# Shear Strength Prediction for SFRC Shear Wall with CFST Columns by Softened Strut and Tie Model

Authors: Peibo You; Shuaiqi Song; Haiyang Zhang; Lijuan Zhang; Ke Shi; Yi Wang
Journal: Advances in Civil Engineering
 (2020)

---

## Abstract

The steel fiber reinforced concrete (SFRC) shear wall with concrete filled steel tube (CFST) columns is an innovative composite structure. In order to calculate the shear strength of SFRC shear wall with CFST columns, the softened strut and tie model (SSTM) of SFRC shear wall with CFST columns was proposed based on the analysis of shear mechanism of SFRC shear wall with CFST columns. The SSTM was composed of diagonal, horizontal, and vertical mechanisms, in which the contributions of concrete, reinforcement, and steel fiber to the shear strength of SFRC web of shear wall were identified. The shear capacities of 24 shear walls were calculated and compared with the available test results, and reasonable agreement was obtained. The results also showed that the steel fibers distributed randomly in concrete could be treated as longitudinal and transverse reinforcement in the shear strength analysis of SFRC web, and the SSTM was reasonable and useful to analyze and predict the shear strength of SFRC shear wall with CFST columns.


---

## Body

## 1. Introduction

Reinforced concrete (RC) shear wall system is a common one to resist lateral force, which has been widely used in the regions of high seismic activity in the past decade. However, due to low tensile strength and large brittleness of normal-strength concrete, the ductility, energy dissipation, and load bearing capacities of RC shear wall are poor. In particular, the transverse dimension of shear wall can be decreased with the application of high strength concrete, but its brittleness increases inevitably. Correspondingly, the higher requirement for the seismic performance of shear wall especially with high strength concrete (HSC) would be put forward. In order to improve the seismic performance of shear wall, the use of RC shear wall with concrete filled steel tube (CFST) columns and steel fiber reinforced concrete (SFRC) shear wall has attracted the interest of structural engineers in recent years.First, theoretical analysis and experimental study [1–4] show that the CFST columns serving as boundary members can provide the effective constraint to reinforced high strength concrete (RHSC) wall, restrain the cracking development, and also bear part of transverse load after the RHSC wall deteriorates gradually, indicating that the RHSC shear wall with CFST columns can form double-channel seismic line and have favorable seismic performance. Secondly, the application of SFRC is a good way to enhance the seismic behavior of conventional shear wall structure, especially the RHSC shear wall [5]. Several researchers [6–10] have conducted the experiments on the seismic performance of SFRC shear wall. In these experiments, the SFRC shear walls were subjected to reversed cyclic loading. As expected, because SFRC has good properties, such as stretching, shear resistance, cracking resistance, toughness [11], fire resistance, durability [12, 13], and earthquake resistance, it can remarkably improve the seismic behavior of RC shear wall and reduce the phenomenon of reinforcement congestion and construction difficulty. The load bearing capacity, ductility, and energy dissipation capacities of SFRC shear walls increase with the increasing of steel fiber volume fraction.In this study, an innovative composite shear wall referred to as SFRC shear wall with CFST columns is introduced, which consists of SFRC web and CFST boundary elements. In this new composite shear wall, SFRC web can mainly improve the ductility and energy dissipation capacities, CFST boundary elements can enhance the load bearing capacity. This kind of composite shear wall will adequately make use of the advantages of SFRC shear wall and CFST columns. In order to more fully understand the seismic behavior of this structure form, six SFRC shear wall with CFST column specimens were tested under constant axial force and reversed cyclic loading. The failure mode of shear wall specimens was analyzed, and the influencing factors, such as the steel fiber volume fraction and concrete strength on the shear strength of shear wall specimens, were studied. Since the load transferring mechanism of shear walls under the combined action of compression, bending, and shearing is very complicated, the researches on the load bearing mechanism and the calculation methods for shear capacity of shear walls are relatively few. In addition, the only existing calculation methods for shear capacity proposed are mostly semiempirical relations based on the experimental results and are lacking in the reasonable theoretical model. Furthermore, the analysis and calculations are not very accurate.The softened strut and tie model (SSTM) for determining the shear strengths of RC members failing in diagonal compressions was proposed by Hwang and Lee, which originated from the strut and tie concept, considered the compression softening characteristics of concrete, and was derived to satisfy equilibrium, compatibility, and constitutive laws of cracked reinforced concrete [14–19]. The SSTM has been used in shear strength prediction of various kinds of concrete members and the accuracy has been checked by comparing calculated shear strengths with experimental data reported in the previous literature [20–24].In this paper, the SSTM of SFRC shear wall with CFST columns was proposed based on the analysis for its shear mechanism, in which the steel fibers distributed randomly in concrete were equivalent to the longitudinal and transverse reinforcements. Furthermore, a new calculation method based on the SSTM for the shear strength of SFRC shear wall with CFST columns, RC shear wall with CFST columns, and SFRC shear wall was established. Finally, the shear capacities of 24 low-rise shear walls were calculated and the calculated values showed a reasonable agreement with the available experimental results [2, 4, 9, 10].


## 2. General Test Situation

### 2.1. Specimen Details

The dimensions and configurations of the shear wall specimens are shown in Figure1. Each specimen mainly consists of five parts: one SFRC web in the centre, two CFST columns which are located in the right and left sides of middle SFRC web, and two RC beams which are located in the top and bottom sides of middle SFRC web. In order to ensure the reliable connections between the CFST columns and the shear wall web, the U-shaped connectors [25] with the same space as the horizontal reinforcing bars are also welded to the wall of steel tubes. The photos of completed steel frame and U-shaped connector are shown in Figures 2(a) and 2(b), respectively.Figure 1
Specimen configuration (unit: mm). (a) Elevation. (b) A-A section. (c) One-dimensional and three-dimensional graph of U-shaped connector.
(a)(b)(c)Figure 2
Steel frame and U-shaped connector. (a) Steel frame. (b) U-shaped connector.
(a)(b)The side length and height of the square CFST columns were 120 mm and 750 mm, respectively. The steel plates for hollow sections of square CFST columns and U-shaped connectors were made by cold-formed steel tubes with the thickness of 3 mm and 2 mm, respectively. The strength grade of reinforcement was HRB400, and the grade of steel plates was Q235B based on the Chinese code. The material properties of steel bar and plate are listed in Table1. The steel fiber is hooked at both ends with length of 35 mm, diameter of 0.55 mm, and aspect ratio of 64.Table 1
Material properties of steel plate and bar.
TypeYield strengthfy (N/mm2)Ultimate strengthfu (N/mm2)Elastic modulusEs (N/mm2)3 mm plate307.67392.001.98 × 1052 mm plate236.67323.201.88 × 105C 6 bar369.17521.601.85 × 105The test parameters included steel fiber volume fractionρsf (0, 0.5%, 1.0%, and 1.5%) and concrete strength grade (CF40, CF60, and CF80). According to the test parameters, six shear wall specimens were cast, tested, and numbered as SS-1.0-00-C60, SS-1.0-05-CF60, SS-1.0-10-CF60, SS-1.0-15-CF60, SS-1.0-10-CF40, and SS-1.0-10-CF80. In the serial number of specimens, the first part SS represents that steel tube and steel fiber; the second part 1.0 refers to the shear span ratio of 1.0, the third part represents the steel fiber volume fraction of the web in shear wall, and the fourth part represents the concrete, in which C represents plain concrete, whilst CF represents SFRC, and the following number denotes target concrete strength of shear wall specimen. The summary of information of each shear wall specimen is listed in Table 2. The mechanical properties of the harden concrete used in the shear wall of all specimens are also listed in Table 2. All the harden concretes reached its target compressive strength.Table 2
Summary of specimen information.
Identification numberρsf (%)Columnsfc (MPa)Columnsft (MPa)Webfc (MPa)Webfft (MPa)Vjh,test (kN)SS-1.0-00-C60055.92.7656.32.81902SS-1.0-05-CF600.555.52.7355.23.67948SS-1.0-10-CF601.055.52.7455.16.151005SS-1.0-15-CF601.556.02.7756.57.881048SS-1.0-10-CF401.038.12.1538.33.78810SS-1.0-10-CF801.064.63.9665.66.841107


### 2.2. Test Scheme

In the test, the load exerted on the specimens included horizontal and vertical one, and the test set-up is shown in Figure3. The horizontal load was cyclic one and applied by an electrohydraulic servo control actuator on the top beam of specimen; the axial compressive force was kept constant and applied by hydraulic Jack. The target axial compression ratio, n, was taken as 0.2 for all specimens. The rolling support between hydraulic Jack and vertical reaction steel frame assured the smooth horizontal slipping of specimens. Rotation control mode was adopted and the typical rotation controlled loading history is shown in Figure 4. A nominal yield rotation angle (θym) was designated to 0.0025 rad according to the research in literature [1], which was corresponding to a nominal yield lateral displacement (Δym) at the loading point, Δym = θym × 750 mm = 1.875 mm. Before θym, the rotation levels of 0.5θym and 0.75θym were imposed. Hereafter, the rotation increment was 0.0025 rad for each rotation level and three cycles were imposed at each rotation level. The tests were conducted under the rotation controlled cyclic loading until the axial compressive force could not be maintained or the lateral force degraded below 85% of peak lateral load.Figure 3
Test set-up. (a) Schematic view. (b) Test scene.
(a)(b)Figure 4
Loading history.


### 2.3. Failure Process

The whole process from the initiation to failure of specimens was observed. During the initial loading period, the inclined cracks gradually appeared and constantly extended under cyclic lateral loading. Then several pairs of visible intercrossing diagonal cracks developed approximately along the diagonal directions of the web in shear wall, the horizontal reinforcing bars in the web of the shear wall yielded successively with the loading, and the diagonal strut mechanism in the web of the shear wall was formed. Hereafter, new cracks seldom occurred and the width of the diagonal cracks at the middle part of the web in shear wall increased rapidly with further loading. The web of the shear wall began to fail in diagonal compression, the concrete in the web was crushed and spalled, and the web lost bearing capacity gradually. Upon further cycles, slight local buckling was observed at the bottom of CFST columns and plastic hinges were formed. Finally, the lateral load rapidly dropped to about 85% of the peak lateral load and the specimen failed. All specimens mainly exhibited a typical diagonal cracking pattern during the test process and showed a shear-dominant failure mode. The failure pattern and crack distribution at the peak lateral load and ultimate state (the lateral force degraded to 85% of the peak lateral load) of all tested specimens are shown in Figure5.Figure 5
Failure pattern and crack distribution of shear wall specimens. (a) Failure pattern at peak state. (b) Failure pattern at ultimate state. (c) Schematic view of crack distribution.
(a)(b)(c)During the test process, the connections between the CFST columns and web were reliable and no significant separation at the interface occurred, which indicated that the performance of the U-shaped connector designed was effective. Moreover, no evident local buckling on the CFST columns of the SFRC shear wall specimens was observed at the ultimate state. The SFRC shear wall specimens had smaller destruction compared with that without steel fibers, which indicated that the SFRC shear wall specimens had the better repairability.


### 2.4. Influence Factors

The shear capacities of all specimens are listed in Table2. It can be seen from Table 2 that the shear performance of specimens is improved obviously with the increase of steel fiber volume fraction from 0% to 1.5%. Compared with the specimen SS-1.0-00-C60, the shear capacity of specimen SS-1.0-05-CF60, SS-1.0-10-CF60, and SS-1.0-15-CF60 is increased by 5.10%, 11.42%, and 16.19%, respectively. In addition, steel fiber could effectively control the crack width and improve the distribution shape of cracks in the web of shear wall. The cracks became much thinner, denser, of more quantity, and wider distribution with the increase of steel fiber volume fraction, whilst the crushing and spalling of concrete were also reduced. For instance, the maximum crack width of RC shear wall specimen numbered SS-1.0-00-C60 was 2.0 mm at the peak load, whilst the maximum crack width of SFRC shear wall specimen numbered SS-1.0-05-CF60, SS-1.0-10-CF60, and SS-1.0-15-CF60 was 1.6 mm, 1.2 mm, and 1.0 mm, respectively. The maximum crack width decreased significantly with the increase of steel fiber volume fraction; the similar results were reported in the literature [9, 10].Furthermore, it can be seen from Table2 that the shear performance of specimens is improved obviously with the enhancing of concrete strength from CF40 to CF80. Compared with the specimen SS-1.0-10-CF40, the shear capacity of SS-1.0-10-CF60 and SS-1.0-10-CF80 is increased by 24.07% and 36.67%, respectively. And as shown in Figure 5, the SFRC shear wall specimens with different concrete strength had the similar failure appearance at the peak and ultimate state under the condition of the same steel fiber volume fraction. It can be seen that the lateral load of SFRC shear wall specimen with higher strength concrete deteriorated more obviously after the peak load, the crushing, and spalling of concrete were more serious at the ultimate state, especially for the specimen SS-1.0-00-CF80.


## 2.1. Specimen Details

The dimensions and configurations of the shear wall specimens are shown in Figure1. Each specimen mainly consists of five parts: one SFRC web in the centre, two CFST columns which are located in the right and left sides of middle SFRC web, and two RC beams which are located in the top and bottom sides of middle SFRC web. In order to ensure the reliable connections between the CFST columns and the shear wall web, the U-shaped connectors [25] with the same space as the horizontal reinforcing bars are also welded to the wall of steel tubes. The photos of completed steel frame and U-shaped connector are shown in Figures 2(a) and 2(b), respectively.Figure 1
Specimen configuration (unit: mm). (a) Elevation. (b) A-A section. (c) One-dimensional and three-dimensional graph of U-shaped connector.
(a)(b)(c)Figure 2
Steel frame and U-shaped connector. (a) Steel frame. (b) U-shaped connector.
(a)(b)The side length and height of the square CFST columns were 120 mm and 750 mm, respectively. The steel plates for hollow sections of square CFST columns and U-shaped connectors were made by cold-formed steel tubes with the thickness of 3 mm and 2 mm, respectively. The strength grade of reinforcement was HRB400, and the grade of steel plates was Q235B based on the Chinese code. The material properties of steel bar and plate are listed in Table1. The steel fiber is hooked at both ends with length of 35 mm, diameter of 0.55 mm, and aspect ratio of 64.Table 1
Material properties of steel plate and bar.
TypeYield strengthfy (N/mm2)Ultimate strengthfu (N/mm2)Elastic modulusEs (N/mm2)3 mm plate307.67392.001.98 × 1052 mm plate236.67323.201.88 × 105C 6 bar369.17521.601.85 × 105The test parameters included steel fiber volume fractionρsf (0, 0.5%, 1.0%, and 1.5%) and concrete strength grade (CF40, CF60, and CF80). According to the test parameters, six shear wall specimens were cast, tested, and numbered as SS-1.0-00-C60, SS-1.0-05-CF60, SS-1.0-10-CF60, SS-1.0-15-CF60, SS-1.0-10-CF40, and SS-1.0-10-CF80. In the serial number of specimens, the first part SS represents that steel tube and steel fiber; the second part 1.0 refers to the shear span ratio of 1.0, the third part represents the steel fiber volume fraction of the web in shear wall, and the fourth part represents the concrete, in which C represents plain concrete, whilst CF represents SFRC, and the following number denotes target concrete strength of shear wall specimen. The summary of information of each shear wall specimen is listed in Table 2. The mechanical properties of the harden concrete used in the shear wall of all specimens are also listed in Table 2. All the harden concretes reached its target compressive strength.Table 2
Summary of specimen information.
Identification numberρsf (%)Columnsfc (MPa)Columnsft (MPa)Webfc (MPa)Webfft (MPa)Vjh,test (kN)SS-1.0-00-C60055.92.7656.32.81902SS-1.0-05-CF600.555.52.7355.23.67948SS-1.0-10-CF601.055.52.7455.16.151005SS-1.0-15-CF601.556.02.7756.57.881048SS-1.0-10-CF401.038.12.1538.33.78810SS-1.0-10-CF801.064.63.9665.66.841107


## 2.2. Test Scheme

In the test, the load exerted on the specimens included horizontal and vertical one, and the test set-up is shown in Figure3. The horizontal load was cyclic one and applied by an electrohydraulic servo control actuator on the top beam of specimen; the axial compressive force was kept constant and applied by hydraulic Jack. The target axial compression ratio, n, was taken as 0.2 for all specimens. The rolling support between hydraulic Jack and vertical reaction steel frame assured the smooth horizontal slipping of specimens. Rotation control mode was adopted and the typical rotation controlled loading history is shown in Figure 4. A nominal yield rotation angle (θym) was designated to 0.0025 rad according to the research in literature [1], which was corresponding to a nominal yield lateral displacement (Δym) at the loading point, Δym = θym × 750 mm = 1.875 mm. Before θym, the rotation levels of 0.5θym and 0.75θym were imposed. Hereafter, the rotation increment was 0.0025 rad for each rotation level and three cycles were imposed at each rotation level. The tests were conducted under the rotation controlled cyclic loading until the axial compressive force could not be maintained or the lateral force degraded below 85% of peak lateral load.Figure 3
Test set-up. (a) Schematic view. (b) Test scene.
(a)(b)Figure 4
Loading history.


## 2.3. Failure Process

The whole process from the initiation to failure of specimens was observed. During the initial loading period, the inclined cracks gradually appeared and constantly extended under cyclic lateral loading. Then several pairs of visible intercrossing diagonal cracks developed approximately along the diagonal directions of the web in shear wall, the horizontal reinforcing bars in the web of the shear wall yielded successively with the loading, and the diagonal strut mechanism in the web of the shear wall was formed. Hereafter, new cracks seldom occurred and the width of the diagonal cracks at the middle part of the web in shear wall increased rapidly with further loading. The web of the shear wall began to fail in diagonal compression, the concrete in the web was crushed and spalled, and the web lost bearing capacity gradually. Upon further cycles, slight local buckling was observed at the bottom of CFST columns and plastic hinges were formed. Finally, the lateral load rapidly dropped to about 85% of the peak lateral load and the specimen failed. All specimens mainly exhibited a typical diagonal cracking pattern during the test process and showed a shear-dominant failure mode. The failure pattern and crack distribution at the peak lateral load and ultimate state (the lateral force degraded to 85% of the peak lateral load) of all tested specimens are shown in Figure5.Figure 5
Failure pattern and crack distribution of shear wall specimens. (a) Failure pattern at peak state. (b) Failure pattern at ultimate state. (c) Schematic view of crack distribution.
(a)(b)(c)During the test process, the connections between the CFST columns and web were reliable and no significant separation at the interface occurred, which indicated that the performance of the U-shaped connector designed was effective. Moreover, no evident local buckling on the CFST columns of the SFRC shear wall specimens was observed at the ultimate state. The SFRC shear wall specimens had smaller destruction compared with that without steel fibers, which indicated that the SFRC shear wall specimens had the better repairability.


## 2.4. Influence Factors

The shear capacities of all specimens are listed in Table2. It can be seen from Table 2 that the shear performance of specimens is improved obviously with the increase of steel fiber volume fraction from 0% to 1.5%. Compared with the specimen SS-1.0-00-C60, the shear capacity of specimen SS-1.0-05-CF60, SS-1.0-10-CF60, and SS-1.0-15-CF60 is increased by 5.10%, 11.42%, and 16.19%, respectively. In addition, steel fiber could effectively control the crack width and improve the distribution shape of cracks in the web of shear wall. The cracks became much thinner, denser, of more quantity, and wider distribution with the increase of steel fiber volume fraction, whilst the crushing and spalling of concrete were also reduced. For instance, the maximum crack width of RC shear wall specimen numbered SS-1.0-00-C60 was 2.0 mm at the peak load, whilst the maximum crack width of SFRC shear wall specimen numbered SS-1.0-05-CF60, SS-1.0-10-CF60, and SS-1.0-15-CF60 was 1.6 mm, 1.2 mm, and 1.0 mm, respectively. The maximum crack width decreased significantly with the increase of steel fiber volume fraction; the similar results were reported in the literature [9, 10].Furthermore, it can be seen from Table2 that the shear performance of specimens is improved obviously with the enhancing of concrete strength from CF40 to CF80. Compared with the specimen SS-1.0-10-CF40, the shear capacity of SS-1.0-10-CF60 and SS-1.0-10-CF80 is increased by 24.07% and 36.67%, respectively. And as shown in Figure 5, the SFRC shear wall specimens with different concrete strength had the similar failure appearance at the peak and ultimate state under the condition of the same steel fiber volume fraction. It can be seen that the lateral load of SFRC shear wall specimen with higher strength concrete deteriorated more obviously after the peak load, the crushing, and spalling of concrete were more serious at the ultimate state, especially for the specimen SS-1.0-00-CF80.


## 3. Shear Resisting Mechanism of SFRC Shear Wall with CFST Columns

It was observed in the test that the failing process of SFRC shear wall with CFST columns can be divided into two stages: at early loading stage, the all bending moments acted on the SFRC shear wall with CFST columns were carried by the CFST columns, nearly all the horizontal shear forces were carried by the web, and the axial forces were shared by the CFST columns and web; at medium and late loading stage, the concrete in the web continuously cracked, resulting in a decrease of the stiffness, and the shear forces gradually transferred from the web to CFST columns. Therefore, the calculation method for shear strength of SFRC shear wall with CFST columns can be defined as(1)Vjh,calc=Vjh+Vcol,where Vjh,calc is the calculated value for shear strength of SFRC shear wall with CFST columns; Vjh is the shear strength of web; and Vcol is the shear strength of CFST columns.The shear strength of CFST columns can be defined as [4](2)Vcol=αv⋅Asc⋅τscy,where αv is the calculation coefficient for shear capacity, αv = 0.97 + 0.2In(ξ), ξ is the confinement factor, ξ = αs fy/fc, fy is the yield strength of steel tube, fc is the axial compressive strength of concrete, αs is the steel ratio of CFST column, αs = As′/Ac′, As′ and Ac′ are the cross-sectional areas of steel tube and concrete, respectively, Asc is the total cross-sectional area of CFST column, and τscy is the shear yield limit, τscy = (0.422 + 0.313αs2.33) ξ0.134 (1.14 + 1.02ξ) fc.The load transferring mechanism of SFRC web under the combined action of compression, bending, and shearing is very complicated, which is similar to RC web in the literature [2, 3]. In the SSTM, the shear strength of the web is assumed to be that of concrete compressive struts, and the concrete in the web can continue to withstand shear force after the tension ties yield. Therefore, the main characteristic of web failure is that the concrete in the web reaches its compressive strength and is crushed and spalled. For diagonal compression failure, the shear strength of the web is defined as the concrete compressive stress on the nodal zone as the concrete reaches its capacity. The concrete bearing capacity is the summation of compressions from the concrete compressive struts in the diagonal, horizontal, and vertical mechanisms [14–19]. Hence, three strut and tie load paths are proposed to simulate the force transfer within the web, including the diagonal, horizontal, and vertical mechanisms, as shown in Figure 6.Figure 6
Shear resisting mechanisms of web. (a) Diagonal. (b) Horizontal. (c) Vertical.
(a)(b)(c)The concrete in the web is subjected to the compression force; thus the diagonal compressive strut is formed, as shown in Figure6(a). Due to the higher tensile strength of SFRC, the diagonal tension forces in the web are mainly carried by the SFRC before the cracking of concrete, the action of reinforcing bars is much smaller at this time. The initial inclined cracks in the web occur when the diagonal tension force is greater than the tensile strength of concrete. After the initial cracking of concrete, the diagonal tension forces in the web are mainly carried by the horizontal and vertical ties. The horizontal ties are composed of horizontal reinforcing bars and steel fibers as shown in Figure 6(b), whilst the vertical ties are composed of vertical reinforcing bars and steel fibers as shown in Figure 6(c). In summary, the reinforcing bars and steel fibers are subjected to tension and the concrete acts as compressive struts after the development of the first cracking pattern in the web; thus a strut and tie action is formed.The diagonal mechanism (Figure6(a)) is a single SFRC diagonal compression strut whose angle of inclination θ is defined as(3)θ=tan−1Hl,where H is the distance from the lateral force loading point to the shear wall base; l is internal lever arm of the vertical shear couple, l = 0.8lw; and lw is the length of the entire web in the direction of the applied lateral force.The effective area of the SFRC diagonal compression strutAstr is defined as(4)Astr=αs×bs,where as is the depth of diagonal strut and bs is the width of diagonal strut that can be taken as the width of the web tw.The depth of the diagonal strutas depends on its end condition provided by the compression zone at the base of the web. It can be intuitively assumed that(5)αs=αw,where aw is the depth of the compression zone at the base of the web, which can be approximately determined by [15](6)αw=0.25+0.85N′Awfc′lw,where N′ is the axial compressive force for the web; Aw is the cross-sectional area of the web, Aw = tw × lw; and fc′ is the compressive strength of a standard concrete cylinder (MPa).The axial compressive force for the webN′ can be calculated as(7)N′=0.85Ectwlw0.85Ectwlw+2Es′As′+2Ec′Ac′N,where N is the axial compressive force for the shear wall specimen; Ec is the elastic modulus of web concrete; 0.85 is the reduction coefficient of axial compressive stiffness for web concrete [4]; Es′ is the elastic modulus of steel tube; and Ec′ is the elastic modulus of concrete inside the steel tube.The horizontal mechanism (Figure6(b)) includes one horizontal tie and two flat struts. Since steel fiber improves the tensile strength of the concrete, which makes the performance of SFRC shear wall different from that of RC shear wall. The horizontal tie is made up of the horizontal reinforcing bars and steel fibers and can be described as the following:(8)Fh=Fs,h+Fsf,h,where Fh is the tension force in the horizontal tie; Fs,h is the tension force of horizontal reinforcing bars, Fs,h = As,h × fs,h; As,h is the sectional area of horizontal reinforcing bars; fs,h is the tensile strength of horizontal reinforcement; Fsf,h is the tension force of horizontal steel fibers, Fsf,h = Asf,h × fsf; Asf,h is the sectional area of horizontal steel fibers; and fsf is the tensile strength of steel fiber.The experimental results indicated that the horizontal reinforcing bars in the web did not fully yield when the web damaged, so the effective coefficient for shear resistance of horizontal reinforcement,η1, is introduced. Thus Fs,h can be defined as Fs,h = η1 × As,h × fs,h, where η1 is taken as 0.75 based on the analysis for the test results in the literature [15].For simplicity, the steel fibers randomly distributed in the three dimensions of the web can be equivalent to horizontal and vertical microreinforcing bars, as shown in Figure7. Thus, the sectional area of horizontal steel fibers, Asf,h, can be defined as(9)Asf,h=nsfAsf,where Asf is the cross-sectional area of a single steel fiber and nsf is the equivalent steel fiber number, which can be calculated as(10)nsf=η2ρsftwHAsfsinθ,where ρsf is the steel fiber volume fraction and η2 is the equivalent coefficient, which can be approximately designated as 0.41 according to the study in the literature [26].Figure 7
Steel fiber ties.The sectional area of horizontal steel fibers,Asf,h, can be determined as(11)Asf,h=0.41ρsftwHsinθ.The vertical mechanism (Figure6(c)) is composed of one vertical tie and two steep struts. Similar to the horizontal tie, the vertical tie includes the vertical reinforcing bars and steel fibers within the web and can be described as the following:(12)Fv=Fs,v+Fsf,v,where Fv is the tension force in the vertical tie; Fs,v is the tension force of vertical reinforcing bars, Fs,v = As,v × fs,v; As,v is the sectional area of vertical reinforcing bars; fs,v is the tensile strength of vertical reinforcement; Fsf,v is the tension force of vertical steel fibers, Fsf,v = Asf,v × fsf; Asf,v is the sectional area of vertical steel fibers; and Asf,v = 0.41ρsftwlw/cos θ based on the above analysis.The research in literature [15] shows that the vertical reinforcing bars are not fully effective in constituting the vertical tie, and the effective coefficient for shear resistance of vertical reinforcing bars, η3, is introduced. Thus, Fs,v can be defined as Fs,v = η3 × As,v × fs,v, where η3 is taken as 0.8 [15].


## 4. Calculation Method for Shear Strength of SFRC Shear Wall with CFST Columns

The SSTM is used to determine the shear strength of web failing in diagonal compression.


### 4.1. Equilibrium Conditions

Based on the proposed strut and tie model for a web of shear wall, the horizontal and vertical shear forces can be calculated as follows:(13)Vjh=Dcosθ+Fh+Fvcotθ,(14)Vjv=Dsinθ+Fv+Fhtanθ,where Vjh is the horizontal shear force; Vjv is the vertical shear force; and D is the compression force in SFRC diagonal strut. The details of force diagram are shown in Figure 8.Figure 8
Strut and tie model for the web of shear wall.The ratios of the horizontal shearVjh assigned among the three resisting mechanisms can be defined as [15](15)Dcosθ:Fh:Fvcotθ=Rd:Rh:Rv,where Rd, Rh, and Rv are the web shear ratios resisted by the diagonal, horizontal, and vertical mechanisms, respectively, which can be defined as the following [15]:(16)Rd=1−γh1−γv1−γhγv,(17)Rh=γh1−γv1−γhγv,(18)Rv=γv1−γh1−γhγv,where γh is the fraction of horizontal shear transferred by the horizontal tie in the absence of the vertical tie and γv is the fraction of vertical shear carried by the vertical tie in the absence of the horizontal tie. The values of γh and γv can be obtained as [27](19)γh=2tanθ−13,for0≤γh≤1,(20)γh=2tanθ−13,for0≤γh≤1.By solving (13)–(15), the values of D, Fh, and Fv can be obtained as(21)D=1cosθ×RdRd+Rh+Rv×Vjh,(22)Fh=RhRd+Rh+Rv×Vjh,(23)Fv=1cotθ×RvRd+Rh+Rv×Vjh.To check whether the web of shear wall reached failure strength, the bearing pressure on the nodal zone (Figure6), where the compressive forces from the diagonal, flat, and steep struts intersect (Figure 8), must be checked. The maximum compressive stress σd,max acting on the nodal zone can be calculated as(24)σd,max=1AstrD+Fhcosθ1−sin2θ2+Fvsinθ1−cos2θ2.


### 4.2. Constitutive Laws

The softened stress-strain relation of cracked SFRC concrete can be represented as the following [28]:(25)σd=ζfc′2εdζε0−εdζε02,forεdζε0≤1,where σd is the average principal stress of concrete in the d-direction; ζ is the softening coefficient; εd is the average principal strain in the d-direction; and ε0 is the concrete cylinder strain corresponding to the cylinder strength fc′, which can be defined approximately as [29](26)ε0=−0.002−0.001fc′−2080,for20MPa≤fc′≤100MPa.The softening coefficientζ can be calculated as [28](27)ζ=5.8fc′11+400εr≤0.91+400εr,where εr is the average principal strain in the r-direction.Whether the shear strength of the web of shear wall reaches the compressive stress, the strain of the concrete diagonal strut can be conformed by the following formula:(28)σd=ζfc′,(29)εd=ζε0.The stress-strain relation of reinforcing bar can be described as(30)fs=Esεs,εs<εy,fs=fy,εs≥εy,where Es is the elastic modulus of reinforcing bars; fy is the yield strength of reinforcing bars; and fs and εs are the stress and strain in mild steel, respectively. fs becomes fs,h or fs,v, Es becomes Es,h or Es,v, εs becomes εs,h or εs,v, and fy becomes fyh or fyv when equation (30) is applied to horizontal or vertical reinforcing bars, respectively.The stress-strain relation of steel fiber can be expressed as(31)fsf=Esfεsf,where Esf is the elastic modulus of steel fiber and fsf and εsf are the stress and strain of steel fiber, respectively.The experiment results show that steel fibers are usually pulled out from concrete rather than broken due to their good tensile strength. Thus, the shear resistant action of the steel fiber depends on the bond strength between steel fiber and concrete. The stress of steel fiber,fsf, should satisfy the relationship as the following:(32)Asffsf≤λsfAspfτsf,max,where τsf,max is the maximum bond strength between steel fiber and concrete, τsf,max = 2.5fct [30]; fct is the matrix tensile strength of SFRC; λsf is the influence coefficient of steel fiber type; the values of λsf for long straight, wave-shaped, and hooked steel fibers are 0.5, 0.75, and 1.0, respectively; Aspf is the surface area of steel fiber, Aspf = πdsflsfo; dsf is the nominal diameter of steel fiber; lsfo is the effective anchorage length of steel fiber, lsfo = 0.25lsf; and lsf is the length of steel fiber.By adopting equation (33), the stress of steel fiber fsf can be defined as(33)fsf≤λsflsfdsfτsf,max.The relationship between forces and strains of the tension ties becomes(34)Fh=Fs,h+Fsf,h=0.75As,hEs,hεs,h+0.41ρsftwHEsfεsf,hsinθ≤Fyh,(35)Fv=Fs,v+Fsf,v=0.8As,vEs,vεs,v+0.41ρsftwlwEsfεsf, vcosθ≤Fyv,(36)εs,h=εsf,h=εh,εs,v=εsf,v=εv,where εs,h and εsf,h are the strains of the reinforcing bars and the steel fibers in the h-direction, respectively; εs,v and εsf,v are the strains of the reinforcing bars and the steel fibers in the v-direction, respectively; εh and εv are the average normal strains in the h- and v-directions, respectively; and Fyh and Fyv are the yielding forces of the horizontal and vertical ties, respectively.


### 4.3. Compatibility Condition

The compatibility conditions adopted herein are given as the following [31]:(37)εr=εh+εh−εdcot2θ,(38)εr=εv+εv−εdtan2θ.


### 4.4. Solution Procedures

The solution procedures are proposed [15, 19], as shown in Figures 9 and 10. The algorithm in Figure 9 starts with a selection of the horizontal shear Vjh and consists of three major steps. Firstly, σd,max is calculated by employing the equilibrium equations. By assuming the strength of the concrete strut is reached, an initial value of the softening coefficient ζ is obtained through ζ=σd,max/fc′. Secondly, the strains of the struts and ties are calculated by applying the corresponding constitutive laws. Finally, the average principal strain in the r-direction εr is calculated with the aid of the compatibility equation. Then, a new value of ζ is calculated by using the softening theoretical formula. If the assumed ζ value is close to the new ζ, the Vjh selected is the shear strength of the web, otherwise back to the iterations.Figure 9
Flow chart showing efficient algorithm.Figure 10
Algorithm for postyielding cases.Furthermore, the procedure in Figure9 is categorized into five types according to varied yielding conditions of the ties [15, 19], as shown in Table 3. Figure 10 presents the force redistribution of the web shears after the yielding of the horizontal tie.Table 3
Calculation type.
TypeType EType YHType YVType YHVType YVHConditionFh < Fyh and Fv < FyvFh = Fyh and Fv < FyvFh < Fyh and Fv = FyvFh = Fyh and then Fv = FyvFv = Fyv and then Fh = Fyh


## 4.1. Equilibrium Conditions

Based on the proposed strut and tie model for a web of shear wall, the horizontal and vertical shear forces can be calculated as follows:(13)Vjh=Dcosθ+Fh+Fvcotθ,(14)Vjv=Dsinθ+Fv+Fhtanθ,where Vjh is the horizontal shear force; Vjv is the vertical shear force; and D is the compression force in SFRC diagonal strut. The details of force diagram are shown in Figure 8.Figure 8
Strut and tie model for the web of shear wall.The ratios of the horizontal shearVjh assigned among the three resisting mechanisms can be defined as [15](15)Dcosθ:Fh:Fvcotθ=Rd:Rh:Rv,where Rd, Rh, and Rv are the web shear ratios resisted by the diagonal, horizontal, and vertical mechanisms, respectively, which can be defined as the following [15]:(16)Rd=1−γh1−γv1−γhγv,(17)Rh=γh1−γv1−γhγv,(18)Rv=γv1−γh1−γhγv,where γh is the fraction of horizontal shear transferred by the horizontal tie in the absence of the vertical tie and γv is the fraction of vertical shear carried by the vertical tie in the absence of the horizontal tie. The values of γh and γv can be obtained as [27](19)γh=2tanθ−13,for0≤γh≤1,(20)γh=2tanθ−13,for0≤γh≤1.By solving (13)–(15), the values of D, Fh, and Fv can be obtained as(21)D=1cosθ×RdRd+Rh+Rv×Vjh,(22)Fh=RhRd+Rh+Rv×Vjh,(23)Fv=1cotθ×RvRd+Rh+Rv×Vjh.To check whether the web of shear wall reached failure strength, the bearing pressure on the nodal zone (Figure6), where the compressive forces from the diagonal, flat, and steep struts intersect (Figure 8), must be checked. The maximum compressive stress σd,max acting on the nodal zone can be calculated as(24)σd,max=1AstrD+Fhcosθ1−sin2θ2+Fvsinθ1−cos2θ2.


## 4.2. Constitutive Laws

The softened stress-strain relation of cracked SFRC concrete can be represented as the following [28]:(25)σd=ζfc′2εdζε0−εdζε02,forεdζε0≤1,where σd is the average principal stress of concrete in the d-direction; ζ is the softening coefficient; εd is the average principal strain in the d-direction; and ε0 is the concrete cylinder strain corresponding to the cylinder strength fc′, which can be defined approximately as [29](26)ε0=−0.002−0.001fc′−2080,for20MPa≤fc′≤100MPa.The softening coefficientζ can be calculated as [28](27)ζ=5.8fc′11+400εr≤0.91+400εr,where εr is the average principal strain in the r-direction.Whether the shear strength of the web of shear wall reaches the compressive stress, the strain of the concrete diagonal strut can be conformed by the following formula:(28)σd=ζfc′,(29)εd=ζε0.The stress-strain relation of reinforcing bar can be described as(30)fs=Esεs,εs<εy,fs=fy,εs≥εy,where Es is the elastic modulus of reinforcing bars; fy is the yield strength of reinforcing bars; and fs and εs are the stress and strain in mild steel, respectively. fs becomes fs,h or fs,v, Es becomes Es,h or Es,v, εs becomes εs,h or εs,v, and fy becomes fyh or fyv when equation (30) is applied to horizontal or vertical reinforcing bars, respectively.The stress-strain relation of steel fiber can be expressed as(31)fsf=Esfεsf,where Esf is the elastic modulus of steel fiber and fsf and εsf are the stress and strain of steel fiber, respectively.The experiment results show that steel fibers are usually pulled out from concrete rather than broken due to their good tensile strength. Thus, the shear resistant action of the steel fiber depends on the bond strength between steel fiber and concrete. The stress of steel fiber,fsf, should satisfy the relationship as the following:(32)Asffsf≤λsfAspfτsf,max,where τsf,max is the maximum bond strength between steel fiber and concrete, τsf,max = 2.5fct [30]; fct is the matrix tensile strength of SFRC; λsf is the influence coefficient of steel fiber type; the values of λsf for long straight, wave-shaped, and hooked steel fibers are 0.5, 0.75, and 1.0, respectively; Aspf is the surface area of steel fiber, Aspf = πdsflsfo; dsf is the nominal diameter of steel fiber; lsfo is the effective anchorage length of steel fiber, lsfo = 0.25lsf; and lsf is the length of steel fiber.By adopting equation (33), the stress of steel fiber fsf can be defined as(33)fsf≤λsflsfdsfτsf,max.The relationship between forces and strains of the tension ties becomes(34)Fh=Fs,h+Fsf,h=0.75As,hEs,hεs,h+0.41ρsftwHEsfεsf,hsinθ≤Fyh,(35)Fv=Fs,v+Fsf,v=0.8As,vEs,vεs,v+0.41ρsftwlwEsfεsf, vcosθ≤Fyv,(36)εs,h=εsf,h=εh,εs,v=εsf,v=εv,where εs,h and εsf,h are the strains of the reinforcing bars and the steel fibers in the h-direction, respectively; εs,v and εsf,v are the strains of the reinforcing bars and the steel fibers in the v-direction, respectively; εh and εv are the average normal strains in the h- and v-directions, respectively; and Fyh and Fyv are the yielding forces of the horizontal and vertical ties, respectively.


## 4.3. Compatibility Condition

The compatibility conditions adopted herein are given as the following [31]:(37)εr=εh+εh−εdcot2θ,(38)εr=εv+εv−εdtan2θ.


## 4.4. Solution Procedures

The solution procedures are proposed [15, 19], as shown in Figures 9 and 10. The algorithm in Figure 9 starts with a selection of the horizontal shear Vjh and consists of three major steps. Firstly, σd,max is calculated by employing the equilibrium equations. By assuming the strength of the concrete strut is reached, an initial value of the softening coefficient ζ is obtained through ζ=σd,max/fc′. Secondly, the strains of the struts and ties are calculated by applying the corresponding constitutive laws. Finally, the average principal strain in the r-direction εr is calculated with the aid of the compatibility equation. Then, a new value of ζ is calculated by using the softening theoretical formula. If the assumed ζ value is close to the new ζ, the Vjh selected is the shear strength of the web, otherwise back to the iterations.Figure 9
Flow chart showing efficient algorithm.Figure 10
Algorithm for postyielding cases.Furthermore, the procedure in Figure9 is categorized into five types according to varied yielding conditions of the ties [15, 19], as shown in Table 3. Figure 10 presents the force redistribution of the web shears after the yielding of the horizontal tie.Table 3
Calculation type.
TypeType EType YHType YVType YHVType YVHConditionFh < Fyh and Fv < FyvFh = Fyh and Fv < FyvFh < Fyh and Fv = FyvFh = Fyh and then Fv = FyvFv = Fyv and then Fh = Fyh


## 5. Experimental Verification

Although the experimental design parameters of SFRC shear wall specimens, RC shear wall with CFST columns specimens, and SFRC shear wall with CFST columns specimens were different, the failure processes of shear wall specimens in this paper and the literature [2, 4, 9, 10] are similar. Based on the experimental results, the failure process generally could be divided into four stages: initial cracking stage, main diagonal cracks forming stage, limit stage, and failure stage. The web of shear wall was separated into a series of diagonal compression rhombic blocks by the intercrossing inclined cracks under gradually increasing lateral load. As the lateral load increased continuously, the concrete rhombic blocks reached ultimate compressive strength and the concrete in the web was crushed and spalled, resulting in obvious shear-dominant failure mode for the shear wall specimens in this paper and the literature [2, 4, 9, 10]. The existence of steel tubes and steel fibers cannot change the ultimate failure mode of shear wall specimens. Therefore, the model presented in this paper can be used to calculate the shear strength of the abovementioned three types of shear walls failing in diagonal compressions in theory.A total of 24 test specimens [2, 4, 9, 10] and their results (Table 4) are used to verify the proposed method, including 6 SFRC shear wall with CFST columns specimens in this paper, 9 RC shear wall with CFST columns specimens, and 9 SFRC shear wall specimens in the available literature [2, 4, 9, 10]. The test specimens selected cover a wide range of structure forms, geometry, loading, steel tube, steel fiber, and reinforcement detailing, as summarized in Table 4. The test specimens considered herein have three major common features: (1) all walls showed a web shear-dominant failure mode; (2) they were one-story isolated walls; (3) all contained both horizontal and vertical reinforcement uniformly distributed basically throughout the web.Table 4
Experimental verification.
SpecimenWebfc (MPa)Columnsfc (MPa)H (mm)tw × lw (mm)a × t (mm)nHorizontal reinforcementfyh (MPa)Vertical reinforcementfyv (MPa)Steel fiberVjh,test (kN)Vjh,calc (kN)Vjh,test/Vjh,calclsf/dsfρsf (%)SS-1.0-00-C6056.355.9750120 × 510120 × 3.00.2C6@100369.1712 C 6369.17——9028481.06SS-1.0-05-CF6055.255.5750120 × 510120 × 3.00.2C6@100369.1712 C 6369.17640.59488961.06SS-1.0-10-CF6055.155.5750120 × 510120 × 3.00.2C6@100369.1712 C 6369.17641.010059311.08SS-1.0-15-CF6056.556.0750120 × 510120 × 3.00.2C6@100369.1712 C 6369.17641.510489471.11SS-1.0-10-CF4038.338.1750120 × 510120 × 3.00.2C6@100369.1712 C 6369.17641.08107951.02SS-1.0-10-CF8065.664.6750120 × 510120 × 3.00.2C6@100369.1712 C 6369.17641.0110710081.10SW1.0-2 [2]42.1442.14740140 × 460159 × 3.70.35A6@120478.73A4@80683.17——872.099740.90CFST-S-S1 [4]39.7139.7192085 × 860120 × 3.00.31A6@120397A6@120397——770.67860.98CFST-S-S2 [4]39.7139.7192085 × 860120 × 3.00.62A6@120397A6@120397——891.48261.08CFST-S-L1 [4]39.7139.7192085 × 1320120 × 3.00.31A6@120397A6@120397——99110440.95CFST-S-L2 [4]39.7139.7192085 × 1320120 × 3.00.62A6@120397A6@120397——126510911.16CFST-C-S1 [4]39.7139.7192085 × 860140 × 2.00.29A6@120397A6@120397——788.57921.00CFST-C-S2 [4]39.7139.7192085 × 860140 × 2.00.58A6@120397A6@120397——895.18341.07CFST-C-L1 [4]39.7139.7192085 × 1320140 × 2.00.29A6@120397A6@120397——987.510530.94CFST-C-L2 [4]39.7139.7192085 × 1320140 × 2.00.58A6@120397A6@120397——122511001.11SW-05-40 [9]21.2—900——0.1A8@1503406 B 14373.5570.57306081.20SW-10-40 [9]26.8—900——0.1A8@1503406 B 14373.5571.07457301.02SW-15-40 [9]25.1—900——0.1A8@1503406 B 14373.5571.57707481.03SW-20-40 [9]26.9—900——0.1A8@1503406 B 14373.5572.08087921.02SW-10-30 [9]17.8—900——0.1A8@1503406 B 14373.5571.07305861.25FSW1 [10]36—600——0.096A6.53106A6.5310641.03354310.78FSW2 [10]33.5—600——0.096A6.53106A6.5310641.53304360.76FSW3 [10]35—600——0.096A6.53106A6.5310642.03404650.73FSW4 [10]34.5—600——0.096A6.53108A6.5310641.03304300.77Total 24, average 1.01, COV 0.14.In Figure11, the experimental shear strengths from 24 walls are compared to their shear strengths predicted by the method presented in this paper. The relationship between the prismatic strength fc and the cylinder strength fc′ is obtained from literature [32]. The last column of Table 4 lists the values of Vjh,test/Vjh,calc ratios to indicate the precision of the proposed model. Figure 11 shows satisfactory results for the comparison of the measured and predicted strengths. The average strength ratio (Vjh,test/Vjh,calc) is 1.01 with a coefficient of variation (COV) of 0.14 (Table 4). It is concluded that the proposed model can accurately predict the shear strengths of the shear walls mentioned above.Figure 11
Correlation of experimental and predicted wall shear strengths.Figure12 shows the effects of steel fiber volume fraction, concrete strength, and axial compression ratio on the shear strength predictions of the proposed method. The steel fiber volume fraction, concrete strength, and axial compression ratio are not observed to have significant influences on the strength ratio (Vjh,test/Vjh,calc), indicating that the main factors which affect the shear wall bearing capacity are reflected in the proposed model.Figure 12
Effects on shear strength predictions. (a) Effect of steel fiber volume fraction on shear strength predictions. (b) Effect of concrete strength on shear strength predictions. (c) Effect of axial compression ratio on shear strength predictions.
(a)(b)(c)


## 6. Conclusions

This paper introduces the tests for an innovative composite shear wall named SFRC shear wall with CFST columns and presents a proposal for determining the shear strength of the shear wall. The proposed SSTM is derived from the concept of struts and ties and satisfies equilibrium, compatibility, and constitutive laws of cracked SFRC. Based on the test results in this paper and available literature and their comparison with the proposed model, the following conclusions can be drawn:(1)
All the SFRC shear wall with CFST columns specimens showed a shear-dominant failure mode in the current tests, and all specimens mainly exhibited a typical diagonal cracking pattern during the testing process. Steel fiber could effectively control the crack width and improve the distribution shape of cracks in the web of shear wall.(2)
The shear mechanism of SFRC shear wall with CFST columns can be described by the SSTM, which consists of diagonal, horizontal, and vertical mechanisms.(3)
The steel fibers distributed randomly in concrete can be treated as longitudinal and transverse reinforcement in the shear strength analysis of SFRC shear wall with CFST columns. And the contribution of steel fiber to the wall shear strength is identified.(4)
Examination of existing experimental data indicated that the proposed model is capable of predicting the shear strengths of SFRC shear wall with CFST columns, RC shear wall with CFST columns, and SFRC shear walls for diagonal compression failures.



---

*Source: 8824308-2020-07-29.xml*