Seismic Performance of Cross-Shaped Partially Encased Steel–Concrete Composite Columns: Experimental and Numerical Investigations

Abstract

Special-shaped partially encased steel–concrete composite (PEC) columns could not only improve the aesthetic effect and room space use efficiency, but also exhibit good mechanical performance under static load when used in multi-story residential and office buildings. However, research on the seismic performance of special-shaped PEC columns is insufficient and urgently needed. To investigate the seismic performance of cross-shaped partially encased steel–concrete composite (CPEC) columns, three CPEC columns were designed and tested under combined constant axial load and lateral cyclic load. The test results show that the CPEC columns had good load capacity and ductility, and that the columns failed because of concrete crushing and steel flange buckling after the yielding of the steel flange. The plump hysteresis loops indicated that the CPEC column also had good energy dissipation capacity. Due to the constraint of hydraulic jacks, increasing the load ratio would decrease the effective length, thereby increasing the load capacity of the CPEC column and decreasing the ductility. A finite element model was also established to simulate the response of the CPEC columns, and the simulated results agree well with the experimental results. Thereafter, an extensive parametric analysis was performed to study the influences of different parameters on the seismic performance of CPEC columns. For the CPEC column with an ideal hinged boundary condition at the top, its lateral load capacity gradually decreases with the growth of the load ratio and link spacing and increases with the rise of the steel yield strength, concrete compressive strength, flange and web thickness, and sectional aspect ratio. This research could provide a basis for future theoretical analyses and engineering application.

1. Introduction

Partially encased steel–concrete composite (PEC) column is a prefabricated composite component, which is composed of H-shaped steel, links, and infilled concrete. PEC columns can be processed in the factory and H-shaped steel can be used as the template for concrete pouring. At the same time, a connection area without pouring concrete is reserved to allow on-site connection, achieving a high degree of prefabrication and assembly, and convenient construction. Currently, many scholars have conducted systematic research on the static mechanical performance [1,2,3] and seismic performance [4,5] of the PEC columns, clarifying the load-bearing mechanism, failure mode, and design method of the PEC columns under different loads, laying a foundation for the design and application of practical engineering. Currently, it has been applied to dozens of projects in China, including the artificial landscape mountains project in Shanghai World Expo Cultural Park and the residential buildings in Wuyue Plaza in Jiaxing.

In multi-story residential and office buildings, the application of PEC columns with regular sections is prone to the convex column phenomena, which reduces the aesthetic effect and room space use efficiency of the building. At the same time, the PEC columns have low load-bearing capacity and complex connections with beams and supports in the weak axis direction. To address the aforementioned issues, special-shaped PEC columns have been proposed, as shown in Figure 1, including T-shaped PEC columns, L-shaped PEC columns and cross-shaped PEC (CPEC) columns. Due to the flexible cross-section, the special-shaped PEC columns can be embedded into the walls, avoiding the occurrence of protruding columns and thereby improving the aesthetic effect and room space use efficiency of the building [6]. Furthermore, the exposed steel flanges in two directions make it easier to connect with steel beams, reducing the construction difficulties and improving the construction efficiency [7].

Currently, there are few studies on the static performance of special-shaped PEC columns. Jamkhaneh and Kafi [8,9] studied the failure modes and stress mechanisms of the octagonal PEC columns under axial compression, eccentric compression, and bending. Zhan et al. [10] investigated the flexural–torsional stability of axially loaded L-shaped PEC column through experimental and numerical analyses and proposed a formula by which to calculate the load-bearing capacity of L-shaped PEC columns considering the effect of the aspect ratio and length-to-thickness ratio of the leg. Liu and Li [11] conducted a series of axial-loaded experiments and numerical studies on CPEC columns in order to analyze the enhancement effects of transverse links on the load capacity and ductility of the CPEC columns. Lai et al. [12] developed a simplified approach to determine the axial load capacity of columns, based on the experimental results from both SRC and PEC columns. Li and Li [13] studied the constraint mechanism of shaped steel and links on concrete under axial compression in the CPEC column. Tao et al. [14] have studied the load-bearing performance of axially compressed T-shaped PEC columns and proposed a method for calculating the load capacity considering the constraints of shaped steel and links on concrete.

Seismic behavior is one of the important mechanical behaviors for a new structure [15,16]. Elnashai and Broderick [4,5] conducted a series of cyclic and pseudo-dynamic experimental investigations on the behavior of the PEC columns and validated the applicability of the PEC columns to the earthquake-resistant design of multi-story structures. However, the research on the seismic performance of the special-shaped PEC columns is currently limited. To fill the gap and reveal the seismic performance of special-shaped PEC columns, the CPEC column is taken as an example and the seismic tests on three CPEC columns with different load ratios were conducted. The test phenomena, failure modes, and the influence of axial load ratio on the performance of the specimens were obtained. Afterwards, a finite element analysis model for the seismic performance of the CPEC columns was established. On the basis of the verification of the reliability of the model, the influences of different parameters and the failure mechanism of specimens were revealed through parametric analyses.

2. Experimental Program

2.1. Specimens Design

Three CPEC columns were tested, for which all the specimens have the same design parameters except for the load ratio. The total height of the specimens was 1400 mm. The sectional width and height of the specimens were both 240 mm, and the width of the steel flange was 120 mm. The thickness of steel flanges and webs was 5 mm. The diameters of links and longitudinal reinforcements were both 6 mm, and the link space was 100 mm. According to previous studies on the axial compressive behavior of special-shaped PEC columns [14], the special-shaped PEC column with the same sectional dimensions and link space of 100 mm exhibited good mechanical performance. The link space was reduced to 50 mm within a range of 250 mm from the top and bottom of the column to enhance the top and bottom part of the column. Steel plates with a thickness of 30 mm were welded to the top and bottom of the specimens to connect them with the loading instruments and ensure that the specimens are uniformly loaded. To prevent local damage at the column ends, stiffeners with a thickness of 12 mm and height of 150 mm were installed at the ends of the specimens. Detailed information of the specimens is shown in Figure 2. As load ratio is one of the most influential parameters in regard to the seismic performance of composite columns, three specimens were designated as CPEC-0.2, CPEC-0.35 and CPEC-0.5, as their load ratios were 0.2, 0.35 and 0.5, respectively. The axial load ratio was defined as N/$N_0$, where N denotes the axial force applied to the test specimens, and $N_0$ represents the axial load capacity of the specimens determined by Equation (1) according to Eurocode 4 [17].

$$N_0=A_{\mathrm{w}}{f}_{\mathrm{v}\mathrm{w}}+A_{\mathrm{f}}{f}_{\mathrm{y}\mathrm{f}}+0.85A_{\mathrm{c}}{f}_{\mathrm{c}\mathrm{o}}^{\prime }+A_{\mathrm{s}}{f}_{\mathrm{s}}$$

(1)

where, $A_{\mathrm{w}}$, $A_{\mathrm{f}}$, $A_{\mathrm{c}}$ and $A_{\mathrm{s}}$ denote the area of the web, the area of the flange, the area of the concrete and the area of the longitudinal reinforcements, respectively, and where $f_{\mathrm{v}\mathrm{w}}$, $f_{\mathrm{y}\mathrm{f}}$, $f_{\mathrm{c}\mathrm{o}}^{\prime }$ and $f_{\mathrm{s}}$ denote the yield strength of the web and the flange, the compressive strength of the concrete and the yield strength of the longitudinal rebar, respectively.

2.2. Material Properties

The material properties of steel plates and reinforcements were tested based on the specification GB/T228.1-2021 [18], and the yield strength (f_y), ultimate strength (f_u), elastic modules (E_s), and the elongation at fracture (δ) are listed in

Table 1. Material properties of steel.

Coupons	Yield Strength fy (MPa)	Ultimate Strength fu (MPa)	Elastic Modulus Es (GPa)	Elongation Δ (%)
5 mm plate	420.6	545.6	207.3	25.4
6 mm longitudinal rebars	465.0	655	206.0	24.0
6 mm links	460.0	650	205.0	23.0

. For the concrete, three cubic samples, each with a dimension of 150 mm, were tested in compliance with the GB/T 50081-2019 [19], and the average measured cubic compressive strength (f_c) is 40.4 MPa.

2.3. Test Setup and Loading Protocols

All specimens were tested under combined constant axial load and lateral cyclic load, and the test setup is shown in Figure 3. A rigid steel base was fixed to the ground by ground anchors with diameters of 60 mm, concrete support, steel rods, hydraulic jack and reaction steel support. The CPEC columns were fixed to the rigid steel base by nine M32 high strength bolts in order to achieve a fixed boundary condition at the bottom of the column. A hydraulic jack was placed on the top plate of the column to apply the axial load combing with the reaction beam, and the axial load was kept constant during the loading process. Two greased polytetrafluoroethylene (PTFE) plates were placed between the hydraulic jack and the reaction beam. The friction coefficient between PTFE plates was extremely small after the application of grease, so that the friction between the hydraulic jack and reaction beam was negligible and the hydraulic jack could move together with the top plate of the CPEC column. An MTS hydraulic actuator with a displacement range of ±250 mm was used to apply the cyclic lateral loading. A clamping device composed of four steel bars and two steel plates was used to connect the hydraulic actuator and the specimen.

During the loading, the designed axial force was applied firstly through the hydraulic jack and was kept constant. A multi-stage cyclic displacement loading protocol recommended by JGJ/T 101-2015 [20] was applied in this test and is depicted in Figure 4. The displacement amplitude at each loading level was determined according to the drift ratio of the specimen. When the drift ratio was below 1/300, each displacement loading repeated once, while it repeated three times for a drift ratio larger than 1/300.

The horizontal displacement of the loading center was measured by a linear variable differential transformer (LVDT) D1, and the horizontal displacement of the bottom plate was measured by LVDT D2, as illustrated in Figure 5. The difference between the measured results of the two LVDTs was the actual deformation of the specimen. Twenty-five strain gauges were attached to measure the strains of the steel, the concrete, the reinforcements and the links at specific locations, as illustrated in Figure 5.

3. Experimental Results and Discussion

3.1. Experimental Phenomena

Taking Specimen CPEC-0.2 as an example to introduce the experimental phenomena. All of the steel and concrete surfaces were numbered to facilitate the description as shown in Figure 5. In the initial stage of loading, the specimen was in the elastic deformation stage, and the strains of the steel column, reinforcements, and concrete, as well as the residual deformation of the specimen, were very small. When the displacement was loaded to 1/70H, tiny horizontal cracks appeared on surface S1 at the heights of approximately 270 mm, 360 mm, 470 mm, and 550 mm from the bottom of the column and developed to surface W3 with the increase of the loading cycles, as shown in Figure 6a. At the same time, horizontal cracks appeared on surface N3 at the heights of 210 mm, 360 mm, and 460 mm, and developed toward surface E1. Surface N1 generated horizontal cracks at the heights of 450 mm and 540 mm which developed to surface W1. When the displacement was loaded to 1/50H, new horizontal cracks were generated on surface N1 at the heights of 240 mm and 350 mm, and the horizontal cracks on surface W1 extended to the entire surface, as shown in Figure 6b. When the displacement increased to 1/40H, the concrete crushed and fell off within the height range of 150–210 mm on surfaces N1, N3, S1, and S3, and a slight bulging of the flange was observed, which detached from the concrete, as shown in Figure 6c. Afterwards, as the loading displacement increased, the concrete detachment and bulging of the flange became more pronounced. When the displacement was loaded to 1/25H, obvious symmetrical flange bulging was observed on surfaces W2 and E2 within the height range of 200 mm to 250 mm, and a large area of concrete detachment was observed within the height range of 150–210 mm on surfaces N1, N3, S1, and S3, as shown in Figure 6d. When the displacement was loaded to 1/22H, concrete crushing on surfaces W1, E1, W3, and E3 within the height range of 150–210 mm extended to surfaces N1, N3, S1, and S3, respectively, as shown in Figure 6e. Finally, the load capacity decreased to below 85% of the peak load, indicating the failure of the specimen.

3.2. Failure Modes

Overall, specimens with different load ratios exhibited similar experimental phenomena and failure modes. In the early stage of loading, many horizontal cracks were generated at the bottom of the specimen, which developed continuously as the number of hysteresis cycles and loading displacement increased. Before the peak load, the concrete gradually peeled off, and the steel flange slightly bulged and detached from the concrete. After the peak load, the steel flange buckled under compression, and the concrete crushed. The failure modes of all specimens are shown in Figure 7. We can see that the top crack is nearly 540 mm, 450 mm and 350 mm from the bottom steel plates of the respective specimens and that the crushing area of the concrete decreased with the increase of the load ratio. This might be because the tensile region and tensile stress of concrete decreased as the load ratio increased. Similar test phenomena have also been reported previously [21,22].

3.3. Load–Displacement (P-Δ) Hysteresis Curves and Skeleton Curves

Load–displacement (P-Δ) hysteresis curves of the CPEC columns under low cyclic loads are important references for analyzing their seismic performance. The hysteresis curves of the CPEC columns are displayed in Figure 8a–c. According to Figure 8, the hysteresis curves of all of the columns can be divided into three stages. In the first loading cycles, the lateral load rose linearly with the increase of the lateral displacement, indicating that the columns were in the elastic deformation stage. As the loading proceeded, the slopes of the load–displacement curves gradually decreased and obvious residual deformation could be observed when the lateral load reduced to zero, which was caused by the elastic–plastic behavior of concrete and the yield of steel flange. Thereafter, the lateral load capacity gradually decreased with the increase of the lateral deformation, due to the damage development of steel and concrete in the CPEC column. Overall, no obvious pinching effect was observed in the spindle-shaped hysteresis loops, indicating that the CPEC column had a good seismic behavior.

The skeleton curves could be determined when connecting the peak load points in each displacement level, as shown in Figure 8d. Then, the yield displacement Δ_y, yield load P_y (point D), maximum displacement Δ_max, maximum load P_max (point E), ultimate displacement Δ_u and ultimate load P_u (point F) could be derived from the skeleton curves and these are listed in

Table 2. Characteristic values of tested CPEC columns.

Specimen ID	Direction	Yield Point		Peak Point		Ultimate Point		$\mathit{\mu }$	θu (%)
		Δy (mm)	Py (kN)	Δmax (mm)	Pmax (kN)	Δu (mm)	Pu (kN)
CPEC-0.2	+	12.92	160.48	27.00	188.90	44.87	160.57	3.47	4.08
	−	12.13	161.05	27.36	190.00	48.58	161.50	4.00	4.42
CPEC-0.35	+	12.51	174.85	21.90	199.14	42.97	169.27	3.44	3.91
	−	12.33	170.18	21.89	198.85	45.61	169.02	3.70	4.15
CPEC-0.5	+	12.94	181.62	21.88	204.00	41.71	173.40	3.22	3.79
	−	12.86	191.93	21.88	211.15	43.76	179.48	3.40	3.98

. The yield displacement and load were determined according to Figure 9 [20]. The ultimate load P_u was defined as 85% of the maximum load P_max, and the ultimate displacement Δ_u was the corresponding displacement. The ductility factor μ and ultimate drift ratio θ_u were used to evaluate the deformation capacity of the columns and were defined, as follows:

μ = Δ_u/Δ_y

(2)

θ_u = Δ_u/H

(3)

According to Table 2, as the load ratio grew from 0.2 to 0.35 and 0.5, the ultimate lateral load capacity of CPEC columns increased by a respective 4.7–5.4% and 8.0–11.1%. This is inconsistent with the results in which the lateral load capacity of the CPEC column gradually decreases with the increase of axial load ratio in Section 4.4, in which the pinned boundary condition is specified at the top of the CPEC column for simplification. This is mainly because the axial force is applied to the top plate of the specimen through a hydraulic jack during the experiment. When lateral displacement occurs, part of the contact areas between the hydraulic jack and the top plate of the specimen will separate due to the rotation of the top steel plate of the specimen. However, as the hydraulic jack was placed on the top steel plate of the specimen directly, the top of the column was not a totally pinned boundary condition, and the jack had certain constraints on the rotation of the top steel plate of the specimen. Furthermore, the increase of the axial load ratio will delay the separation between jacks and top plate and enhance the rotational constraint effect, thereby reducing the calculated length of CPEC column and improving its lateral load capacity [23]. Furthermore, when the load ratio was raised from 0.2 to 0.35 and 0.5, the ductility factor decreased by a respective 1.1–7.6% and 7.2–15.0%, and the ultimate drift ratio decreased by a respective 4.2–6.1% and 7.0–10.0%. However, due to the composite action between steel and concrete, the CPEC columns exhibited good deformation capacity. Even specimen CPEC-0.5 with the highest axial compression ratio had a ductility factor and ultimate drift ratio of 3.22–3.4 and 3.79–3.98%, respectively, which could satisfy the seismic design requirements of structures.

3.4. Strength Degradation

Due to the damage accumulation, the peak load of each load cycle at the same displacement level decreases gradually, a phenomenon which is defined as strength degradation. Strength degradation can be evaluated by strength degradation factor λ_i, defined in Equation (4), where $F_j^1$ and $F_j^i$ are the peak load of the first load cycle and the last load cycle at the jth displacement level, respectively.

$${\lambda }_i=\frac{F_j^i}{F_j^1}$$

(4)

Figure 10 presents the relationship curves between the strength degradation factor and the displacement at the loading center. It is shown that the strength degradation factor of each specimen is close to 1.0 in the early stage of loading. This is because the specimen was in elastic state at this time, and there was little damage accumulation in the first few loading cycles. Thereafter, the strength degradation factors gradually decreased, indicating that the accumulation of material damage began to affect the load capacity of the specimen. Overall, the strength degradation factors of the CPEC specimen were 0.93 to 1.0, indicating that the strength degradation of the CPEC column was small and stable, which reflects the excellent load-bearing capacity of the CPEC column. In addition, it could be observed that the strength degradation of the CPEC column slightly intensified as the load ratio increased, but that the impact was relatively minor.

3.5. Stiffness Degradation

The stiffness of the CPEC columns would gradually weaken as the loading displacement and cycle numbers increased, and this can be evaluated by the secant stiffness defined in Equation (5), where K_i was the secant stiffness at the ith loading cycle. Δ_i+ and Δ_i- are the maximum lateral displacements at the ith loading cycle, and P_i+ and P_i- are the lateral loads respectively corresponding with Δ_i+ and Δ_i-. The stiffness degradation of CPEC specimens with different load ratios is shown in Figure 11.

$$K_{\mathrm{i}}=\frac{\left|P_{\mathrm{i}+}\right|+\left|P_{\mathrm{i}-}\right|}{\left|{∆}_{\mathrm{i}+}\right|+\left|{∆}_{\mathrm{i}-}\right|}$$

(5)

It has been shown that the overall equivalent stiffness of all specimens decreased with the increase of loading displacement, and that the stiffness degradation develops rapidly from the yield point to the maximum point. This is mainly due to the rapid development of concrete cracks under tension and the plastic deformation of concrete under compression. After the maximum point, the concrete gradually crushed and peeled off. The steel flange made a major contribution to the stiffness of the columns, and the stiffness of the specimen decreased slower. Furthermore, due to the symmetry of the cross section, the stiffness degradation curves of the CPEC columns in two directions showed a generally symmetrical pattern. The overall performance of the specimen showed that the initial stiffness and the degradation speed of the CPEC columns increased as the load ratio increased. This is mainly due to the larger area of the concrete compression zone in the early stage with the higher load ratio. Meanwhile, the higher load ratio would also lead to earlier and more severe concrete crushing and steel flange buckling, which results in a faster stiffness degradation. A similar influence has also been obtained in the pseudo-dynamic test of composite columns [24].

3.6. Energy Dissipation

To evaluate the energy dissipation ability of the specimens, the total dissipated energy E_sum [20] is used, which is expressed as Equation (6), where S_ABCDA is the area of the hysteresis loop at the ith displacement level according to Figure 12 and n is the cycle number.

$$E_{sum}={\displaystyle \sum _{i=1}^n{S}_{ABCDA}}$$

(6)

As shown in Figure 13, as the axial load ratio increased and the specimen entered the elastic–plastic stage earlier, so the energy dissipated under the same displacement is slightly greater. However, specimens with smaller axial load ratio have better deformation ability, larger failure displacement, and thereby achieve greater plastic deformation to consume energy. One can see, in Figure 13, that the specimen with load ratio of 0.35 exhibited the highest cumulative energy consumption, while the specimen with load ratio of 0.5 had the lowest cumulative energy consumption.

3.7. Strain Analysis

The strain development of the steel flange could reflect the mechanical characteristics of the specimen, and strain gauges 2, 4, and 23 in Specimen CPEC-0.35 were selected as typical measuring points for analysis. As shown in Figure 5, strain gauge 2 was located on the outer side of the flange perpendicular to the loading direction, strain gauge 4 was located on the outer side of the flange parallel to the loading direction, and strain gauge 23 was in the middle of the transverse link. The strain–displacement curves of three measuring points are shown in Figure 14.

According to Figure 14a, it can be seen that the steel flange at strain gauge 2 was in compression when loaded along the positive direction while in tension under the reverse loading. In the initial stage, the specimen was in the elastic stage, and the compressive and tensile strains increased linearly until reaching the yield strain. Then, as the lateral displacement developed, the strain of the steel flange began to increase significantly. Before the peak displacement, the compressive strain of the flange gradually decreases and is transformed into tensile strain, indicating that local buckling has occurred on the steel flange at that time.

The flange at strain gauge 4 was in tension when loaded along the positive direction while in compression under the reverse loading as shown in Figure 14b. Before the peak load of the skeleton curve, the flange had already yielded, with a significant increase in tensile and compressive strain. However, the strain of the steel flange at strain gauge 4 was basically symmetrical during the seismic loading and the decrease of the compressive strain was not observed, indicating that the steel flange parallel to the loading direction did not buckle in the whole loading process.

Observing the strain development of strain gauge 23 in Figure 14c, the strain development of the link was relatively slow and small in the initial stage. At the peak displacement, the link at the strain gauge 23 had not yielded. After the peak displacement, the concrete expanded and fell off under compression, and, at the same time, the steel flange buckled, causing the strain of the link to rapidly increase and exceed the yield strain. This is consistent with the experimental phenomenon in this article and the research results of the compressive performance of PEC columns [14].

4. Finite Element Analyses

4.1. Establishment of the FE Model

As the PEC columns have been successfully modelled by the finite element software ABAQUS 6.14 [3,10], an FE model of the CPEC column was created using ABAQUS, including the steel section, concrete, top and bottom steel plates, stiffeners, clamps, and hydraulic jacks, as shown in Figure 15. According to previous research [3,10], a shell element (S4R) is adopted for steel section and nine integral points are selected in the thickness direction to increase accuracy. According to the mesh sensitivity analysis in Appendix A, the element size is taken as 30 mm. A solid element (C3D8R) was adopted for the concrete, the bottom and top plates, steel stiffeners and clamping devices. The truss element (T3D2) was utilized for the reinforcements and links. Furthermore, to accurately simulate the constraint of the hydraulic jacks on the rotation of the top plate of the specimens, a hydraulic jack was also included in the FE model and was simulated with a C3D8R element.

The steel section was connected to the top and bottom steel plates by shell-to-solid coupling constraints. The links, clamping device, and stiffeners are connected to the steel section by TIE constraints. Surface-to-surface contact is used between the steel section and concrete, in which hard contact allowing separation after contact is defined with a penalty friction coefficient of 0.3. Surface-to-surface contact is also defined between the top steel plate and the hydraulic jack, with a friction coefficient of 0.6. As the separation of the partial contact area between the hydraulic jack and the top steel plate of the specimen during the loading process was observed, separation after contact is also allowed. Longitudinal reinforcements and links are embedded in concrete.

To simulate the actual boundary conditions during the experiment, a fixed boundary condition is applied to the bottom plate of the column. The external axial load is exerted to a reference point RP1, which is coupled with the top surface of the hydraulic jack. In order to ensure that the vertical loads can move with the column, the rotational boundary conditions of the RP1 are constrained but allowed to undergo translational motion. Similarly, RP2 is coupled with the clamping device to apply horizontal displacement. The out-of-plane translation degrees of RP1 and RP2 are both restricted.

4.2. Material Properties

The steel material constitutive relationships proposed by Han et al. [25], considering the yield plateau and second plastic flow stages, were adopted for the steel section. A bilinear model, in which the hardening modulus is defined as 1% of the elastic modulus, was used for the steel reinforcements and links. Kinematic hardening is adopted for steel as it can consider the Bausinger effect under cyclic loading. The measured mechanical properties were applied in the model.

The concrete damaged plasticity model was adopted for the concrete in the CPEC column, with a dilation angle and a viscosity parameter of 30° and 0.0005, respectively [26]. Other parameters were defined as default. The stress–strain relationship of the concrete under compression is defined according to following Equations (7) through to (11) [27].

$${\sigma }_c=\frac{f_{c0}^{\prime }xr}{r-1+x^r}$$

(7)

$$r=\frac{E_{\mathrm{c}}}{E_{\mathrm{c}}-E_{\sec }}$$

(8)

$$x=\frac{{\epsilon }_{\mathrm{v}}}{{\epsilon }_{\mathrm{c}0}}$$

(9)

$$E_c=5000\sqrt{f_{co}^{\prime }}$$

(10)

$$E_{\sec }=\frac{f_{\mathrm{c}0}^{\prime }}{{\epsilon }_{\mathrm{c}0}}$$

(11)

where, σ_c and f′_co are the compressive stress and strength of concrete, respectively; ε_v and ε_co are the compressive strain and peak strain of concrete, respectively; ε_co = 0.002 when f′_co ≤ 28 MPa; and ε_co = 0.003 for f′_co > 82 MPa. ε_co performs linear interpolation within the middle range, E_c is the tangent modulus of concrete, and E_sec is the tangent elastic modulus at the peak point of concrete.

The nonlinear tensile stress–strain relationships recommended in GB 50010-2010 [28] were applied for concrete in tension, which had been adopted widely in existing investigations [29]. The stress–strain relationship curve of the concrete under uniaxial load cycle (tension–compression–tension) is shown in Figure 16. It is shown that the damage variables are considered in the unloading stiffness compared with the initial elastic modulus. The damage variables in compression and tension were considered and numbered as d_c and d_t. The calculation methods proposed by Birtel and Mark [30], and as listed in Equations (12) through to (15), were adopted. Furthermore, the stiffness recovery factors w_t and w_c were taken as the default values, as follows: w_t = 0 and w_c = 1.

$$d_{\mathrm{c}}=\frac{\left(1-b_{\mathrm{c}}\right)\cdot {\epsilon }_{\mathrm{c}}^{\mathrm{in}}\cdot E_{\mathrm{c}}}{{\sigma }_{\mathrm{c}}+\left(1-b_{\mathrm{c}}\right)\cdot {\epsilon }_{\mathrm{c}}^{\mathrm{in}}\cdot E_{\mathrm{c}}}$$

(12)

$$d_{\mathrm{t}}=\frac{\left(1-b_{\mathrm{t}}\right)\cdot {\epsilon }_{\mathrm{t}}^{\mathrm{in}}\cdot E_{\mathrm{c}}}{{\sigma }_{\mathrm{t}}+\left(1-b_{\mathrm{t}}\right)\cdot {\epsilon }_{\mathrm{t}}^{\mathrm{in}}\cdot E_{\mathrm{c}}}$$

(13)

$${\epsilon }_{\mathrm{c}}^{\mathrm{in}}={\epsilon }_{\mathrm{c}}-\frac{{\sigma }_{\mathrm{c}}}{E_{\mathrm{c}}}$$

(14)

$${\epsilon }_{\mathrm{t}}^{\mathrm{in}}={\epsilon }_{\mathrm{t}}-\frac{{\sigma }_{\mathrm{t}}}{E_{\mathrm{c}}}$$

(15)

where ε_cⁱⁿ and ε_tⁱⁿ are inelastic compressive and tensile strains of concrete, respectively and where b_c and b_t are the ratios of the plastic strains to the inelastic strains under compressive and tensile state, respectively, which were taken as 0.7 and 0.1, respectively.

4.3. Validations

The FE model is validated by the tested hysteresis curves, skeleton curves and failure modes. Comparing the load–displacement hysteresis curves and skeleton curves simulated by FE analyses with the experimental results, as shown in Figure 17, it can be seen that the load capacities of the specimen match well, but that the initial slope, the elastic–plastic stage of the skeleton curves, is higher than the test results, and that the simulated hysteresis curves are plumper than the tested curves. This may be caused by the slips between each component of the test setup and the deviation of concrete material properties. Similar validation phenomena also exist in many finite element analyses of the seismic performance of composite columns [23,31]. Further, the descending portion of the simulated skeleton curves is smoother than the experimental results. This may be due to the peeling of the concrete after crushing in the experiment, which reduces the constraint on the steel flange, makes the buckling of steel flange more severe and reduces the load capacity faster. However, it is difficult to simulate the peeling of concrete in an FE simulation. In addition, the comparison of the failure modes of the CPEC column is shown in Figure 18. The specimens exhibit similar failure characteristics such as bulging of the steel flange above the stiffener and concrete crushing in both the test and simulation. Therefore, the overall agreement between the experimental and simulated results is acceptable, and the modeling method can be used to simulate the seismic performance of CPEC columns.

4.4. Parametric Analysis

After validating the experimental results, a parametric analysis was conducted to examine the seismic behavior of the CPEC columns. To accurately simulate the influence of various parameters on the seismic performance of CPEC columns, a full-scale finite element model is used in parameter analyses. For the standard model, the sectional width and height are both 400 mm, and the width of the steel flange is 200 mm. The thickness of steel flanges and webs is set as 10 mm. The diameters of links and longitudinal reinforcements are both 6 mm, and the link space is 200 mm. The yield strength of steel is 335 MPa, and the cubic compressive strength of concrete is 30 MPa. Furthermore, monotonic loading was applied and the skeleton curves were analyzed in parametric analyses to improve the calculation efficiency, as the calculation of CPEC columns under cyclic loading is time-consuming, and the load capacity and ductility are the most important features in the design.

In addition, to eliminate the constraint effect of the hydraulic jack on the rotation of the top part of the column and make the force on the column clearer, the hydraulic jacks, bottom plates, top steel plates, stiffeners and clamping device were not included in the parametric analysis. The top and bottom sections of the CPEC column are directly coupled to the reference points. A fixed boundary condition is applied to the bottom reference point, while a hinged boundary condition is applied to the top reference point. The vertical load and lateral displacement are also applied to the top reference point. The constitutive relationships and interactions are consistent with the definition in Section 4.1 and Section 4.2. On the basis of the standard model, the influences of parameters such as load ratio, link spacing, steel yield strength, cubic concrete compressive strength, and steel flange and web thickness on the seismic behavior of CPEC columns are investigated. The detailed parameters of each specimen are shown in

Table 3. Detailed parameters of specimen.

Coupon ID	b × h × bf × tw × tf (mm)	fy (MPa)	fc (MPa)	n	s (mm)	NFE (kN)
CPEC-n-0.1	400 × 400 × 200 × 10 × 10	355	30	0.1	200	290.38
CPEC-n-0.2	400 × 400 × 200 × 10 × 10	355	30	0.2	200	279.82
CPEC-n-0.3	400 × 400 × 200 × 10 × 10	355	30	0.3	200	267.04
CPEC-n-0.4	400 × 400 × 200 × 10 × 10	355	30	0.4	200	249.74
CPEC-n-0.5	400 × 400 × 200 × 10 × 10	355	30	0.5	200	225.55
CPEC-n-0.6	400 × 400 × 200 × 10 × 10	355	30	0.6	200	190.99
CPEC-n-0.7	400 × 400 × 200 × 10 × 10	355	30	0.7	200	141.09
CPEC-n-0.8	400 × 400 × 200 × 10 × 10	355	30	0.8	200	91.79
CPEC-fy-235	400 × 400 × 200 × 10 × 10	235	30	0.3	200	204.15
CPEC-fy-355	400 × 400 × 200 × 10 × 10	355	30	0.3	200	267.04
CPEC-fy-420	400 × 400 × 200 × 10 × 10	420	30	0.3	200	298.04
CPEC-fy-460	400 × 400 × 200 × 10 × 10	460	30	0.3	200	316.65
CPEC-fc-30	400 × 400 × 200 × 10 × 10	355	30	0.3	200	267.04
CPEC-fc-40	400 × 400 × 200 × 10 × 10	355	40	0.3	200	284.44
CPEC-fc-50	400 × 400 × 200 × 10 × 10	355	50	0.3	200	301.72
CPEC-tf-6	400 × 400 × 200 × 6 × 6	355	30	0.3	200	188.89
CPEC-tf-10	400 × 400 × 200 × 10 × 10	355	30	0.3	200	267.04
CPEC-tf-12	400 × 400 × 200 × 12 × 12	355	30	0.3	200	304.81
CPEC-h/bf-2	400 × 400 × 200 × 10 × 10	355	30	0.3	200	267.04
CPEC-h/bf-2.5	500 × 500 × 200 × 10 × 10	355	30	0.3	200	364.10
CPEC-h/bf-3	600 × 600 × 200 × 10 × 10	355	30	0.3	200	476.08
CPEC-s-100	400 × 400 × 200 × 10 × 10	355	30	0.3	100	282.45
CPEC-s-200	400 × 400 × 200 × 10 × 10	355	30	0.3	200	267.04
CPEC-s-400	400 × 400 × 200 × 10 × 10	355	30	0.3	400	257.97
CPEC-1	400 × 400 × 200 × 10 × 10	355	30	0.1	200	290.38
CPEC-2	400 × 400 × 200 × 10 × 10	355	40	0.8	200	98.95
CPEC-3	400 × 400 × 200 × 10 × 10	355	40	0.5	200	241.56
CPEC-4	400 × 400 × 200 × 6 × 6	460	40	0.3	200	236.49
CPEC-5	400 × 400 × 200 × 12 × 12	355	50	0.3	200	338.77
CPEC-6	400 × 400 × 200 × 12 × 12	460	30	0.8	200	104.83
CPEC-7	400 × 400 × 200 × 6 × 6	235	30	0.5	200	135.25
CPEC-8	500 × 500 × 200 × 12 × 12	235	30	0.8	200	60.25
CPEC-9	600 × 600 × 200 × 10 × 10	355	50	0.1	200	572.43
CPEC-10	500 × 500 × 200 × 6 × 6	460	30	0.5	200	263.8
CPEC-11	600 × 600 × 200 × 10 × 10	235	40	0.3	200	405.32
CPEC-12	400 × 400 × 200 × 10 × 10	355	40	0.8	100	105.35
CPEC-13	400 × 400 × 200 × 10 × 10	235	50	0.3	100	254.24
CPEC-14	400 × 400 × 200 × 10 × 10	355	40	0.1	400	300.54
CPEC-15	400 × 400 × 200 × 10 × 10	460	30	0.5	400	246.54

.

4.4.1. Load Ratio

The influence of the load ratio on the behavior of the CPEC column was investigated, as shown in Figure 19a. The results indicate that the lateral load capacity and corresponding peak displacement of the CPEC column decreased with the increase of load ratio. The main reason is that the larger the axial load ratio, the earlier the steel and concrete enter the plastic stage, and the earlier the concrete reaches the ultimate strain, resulting in a poorer horizontal load capacity and ductility of the specimen [32]. The lateral capacity of the CPEC columns with load ratios of 0.1 and 0.2 increased by 8.7% and 4.8%, respectively, when compared with that of the specimen with load ratio of 0.3. With the increase of load ratio from 0.3 to 0.4, 0.5, 0.6, 0.7, and 0.8, the lateral load capacity decreased by 6.5%, 15.5%, 28.5%, 47.2% and 65.6%, respectively. Furthermore, the initial stiffness of the CPEC column was scarcely influenced by load ratio.

4.4.2. Steel Yield Strength

Figure 19b displays the impact of steel yield strength on the performance of the CPEC column. It is shown that the load capacity of the CPEC column significantly increases with the growth of the steel yield strength. The columns with f_y of 235 MPa, 355 MPa, and 420 MPa showed obvious degradations of 35.5%, 15.7% and 5.9% in lateral load capacity, respectively, compared with that with f_y of 460, due to the decreased contribution of steel section and the decreased confined concrete compressive strength. Furthermore, the steel yield strength has little influence on the initial lateral stiffness according to Figure 19b, because the elastic modulus stays constant.

4.4.3. Concrete Compressive Strength

Figure 19c depicts the effect of concrete compressive strengths on the behavior of the CPEC column. It is shown that the concrete compressive strength has a slight influence on the elastic stiffness, load capacity, and peak displacement. This is because the concrete strength has a slight influence on the neutral axis location and the resultant compressive force, which is similar to the composite beams [26,29]. The load capacities of specimens with cubic compressive strengths of 40 MPa and 50 MPa increase by 6.5% and 13%, respectively, compared with that with cubic compressive strength of 30 MPa.

4.4.4. Flange and Web Thickness

The responses of the CPEC columns with flange thickness ranging from 6 mm to 12 mm are also explored in Figure 19d. It is shown that, with the increase of flange and web thickness, the elastic stiffness and load capacity of the column are significantly improved, which is similar to the influence of steel yield strength. When the flange and web thickness increase from 6 mm to 10 mm, the load capacity increases by 41.4%. The load capacity increases by 14.1% when the thickness increases from 10 mm to 12 mm.

4.4.5. Sectional Aspect Ratio

The influence of the sectional aspect ratio (h/b_f) on the behavior of the CPEC column are also investigated in Figure 19e. It is indicated that the horizontal load capacity and initial stiffness significantly increased as the sectional aspect ratio rose. The load capacities of the CPEC columns with aspect ratios of 2.5 and 3.0 are 36.3% and 78.3% higher than that of the specimen with an aspect ratio of 2.0, respectively. Furthermore, the influence of the shear effect on the deformation and failure mode of the cyclic loaded columns decreases as the length to cross-sectional height of the column increases [33,34,35].

4.4.6. Link Spacing

As shown in Figure 19f, the effect of the link spacing on the lateral performance of the CPEC columns is displayed. It can be seen that reducing the link spacing will improve the lateral load capacity and ductility of the columns, but that the effect is relatively small and has little effect on the initial stiffness. The lateral load capacity of the specimen with link spacing of 100 mm increased by 5.8% compared with that with link spacing of 200 mm. The lateral load capacity of the specimen with link spacing of 400 mm was reduced by 3.4% compared with that with link spacing of 200 mm. Reducing the link spacing could enhance the confinement of the steel section on the infilled concrete, thereby improving the concrete compressive strength and the load capacity [14].

To quantify the respective influence of all of the parameters on the load capacities of the CPEC columns under combined axial and lateral loads, ANOVA and the F test were performed [26,36,37], and the parameter with a larger F value is believed to be more critical. It is noted that machine learning techniques with accelerated computational efficiency could also be used to effectively estimate the main parameters [38], especially when using the advanced models. Liu and Lu [39] have proposed a computational framework using surrogate models to systematically and comprehensively address a number of related stochastic multi-scale issues in composites design. Wang et al. [40] have proposed a novel gradient-based neural network model with an activated variable parameter to solve time-varying constrained quadratic programming problems with accelerated convergence rate. A general time series forecasting framework, called the deep non-linear state space model, is proposed by Du et al. [41] to predict probabilistic distribution based on estimated underlying unknown processes from historical time series data, and the accuracy of probability forecasts was improved. However, thousands of data are usually required to train the model when applying the machine learning techniques. As the model number is limited in the above parametric analysis, ANOVA and the F test were conducted by SPSS 27 and the results are shown in

Table 4. Analysis of variance of the load capacity.

Parameters	Degree of Freedom, f	Sum of Squares, SSA	Variance, VA	F Value	Contribution (%)
n	3	107,586	35,862	132.9	33.34
fy (MPa)	2	24,475	12,237	45.4	11.38
fc (MPa)	2	6239	3120	11.6	2.90
tf (mm)	2	9784	4892	18.1	4.55
h/bf	2	101,648	50,824	188.4	47.25
s (mm)	2	709	355	1.3	0.33
Error	20	5396	270	N/A	0.25
Total	34	255,836	N/A	N/A	100.00

. It was found that the sectional aspect ratio and load ratio were the most critical parameters, and had influence percentages of 47.25% and 33.34%, respectively, followed by the steel yield strength, steel plate thickness and concrete compressive strength. The link spacing was found to have a negligible influence on the load capacity of the CPEC column with an influence contribution of 0.33%. That is to say, the load capacity of a CPEC column is not sensitive to the link spacing in the studied parameter ranges.

5. Conclusions

(1) CPEC columns failed by concrete crushing and steel flange buckling after the steel flange yielded under the combined constant axial load and lateral cyclic loading. CPEC columns exhibited good energy dissipation capacity due to the plump hysteresis loops.

(2) Due to the constraint of hydraulic jacks on the top plate of the column, the lateral load capacity increased while, as the load ratio increased, the ductility and ultimate drift ratio decreased. The ductility factor and ultimate drift ratio were larger than 3.0 for all specimens.

(3) A finite element model considering the hydraulic jacks was established to simulate the response of the columns. In addition, the simulated hysteresis curves, skeleton curves and failure modes agreed well with the experimental results.

(4) For the CPEC column with an ideal hinged boundary condition at the top, lateral load capacity gradually decreases with the growth of axial load ratio and link spacing, and increases with the rise of steel yield strength, concrete compressive strength, steel plate thickness, and section aspect ratio.

(5) Among the analyzed parameters, the sectional aspect ratio and load ratio are the most critical parameters, followed by the steel yield strength, steel plate thickness and concrete compressive strength, while the link spacing has a negligible influence on the load capacity of CPEC column.