1 Introduction

The concrete-filled steel tube (CFST) has the advantages of high bearing capacity, excellent seismic ability, superior ductility and easy construction, therefore, it has been widely applied in engineering practice [1, 2]. With the development of building structures towards high-rise and heavy load, the application of conventional concrete-filled steel tube is limited due to the exceeding axial compression ratio and poor corrosion resistance. Then, the section steel reinforced concrete-filled steel tubular (SSRCFST) column, by inserting section steel into concrete-filled steel tube, has been utilized in recent years [3]. Stainless steel materials have many advantages, such as elegant appearance, high ductility, good corrosion resistance, impact resistance and fire resistance, leading to no corrosion protection and minimal maintenance requirements, which reduce the life-cycle costs and considerably extended design lives [4]. Thus, in this paper, a section steel reinforced concrete-filled stainless steel tubular (SSRCFSST) structural member is proposed, which owns both the advantages of stainless tube and the SSRCFST. The enhanced steel ratio due to the usage of encased section steel satisfies the design requirement of specification. It can reduce the cross-sectional dimensions of a member while still maintaining high bearing capacity, durability, and aesthetic requirements. The commonly used cross section form contains round and square (Fig. 1). Similar to ordinary concrete-filled steel tube, the circular steel tube has better restraint effect on concrete, but the rectangular cross section has plain surface which has easy joint construction [5].

Fig. 1
figure 1

Cross section of the specimen

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Some researchers have conducted studies on SSRCFST, concrete-filled stainless steel tube (CFSST) and section steel reinforced concrete-filled stainless steel tube (SSRCFSST). Wang [6] put forward in early time that section steel reinforced concrete-filled circular steel tubular (SSRCFCST) composite column can be acted as a new type of high load-carrying column, and the influence of confinement index, structural steel index and slenderness ratio on axial compression bearing capacity was studied. Based on these studies, a numerical calculation model for axial compression bearing capacity was raised. Zhu et al. [7] conducted axial compression experiments on 18 square steel tubular columns filled with section steel-reinforced self-consolidating high-strength concrete, and the experiments results showed that the rectangular steel tube can improve the bearing capacity and ductility. The encased section steel can limit the shear cracks of core concrete, and it has high influence on the failure pattern of member after peak load. Chang et al. [8] performed finite element analysis on the compression bending hysteresis performance of SSRCFCST, and analysis results showed that the encased section steel can provide restraint on core concrete and improve the deformation behavior of member. Cai et al. [3] proposed the finite element model of SSRCFCST and studied the influence of material consumption and strength on axial compression bearing capacity. According to the study results, a calculation method for axial compression bearing capacity was proposed.

Ding et al. [9] conducted experimental studies on 6 SSRCFRST members and established a finite element model based on test data. The results showed that the section steel can effectively prevent the propagation of concrete cracks. Zhang et al. [10] performed axial compression performance test on 8 SRCFCST specimens and analyzed the characteristics of load–displacement curve, ultimate bearing capacity, ductility, strength-weight ratio (strength/weight) and strain ratio. Wang et al. [11] conducted experimental and theoretical analysis on the mechanical performance of SRCFST under compression and torsion load and explored the influence of section steel type, axial compression ratio, loading path and slenderness ratio. Shi et al. [12] conducted pure bending test on 6 SRCFCST specimens, and the results showed that no localized buckling phenomenon occurred in the member, showing relatively good ductility. Further, based on the test and finite element analysis results, a simplified calculation formula for bending bearing capacity was raised. Wang et al. [13] conducted compression-bending-shear test on 10 SRCFRST members, and studied the influence of shear span ratio, axial compression ratio and section steel form on the bearing capacity and failure pattern.

Tao et al. [14] performed fire resistance test on 12 CFSST columns and analyzed the influence of cross-section shape, axial compression ratio and inner rebar. Qiao et al. [15] studied the axial compression performance of 20 concrete-filled stainless-carbon steel tubes, and the test indicated that the encased carbon steel can greatly improve the axial compression bearing capacity and ductility, with high residual bearing capacity. In addition, Azad et al. [16] performed axial compression test on 3 different types of CFSST members and determined their critical slenderness ratios. Ye et al. [17, 18] conducted tensile mechanical performance study on 20 CFSST members and explored the influence of steel type, eccentricity, steel ratio, concrete strength and interface contact conditions on tensile performance. The test indicated that the stainless steel can well collaborative work with concrete, showing certain ductility. Further, the correctness of current design specifications was verified, and based on test data, a proper finite element model was proposed. Moreover, Zhou et al. [19] performed hysteresis performance test on 24 concrete-filled double-skin stainless steel tubes, and analyzed the influence of axial compression ratio, outer tube thickness, hollow ratio, and concrete strength. The test results showed that the members own good ductility and energy dissipation performance.

Liao et al. [20] conducted compression bending hysteresis performance test on 10 concrete-filled rectangular stainless steel tubes (CFRSST) and analyzed the influence of axial compression ratio, section shape and concrete type on their performance. Further, the bearing capacity, ductility, stiffness and energy dissipation ability were explored. Zheng et al. [21] performed compression bending hysteresis performance test on circular double-tube concrete filled stainless steel tubular columns and analyzed the influence of axial compression ratio, diameter ratio of inner tube to outer tube, section type on its performance. Yang et al. [22] conducted experimental and numerical analysis on 18 CFSST members and explored the influence of section type, steel ration and axial compression ratio on its performance. Based on test data, a finite element model which can accurately analyze the member performance was proposed. Zheng [23] performed test study on hysteresis performance of CFRSST members and analyzed the influence of axial compression ratio, diameter ratio of inner tube to outer tube and column type on lateral load–displacement skeleton.

According to above summary, it is shown that the current studies are mainly focused on the axial compression and hysteresis performance of SRCFST or CFSST, while the study on SSRCFRSST column is rare. The CFST members are usually applied in the seismic fortification area and the engineering structures bearing dynamic load, therefore, the seismic performance directly affects its application, and it is necessary to study the hysteresis performance of this type of member. Set against the above background, this paper aims to investigate the overall mechanical behaviors of SSRCFRSST columns under cyclic loading. For the SSRCFRSST column, an I-shaped section steel is encased in concrete-filled rectangular stainless steel tubular column, as shown in Fig. 1b. In the next section, ABAQUS software is used to establish the SSRCFRSST, and its bearing performance in the condition of concrete damage under repeated load is simulated. Section 3 shows the working mechanism of the SSRCFRSST, and the failure pattern, full hysteretic process of load–displacement and the load–displacement skeleton curve are analyzed. In Sect. 4, we present the influence of relevant parameters on its mechanical properties. Through mechanism analysis and parameter analysis, accompanied by existing research results, a simplified hysteretic model of the load–displacement of member is proposed in the Sect. 5. Section 6 is the conclusion of the investigation.

2 Finite element model

2.1 Constitutive model of materials

In this paper, the ABAQUS software package is used to establish the finite element model for SSRCFRSST. ABAQUS has a variety of material model libraries, which can simulate the mechanical properties of metal, concrete, polymer materials, etc., accurately define the interaction relationship between various constituent materials, correctly apply boundary conditions and loads, and can solve the static and dynamic analysis problems of structures perfectly. The steel is assumed to behave as a double fold line follow-up strengthening model under the monotonic loading, which consists of elastic linear stage and plastic strengthening linear stage. The model is determined by coordinate origin, point (εy, fy) and point (εu, fu). The slopes of the first and second stages are elastic modulus and tangent modulus respectively, and the tangent modulus value is between zero and elastic modulus [1]. In order to accurately simulate the mechanical performance of members under repeated load, the stainless steel tube and carbon steel section steel both adopt the double fold line follow-up strengthening model considering Bauschinger effect. At the elastic stage, the member is loaded and unloaded along Es, while at the strengthening stage, the member is loaded and unloaded along 0.01Es, and when the steel strength exceeds the ultimate strength fu, it no longer increases, as shown in Fig. 2. The value of the strength indexes of stainless steel fyo, fuo and the strength indexes fyi, fui of encased carbon steel section steel can be referred to References [24, 25]. Research results indicate that the beneficial effect of cold bending effect in the corner region of square steel tube is little [26], thus it is not considered in this paper.

Fig. 2
figure 2

The steel σ−ε relations under cyclic loading

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Concrete adopts damage plasticity model, the expansion angle is 30°, the flow eccentricity parameter is 0.1, the stress ratio fb0/fc0 is 1.16, and K is 2/3. The compression constitutive model of concrete adopts the CFST model proposed in Reference [27]:

$$y = \left\{ {\begin{array}{*{20}c} {2 \cdot x - x^{2} } \\ {\frac{x}{{\beta \cdot \left( {x - 1} \right)^{\eta } + x}}} \\ \end{array} } \right.\begin{array}{*{20}c} {} & {} & {} & {\begin{array}{*{20}c} {(x \le 1)} \\ {(x > 1)} \\ \end{array} } \\ \end{array}$$
(1)

where, x = ε/ε0, y = σ/σ0, ε0 = εc + 800ξ0.2 × 10–6, ξ = fyAs/(fckAc). σ0 = fc'; εc = (1300 + 12.5fc') × 10–6; η = 1.6 + 1.5/x; β = (fc')0.1/[1.2(1 + ξ)0.5].

The tensile constitutive model of concrete adopts the model proposed in Reference [28]:

$$y = \left\{ \begin{gathered} 1.2x - 0.2x^{6} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x \le 1 \hfill \\ \frac{x}{{0.31\sigma_{{{\text{t0}}}}^{{2}} \left( {x - 1} \right)^{1.7} + x}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x > 1 \hfill \\ \end{gathered} \right.$$
(2)

where, x = ε/εt, y = σ/σt0, σt0 = 0.26 × (12.5fc')2/3, εt = 43.1σt0 × 10–6.

Under repeated load action, the concrete cracks alternatively open and close. In this model, the damage factor dc and dt are introduced to modify the initial modulus matrix D0el. ωc and ωt are combined to define the accumulated damage and stiffness recovery, as shown in Fig. 3.

$$\sigma = \left( {1 - d} \right)D_{0}^{el} :\left( {\varepsilon - \varepsilon^{pl} } \right)$$
(3)
Fig. 3
figure 3

The concrete constitutive relations under cyclic loading

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2.2 Contact, boundary conditions and element type

Due to the symmetry of member and load, one half of the model can be used for analysis, as shown in Fig. 4. The member and corresponding reference points are coupled, and the constant axial force N0 is applied on the reference points at end, and the lateral repeated displacement load is imposed on the middle span [5]. The interfaces between stainless steel tube and concrete, section steel and concrete both adopt surface to surface contact. Detailedly, at normal direction, the hard contact is set, at tangent direction, the penalty function of "Coulomb friction" is set, and the friction coefficient μ adopts 0.25 [14] and 0.6 [1] respectively. The stainless steel tube, section steel and concrete all use 8-node reduced integral solid element C3D8R [13]. Because the concrete filled stainless steel tube is not sensitive to geometric imperfections, the influence of the geometric imperfections of thin stainless steel tubes is ignored in the finite element model [29]. Different element sizes are tried to find a reasonable mesh that ensure the calculation accuracy and improve the calculation efficiency. It is found that a mesh size of 1(length):1(width):2(depth) approximately for solid elements and 1(width):2(depth) approximately for shell elements, can achieve accurate results. Through comparative analysis, the width size of the element is selected as 1/12 of the column section width.

Fig. 4
figure 4

The finite element model

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2.3 Experimental description

The performances of concrete filled steel tubular members under monotonic and reciprocating loads were simulated to validate the precision of the finite element model. The necessary description and key points of the experiment are briefly given below. The test setup is shown in Fig. 5. The pin-pin boundary condition was adopted for the specimen. During the whole loading process, a constant axial force N0 was applied to the end of the column by a horizontal hydraulic jack, and a lateral repeated displacement load was applied to the mid-span of the specimen by an electro-hydraulic servo loading system. The specimen was connected to the loading system by a rigid fixture. The lateral braces were set to prevent the out of plane displacement. During the test, the hydraulic servo loading system automatically recorded the lateral load (P)–lateral displacement (Δ) hysteresis curve data, and set two displacement sensors to check the mid-span deflection. Two displacement sensors were applied at 1/4L and 3/4L along the length of the specimen, and one was placed at the bottom of the end plate to measure the axial deformation. The specimen was preloaded and unloaded firstly to avoid the influence of the gap between loading settings and specimen, and to check the workability of measuring instruments. Force control was adopted before yielding of specimen and displacement control was adopted after that. The force and displacement control were recommended by ACT-24[30].

Fig. 5
figure 5

Schematic diagram of the test set-up (unit: mm)

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2.4 Model verification

In this section, the test data in References [13] 23 and [31] are used to verify the above finite element model. The detailed information of members is listed in Table 1, in which N0 is the axial force applied on the member, fyo is the yield strength of outer steel tube, fyi is the yield strength of encased section steel, fcu is the cubic compression strength of concrete, Pu,T is the measured lateral bearing capacity from test, Pu,F is the lateral bearing capacity obtained by finite element calculation. Figure 6 shows the comparison of the load–displacement curve measured from test and obtained by numerical calculation. From this figure, it can be seen that the peak and shape of calculated load–displacement are both in consistence with the measured results, indicating that the finite element model can be used for theoretical analysis.

Table 1 Information on specimens for verification
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Fig. 6
figure 6

Comparison between experimental and FEA results

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3 Mechanism analysis

The main parameters of the typical member include: the dimension of stainless steel tube Bo × Bo × to = 600 mm × 600 mm × 6 mm, the dimension of encased section steel Bi × Hi × ti = 280 mm × 400 mm × 8 mm (The meanings of Bo, to, Bi, Hi and ti are detailed in Fig. 1b), the member length L = 6000 mm, the nominal yield strength of stainless steel fyo = 280 MPa, the carbon steel section steel adopts Q355, concrete adopts C60, the axial compression ration n is 0.4 (n = N0/Nu, N0 is the constant axial force, Nu is the predicted axial force).

3.1 Failure pattern

Figure 7 plots the comparison of the typical outside failure pattern of numerical model and tested members under repeated load. From this figure, it can be seen that they present high similarity. Detailedly, the members show bending failure pattern, and the stainless steel appear obvious drumming at the repeated load position in the middle span. Besides, the localized drumming exacerbates with the increasing lateral displacement and load.

Fig. 7
figure 7

Exterior failure modes under cyclic loading load

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3.2 Analysis on full hysteretic process of load–displacement

The P−Δ hysteretic curve of SSRCFRSST is shown in Fig. 15. From this figure, it is shown that the hysteretic curve is full, without obvious pinch phenomenon, indicating that the energy dissipation of this type of member is good. Besides, there is no obvious slide among the steel tube, section steel and core concrete. In order to deeply study the hysteretic performance of this kind of member, several key points on the hysteretic curve are selected to plot the characteristic curve (Fig. 8). For convenience, the hysteretic curve is divided into 6 stages, as follows:

  1. 1.

    OA section. In this stage, the load is basically in linear relationship with displacement, and each component material is in elastic state, with large stiffness. With the increasing lateral displacement, initial cracks begin to appear in the concrete at the tension side, and the stiffness shows certain decreasing trend.

  2. 2.

    AB section. The linear relationship between load and displacement no longer remains, and the section material enters the elastic–plastic stage. The tensile zone of concrete gradually enlarges, and the stainless steel tube gradually yields, with the longitudinal strain on the tensile side becoming large and the stiffness decreasing significantly

  3. 3.

    BC section. The member is unloaded from point B, and it can be seen that during the unloading process, the load and displacement is basically in linear relationship. The stress of each material gradually decreases, and since the member has entered elastic–plastic stage, the lateral displacement is not zero when the load decreases to zero, and the member appears large plastic residual deformation.

  4. 4.

    CD section. With continuous loading to the other direction, before point D, the load is basically in linear relationship with displacement. The tension and compression zones are replaced, and the concrete in the new tension zone begins to crack.

  5. 5.

    DE section. With load continuously increases, each material again enters the elastic–plastic stage, and the stainless steel yields. The strain inside concrete continuously increases, cracks continuously develop, and the stiffness of member further decreases.

  6. 6.

    EF section. The encased section steel begins to yield. In this stage, the bearing state is basically similar to that in stage BE, the stiffness of member gradually decreases.

Fig. 8
figure 8

The P−Δ characteristic hysteretic curve

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3.3 Analysis on load–displacement skeleton curve

Figure 9 shows the P−Δ skeleton of the typical case of SSRCFRSST, and this curve is formed by peak points of the hysteresis loops at all levels. From this figure, it can be seen that the loading curve is antisymmetry at positive and negative directions, therefore, only the skeleton at positive direction is taken for analysis. The skeleton curve includes three sections: the elastic section (OD), where each material is in the elastic stage at point D; the strengthening section (DE), where the stainless steel tube gradually yields and the member reaches peak bearing capacity at point E; and the softening section (EF), where the bearing capacity of the member reduces to 85% of the peak bearing capacity at point F.

Fig. 9
figure 9

The P−Δ skeleton of typical column

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Figure 10 shows the longitudinal stress distribution at each characteristic point of the most unfavorable section. From this figure, it can be seen that at point D the longitudinal stress of concrete is in linear relationship with its distance to the neutral axis. The concrete first produces initial cracks, and the outermost fiber of the stainless steel tube begins to yield. With the increasing load, the neutral axis begins to move towards the compression zone, and the plastic zone gradually expands. Due to the combined restraint of the stainless steel tube and carbon steel section steel, the maximum compression stress at the compression zone exceeds fc' (fc' is the cylinder compression strength), and the stainless steel enters the strengthening stage. At point F, the concrete cracks further develop, and the neutral axis continues to move upwards, along with the edge concrete of compression zone entering the softening stage.

Fig. 10
figure 10

Longitudinal stress contour line of concrete in the middle span

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There exits large contact stress between stainless steel tube and concrete. Figure 11a shows the points at different positions. From the mechanism analysis results, it can be seen that the contact stress of points 1 and 2 is nearly 0, with a maximum value of 0.10 MPa. The contact stress of points 3, 6 and 7 is larger, reaching 1.11 MPa, 1.15 MPa and 0.92 MPa, and the contact stress at corner is larger. Figure 11b presents the entire curve of the relationship between contact stress p and the outer edge strain ε at the positions 4 and 5 in the middle span section. It can be seen that with the increasing strain, the contact stress gradual increases.

Fig. 11
figure 11

Contact stress distribution

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4 Parametric analysis

The factors that influence the hysteretic performance of SSRCFRSST columns include the concrete strength (fcu), strength of stainless steel (fyo), strength of section steel (fyi), steel ratio of stainless steel tube (αso), steel ratio of section steel (αsi), axial compression ratio (n) and slenderness ratio (λ). For simplification, single variable method is used based on the above typical cases.

4.1 Concrete strength (f cu)

Figure 12a shows the influence of concrete strength (20–100 MPa) on P−Δ skeleton of members. From this figure, it can be seen that with the increasing fcu, the bearing capacity and elastic stiffness greatly increases. When fcu increases from 60 to 80 MPa, the bearing capacity increases 22.0%, but the descending stage becomes steeper. The reason is that the higher the concrete strength, the less restraint of stainless tube on concrete, inducing smaller ductility of member. Therefore, concrete strength has different influence on bearing capacity and ductility of member, and the adverse influence should be considered when adopting high strength concrete at designing.

Fig. 12
figure 12

Influence of parameters on P−Δ skeletons

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4.2 Strength of stainless steel (f yo)

Figure 12b shows the influence of strength of stainless steel tube fyo (220–440 MPa) on P−Δ skeleton of members. From this figure, it can be seen that with the increasing fyo, the bearing capacity greatly increases. When fcu increases from 220 to 440 MPa, the bearing capacity increases 28.9%. The reason is that the stainless steel tube bears more load and provides better restraint effect on concrete. But the strength of stainless steel tube has little effect on the stiffness at elastic and descending stages and the curve shape.

4.3 Strength of section steel (f yi)

Figure 12c shows the influence of strength of section steel fyi (235–460 MPa) on P−Δ skeleton of members. From this figure, it can be seen that with the increasing fyi, the bearing capacity and stiffness at elastic stage has little changes, only the descending stage is influenced on certain degree. The reason is that the encased section steel is closed to the neutral axis, and its strain is relatively small under loading, which results in the strength of section steel not being fully utilized.

4.4 Steel ratio of stainless steel tube (α so)

Figure 12d shows the influence of steel ratio of stainless steel tube αso (0.04–0.16) on P−Δ skeleton of members. From this figure, it can be seen that with the increasing αso, the bearing capacity and stiffness at elastic stage greatly increase. When αso increases from 0.04 to 0.08, the bearing capacity increases by 35.7%. The reason is that the stainless steel tube bears more load and provides better restraint effect on concrete, but the steel ratio of stainless steel tube has little effect on the stiffness at elastic and curve shape at descending stages. Therefore, the bearing capacity of member can be improved by increasing steel ratio of stainless steel tube at designing.

4.5 Steel ratio of section steel (α si)

Figure 12e shows the influence of steel ratio of section steel αsi (0.02–0.08) on P−Δ skeleton of members. From this figure, it can be seen that αsi has little influence on bearing capacity and stiffness at elastic stage. When αsi increases from 0.02 to 0.08, the bearing capacity increases by 6.0%. The steel ratio of section steel shows certain influence on the stiffness at descending stage of member and ductility. The reason is that the encased section steel assists the stainless steel tube to provide stronger restraint on concrete, which delays the development of concrete cracks and improves the member ductility.

4.6 Axial compression ratio (n)

Figure 12f shows the influence of axial compression ratio n (0.02–0.8) on P−Δ skeleton of members. From this figure, it can be seen that n has significant influence on the skeleton shape, and the larger the n, the poorer the ductility. When n increases from 0.02 to 0.1, 0.2 and 0.3, the lateral bearing capacity increases by 3.4%, 5.5% and 0.7% respectively. When n reaches 0.7 and 0.8, the lateral peak load decreases by 21.0% and 38.5% respectively. The axial compression ratio has little influence on elastic stiffness of member, but it greatly influences the shape of descending stage. When n is small, the second-order effect of the member is not obvious, and the descending stage is relatively flat. While with the increasing n, the decreasing magnitude of the descending stage becomes larger. When n exceeds 0.4, a larger axial force induces the second-order effect becomes gradually obvious.

4.7 Slenderness ratio (λ)

Figure 12g shows the influence of slenderness ratio λ (20–80) on P−Δ skeleton of member. From this figure, it can be seen that with the increasing λ, bearing capacity, stiffness at initial stage and strengthening all obviously decreases. With the increasing λ, the descending stage becomes more flat, and when λ increases from 20 to 35, the bearing capacity reduces by 48.6%. The reason is that the longer the member, the more obvious the second-order effect, and the larger the adverse effect.

4.8 Sensitivity analysis

In this section, the sensitivity analysis is conducted using the sensitivity index (SI) to assess the relative contributions of all parameters to the lateral bearing capacity and corresponding displacement. And the SI can be calculated as follows [32]:

$${\text{SI}}_{j} = \frac{{V_{\max } \left( {x_{j} } \right) - V_{\min } \left( {x_{j} } \right)}}{{\sum\nolimits_{i = 1}^{k} {\left( {V_{\max } \left( {x_{i} } \right) - V_{\min } \left( {x_{i} } \right)} \right)} }} \times 100$$
(4)

where, Vmax(xj) and Vmin(xj) are respectively the maximum and minimum values of the jth variable, and the other values are held constant at their mean values, and k represents the number of variables involved. The sensitivity analysis results of lateral bearing capacity and corresponding displacement are shown in Fig. 13.

Fig. 13
figure 13

Relative contributions of the parameters

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5 Simplified calculation

5.1 Bearing capacity calculation for compression bending member

Han [1] proposed a simplified calculation formula for the bearing capacity of the compression bending rectangular concrete-filled steel tubular (CFST) members:

$$\left\{ {\begin{array}{*{20}l} {\frac{N}{{\phi \cdot N_{{\text{u}}} }} + \frac{a}{d} \cdot \frac{M}{{M_{{\text{u}}} }} = 1} \hfill & {{N \mathord{\left/ {\vphantom {N {N_{{\text{u}}} }}} \right. \kern-0pt} {N_{{\text{u}}} }} \ge 2\phi_{{}}^{{3}} \cdot \eta_{{0}} } \hfill \\ { - b \cdot \left( {\frac{N}{{N_{{\text{u}}} }}} \right)^{2} - \frac{c \cdot N}{{N_{{\text{u}}} }} + \frac{1}{d} \cdot \frac{M}{{M_{{\text{u}}} }} = 1} \hfill & {{N \mathord{\left/ {\vphantom {N {N_{{\text{u}}} }}} \right. \kern-0pt} {N_{{\text{u}}} }} < 2\phi_{{}}^{{3}} \cdot \eta_{{0}} } \hfill \\ \end{array} } \right.$$
(5)

where, a = 1 − 2φ2·η0; b = (1 − ζ0) /(φ3·η02); c = 2(ζ0 − 1) /η0; d = 1–0.25·N/NE. The other parameters can be referred to Reference [1].

The bending capacity Mu of SSRCFRSST can be calculated by using Eq. (6):

$$M_{{\text{u}}} = \gamma_{{\text{m}}} W_{{{\text{scm}}}} f_{{{\text{scy}}}} + W_{{\text{i}}} f_{{{\text{yi}}}}$$
(6)
$$\gamma_{{\text{m}}} = 1.04 + 0.48\ln \left( {\xi + 0.1} \right)$$
(7)
$$f_{{{\text{scy}}}} = \left( {1.18 + 0.85\xi } \right)f_{{{\text{ck}}}}$$
(8)

where, Wscm is the flexural section modulus of RCFST; Wi is section modulus of the bending moment plane of encased section steel.

The axial compression strength bearing capacity Nu of SSRCFRSST can be calculated by Eq. (9):

$$N_{{\text{u}}} = 0.635 \times \left( {1.21\xi + 1} \right)f_{{\text{c}}} A_{{\text{c}}} + 0.365f_{{\text{c}}} A_{{\text{c}}} + f_{{{\text{yi}}}} A_{{{\text{si}}}}$$
(9)

Equations (5)–(9) can be applied to calculate the bearing capacity of SSRCFRSST under static load, but the Bauschinger effect of steel is not considered, inducing a relative low prediction for bearing capacity under repeated loading. In the present work, a simplified P−Δ hysteretic model of SSRCFRSST is established based on the above equations and the results of parameter analysis.

5.2 P–Δ skeleton model

Based on the parameter analysis and the method referred in Reference [1], a trilinear P−Δ skeleton model of SSRCFRSST member is proposed, as shown in Fig. 14. In this figure, point B represents the peak point Py of the skeleton, and point A has a load of 0.6Py. The key parameters of this model include the stiffness at elastic stage (Ka), the peak load (Py), displacement at peak point (Δp) and stiffness at softening stage (KT), the application scope of each parameter is: fcu = 20 ~ 100 MPa, fyo = 220 ~ 440 MPa, fyi = 235 ~ 460 MPa, αso = 0.04 ~ 0.16, αsi = 0.02 ~ 0.08, n = 0.02 ~ 0.8, and λ = 20 ~ 80.

Fig. 14
figure 14

P−Δ hysteretic model

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From parameter analysis, it is found that the axial compression ratio n has little influence on the stiffness of SSRCFRSST compression bending member at elastic stage, therefore, its stiffness can be calculated using a similar method to the pure bending member, and the expression is:

$$K_{{\text{a}}} = {{3K_{e} } \mathord{\left/ {\vphantom {{3K_{e} } {L_{1}^{3} }}} \right. \kern-0pt} {L_{1}^{3} }}$$
(10)

where, Ke = EsoIso + 0.6EcIc + EsiIsi; Iso, Ic, Isi is the sectional inertia moment of stainless steel tube, concrete and encased section steel respectively; L1 = Lc/2, Lc is the calculation length of member.

The peak load Py obeys the Eq. (11):

$$P_{{\text{y}}} = \left( {5.874n^{3} - 3.844n^{2} + 0.567n + 1.934} \right){{M_{y} } \mathord{\left/ {\vphantom {{M_{y} } {L_{1} }}} \right. \kern-0pt} {L_{1} }}$$
(11)

where, My is the bending bearing capacity of compression bending member, which can be obtained by Eq. (5).

The displacement Δp at peak point is calculated according to Eq. (12):

$$\Delta_{{\text{p}}} = \left( {0.614 - 0.369\xi^{2} + 0.519\xi - 0.56n + 0.105e^{1.839r} } \right)\frac{{P_{{\text{y}}} }}{{K_{a} }}$$
(12)

where, r = λ/35.

The stiffness KT at softening stage can be calculated based on Eq. (13):

$$K_{{\text{T}}} = 0.319 \times \left( {\frac{{f_{{{\text{cu}}}} }}{60}} \right)^{0.237} \left( { - 0.02\lambda + 1.689} \right)f\left( n \right)K_{{\text{a}}}$$
(13)
$$f\left( n \right) = \left\{ {\begin{array}{*{20}c} {{2}{\text{.155}}n{ + 0}{\text{.246}}} & {n \le 0.5} \\ {{342}{\text{.66}}n^{{{10}{\text{.03}}}} } & {n > 0.5} \\ \end{array} } \right.$$
(14)

5.3 P−Δ hysteretic model

Figure 14 shows the P−Δ hysteretic model of traditional concrete-filled steel tubular compression bending members [1], which can also applied for SSRCFRSST members. The loading and unloading criteria is described as follows: when it is unloaded from the point 1, the unloading path would follow the original stiffness Ka at elastic stage, and it is reverse loaded to point 2, the load of which is 0.2 time of point 1. Then the reverse loading would obey the path 23’, in which point 3’ is on the extended line of OA, and the load of point 3’ is the same as point 1 (or point 3). The situation is similar when unloading from point 4, and the load of point 5 is 0.2 time of point 4, while that of point D’ is the same as point 4 (or point D). The complete loading path mainly includes O–A–1–2–3'–1'–2'–3–1–B–4–5–D'–4'–5'–D–4–C, and the hysteretic model is centrosymmetry about point O.

The P-Δ hysteretic curve obtained by the above hysteretic model is compared with the measured finite element result, as shown in Fig. 15. It can be seen that these two curves are basically coincide with each other. Therefore, this hysteretic model can be used to perform the elastic–plastic analysis of SSRCFRSST.

Fig. 15
figure 15

Comparison between FE and simplified formula curves

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6 Conclusions

In this paper, the hysteretic performance of encased SSRCFRSST is numerically analyzed, and the working mechanism of this type of member is obtained by analyzing typical members. Besides, the relative factors influencing its bearing performance is studied by parameter analysis. The following conclusions can be drawn:

  1. 1.

    Through the numerical analysis model established by ABAQUS finite element software, the hysteretic performance of SSRCFRSST is sufficiently analyzed. The results indicate that the encased section steel can effectively improve the strength of core concrete, resulting in a full hysteretic curve, strong energy dissipation and good deformation resistance.

  2. 2.

    The parameter analysis shows that the bearing capacity of encased SSRCFRSST increases with the increasing concrete material, and it decreases with the increasing slenderness ratio. The strength of section steel has little effect on the bearing capacity of member. Under small axial compression ratio condition, the bearing capacity increases with the increasing axial compression ratio, and when the axial compression ratio is larger than 0.4, the bearing capacity decreases with the increasing axial compression ratio.

  3. 3.

    The strength of each constituent material has little influence on the shape of skeleton of SSRCFRSST, while the axial compression ratio shows large influence on the shape of descending stage of skeleton, and the larger the axial compression ratio, the steeper the descending stage of curve. Besides, the slenderness ratio has a high influence on the shape of skeleton, and the larger the slenderness ratio, the lower the elastic stiffness, resulting in a more plain descending stage of curve.

  4. 4.

    A simplified load–displacement hysteretic model of encased SSRCFRSST members is proposed, which shows good agreement with the results of numerical results, with high accuracy.

A further point to note is that the current study employed rectangular cross section members. The FEA model and the analysis method should be extended to members with other types of sections, such as circular or round end ones. We can also carry out corresponding research on this type of members with multiple section steels. Besides, fire hazard has been one of the primary concerns for building design, the fire resistance of the SSRCFRSST will be a focus of the future research.