Study on the Stress Performance of an Incompletely Prefabricated Profile Steel Reactive Powder Concrete Column in High-Rise Modular Buildings

Abstract

In this study, 11 PSRPC column specimens were designed and fabricated in order to investigate the stressing performance of incompletely prefabricated steel reactive powder concrete (PSRPC) columns and were subjected to static loading tests. The effects of the strength grade and steel content of reactive powder concrete (RPC) on the axial compression performance of PSRPC columns and the effects of the steel content and eccentricity on the eccentricity performance of PSRPC columns were investigated. The test results show that the external prefabricated RPC, the internal steel section and the internal cast-in-place RPC of the PSRPC column work well together. The damage morphology of the specimens and the load–displacement and strain relationships were analysed, and the damage mechanism and the main factors influencing the compressive load capacity of the PSRPC columns were investigated. The test results are compared with the calculation results of the Chinese code (JGJ 138-2016) and the European and American codes (AISC-LRFD), and the applicability of the existing codes is discussed. Based on the hoop restraint action and limit state design method, formulae for calculating the axial and partial compression load capacity of PSRPC columns were established. The calculated values were in good agreement with the test values, and the calculation errors were within 10%.

1. Introduction

Today, there is already a wide range of applications for steel and concrete composite structures [1,2,3,4,5] and precast concrete structures [6,7]. A new type of material has emerged based on steel and concrete composite structures and precast concrete structures— incomplete prefabricated steel-reactive powdered concrete (PSRPC). Compared to traditional steel and concrete, PSRPC materials do not require the on-site tying of reinforcement, formwork and dismantling, which simplifies on-site construction processes, shortens the construction cycle, improves construction efficiency and reduces construction energy consumption. Refs. [8,9] respond to the national concept of sustainable construction.

Due to their superior performance, steel and concrete composite structures (SRC) have been widely used in high-rise buildings, large-span structures and offshore structures. Refs. [10,11,12] SRC structures have high strength and stiffness, and the concrete protects the steel sections from impact, fire damage and chemical attack. With the advancement of assembled construction, prefabricated SRCs (PSRCs) were born, initially in shopping malls and then in an increasing number of countries [13]. However, the high weight of PSRC materials, the difficulty of transporting and installing them, the dense arrangement of reinforcement and the complexity of the connections between the materials limited their use, which led to the creation of PSRPC materials.

Partially precast composite structures have been studied by scholars in China. Yang Yong [14,15] proposed a partially precast honeycomb section steel–concrete beam (CPSRC beam) and carried out tests on the bending and shear performance of CPSRC beams, finding that the precast section concrete of CPSRC beams could work in coordination with the cast-in-place section concrete and that the force performance did not differ much from that of cast-in-place beams. Yu Yunlong [16] studied the force performance of three different cross-sectional forms of partially precast section concrete beams (PPSRC beams, HPSRC beams, PPCSRC beams and PPCSRC beams) and established a calculation method for beam stiffness and deformation. Jintao Zhang [17] conducted shear performance tests on nine PPSRC beams, investigated the effect of precast shell strength and shear-to-span ratio on the shear bearing capacity of solid PPSRC beams and hollow PPSRC beams, and proposed a formula for calculating the shear bearing capacity. Gong Zhichao [18] studied the effects of eccentricity, heating time and concrete type on the damage pattern of PPSRC columns after fire exposure, testing six PPSRC columns under eccentric compression after fire exposure and one PPSRC column under eccentric compression at room temperature. The greater the eccentricity and the longer the heating time were, the lower the residual load capacity of the PPSRC column.

In this study, a new type of PSRPC column is designed. The idea was to first fabricate a prefabricated RPC shell in the factory, which consisted of an external RPC, hoop reinforcement and an internal H-beam. Corrugated steel plates were welded to the flanks of the internal H-beam to form the internal cavity. The precast RPC shell was then maintained to the required strength, transported to the construction site and assembled before the internal RPC was poured to form the complete PSRPC column, thereby compensating for the complex assembly and poor integrity of conventional fully precast steel and concrete columns. In the PSRPC column, the precast shell is made of high-performance reactive powder concrete with good durability, while the internal cast-in-place concrete can be designed as a hollow section with plain concrete or uncast concrete.

Research has been carried out on the performance of PSRPC beams and PSRPC columns after fire exposure, but less research has been carried out on the performance of PSRPC columns under static loads. Therefore, this study investigates the behaviour of PSRPC columns under axial and deflective loads, discusses the applicability of existing codes to PSRPC columns, presents the design theory of PSRPC columns and establishes the calculation method for PSRPC columns.

2. Experimental Program

2.1. Specimen Design

Eleven PSRPC column specimens were designed and divided into two batches (Figure 1). The first batch of 5 specimens was used to investigate the effect of the RPC strength grade and steel content on the performance of a foot PSRPC column under axial force, and the second batch of 6 specimens was used to investigate the effect of steel content and eccentricity distance e0 on the performance of a PSRPC column under eccentric force. For the first batch of specimens, three different H-beam flanges and web thicknesses of 6 mm, 8 mm and 10 mm were designed, and the three RPC strength classes were R120, R150 and R180. For the second batch of specimens, the H-beam flange and web thicknesses were set. In order to facilitate the eccentricity test, a bull’s leg was set at the end of the column of the eccentric specimen, and a steel plate was set inside the bull’s leg for reinforcement. To prevent stress concentration at the end of the specimen, an encrypted zone of 100 mm between the encrypted zone and 200 mm between the nonencrypted zone was set up at both ends of the specimen. Two-millimeter-thick corrugated steel plates were welded on both sides of the flange of the H-beam to form a joint with the H-beam. There are two confined spaces for the steel elements, and the specimen parameters are shown in

Table 1. Main parameters of the specimens.

Batch	Specimen Number	Specimen Size (mm × mm × mm)	RPC Strength Grade/MPa	Size of Section Steel	Eccentricity (Rate) e0/mm	Steel Ratio (%)
Ⅰ	Ⅰ-PSRPC-1	350 × 300 × 3000	120	H250 × 200 × 6 × 6	-	4.1
	Ⅰ-PSRPC-2	350 × 300 × 3000	150	H250 × 200 × 6 × 6	-	4.1
	Ⅰ-PSRPC-3	350 × 300 × 3000	180	H250 × 200 × 6 × 6	-	4.1
	Ⅰ-PSRPC-4	350 × 300 × 3000	180	H250 × 200 × 8 × 6	-	4.4
	Ⅰ-PSRPC-5	350 × 300 × 3000	180	H250 × 200 × 10 × 6	-	5.1
Ⅱ	Ⅱ-PSRPC-1	350 × 300 × 3000	180	H250 × 200 × 6 × 6	120 (0.4)	4.1
	Ⅱ-PSRPC-2	350 × 300 × 3000	180	H250 × 200 × 10 × 10	120 (0.4)	6.4
	Ⅱ-PSRPC-3	350 × 300 × 3000	180	H250 × 200 × 14 × 14	120 (0.4)	8.7
	Ⅱ-PSRPC-4	350 × 300 × 3000	180	H250 × 200 × 6 × 6	180 (0.6)	4.1
	Ⅱ-PSRPC-5	350 × 300 × 3000	180	H250 × 200 × 10 × 10	180 (0.6)	6.4
	Ⅱ-PSRPC-6	350 × 300 × 3000	180	H250 × 200 × 14 × 14	180 (0.6)	8.7

.

2.2. Mechanical Properties of Materials

The tests were carried out on RPC specimens that were cast under the same conditions and with reference to the relevant code requirements [19,20]. The mechanical properties of the RPC specimens were tested, as shown in

Table 2. Mechanical properties of RPC.

RPC Strength Grade	fcu/MPa	fc/MPa	fts/MPa	Ec/MPa
R120	124.3	106.5	8.0	41.6
R150	150.6	128.3	9.0	45.6
R180	180.3	146.5	9.6	50.3

Note: f_cu is the RPC cubic compressive strength, f_c is the RPC axial compressive strength, f_ts is the RPC splitting tensile strength and E_c is the RPC modulus of elasticity.

.

Some sections and bars were reserved during the fabrication of the specimens, and the material properties of the bars and sections used for the tests were tested in accordance with the provisions of GBT228.1-2010 [21]. The test results are shown in

Table 3. Mechanical property index of steel.

Steel Categories	Thickness or Diameter/mm	Steel Grade	fy/MPa	fu/MPa	Ec/MPa
Hoop reinforcement	8	HRB400	396.5	539.6	1.92 × 105
H-beams	6	Q345B	342	513	2.06 × 105
H-beams	8	Q345B	345	523	2.06 × 105
H-beams	10	Q345B	352	549	2.06 × 105
H-beams	14	Q345B	350	565	2.08 × 105

Note: f_y is the yield strength of the steel, f_u is the tensile strength of the steel and E_c is the modulus of elasticity of the steel.

.

2.3. Specimen Making

The PSRPC columns were fabricated in a factory prefabricated form, with the factory prefabricated part of the RPC shell being fabricated before the internal concrete was poured. The exact steps of fabrication are as follows.

• The fabricated H-beam was sandpapered and polished to remove surface impurities, as shown in Figure 2a;
• The bias specimen was welded with 4 steel plates at each end, with H-beam strain gauges, and then corrugated steel plates were welded to the H-beam flange in order to form the external RPC inner membrane;
• The hoops were placed over the sections and welded to the sections at the four corners in order to form the internal section skeleton, as shown in Figure 2b. The hoop strain relief was then applied, and the external prefabricated RPC formwork was produced, as shown in Figure 2c;
• The external precast RPC was poured and then maintained at room temperature;
• After the initial setting of the external precast RPC, the internal RPC was poured to form the PSRPC column, and the specimens were fabricated as shown in Figure 2d.

2.4. Loading and Testing Solutions

The test loading was carried out using a 20,000 kN four-column electrohydraulic servo pressure tester. Figure 3a shows a schematic view of the loading device, and Figure 3b shows a schematic diagram of the test loading device. In order to avoid predamage of the specimen ends, the specimen ends were reinforced by placing 30-mm-thick steel plates at both ends of the specimen. The specimen was geometrically aligned using an optical centering device prior to loading. After alignment, the specimen was preloaded to 20% of the measured ultimate load in order to ensure that the test apparatus and equipment was functioning correctly. Displacement transducers D1–D9 and RPC strain gauges were arranged as shown in Figure 3a. D1–D7 were used to measure the horizontal lateral deflection of the eccentric specimen, and D8–D9 were placed on the steel plate at the bottom of the test machine in order to measure the axial deflection of the specimen. The arrangement of the hoop and section strain gauges is shown in Figure 3b.

3. Test Results and Analysis

3.1. Experimental Phenomena

For the axial compression specimens, the damage process was similar for all five specimens. The damage pattern is illustrated by the example of I-PSRPC-3, where the specimen is in the elastic phase at the beginning of the loading period and no cracks are evident. When loaded to 30% of the ultimate load (P_max), a small crack appears at the top end of the column. When loaded to 75% P_max, small cracks appear at both ends of the column in different directions. As the load continues to 90% P_max, the crack extends from the end to the middle of the column as the load increases, and the RPC at the lower end of the column spalls off extensively. When the load reaches P_max, several vertical main cracks can be seen on the surface of the column, the external RPCs are spalling off, and the load-carrying capacity of the specimen is reduced, with damage occurring at the bottom end of the specimen.

For the eccentric specimens, the damage pattern of PSRPC columns under eccentric loading is similar to that of ordinary reinforced concrete columns under eccentric loading. The test results show that the main influencing factor on the damage pattern is the eccentricity distance, which is illustrated by II-PSRPC-2 and II-PSRPC-5. The eccentricity of II-PSRPC-2 is 0.4, and the specimen deforms elastically without significant changes in the early stages of loading. When the load reaches 48% P_max, small cracks appear in the middle of the column on the tensile side, and longitudinal diagonal cracks appear at the end of the column. As the load continues to increase, the cracks continue to extend, accompanied by the tearing sound of the RPC, and the lateral deflection increases significantly. When the load reaches 80% P_max, it is clear that the column buckles, with the RPCs at the top and bottom ends spalling off and the RPCs in the middle bulging out. When the load approaches P_max, the central RPC is crushed, the main crack penetrates the cross-section, the specimen shows a “broken” state, and the load-carrying capacity of the specimen begins to decrease. When loaded to 20% P_max, a horizontal crack appears in the middle tensile zone of the specimen. When loaded to 36% P_max, a longitudinal crack appears at the upper end of the specimen. When loaded to 70% P_max, RPC spalling in the upper compressive zone is evident, accompanied by a “tearing” sound of the RPC, and the horizontal cracks increase and extend to both sides of the cross-section. When loaded to the ultimate load, the RPC in the compressive zone breaks up, the RPC in the tensile zone cracks, and the load capacity of the specimen begins to decrease.

3.2. Load–Displacement/Strain Analysis

The load–displacement curves of the axially compressed specimens are shown in Figure 4. In the elastic stage, the specimen is slightly deformed, the displacement changes linearly with increasing load, and only a few cracks appear at this time. When the load reaches 75% P_max, the specimen enters the elastoplastic state, the stiffness of the specimen decreases, the displacement increases faster and the load–displacement relationship becomes nonlinear. When the load reaches P_max, the specimen enters the damage phase, the section yields, and the load carrying capacity of the specimen begins to decrease. Figure 5 shows the relationship between the load and the strains in the section flange, section web, RPC and hoop. The section flange and web reach their yield stress before P_max, and the hoop bar yields close to P_max, indicating that the hoop bar plays an important role in the ultimate stage. The load–strain relationship curves for the section flange, section web and RPC are approximately the same, indicating that the section and RPC may act in concert.

Figure 6 shows the load–transverse displacement curves for each point of the deflected specimen, from which the transverse displacements at each point are symmetrically distributed along the cross section in the column towards the ends. The greater the eccentricity is, the greater the transverse displacement of the specimen. Figure 7 shows the relationship between the eccentric load and the transverse displacement in the middle of the specimen. The higher the steel content is, the higher the ultimate displacement of the specimen. The eccentricity has a significant effect on the ultimate load capacity of the specimen; the higher the eccentricity is, the lower the ultimate load of the specimen. For II-PSRPC-1 and II-PSRPC-4, II-PSRPC-2 and II-PSRPC-5, and II-PSRPC-3 and II-PSRPC-6, the steel content and RPC strength. The eccentricity increases from 0.4 to 0.6 with the same grade, and the ultimate load decreases by 27.07%, 25.14%, and 24.55%, respectively.

3.3. Analysis of Influencing Factors

3.3.1. Impact of RPC Strength Levels

The ultimate loads of the specimens are 11,036 kN, 12,125 kN, and 12,996 kN, and Figure 8 shows that the RPC strength grade has a significant effect on the load capacity of the specimens. The ultimate loads of the specimens were 11,036 kN, 12,125 kN, and 12,996 kN, respectively. The higher the RPC strength grade, the greater the slope of the elastic phase of the load–displacement curve, the slower the rate of increase in displacement and the steeper the decreasing section of the load capacity. This indicates that the higher the RPC strength grade is, the higher the stiffness of the specimen and the higher the ultimate load-carrying capacity. From the damage characteristics, it can be seen that increasing the strength of RPC can enhance the bonding force of concrete, and the better the bonding performance of RPC and steel sections, the more it inhibits the development of cracks, and the fewer the cracks that will be produced in the specimen. After the ultimate load, the ductility of the high-strength RPC specimens is poor due to the increased brittleness of the high-strength RPC, and the specimens show brittle damage.

3.3.2. Effect of Steel Content

Batch II eccentric specimens II-PSRPC-1, II-PSRPC-2 and II-PSRPC-3 have the same cross-sectional dimensions, eccentricity and RPC strength class and contain 4.1%, 6.4%, and 8.7% steel, respectively. The same is true for specimens II-PSRPC-4, II-PSRPC-5 and II-PSRPC-6. As shown in Figure 9, the ultimate load of the specimens increases with increasing steel content. The slope of the rising section of the load–displacement curve of the deflected specimen increases as the steel content increases, the displacement increases at a slower rate and the load capacity decreases at a slower rate, as shown in Figure 7. As the thickness of the steel section increases, the contact surface between the section and the RPC increases, which improves the chemical bond between the section, and the RPC and increases the restraint effect of the section and the hoop on the RPC in the core area, thus increasing the stiffness and ductility of the specimen. Figure 9 shows that increasing the steel content improves the load bearing properties of the specimen but is not the most important factor affecting the load bearing capacity.

3.3.3. Effect of Eccentricity

The ultimate load capacity of the two groups of specimens is shown in Figure 10, which shows that the eccentricity has a significant effect on the ultimate load capacity of the specimens. From the damage characteristics, the lateral displacement in column II-PSRPC-1 was 35.2 mm when the ultimate load capacity was reached, and the lateral displacement of II-PSRPC-1 was 35.2 mm when the ultimate load capacity was reached. The damage pattern of the specimens did not differ much, but the ultimate loads did vary considerably. The greater the eccentricity, the greater the width of the main crack and the greater the number of cracks in the specimens.

4. Calculation of the Axial Compression Load Capacity of PSRPC Columns Based on Hoop Restraint Action

4.1. Calculation Methods Already Available

There are three main methods for calculating the prevailing international positive section load capacity of steel and concrete columns. First, the Code for Design of Combined Structures (JGJ 138-2016) issued by the Ministry of Housing and Urban–Rural Development of China [22] stipulates the calculation method. Its proposed section steel load-carrying capacity of concrete columns is calculated in accordance with the assumption of a flat section, ignoring the tensile strength of the concrete; taking into account the stiffening effect of the concrete, it is assumed that the sections with the required width to thickness ratio will not buckle locally during loading. The axial compressive load capacity of the combined column in the code is calculated according to Formula (1):

$$N=0.9\phi (f_c{A}_c+f_y{A}_s+f_a{A}_a)$$

(1)

where

• f_c is the axial compressive strength of concrete (MPa);
• A_c—cross-sectional area of the concrete (mm²);
• f_y, f_a—yield strength of longitudinal bars and sections;
• A_a—cross-sectional area of longitudinal bars and sections;
• φ—the axial compression stability factor of the combined steel and concrete column, determined from the length to the slenderness table;

$$\lambda =\frac{l_0}{i}$$

(2)

• l₀—calculated length of the member (mm)
• i—minimum radius of gyration of the section (mm)

$$i=\sqrt{\frac{E_c{I}_c+E_a{I}_a}{E_c{A}_c+E_a{A}_a}}$$

(3)

• E_c, E_a—modulus of elasticity of concrete, steel sections (MPa)
• I_c, I_a—moment of inertia of the concrete and steel sections (mm⁴)

The second calculation method for steel structures, mostly adopted in Europe and the USA, considers the reinforcement of the strength and stiffness of the section columns by adding an expansion factor to the external concrete or replacing the concrete with an equivalent steel structure and calculating the load bearing capacity in accordance with the steel design method, such as the American Institute of Steel Construction AISC-LRFD (2005) Code [23], which uses the following calculation formula:

$${\lambda }_c\le 1.5P=A_s\left({0.685}^{{\lambda }_c^2}\right)[F_y+0.7F_{yr}\left(A_r/A_s\right)+0.6f_c^{\prime }\left(A_c/A_s\right)]$$

(4)

$${\lambda }_c>1.5P=A_s\left(\frac{0.877}{{\lambda }_c^2}\right)[F_y+0.7F_{yr}\left(A_r/A_s\right)+0.6f_c^{\prime }\left(A_c/A_s\right)]$$

(5)

in the formula,

• λ_c—slenderness ratio parameter, ${\lambda }_c=\frac{Kl}{r\pi }\sqrt{\frac{F_y}{E_m}}$;
• K—effective length factor;
• R—rotational radius of the buckling section;
• L—free length of steel-reinforced concrete columns;
• E_m—the converted elastic modulus of the steel-reinforced concrete column;
• E, E_c—elastic modulus of steel and concrete;
• A_s, A_r, A_c—section area of steel, longitudinal reinforcement and concrete;
• F_y, F_yr, $f_c^{\prime }$—yield strength of steel, longitudinal reinforcement and compressive strength of concrete.

The third, the Soviet theory of equating steel sections to reinforcing bars and then calculating them in accordance with reinforced concrete structures, is that the sections can be deformed and fully bonded to the concrete in a coordinated manner.

In accordance with the specifications JGJ 138-2016 [22] and AISC-LRFD [23], the positive section compressive load bearing capacity of the PSRPC axial compression column was calculated, and the comparison between the calculated results and the test results is shown in Figure 11. The calculation results of our code JGJ 138-2016 [22] are closer to the test values, and the calculation results of both codes are on the conservative side.

4.2. Calculation of Load-Carrying Capacity Based on Hoop Restraint Action

The calculation methods in the existing codes do not take into account the effect of restraint on the force performance of concrete. Mander [24] considered that hoop bars have a lateral restraint on concrete, which can put concrete in a three-way compressive stress state and can improve the axial compressive strength of concrete. His proposed model of hoop-constrained concrete columns considers that the restraint boundary curve in the plane of hoop-constrained concrete and between the hoop bars on both sides is parabolic, assuming an initial angle of 45 degrees. According to the lateral restraint action, the concrete can be divided into two different zones—the invalid constraint zone and the valid constraint zone [25]—as shown in Figure 12.

The RPC has no transverse restraint and is in a state of unidirectional compressive stress; therefore, the axial compressive strength of the RPC can be used. As section H is an open section, the restraint of the section is not considered. The restraining effect of the hoop on the RPC in the effective restraint zone is calculated as follows.

Effective restraint RPC area at hoop section A_e0:

$$A_{e0}=b_s{h}_s-2\times \frac{w_0^2}{6}$$

(6)

Effective restraint area of RPC at the midpoint of the two hoop layers:

$$A_e=(b_s{h}_s-2\times \frac{w_0^2}{6})\left(1-\frac{s^{\prime }}{2b_s}\right)\left(1-\frac{s^{\prime }}{2h_s}\right)$$

(7)

Effective restraint factor of hoop reinforcement on RPC in the effective restraint zone K_e:

$$K_e=\frac{A_e}{b_s{h}_s}$$

(8)

The effective restraint zone is subjected to the lateral compressive stress in the y-direction of hoop f_l:

$$f_l=K_e\rho f_{yh}$$

(9)

$$\rho =\frac{V_s}{s^{\prime }bh}$$

(10)

in the formula:

• f_yh—standard value of the tensile strength of the hoop reinforcement;
• ρ—the volume hoop ratio between the centerline of two adjacent hoop layers;
• V_s—hoop volume;
• $s^{\prime }$—distance between the centerline of two adjacent layers of hoop reinforcement.

Concrete strength improvement factors:

$$k=-1.254+2.254\sqrt{1+\frac{7.94f_l}{f_{co}}}$$

(11)

in the formula:

• f_co—axial compressive strength of unconstrained RPC.
• Axial compressive strength of hoop restrained RPC f_ce:

$$f_{ce}=k{f}_{co}$$

(12)

According to specification JGJ138-2016 [22], the ultimate load on the PSRPC column is shared by the steel section, the unrestrained zone RPC and the restrained zone RPC, and the modified calculation formula is proposed as follows:

$$\begin{gathered} N_{max}=0.9\phi \left(N_a+N_{co}+N_{ce}\right) \\ =0.9\phi \left(f_a{A}_a+f_{co}{A}_{co}+f_{ce}{A}_{eo}\right) \end{gathered}$$

(13)

in which:

• N_a—the axial force carried by the profile;
• N_cn—the axial force assumed by the RPC in the invalid constraint zone;
• N_ce—the axial force assumed by the RPC in the effective constraint zone.

As shown in Figure 13, the test results of the axial compression specimens were compared with the calculated results according to Equations (3)–(13). The results show that the deviation between the test results and the calculation results of the modified formula is within 10%, and the modified formula considering the hoop restraint effect can predict the ultimate load capacity of such specimens more accurately.

5. Limit State Design Method PSRPC Calculation of Column Bias Load Capacity

5.1. Basic Assumptions

There have been many studies on the deflected bearing capacity of combined steel and concrete columns [26,27,28]. In order to simplify the calculation process, the following assumptions are made.

(1)

The tensile strength of the concrete is not considered;

(2)

The average strain is consistent with the flat section assumption;

(3)

Concrete compressive zone stress is graphically equivalent to rectangular stress; compressive zone edge concrete ultimate compressive strain ${\epsilon }_{\mathrm{c}\mathrm{u}}=0.003$;

5.2. Calculation of Load-Carrying Capacity

In this study, bending damage occurred in specimens II-PSRPC-1 to 6, with the section flange in the tension zone breaking before the concrete in the compression zone, and the section web did not reach its yield strength. Due to the uncertainty of the location of the neutral axis, the section may or may not reach its yield strength during loading, so the following two scenarios are derived based on the location of the neutral axis, as shown in Figure 14.

(1) As shown in Figure 14a, when $1.2a_a^{\prime }\le x\le x_b$, it is known from the assumption of a plane section that the compression flange of steel reaches the yield strength. Since the section is symmetrically reinforced, $f_a=f_a^{\prime }$., $a_a=a_a^{\prime }$, $a_s=a_s^{\prime }$, According to the equilibrium condition of force $\sum x=0$, we can obtain:

$$\begin{array}{l}{N}_{cu}=f_{c,out}b{h}_2+2f_{c,out}{b}_2\left(x-h_2\right)+f_{c,in}{b}_1\left(x-h_2\right)\\ +A_{af}^{\prime }{f}_{yf}^{\prime }+\left(x-a_a^{\prime }-\frac{t_f^{\prime }}{2}-d\right)t_w^{\prime }{f}_{aw}^{\prime }-A_{af}{f}_{af}\\ -(h-x-a_a-\frac{t_f}{2}-d)t_w{f}_{aw}\end{array}$$

(14)

$$x=\frac{N_{cu}-f_{c,out}b{h}_2+2f_{c,out}{b}_2{h}_2+f_{c,in}{b}_1{h}_2+t_w^{\prime }{f}_{yw}^{\prime }h}{2f_{c,out}{b}_2+f_{c,in}{b}_1+2t_w^{\prime }{f}_{yw}^{\prime }}$$

(15)

According to the stress–strain distribution diagram in Figure 14a, the moment of the neutral axis is obtained from the moment balance equation:

$$\begin{array}{l}{N}_{cu}\left(e_0-\frac{h}{2}+x\right)=f_{c,out}b{h}_2\left(x-\frac{h_2}{2}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{2}{f}_{c,out}{b}_2{(x-h_2)}^2+\frac{1}{2}{f}_{c,out}{b}_1{(x-h_2)}^2+A_{af}^{\prime }{f}_{yf}^{\prime }\left(x-a_a^{\prime }\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+t_w^{\prime }{f}_{yw}\left[\frac{{(x-a_a^{\prime }-\frac{t_f}{2})}^2}{2}-\frac{d^2}{6}\right]+A_{af}{f}_{yf}\left(h-x-a_a\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+t_w^{\prime }{f}_{yw}\left[{(h-x-a_a-\frac{t_f}{2})}^2-\frac{d^2}{6}\right]\end{array}$$

(16)

$$x_b=\frac{0.8}{1+\frac{f_a}{2E_s{\epsilon }_{cu}}}(h-a_a^{\prime })$$

(17)

(2) As shown in Figure 14a, when $a_a^{\prime }\le x\le 1.2a_a^{\prime }$, it is known from the assumption of a plane section that the compression flange of steel reaches the yield strength. Since the section is symmetrically reinforced, $\sum x=0$. According to the equilibrium condition of force x = 0, we can obtain:

$$\begin{array}{l}{N}_{cu}=f_{c,out}b{h}_2+{2f}_{c,out}{b}_2\left(x-h_2\right)+f_{c,in}{b}_1\left(x-h_2\right)\\ -\left(h_1-2t_f\right)t_w{f}_{yw}-A_{af}{f}_{yf}\end{array}$$

(18)

$$x=\frac{N_{cu}-f_{c,out}b{h}_2+2f_{c,out}{b}_2{h}_2+f_{c,in}{b}_1{h}_2+\left(h_1-2t_f\right)t_w^{\prime }{f}_{yw}+A_{af}{f}_{af}}{2f_{c,out}{b}_2+f_{c,in}{b}_1}$$

(19)

From the moment balance, the moment of the neutral axis is obtained:

$$\begin{array}{l}{N}_{cu}\left(e_0-\frac{h}{2}+x\right)=f_{c,out}b{h}_2\left(x-\frac{h_2}{2}\right)+f_{c,out}{b}_2{\left(x-h_2\right)}^2+\\ \frac{1}{2}{f}_{c,in}{b}_1{\left(x-h_2\right)}^2+A_{af}{f}_{af}\left(h-a_a-x\right)+\\ (h_1-2t_f)t_w^{\prime }{f}_{yw}(\frac{h}{2}-x)\end{array}$$

(20)

in the formula:

• x—height of the concrete compression zone;
• f_c,in, f_c,out—axial compressive strength of RPC in effective and unconstrained zones of stirrups;
• $f_{af}^{\prime }$, $A_{af}^{\prime }$—yield strength and section area of the steel flange in the compression zone;
• f_af, A_af—yield strength and section area of steel flange in tensile zone;
• t_f, t_w—steel flange thickness and web thickness;
• $f_{aw}^{\prime }$, $A_{aw}^{\prime }$—yield strength and section area of steel web in compression zone;
• f_aw, A_aw—yield strength and section area of steel web in tension zone;
• d—height of unyielding steel.

According to the above calculation method, the calculation results and test errors of the eccentric compression bearing capacity of each specimen are shown in Figure 15.

6. Conclusions

In this paper, the basic force performance of incomplete prefabricated steel-reactivated powder concrete (PSRPC) columns under axial and eccentric loads is investigated. Eleven PSRPC column elements were fabricated. The effects of eccentricity and steel content on the eccentricity of PSRPC columns were investigated in six eccentric specimens. On the basis of the experimental study, the load carrying capacity calculation equations for PSRPC columns were proposed, and the following conclusions were obtained.

(1)

The new PSRPC columns proposed in this paper, with their external prefabricated RPC, internal steel sections and internal cast-in-place RPC, can act in harmony with each other and have a strong ultimate compressive load-bearing capacity. This new construction method is reasonable and has good promotion prospects.

(2)

All specimens had good deformation properties until the concrete was crushed and spalled; the RPC strength had a significant effect on the axial load capacity of the PSRPC columns, and the eccentricity had a significant effect on the load capacity of the eccentrically compressed specimens.

(3)

The eccentricity is the main factor in determining whether the flange of a steel section in the compression zone yields or not. When the eccentricity is 0.4, the steel flange in the compression zone yields, and when the eccentricity is 0.6, the steel flange in the compression zone does not yield.

(4)

When the eccentricity is 0.6, the flange of the section in the compression zone does not yield.

(5)

For the axial compression specimens, the lateral restraint of the concrete by the hoop bars is considered, and the calculation formula for the axial compression load capacity of PSRPC columns based on the restraint of hoop bars is proposed in conjunction with Mander’s hoop concrete restraint theory. The calculation results are in good agreement with the test results, with errors within 10%.

(6)

For the deflected specimens, a formula for calculating the deflected load capacity of PSRPC columns based on the limit state design method was proposed, and the calculation results were compared with the test results with good accuracy, with a maximum deviation of 6.12%.