Study on Seismic Behavior of Cross-Shaped-Steel-Reinforced RPC Columns

Abstract

Based on the hysteretic tests of steel-reinforced reactive powder concrete (RPC) columns and reinforced RPC columns, the finite element numerical models of these two kinds of RPC columns were established by OpenSees (2016). The feasibility of the model was verified by comparing the results of tests and simulation. On this basis, the nonlinear analysis of seismic performance of cross-shaped-steel-reinforced RPC columns was carried out. The influences of different factors such as longitudinal reinforcement ratio, steel sectional resistance moment, RPC grade, steel strength and section form of shape steel on the hysteretic performance were investigated. Finally, the hysteretic model of cross-shaped-steel-reinforced RPC columns was established. The results showed that, compared with H-shaped-steel-reinforced RPC columns, the peak bearing capacity of cross-shaped-steel-reinforced RPC columns was increased by 21.2%, but the displacement ductility was obviously reduced. With the increase of slenderness ratio, the lateral stiffness and horizontal bearing capacity of cross-shaped steel RPC columns decreased rapidly. In addition, the peak load was improved with the increase of RPC strength, steel sectional resistance moment and longitudinal reinforcement ratio. The hysteretic model was consistent with the simulation results, which can effectively predict the hysteretic characteristics of cross-shaped-steel-reinforced RPC columns. The research results can provide a theoretical basis for the engineering design and application of cross-shaped-steel-reinforced RPC columns.

1. Introduction

In recent years, steel-reinforced concrete (SRC) structures have been widely used in high-rise, long-span and heavy-duty buildings. The evaluation of earthquake collapse risk of existing buildings is constantly updated [1], and new progress has been made in the internal force transmission of composite structures [2]. However, with the continuous advancement of sustainable development, the performance requirements of structure have been improved from safety to high durability, high earthquake resistance and high environmental protection [3]. In freezing and erosive environments, the low tensile properties and relatively poor durability of normal concrete (NC) lead to serious cracks or damage in SRC structures before the end of the design service life. Reactive powder concrete (RPC) is a kind of ultra-high-performance concrete (UHPC), which has ultra-high compressive strength, excellent tensile strength and durability [4,5]. When it is applied to an SRC structure to replace NC, it can not only improve the bearing capacity of the structure and reduce the cross-section size, cement consumption and carbon emissions, but also prolong the service life and reduce the maintenance cost.

At present, scholars at home and abroad have performed many tests on the seismic performance of RPC columns. Aboukifa et al. [6] conducted seismic tests on UHPC columns and found that the higher the reinforcement ratio and reinforcement grade, the better the seismic performance of such columns. Deng et al. [7] found that the ductility and shear capacity of RPC columns were greatly affected by the axial compression ratio through finite element analysis (FEA). Qian et al. [8] carried out low-cycle cyclic loading tests on UHPC columns strengthened with glass-fiber-reinforced plastic (GFRP) bars, and found that cracks can be effectively suppressed by GFRP bars, but that the seismic performance of UHPC columns was adversely affected. He et al. [9] carried out seismic tests on high-strength steel-bar-reinforced RPC columns and found that RPC can be effectively confined by high-strength steel bars, and that the initial stiffness, ductility and bearing capacity of the specimens was improved. Abbassi et al. [10] studied the seismic performance of RPC columns strengthened with FRP based on FEA. The results showed that this kind of column had good seismic performance and ductility, and the peak load increased with the increase of RPC strength.

Furthermore, the mechanical properties of cross-shaped-steel-reinforced concrete columns have also attracted a lot of attention. Pan et al. [11] put forward a cyclic axial stress–strain model of a cross-section steel-concrete-filled steel tube and verified the accuracy of the model through test results. Jiang et al. [12] carried out a low-cycle cyclic loading test and FEA on giant cross-shaped-steel-reinforced concrete columns, and found that steel content and axial compression ratio have significant influence on the seismic performance of such columns. Nguyen et al. [13] put forward the fracture criterion of cross-shaped-steel-reinforced concrete columns based on FEA. Mostafa et al. [14] carried out seismic tests on three new types of cross-shaped-steel-reinforced high-strength lightweight aggregate concrete columns, and found that such columns demonstrate good seismic performance.

Moreover, remarkable achievements have been made in the seismic performance of steel RPC columns at home and abroad. Ji et al. [15] carried out the seismic performance test and FEA of steel-reinforced RPC columns wrapped with GFRP tubes, and put forward the hysteretic model of such columns. Ma et al. [16] carried out the low-cycle cyclic loading test on steel-reinforced UHPC columns, and the results showed that with the increase of axial compression ratio, steel-reinforced UHPC columns showed bending failure. Zhang et al. [17] investigated the influence of different parameters on the seismic performance of steel-reinforced RPC columns through tests and established the flexural capacity formula of such columns.

In practical engineering and structural design, different from H-shaped-steel-reinforced RPC columns, the influence of side webs should be considered in the calculation of the normal section bearing capacity of cross-shaped-steel-reinforced RPC columns under uniaxial eccentric compression. In addition, the vertical H-shaped steel will also affect the energy consumption and ductility of cross-shaped-steel-reinforced RPC columns. At present, there are few reports on the seismic behavior of cross-shaped-steel-reinforced RPC columns. Therefore, the numerical model of such columns will be established in this paper, and the influence of RPC grade, axial compression ratio, steel sectional resistance moment, longitudinal reinforcement ratio, stirrup ratio, slenderness ratio and section form of shape steel on seismic performance will be investigated. Finally, a horizontal load-displacement hysteretic model of cross-shaped-steel-reinforced RPC columns will be established.

2. Numerical Model of Cross-Shaped-Steel-Reinforced RPC Column

2.1. Basic Assumption

The software OpenSees (2016, University of California, Berkeley, CA, the U.S.) was used to analyze the hysteretic behavior of cross-shaped-steel-reinforced RPC columns. The basic assumptions were as follows: (1) Plane section assumption: the cross section of the member is always a plane from stress to failure, and the strain distribution is a straight line. (2) The relative slip between longitudinal reinforcement, section steel and RPC is not considered. (3) The members have sufficient shear bearing capacity. (4) Changes in stress and strain caused by temperature and humidity are not considered.

2.2. Constitutive Relationship of Materials

The Steel02 model provided by OpenSees was adopted as the constitutive relation of steel and reinforcement. The bilinear follow-up strengthening model proposed by Menegotto and Pinto [18] was adopted as the skeleton curve of Steel02, and the Bauschinger effect was considered in unloading. Figure 1 is a schematic diagram of the stress–strain (σ_s–ε_s) curve of the Steel02 model under cyclic load. In Figure 1, f_y, ε_y and E_s are yield strength, yield strain and elastic modulus, respectively.

The Concrete07 model provided by OpenSees was adopted as stress–strain (σ_c−ε_c) curve of RPC, which is based on the Chang-Mander model [19], modified by Waugh [20]. The stress–strain relationship is shown in Figure 2, in which f_c and ε₀ are the peak axial compressive stress and strain, respectively, and f_t and ε_t are the peak axial tensile stress and strain, respectively. The calculation method can be found in reference [21]. ε⁻_cr and ε⁺_cr are the strains at the starting point of the straight descending branch in the compression and tension curves, respectively. ε_sp and ε_crt are the ultimate strains in compression and tension, respectively. When considering the effect of the stirrup constraint, the enhanced peak compressive stress f_cc and peak compressive strain ε_cc were calculated according to reference [22].

2.3. Establishment of Numerical Model

As shown in Figure 3, the numerical model of cross-shaped-steel-reinforced RPC columns was established. The part from the anti-bending point to the bottom of the column was selected as the model, and the numerical model was divided into five elements along the column axis. The nonlinear beam–column element considering distributed plasticity was adopted to the elements. The division of column section is shown in Figure 3b. To consider the constraint of stirrups, the RPC section was divided into two parts: the core area and the cover concrete. The boundary condition of column bottom was fixed-bearing constraint, in which rotation and displacement in plane were not considered. The top of the column was free, and horizontal cyclic load ± P and vertical concentrated force N₀ were applied here.

3. Verification of Numerical Model

To verify the accuracy and rationality of the above model, H-shaped-steel-reinforced RPC columns and reinforced RPC columns were selected to simulate hysteretic performance. The simulation results were compared with the test results. The simulation results can provide a basis for the effectiveness of the numerical model of the cross-shaped-steel-reinforced RPC columns.

3.1. Reinforced RPC Column

Deng et al. [7] carried out the hysteretic test of reinforced RPC columns under cyclic horizontal load. Two typical test column specimens DZ and DGZ were selected to simulate the hysteretic behavior. The size and reinforcement of the specimens are shown in Figure 4. The clear height of the column was 500 mm, and the section size was 250 mm × 250 mm. The strength grades of steel bars in specimens DZ and DGZ were HRB400 and HRB500, respectively. The measured yield strengths f_y of C10 and C16 steel bars were 472 N/mm² and 436 N/mm², respectively. The measured yield strengths f_y of D10 and D16 steel bars were 606 N/mm² and 551 N/mm², respectively. According to GB50010-2010 [23], the elastic modulus E_s of steel bars was 2.0 × 10⁵ N/mm². The measured axial compressive and tensile strength of RPC were 120 N/mm² and 19.8 N/mm², respectively. The elastic modulus E_c was 4.32 × 10⁴ N/mm², and the peak compressive strain ε₀ and peak tensile strain ε_t were 0.0033 and 0.0006, respectively.

According to the test parameters, the FEA of reinforced RPC columns was carried out by OpenSees. The constitutive relationships of RPC and steel bars were the same as those in Section 2.2. Figure 5 and Figure 6 show the comparison of hysteretic curves and skeleton curves between test and simulation, respectively.

Table 1. Comparison of characteristic values of test and simulated hysteretic curves.

No.	Direction	${\mathit{P}}_{\mathbf{m}}^{\mathbf{T}}$/kN	${\mathit{P}}_{\mathbf{m}}^{\mathbf{C}}$/kN	${\mathit{P}}_{\mathbf{m}}^{\mathbf{T}}/{\mathit{P}}_{\mathbf{m}}^{\mathbf{C}}$	${\mathit{K}}_{\mathbf{y}}^{\mathbf{T}}$ /kN·mm−1	${\mathit{K}}_{\mathbf{y}}^{\mathbf{C}}$ /kN·mm−1	${\mathit{K}}_{\mathbf{y}}^{\mathbf{T}}/{\mathit{K}}_{\mathbf{y}}^{\mathbf{C}}$	${\mathit{K}}_{\mathbf{d}}^{\mathbf{T}}$ /kN·mm−1	${\mathit{K}}_{\mathbf{d}}^{\mathbf{C}}$ /kN·mm−1	${\mathit{K}}_{\mathbf{d}}^{\mathbf{T}}/{\mathit{K}}_{\mathbf{d}}^{\mathbf{C}}$
DZ	+	+543.09	+559.67	0.97	+6.88	+6.00	1.15	6.91	8.32	0.83
	−	−555.01	−549.26	1.01	−7.68	−8.00	0.96	8.63	8.76	0.99
DGZ	+	+577.00	+585.56	0.99	+7.84	+8.00	0.98	5.92	9.14	0.65
	−	−587.11	−581.12	1.01	−8.44	−8.00	1.06	6.01	8.88	0.68

shows the comparison of characteristic values of hysteretic curves between test and simulation, where + and − indicate that the loading direction is positive and negative respectively. In the table, $P_{\mathrm{m}}^{\mathrm{T}}$ and $P_{\mathrm{m}}^{\mathrm{C}}$ were the horizontal peak loads of test and simulation, respectively. $K_{\mathrm{y}}^{\mathrm{T}}$ and $K_{\mathrm{y}}^{\mathrm{C}}$ were the yield stiffness of the test and simulation, respectively. $K_{\mathrm{d}}^{\mathrm{T}}$ and $K_{\mathrm{d}}^{\mathrm{C}}$ were the stiffness of descending branch of the test and simulation, respectively. It can be seen that the hysteretic curves of simulation were close to the test results. The ratio $P_{\mathrm{m}}^{\mathrm{T}}/P_{\mathrm{m}}^{\mathrm{C}}$ of peak load between test and simulation was ranging from 0.97 to 1.01, with an average value of 0.99. The ratio $K_{\mathrm{y}}^{\mathrm{T}}/K_{\mathrm{y}}^{\mathrm{C}}$ of yield stiffness between test and simulation was between 1.02 and 1.04, with an average value of 1.03. The ratio $K_{\mathrm{d}}^{\mathrm{T}}/K_{\mathrm{d}}^{\mathrm{C}}$ of stiffness of descending branch between test and simulation was ranging from 0.66 to 0.91, with an average of 0.79. Generally speaking, the hysteretic curves of test and simulation were in good agreement, which shows that it was feasible to simulate the hysteretic behavior of reinforced RPC columns with OpenSees software. That provided a research basis for the subsequent analysis of the hysteretic behavior of cross-shaped-steel-reinforced RPC columns.

3.2. Steel-Reinforced RPC Column

To further explore the applicability of the finite element model, the hysteretic performance of four H-shaped steel RPC columns that had been investigated in the previous research was simulated by OpenSees [17]. The test column specimens were numbered C1~C4, and the size and section reinforcement are shown in Figure 7. The clear height of the column was 1100 mm, and the distance from the bottom to the loading point was 1000 mm. Column section size was 200 mm × 200 mm. The shape steel in the column was welded with Q235 steel plates. The measured yield strengths of steel with thickness of 8 mm and 10 mm were 289 N/mm² and 248 N/mm², respectively. The elastic modulus was 2.06 × 10⁵ N/mm² according to GB/T 228.1-2010 [24]. The longitudinal bars and stirrups were the grade of HRB400. The measured yield strengths of C8 and C18 steel bars were 463 N/mm² and 490.9 N/mm², respectively. The elastic modulus of steel bars was 2.0 × 10⁵ N/mm² according to GB/T 228.1-2010 [24]. The measured axial compressive strength and tensile strength of RPC were 122.6 N/mm² and 11.0 N/mm², respectively. The elastic modulus was 4.3 × 10⁵ N/mm² according to DBJ43/T 325-2017 [25]. In the simulation, the constitutive relations of RPC, shape steel and reinforcement were the same as those in Section 2.2.

Figure 8 and Figure 9 show the comparison of hysteretic curves and skeleton curves between the test and simulation, respectively. It can be seen that the simulation results were basically consistent with the test results.

Table 2. Comparison of characteristic values of test and simulation hysteretic curves.

No.	Direction	${\mathit{P}}_{\mathbf{m}}^{\mathbf{T}}$/kN	${\mathit{P}}_{\mathbf{m}}^{\mathbf{C}}$/kN	${\mathit{P}}_{\mathbf{m}}^{\mathbf{T}}/{\mathit{P}}_{\mathbf{m}}^{\mathbf{C}}$	${\mathit{K}}_{\mathbf{y}}^{\mathbf{T}}$ /kN·mm−1	${\mathit{K}}_{\mathbf{y}}^{\mathbf{C}}$ /kN·mm−1	${\mathit{K}}_{\mathbf{y}}^{\mathbf{T}}/{\mathit{K}}_{\mathbf{y}}^{\mathbf{C}}$	${\mathit{K}}_{\mathbf{d}}^{\mathbf{T}}$ /kN·mm−1	${\mathit{K}}_{\mathbf{d}}^{\mathbf{C}}$ /kN·mm−1	${\mathit{K}}_{\mathbf{d}}^{\mathbf{T}}/{\mathit{K}}_{\mathbf{d}}^{\mathbf{C}}$
C1	+	111.25	112.47	0.99	6.75	6.13	1.10	0.64	0.67	0.96
	−	−106.00	−108.83	0.97	5.41	5.61	0.96	0.46	0.36	1.28
C2	+	123.65	124.74	0.99	6.92	6.52	1.06	0.73	0.54	1.35
	−	−119.54	−121.77	0.98	6.71	6.62	1.01	0.43	0.47	0.91
C3	+	142.75	149.71	0.95	10.83	9.3	1.16	1.76	1.98	0.89
	−	−135.59	−149.1	0.91	10.55	9.3	1.13	2.15	1.62	1.33
C4	+	145.28	149.68	0.97	10.31	9.28	1.11	1.48	1.46	1.01
	−	−139.05	−149.07	0.93	9.14	9.3	0.98	1.34	1.43	0.94

gives a comparison of the characteristic values of the hysteretic curves between the test and simulation. It can be seen that the ratio $P_{\mathrm{m}}^{\mathrm{T}}/P_{\mathrm{m}}^{\mathrm{C}}$ of the peak load between test and simulation was between 0.91 and 0.99, with an average value of 0.96. The ratio $K_{\mathrm{y}}^{\mathrm{T}}/K_{\mathrm{y}}^{\mathrm{C}}$ of yield stiffness between test and simulation was ranging from 0.96 to 1.16, with an average value of 1.07. The ratio $K_{\mathrm{d}}^{\mathrm{T}}/K_{\mathrm{d}}^{\mathrm{C}}$ of stiffness of descending branch between test and simulation was ranging from 0.89 to 1.33, with an average value of 1.16. It can be seen that the numerical model of H-shaped-steel-reinforced RPC columns established by OpenSees was feasible, which can provide a solid foundation for the analysis of hysteretic performance of cross-shaped-steel-reinforced RPC columns.

4. Hysteretic Behavior of Cross-Shaped-Steel-Reinforced RPC Columns

4.1. Analysis Model

To study the seismic performance of cross-shaped-steel-reinforced RPC columns, 12 numerical specimens were established by OpenSees, numbered C-1~C-12. The cross-sectional dimensions were 250 mm× 250 mm. The main investigation factors were section form of shape steel (cross-shaped steel and H-shaped steel), axial compression ratio, RPC strength, section resistance distance of shape steel, reinforcement ratio, stirrup ratio and slenderness ratio.

Table 3. Main parameters of specimens.

No.	Shape Steel Size/mm	Section Resistance Distance Wss/cm3	Longitudinal Bar	Reinforcement Ratio ρ/%	RPC Strength Grade /MPa	Axial Compression Ratio n0	Stirrup	Volume Stirrup Ratio ρsv/%	Column Length l /m	Slenderness Ratio
C-1	2 × H150 × 75 × 5 × 7	86.1	418	1.63%	C120	0.6	8@80	1.26	1.2	9.6
C-2	2 × H150 × 75 × 5 × 7	86.1	418	1.63%	C120	0.4	8@80	1.26	1.2	9.6
C-3	2 × H150 × 75 × 5 × 7	86.1	418	1.63%	C120	0.2	8@80	1.26	1.2	9.6
C-4	2 × H150 × 75 × 5 × 7	86.1	418	1.63%	C140	0.2	8@80	1.26	1.2	9.6
C-5	2 × H150 × 75 × 5 × 7	86.1	418	1.63%	C160	0.2	8@80	1.26	1.2	9.6
C-6	2 × H150 × 75 × 5 × 7	86.1	422	2.43%	C120	0.2	8@80	1.26	1.2	9.6
C-7	2 × H175 × 90 × 5 × 8	134.2	422	2.43%	C120	0.2	8@80	1.26	1.2	9.6
C-8	2 × H150 × 75 × 5 × 7	86.1	422	2.43%	C120	0.2	8@150	0.67	1.2	9.6
C-9	2 × H150 × 75 × 5 × 7	86.1	422	2.43%	C120	0.2	10@80	1.96	1.2	9.6
C-10	2 × H150 × 75 × 5 × 7	86.1	418	1.63%	C120	0.4	8@80	1.26	0.8	6.4
C-11	2 × H150 × 75 × 5 × 7	86.1	418	1.63%	C120	0.4	8@80	1.26	1.6	12.8
C-12	1 × H150 × 75 × 5 × 7	86.1	418	1.63%	C120	0.2	8@80	1.26	1.2	9.6

gives the main parameters of the specimens. The strength of shape steel was 355 MPa, and the strength of longitudinal bars and stirrups was 400 MPa. The axial compressive and tensile strength of RPC were calculated according to reference [21].

4.2. Hysteretic Curve

Figure 10 shows the hysteretic curves of the 12 specimens. It can be seen that: (1) At the initial stage of loading, the area of the hysteretic curve was small, and it can return to the original point when unloading, indicating that the specimens were in the elastic stage. (2) With the increase of displacement, the area of hysteretic curve gradually increased, the slope of the curve gradually decreased and the residual deformation became larger. At this point, the specimens were in the elastic-plastic stage. (3) After the peak load was reached, the bearing capacity of the specimens began to decrease and the horizontal displacement changed quickly. After unloading, the residual deformation continued to increase.

4.3. Skeleton Curve

The skeleton curves of the 12 specimens are shown in Figure 11, and the influences of different parameters are as follows: (1) As shown in Figure 11a, with the increase of axial compression ratio, the peak load increased, the displacement corresponding to the peak load decreased and the descending branch after the peak load became steeper. (2) As can be seen from Figure 11b, when the RPC strength was less than 140 MPa, with the increase of RPC strength, the initial stiffness and peak load of the specimen increased. However, when the RPC strength was higher than 140 MPa, the improvement of RPC strength had little effect on the seismic performance of the specimens. (3) As can be seen from Figure 11c, with the increase of reinforcement ratio and steel sectional resistance moment, the initial stiffness and peak load of the specimen was increased obviously, and the skeleton curve shape was similar. (4) From Figure 11d, it can be seen that the increase of volume stirrup ratio had little effect on the peak load, but it could improve the stiffness of the descending branch. (5) As can be seen from Figure 11e, with the increase of slenderness ratio of the specimen, the initial stiffness and peak bearing capacity of the specimens were obviously reduced, but the slope of the descending branch became gentle. (6) As can be seen from Figure 11f, compared with the H-shaped steel RPC column, the peak load of the cross-shaped-steel-reinforced RPC column was increased by 21.20%. But the descending branch of the cross-shaped-steel-reinforced RPC column was steeper, which indicated that the ductility was reduced to some extent.

4.4. Displacement Ductility

The yield point was determined by energy equivalence method [26].

Table 4. Displacement ductility.

No.	Direction	${\mathit{P}}_{\mathbf{y}}$ /kN	${\Delta }_{\mathbf{y}}$ /mm	${\mathit{P}}_{\mathbf{m}}$ /kN	${\Delta }_{\mathbf{u}}$ /mm	$\mathit{\mu }$	No.	Direction	${\mathit{P}}_{\mathbf{y}}$ /kN	${\Delta }_{\mathbf{y}}$ /mm	${\mathit{P}}_{\mathbf{m}}$ /kN	${\Delta }_{\mathbf{u}}$ /mm	$\mathit{\mu }$
C-1	+	227.45	13.74	260.15	36.30	2.64	C-7	+	256.81	15.84	290.50	45.07	2.85
	−	226.79	13.76	259.36	36.59	2.66		−	255.96	15.85	289.70	45.91	2.90
C-2	+	220.30	15.12	251.96	38.64	2.56	C-8	+	229.50	16.14	261.63	41.58	2.58
	−	218.49	15.14	251.37	39.00	2.58		−	229.43	16.14	260.84	42.38	2.63
C-3	+	208.05	16.53	240.01	42.37	2.56	C-9	+	230.00	16.15	263.36	46.39	2.87
	−	208.23	16.53	239.31	43.31	2.62		−	229.78	16.15	262.54	47.27	2.93
C-4	+	221.47	17.65	258.98	44.02	2.49	C-10	+	336.10	7.14	377.86	17.18	2.41
	−	221.54	17.66	258.11	45.12	2.56		−	333.76	7.21	376.61	17.35	2.41
C-5	+	227.00	17.91	266.90	40.97	2.29	C-11	+	162.17	25.98	187.67	69.29	2.67
	−	226.79	17.87	265.91	41.82	2.34		−	161.38	26.10	187.60	69.60	2.67
C-6	+	229.92	16.16	262.8	43.88	2.72	C-12	+	170.84	13.99	199.44	61.87	4.42
	−	229.79	16.16	261.99	44.78	2.77		−	167.40	13.63	196.04	63.47	4.65

gives the yield strength P_y, yield displacement Δ_y, peak load P_m, displacement ductility coefficient u and other characteristic values of the specimens. It can be seen that the ductility increased with the increase of axial compression ratio, reinforcement ratio, stirrup ratio and slenderness ratio. However, with the increase of RPC strength and steel sectional resistance moment, the ductility decreased obviously. The ductility of the specimen with H-shaped steel was obviously larger than that of the specimen with cross-shaped steel, and the average ductility had increased from 2.65 to 4.54, which is an improvement of about 71%.

4.5. Stiffness Degradation

The stiffness degradation of the specimen is taken as secant stiffness, and the secant stiffness K_i under each cyclic load is calculated according to Formula (1), where F_i and X_i are the i-th peak point load and displacement, respectively.

$$K_{\mathrm{i}}={\displaystyle \frac{\left|+F_{\mathrm{i}}\right|+\left|-F_{\mathrm{i}}\right|}{\left|+X_{\mathrm{i}}\right|+\left|-X_{\mathrm{i}}\right|}}$$

(1)

Figure 12 shows the stiffness degradation curves of 12 simulated specimens. As can be seen from Figure 12a, before reaching the peak load, the secant stiffness of the specimen increased with the increase of axial compression ratio. After reaching the peak load, with the increase of displacement, the secant stiffness of specimens tended to be consistent. It can be seen from Figure 12b that the secant stiffness increased with the increase of RPC strength before RPC strength reached 140 MPa, but when RPC strength was higher than 140 MPa, the increase of RPC strength had little effect on secant stiffness. As can be seen from Figure 12c, the secant stiffness increased with the increase of reinforcement ratio and sectional resistance moment, and the degradation trend was basically the same. It can be seen from Figure 12d that the stiffness degradation curves under different volume stirrup ratios were close, which indicated that the change of volume stirrup ratio had little effect on the stiffness degradation. As can be seen from Figure 12e, with the increase of slenderness ratio, the initial secant stiffness and stiffness degradation rate of the specimen decreased obviously. At the later stage of loading, the secant stiffness of the specimens tended to be the same. As can be seen from Figure 12f, the secant stiffness of the specimen with cross-shaped steel and the specimen with H-shaped steel were not much different at the initial stage of loading. With the increase of displacement, the secant stiffness of the specimen with cross-shaped steel was greater than that of the specimen with H-shaped. At the later stage of loading, the secant stiffness of the two specimens was basically the same.

4.6. Energy Dissipation Capacity

In this paper, the energy dissipation coefficient E was used to measure the energy dissipation capacity of the specimens. The energy dissipation coefficient is calculated according to Formula (2), where S_ABCDE is the area of hysteresis loop of the specimen, and S_OBF and S_ODG are the areas of the triangle corresponding to the upper and lower vertices of the hysteretic loop, as shown in Figure 13.

$$E={\displaystyle \frac{S_{\mathrm{ABCDE}}}{S_{\mathrm{OBF}}{+S}_{\mathrm{ODG}}}}$$

(2)

The energy dissipation coefficients of the 12 specimens are shown in Figure 14. It can be seen as follows: (1) Comparing the specimens C-1~C-3, with the increase of axial compression ratio, the energy dissipation capacity of the specimens was improved in the middle stage of loading, but the total energy dissipation capacity was basically the same. (2) Comparing the specimens C-3~C-5, the energy dissipation capacity decreased with the increase of RPC strength, which may be due to the increase of initial stiffness. (3) As can be seen from Figure 14c, with the increase of reinforcement ratio and steel sectional resistance moment, the energy dissipation capacity increased obviously. (4) Comparing the specimens C-6, C-8 and C-9, the increase of volume stirrup ratio has little effect on the energy dissipation capacity of the specimens. (5) As can be seen from Figure 14e, with the increase of the slenderness ratio of the specimens, the energy dissipation capacity of the specimens under each displacement was obviously reduced, and the total energy dissipation was also obviously reduced. (6) Comparing the specimens C-3 and C-12, it can be seen that the total energy dissipation capacity of the specimen with cross-shaped steel was better than that of the specimen with H-shaped steel.

5. Hysteretic Model

5.1. Skeleton Curve

The degenerate trilinear model was adopted as the skeleton curve model. It is necessary to determine three characteristic points: yield point Y, peak point M and failure point U, as shown in Figure 15a. According to the results of numerical regression, the yield stiffness K_y and yield load P_y of cross-shaped-steel-reinforced RPC columns can be calculated according to Formulas (3) and (4), respectively. The stiffness of descending branch can be calculated according to Formula (5), where E_aI_a and E_cI_c are the stiffness of shape steel and RPC, respectively. l is the length of column and n₀ is the axial compression ratio.

$$K_{\mathrm{y}}=3[E_{\mathrm{a}}{I}_{\mathrm{a}}+(0.44+0.3646n_0^2\left)E_{\mathrm{c}}{I}_{\mathrm{c}}\right]/l^3$$

(3)

$$P_{\mathrm{y}}=(-0.052n_0^2+0.054n_0+0.86)P_{\mathrm{m}}$$

(4)

$$K_{\mathrm{d}}=(-0.334n_0^2+0.289n_0+0.116)K_{\mathrm{y}}$$

(5)

5.2. Hysteresis Rule

Figure 15b shows the hysteretic rule of cross-shaped-steel-reinforced RPC columns. The numbers from small to large indicate the walking route of the model in the process of forward and backward loading and unloading. The specific hysteretic rules are as follows: (1) Before the horizontal load reaches yield load P_y, there is no stiffness degradation or residual deformation, and the specimen is loaded and unloaded according to the walking route of 0–1 and 1–0, respectively. (2) When the restoring force is between the P_y and the peak load P_m, the incremental stiffness of the specimen after yield is taken as loading stiffness. The unloading stiffness K_un is the reduction of the initial stiffness, which is calculated according to Formula (6), where Δ_un is the maximum displacement experienced. (3) When the restoring force exceeds the peak load P_m, the stiffness K_d is used as the loading stiffness. The unloading stiffness continues to be determined according to Formula (6). (4) When the specimen is subjected to backward load, if the yield load is not exceeded, the path follows from P = 0 to the yield load P_y. Once the load exceeds the yield load, the path moves from P = 0 to the maximum load point ever passed in the backward direction. When forward reloading after backward unloading, the path goes directly from P = 0 to the maximum load point ever passed in the forward direction.

$$K_{\mathrm{un}}=1.291{\left({\displaystyle \frac{{\Delta }_{\mathrm{un}}}{{\Delta }_{\mathrm{y}}}}\right)}^{-0.7515}{K}_{\mathrm{y}}$$

(6)

5.3. Comparison between Simulated Hysteretic Curve and Hysteretic Model

The 12 cross-shaped-steel-reinforced RPC specimens were predicted by the above hysteretic model. The comparison between the hysteretic curves predicted by hysteretic model and that calculated by numerical simulation is shown in Figure 16. It can be found that the two results are in good agreement, which shows that the hysteretic model established in this paper is reliable.

6. Discussion

This study was mainly based on the existing literature; after establishing the finite element model and verifying its feasibility and rationality, the nonlinear parameter analysis was carried out. The research results are reliable to some extent, but related tests are still needed.

7. Conclusions

Based on the nonlinear beam–column element, the numerical model of cross-shaped-steel-reinforced RPC columns was established by OpenSees. The hysteretic behavior of such columns under horizontal cyclic load was studied, and the following conclusions are drawn:

(1)

The calculated results of the models of reinforced RPC columns and steel-reinforced RPC columns were in good agreement with the test results. The average difference of peak load, yield stiffness and stiffness of descending branch was 2.75%, 5.50% and 1.5%, respectively. It shows that the established numerical model had high reliability and can provide theoretical support for subsequent parameter analysis.

(2)

The results of hysteretic analysis show that the cross-shaped-steel-reinforced RPC column had good seismic performance. Steel sectional resistance moment and reinforcement ratio were two important factors that affected the hysteretic behavior of cross-shaped-steel-reinforced RPC columns. With the increase of steel sectional resistance moment and reinforcement ratio, the peak load and energy dissipation capacity were significantly improved, but it has little effect on ductility.

(3)

Compared with H-shaped-steel-reinforced RPC columns, the peak bearing capacity of cross-shaped-steel-reinforced RPC columns was increased by 21.2%, but the ductility decreased. The total energy dissipation capacity of the specimen with cross-shaped steel was better than that of the specimen with H-shaped steel.

(4)

According to the results of parametric analysis, the hysteretic model of cross-shaped-steel-reinforced RPC columns was established. The hysteretic curves predicted by the hysteretic model were basically consistent with the numerical analysis results. The hysteretic model provides a theoretical basis for the hysteretic analysis of cross-shaped-steel-reinforced RPC columns.

(5)

For cross-shaped steel columns, RPC with 140 MPa is an economical choice. In this case, the axial compression ratio and slenderness ratio can be reasonably improved according to the use of the building. Cross-shaped steel columns are recommended in disaster-prone areas where the direction of earthquake load cannot be predicted, because when the earthquake load is perpendicular to the web of the H-beam, its seismic performance will be difficult to give full play.

(6)

The conclusions were summarized according to the results of parameter analysis, which was based on the numerical model established by the existing literature. The conclusions are reliable to some extent, but related tests are still needed for further verification.