% of the maximum load. The loading system [65] is shown in Figure 8.Figure 8
Loading system.In this paper, the loading system shown in Figure8 is adopted by ABAQUS software, and seismic analysis is carried out for 6 working conditions when the control axial pressure ratios (u) of frame column before and after strengthening are 0.3, 0.6, and 0.9, respectively. The analyzed stress field is shown in Figure 9. When u equals to 0.9 of the strengthened frame column. The material strength of both reinforcement and concrete of the frame column after strengthening is fully developed.Figure 9
Stress field: (a) stress field of reinforcement skeleton; (b) stress field of concrete; (c) stress field of steel plate.
(a)
(b)
(c)(1) P-Δ Hysteretic Curve. The P-Δ hysteretic curves of frame columns before and after strengthening in 6 working conditions are shown in Figure 10. Before strengthening, the shapes of the frame columns’ P-Δ hysteresis loops under three axial compression ratios are all bow with “rheostriction,” indicating that the circular RC frame column with spiral stirrups has a strong energy dissipation capacity. After strengthening, the shapes of the frame column’s hysteresis loop under three axial compression ratios are fusiform, which is fuller than the bow before strengthening, and the ultimate load and ultimate displacement are significantly improved from that before strengthening, indicating that the strengthened frame column has a strong energy dissipation ability and good seismic performance. In all hysteresis curves, the hysteresis loops of each grade of load repeated three times basically coincide. The hysteresis curves are stable, indicating no obvious stiffness degradation phenomenon [66].Figure 10
Hysteretic curves: (a) before reinforcement (u = 0.3); (b) before reinforcement (u = 0.6); (c) before reinforcement (u = 0.9); (d) after reinforcement (u = 0.3); (e) after reinforcement (u = 0.6); (f) after reinforcement (u = 0.9).
(a)
(b)
(c)
(d)
(e)
(f)(2) Skeleton Curves and Ductility Coefficients. The skeleton curves corresponding to all hysteresis curves before and after strengthening are shown in Figure 11. With the increase of axial compression ratio, the peak loads corresponding to all skeleton curves increase and the ultimate displacements decrease, which indicates that increasing axial pressure ratio can improve the horizontal resistance of the column but reduce the ductility of the column. Compared with that before strengthening, both the peak load and limit displacement increase to different extent after strengthening, indicating that the energy dissipation capacity is obviously enhanced. Skeleton curves can be divided into two categories: one with a descending stage and the other without. The skeleton curves before strengthening have obvious descending stage. A more obvious descending is occurred with increasing the axial compression ratio. After strengthening, the skeleton curves have no descending stage when u equals to 0.3 or 0.6; when u = 0.9, it is the skeleton curve with descending stage. Whether the skeleton curve has a descending stage or not can reflect the change of ductility coefficient and the following is a brief discussion on the change of ductility coefficient.Figure 11
Skeleton curves: (a) before reinforcement; (b) after reinforcement.
(a)
(b)Ductility coefficient (μ) is the ratio of displacement when the horizontal load is reduced to 85% of the maximum load to yield displacement; it is the index of ductility deformation ability of structure and can reflect the seismic performance of structure. The influence of axial compression ratio on ductility coefficient before and after strengthening of frame column is shown in Figure 12. Before strengthening, the ductility coefficient decreases with increasing the axial compression ratio. After strengthening, the ductility coefficient is infinite when the axial compression ratio is less than 0.6. When the axial compression ratio is greater than 0.6, the ductility coefficient also decreases with the increase of the axial compression ratio. Moreover, the ductility coefficient curve after strengthening is much higher than that before strengthening, proving that the strengthened frame column has good plastic deformation capacity and good seismic performance.Figure 12
The influence of axial compression ratio on ductility factor.(3) Analysis of Energy Consumption. There is no unified standard to evaluate the energy dissipation capacity of components till now. The abovementioned hysteretic curve, skeleton curve, and ductility coefficient can be used to evaluate the energy dissipation capacity qualitatively instead of quantitatively. Generally, equivalent viscous damping coefficient and energy dissipation coefficient (the energy dissipation coefficient is 2π of the equivalent viscous damping coefficient) are used to quantitatively determine the energy dissipation capacity of components [67]. The formula of equivalent viscous damping coefficient is as follows:(6)he=12π⋅ShSOAB+SOCD,and the symbols in the formula are shown in Figure 13: Sh represents the area of hysteretic loop; SOAB is the area of triangle OAB; and SOCD is the area of triangle OCD.Figure 13
Hysteretic loop.The relation curve between the equivalent viscous damping coefficienthe and the recurrence number n of horizontal load is the energy dissipation curve, as shown in Figure 14. As can be seen from Figure 14, when n is less than 12, the greater the axial pressure ratio u is, the higher he is, indicating that increasing the axial pressure ratio is beneficial to the energy consumption of members at the early loading stage. When n is greater than 12, the larger u is, the smaller he is, indicating that increasing axial compression ratio is unfavorable to energy dissipation of members at the later loading stage. Therefore, the axial compression ratio of columns shall be limited, regarding the earthquake resistance. Before strengthening, with the increase of n, he presents an upward trend; when n is less than 6, with the members in the elastic stage, he is small and increases slowly, and energy dissipation capacity is low; when n is within the range of 6∼18, with the members in the elastoplastic stage, he increases fast, and the energy dissipation capacity increases rapidly; when n is greater than 18, with the members in the plastic stage, he slowly increases to the maximum value, and the energy dissipation capacity reaches a high degree. After strengthening, when n is less than 12, he increases fast and energy dissipation capacity increases rapidly; when n is greater than 12, he reaches the maximum value, and the energy dissipation capacity of members no longer increases, and the energy dissipation capacity of members with u = 0.9 slightly decreases. Compared with that before strengthening, he after strengthening has been obviously improved, which proves that the frame column after strengthening has good energy dissipation capacity and seismic performance.Figure 14
Energy-absorbed curve: (a) before reinforcement; (b) after reinforcement.
(a)
(b)


## 4.4. Validation of FEM Model

In order to fully verify the accuracy and effectiveness of the proposed model, the FEM modeling method introduced in Section4.2 is used to conduct modeling analysis on the samples in similar literature [41, 42, 45, 68, 69], and the analysis results are compared with the test data. The following is a brief introduction to the experimental overview of literature [41, 42, 45, 68, 69]:In literature [41, 42], Priestley et al. conducted a systematic study on bridge piers strengthened with externally wrapped steel and introduced a set of design methods for strengthening, which guided a large number of bridge strengthening projects in California and other regions. In order to study the brittle shear failure mode of the built circular bridge piers, four models of the piers strengthened with external steel plates were made, with the model scale of 1 : 2.5. They carried out shear tests in the horizontal direction with the control axial compression ratios of 0.06 and 0.18, respectively. The research showed that the shear and flexural capacity of the strengthened column could be greatly improved by the externally wrapped steel strengthening method. Wang [45] made four RC columns strengthened by circular steel tubes, of which two were strengthened by steel pipes without initial load, and the other two were strengthened by steel pipes with initial axial load of 500 kN and 1000 kN, respectively. The gaps between steel pipes and RC column were filled with cement mortar material of the same grade and high-strength nonshrinkage grouting material, respectively. After the axial compression tests on 4 specimens, the results showed that the RC column strengthened by circular steel pipes could greatly improve the bearing capacity and ductility. There was no slippage between the outer steel pipes and the core RC columns. The two deformed with coordination and worked together.After searching, there are relatively few studies on the experimental research of circular RC columns strengthened with steel plates. In a broad sense, the circular RC columns strengthened by externally wrapped steel plates belong to the category of circular tube confined RC (hereinafter referred to as CTRC) columns [68, 69], and the research on CTRC columns is relatively extensive. The test in literature [68] took the frame columns as the research object and assumed that the columns were fixed at both ends. First, the columns were subjected to a constant vertical axial pressure load controlled by the axial pressure ratio, and then repeated horizontal load was applied at the top of the columns to simulate the seismic load. A total of 4 CTRC column specimens were made, one filled with normal strength concrete (C30) and three filled with high-strength concrete (C60). The test results showed that the CTRC columns had good flexural capacity, ductility, and energy dissipation capacity, which effectively improved the brittle failure of high-strength RC columns under high axial compression ratio. Zhou et al. [69] made 16 CTRC column specimens and conducted eccentric compression test on them with slenderness ratio, eccentricity, and steel content ratio as parameters. According to the mechanical characteristics, the specimens were divided into two modes, A and B: a model steel pipe wall connected up and down (only slotted at both ends of the specimens); in mode B, the middle part of the steel pipe wall was also slotted, which divided the steel pipe wall into upper and lower parts. The test results showed that the long CTRC columns had good ductility and still maintained a high bearing capacity after the peak; with the increase of slenderness ratio and eccentricity, the stiffness of specimens decreased, but ductility increased; mode A and B had no obvious influence on the bearing capacity and ductility of the specimens. The reinforcement diagram of specimens in literature [41, 42, 45, 68, 69] is shown in Figure 15, and other test parameters are shown in Table 3.Figure 15
Reinforcement diagram of the specimens in literature: (a) literature [41, 42]; (b) literature [45]; (c) literature [68]; (d) literature [69].
(a)
(b)
(c)
(d)Table 3
Specimen parameters and comparison of ultimate bearing capacity.
Specimens
e (mm)
u
L (mm)
D (mm)
t (mm)
f
y (MPa)
f
yz (MPa)
f
yh (MPa)
f
c (MPa)
P
ut (kN)
P
uf (kN)
P
uf/Put
Loading methods
Data sources
C2R
—
0.06
1220
610
4.8
348
324
359
34
734
639
0.87
Horizontal shear
Literature [41, 42]
C4R
—
0.06
1220
610
4.8
348
324
324
35
957
813
0.85
C6R
—
0.18
1220
610
3.2
286
469
324
40
1024
850
0.83
C8R
—
0.18
915
610
3.2
286
469
324
31
1228
1105
0.90
ZY2-1
—
—
1200
280
4.0
433
400
380
36
4358
3879
0.89
Axial compression
Literature [45]
ZY2-2
—
—
1200
280
4.0
433
400
380
36
4594
4181
0.91
ZY3-1
—
—
1200
280
4.0
433
400
380
36
4700
4277
0.91
ZY3-2
—
—
1200
280
4.0
433
400
380
36
4581
4260
0.93
CTRC-60-8
—
0.8
1200
226
2.7
436
506
305
53
232
218
0.94
Horizontal circulation
Literature [68]
CTRC-60-6
—
0.6
1200
226
2.7
436
506
305
53
199
186
0.93
CTRC-60-3
—
0.3
1200
226
2.7
436
506
305
53
139
133
0.96
CTRC-30-8
—
0.8
1200
226
2.7
436
506
305
39
164
150
0.92
C200-6-25-a
25
—
1200
200
1.5
364
477
286
56
1926
1887
0.98
Eccentric compression
Literature [69]
C200-6-25-b
25
—
1200
200
1.5
364
477
286
56
1758
1688
0.96
C200-6-50-a
50
—
1200
200
1.5
364
477
286
56
1317
1225
0.93
C200-6-50-b
50
—
1200
200
1.5
364
477
286
56
1238
1201
0.97
C200-10-25-a
25
—
2000
200
1.5
364
477
286
56
1602
1522
0.95
C200-10-25-b
25
—
2000
200
1.5
364
477
286
56
1520
1490
0.98
C200-10-50-a
50
—
2000
200
1.5
364
477
286
56
1121
1054
0.94
C200-10-50-b
50
—
2000
200
1.5
364
477
286
56
1136
1102
0.97
C240-6-25-a
25
—
1440
240
1.5
364
477
286
56
2518
2468
0.98
C240-6-25-b
25
—
1440
240
1.5
364
477
286
56
2464
2292
0.93
C240-6-50-a
50
—
1440
240
1.5
364
477
286
56
1957
1879
0.96
C240-6-50-b
50
—
1440
240
1.5
364
477
286
56
1872
1685
0.90
C240-10-25-a
25
—
2400
240
1.5
364
477
286
56
2219
2175
0.98
C240-10-25-b
25
—
2400
240
1.5
364
477
286
56
2212
2124
0.96
C240-10-50-a
50
—
2400
240
1.5
364
477
286
56
1909
1756
0.92
C240-10-50-b
50
—
2400
240
1.5
364
477
286
56
1687
1653
0.98
Note. e is the eccentricity; u is the axial compression ratio; L, D, and t represent the steel pipe length, outer diameter, and wall thickness, respectively; fy, fyz, and fyh are the measured yield strength of steel pipe, longitudinal bar, and stirrup, respectively; fc is the compressive strength of concrete; Put represents the test value of bearing capacity; Puf is the FEM simulation value of bearing capacity.The analysis results are compared with the test data, as shown in Table3 and Figure 16. As can be seen from Table 3 and Figure 16, the results of FEM simulation are all smaller than the experimental data in the literatures. The biggest error is in the comparison with literature [41, 42], with Puf/Put of 0.83. The probable reasons may be the following: differences between Chinese and American codes on the value of material strength, where American codes have higher reliability and safety; differences on the structure types, where literature [41, 42] studied piers of bridges, while the FEM simulation in this paper was RC columns of buildings; and differences in time, where experiment in literature [41, 42] was carried out in the early 1990s, with a history of nearly 30 years. Compared with literature [45, 68, 69], Puf/Put is between 0.89 and 0.98. The FEM simulation results are consistent with the experimental data, and the simulation results tend to be safer. The above analysis can fully prove the accuracy and applicability of the FEM model adopted in this paper.Figure 16
Comparison of ultimate bearing capacity between FEM simulation value and experimental value [41, 42, 45, 68, 69].


## 5. Conclusions

(1)
Compared with the traditional steel strengthening methods, the strengthening method with integral externally wrapped cold-formed steel plates introduced in this paper improves the bearing capacity of RC columns greater and influences the section size less, with the characteristic of simpler construction technology of strengthening and fast construction speed. After strengthening, the bearing capacities of the normal section of RC columns in this project were about 80% higher than that before strengthening. The diameter of the circular section was increased by 5 mm. The total construction period of strengthening and layer-adding extension was only 33 days (during the summer vacation). The strengthening effect met the requirements of party A, and it did not affect the normal teaching of the school.
(2)
The formula for calculating the bearing capacity of normal section of circular RC columns strengthened with externally wrapped steel plates was derived. In addition, ABAQUS software was used to carry out the vertical monotone static loading analysis of full-size RC columns before and after strengthening in the engineering project. The calculation results of the formula were in good agreement with the analysis and calculation results of ABAQUS software, indicating that the modeling and analysis of such strengthened members simulated by ABAQUS software could meet the engineering requirements.
(3)
ABAQUS software was used to carry out the horizontal low-cyclic reversed loading analysis of full-size RC columns before and after strengthening in the engineering project. The results showed that the hysteretic curve was fuller, and the ductility coefficient and equivalent viscous damping coefficient were larger after strengthening than before, which thereby proved that the circular RC columns strengthened with externally wrapped steel plates had good plastic deformation ability and notable seismic performance.
(4)
It is recommended that this strengthening method be used in RC structures in the following situations: (1) areas with high seismic fortification intensity; (2) members requiring a substantial increase in carrying capacity; (3) little effect on the appearance size of members; (4) little effect on the usable floor area of buildings; and (5) original understructure needing for adding-layer expansion.
(5)
It has been four years since the completion of the strengthening in the project. After the later monitoring and feedback from party A, no abnormal phenomena have been found, proving that the strengthening effect is fine [70].



---

*Source: 2515091-2019-08-26.xml*