The fine finite element (FE) models of PSRHSCBs are established by ABAQUS software, as shown in Figure 3. Concrete, H-shaped steels and cushion blocks are established by element C3D8R, and longitudinal steel bars, stirrups and prestressing tendons are established by element T3D2 (Ji et al., 2020). During the process of FE modeling, reinforcement cage, H-shaped steel, and prestressing tendons are embedded in the entire model. The interfacial contact between the H-shaped steel and concrete consisted of hard contact in the normal direction and friction contact with coulomb friction coefficient of 0.5 in the tangent direction. Internal prestressing tendons were embedded in the concrete area at specified position. The two reference points (RP-1 and RP-2) are established at a third of the beams respectively. The RP3 is located on the left side of the beams, and the rotation was set along the x-axis (U ₁ = U ₂ = U ₃ = UR ₂ = UR ₃ = 0). While RP4 is located on the right side of the beams, and the translation is set along the z-axis and the rotation is set along the x-axis (U ₁ = U ₂ = UR ₂ = UR ₃ = 0). For static analysis, prestressing effect is applied through the temperature reduction method firstly, then the vertical load is applied according to loading rules. During the quasi-static analysis, vertical cyclic load was applied according to cyclic loading rules.

To verify the rationality of the finite element modeling method, 6 test specimens from reference (Li, 2007), 3 test specimens from reference (Wang et al., 2012), and 2 test specimens from reference (Xue et al., 2007) are selected for finite element simulation, and the specific parameters of existing test specimens are shown in Table 4.

The static analysis of 9 specimens is carried out, and the load-displacement relationship curves of beams under static load are obtained, as shown in Figure 4. It can be seen from Figure 4 that the beams have experienced the elastic stage, the yield stage and the failure stage during the loading process. Since the trend of the load-displacement curves of the finite element simulation is consistent with the trend of the test load-displacement curves, the simulated results are in good agreement with the test results, which confirms the accuracy of the constitutive model of the materials. Comparing the numerical simulation value of the ultimate bearing capacity with the experimental ultimate bearing capacity, it is found that the error between the simulated value and the experimental value is small, and the average relative error is about 5%, as shown in Table 4. Then, the quasi-static analysis of the two test specimens is carried out, and the overall trends of the simulated skeleton curve and hysteresis curve are in good agreement with the experimental results, as shown in Figure 5, which verifies the rationality of the finite element modeling method in this paper.

The specimens are subjected to displacement loading, and the cyclic displacement loading is applied at three points along the beams. The mechanical model of PSRHSCBs is shown in Figure 6A, and the loading scheme describing cyclic displacement loading is shown in Figure 6B. Each level of load undergoes a cycle before reaching the yield displacement (denoted as δ). After completing two cycles and reaching the yield displacement load, the specimens are loaded with a multiple of the yield displacement load until they are destroyed. Where n denotes the number of loading cycles.

The hysteretic curves of PSRHSCBs are shown in Figure 7. It can be seen from that the hysteretic curves of 13 PSRHSCBs show a relatively full shuttle shape, indicating that the PSRHSCBs have excellent seismic behavior. In the process of applying cyclic load, the process from the elastic stage to yield and finally to bearing capacity decline can be clearly obtained. By observing the hysteretic curves shown in Figure 7, it can be found that the e _H and p have a great influence on the hysteretic curves of PSRHSCBs. With the increasing of e _H, the bearing capacity of the beam degrades obviously. With the increase of p, the peak load and corresponding displacement of the hysteresis curve increase obviously, and the ductility increases significantly. This is because when the resistance of the beam section remains unchanged, the increase of p, which makes the bearing capacity of the beam increase.

The load-displacement (P-Δ) skeleton curves of 13 groups of PSRHSCB specimens are shown in Figure 8, and the skeleton curves of all specimens conform to the rule of three broken lines in the elastic stage, elastic-plastic stage, and descending stage. When the L/H decreases from 9.375 to 8.300 and 7.500 in turn, the bearing capacity of composite beams under positive and negative loads increases gradually, as shown in Figure 8A. It can be seen from Figure 8C, with the increasing of p _v, the initial stiffness of the specimens gradually decreases, the peak load increases greatly, and the falling section of the skeleton curves tends to be flattened, which is showing excellent load holding capacity. It can be seen from Figure 8D that with the increasing of e _H, the bearing capacity of PSRHSCBs decreases significantly. Figures 8B, E, F show that the influences of f _cu, p, and σ _con on the peak load of the skeleton curves for PSRHSCBs are not obvious.

During repeated loading of structures, the degradation of stiffness can be expressed by secant stiffness (K _i) (Ji et al., 2021a), which could be expressed by Eq. 9. K i = + F i + − F i + Δ i + − Δ i (9)

Where F _i denotes the peak horizontal load of the first cycle under the i-th control displacement and Δ _i refers to the corresponding displacement.

The secant stiffness comparisons of 13 PSRHSCBs with different parameters are shown in Figure 9. The K _i decreases with the increasing of horizontal displacement (Δ), and its decreasing range gradually slows down with the increase of Δ. It can be seen from Figure 9 that with the decreasing of σ _con and L/H, the initial stiffness and declining degree of secant stiffness increase gradually. When σ _con reaches 0.7f _ptk, the deformation resistance decreases seriously, which indicates that σ _con should not less than 0.7f _ptk . The e _H has a significant effect on the secant stiffness of specimens. With the increase of e _H, the initial stiffness of the specimens increases obviously, and the stiffness degradation phenomenon becomes faster. The effects of f _cu, p _v, and p on the stiffness degradation of specimens are not obvious.

The energy dissipation capacity is applied to evaluate the ability to absorb the energy released in the earthquake of the structure (Cai et al., 2021). The energy dissipation coefficient (E) is used to evaluate the energy dissipation capacity of PSRHSCB specimens, and the energy dissipation coefficient refers to the ratio of the shadow area to the triangle DFO and the triangle BEO area, as shown in Figure 10 (1).

The energy dissipation coefficient can be calculated as follows in Eq. 10: E = S A B C + C D A S Δ O B E + Δ O D F (10)

Figure 10 (2) shows the horizontal displacement (Δ) versus the energy dissipation coefficient (E) relationships of the 13 PSRHSCBs under the first reciprocating load of each stage. It can be found from Figure 10 (2) that the energy dissipation capacity values of the 13 PSRHSCBs are substantially similar.

The energy dissipation capacity of PSRHSCBs is shown in Figure 10 (2). With the increase of f _u, the energy dissipation capacity of the specimen decreases gradually, while with the increase of L/H and p _v, the energy dissipation capacity of the specimen increases gradually. It can be seen from Figure 10 (2) (D) that E _H has a great influence on the energy dissipation capacity of PSRHSCBs.When the H-shaped steels are eccentrically arranged, the energy dissipation capacity is poor. It can be seen from Figures 10 E, F (2) that p and σ _con have little effect on the energy dissipation capacity of PSRHSCBs. The E of most of the specimens in this paper can reach more than 2.0, indicating that PSRHSCBs have excellent energy dissipation capacity.

Ductility is regarded as the main parameter to measure the deformation performance of the specimens (Ji et al., 2021b), and it can be explained by the ductility coefficient (μ), which can be calculated by Eq. 11. μ = Δ u / Δ y (11)

Where Δ _u refers to the ultimate displacement and Δ _y denotes the yield displacement.

Based on the P-Δ curves, the equivalent yield point can be obtained by finding the corresponding point of 0.7 times as P_max on the curves, and the yield load (P _y) can be calculated, as shown in Figure 11 (1) The peak displacement (Δ _max), yield displacement, ultimate displacement, and ductility coefficient of the specimens are shown in Table 5. The relationships between μ and L/H, f _cu, p _v, e _H, σ _con, and p of the PSRHSCB specimens are shown in Figure 11 (2). It can be seen from Figure 11 (2) that the μ of the PSRHSCB_S increases gradually with the decreasing of L/H and f _cu. When the L/H decreases from 9.375 to 8.300 and 7.500 in turn, µ increases from 3.091 to 3.100 and 3.716, which increases by 3.16% and 20.23%, respectively. When the f _cu decreases from 80Mpa to 70 Mpa and 60 Mpa in turn, µ increases from 2.663 to 3.007 and 3.091, which increases by 12.92% and 16.07%, respectively. So, it can be found that L/H and f _cu have significant effects on the ductility of specimens. By consulting the relevant literature, the ductility coefficient of reinforced concrete seismic structures in general engineering is in the range of 3–4. Among the ductility coefficients of 13 specimens calculated by the R-park method, 7 specimens are greater than 3, and the remaining 6 specimens are above 2.6. The six components that do not reach 3 are due to the excessive concrete strength (PSRHSCB-5), the excessive tension control stress (PSRHSCB-7), and the torsion of steel eccentricity (PSRHSCB-8,9,10,11).

The failure modes of PSRHSCBs under low cyclic vertical loading are analyzed in this paper. The bending failure at the mid-span section of the beam is the mainly failure mode of the specimen, and the maximum strain appears at the mid-span of the beam. Finally, the steel reinforcement cage, H-shaped steel, and prestressing tendons reach the ultimate strain, and the composite beam is destroyed as a whole. The stress distribution of several representative specimens is shown in Figure 12. As shown in Figure 12, the maximum stress of the steel reinforcement cage and the H-shaped steel is occurred in the middle of the beams.

To establish a load-displacement restoring force model for PSRHSCBs, the ratio coefficient of section steel stiffness to total section stiffness (γ) is introduced, the calculation formulas are as follows (Ji, 2008): γ = B s s B s (12) B s = B p r c + B s s (13)

Where, B _ss the sectional stiffness of H-shaped steel, B _s is the total stiffness of the beam section, and B _prc is the sectional stiffness of reinforced concrete. B p r c = 0.85 E 0 I 0 (14)

Where, E ₀ is the elastic modulus of concrete, I ₀ is the converted section moment of inertia of reinforced concrete. B s s = E s s I s s + A s s y s s − x 0 2 (15)

Where E _ss is the elastic modulus of the section of steel, I _ss is the moment of inertia of the section of the steel to its centroid axis, y _ss is the distance from the centroid of the section of the steel to the top surface, A _ss is the section area of section steel, and x ₀ is the distance from the neutral axis to top surface in actual cross-section, calculated according to reference (Zhao et al., 2001).

The γ of 13 specimens is calculated using the above formulas. The values of γ are listed in Table 6.

Through the stress nephograms of the finite element model of PSRHSCBs, it can be seen that the H-shaped steel flange yields under both tension and compression. Therefore, the flexural capacity of the normal section of PSRHSCBs is calculated by using the formula proposed in the literature (Ji, 2008). The stress diagram of the H- shaped steel with uniform yield at the flange under tension and compression is shown in Figure 13.

According to the characteristics of skeleton curves, the trilinear skeleton curve model of PSRHSCBs is obtained by statistical regression, as shown in Figure 14 (1). Regression is required to determine 5 pairs of reference points: peak load P _u(P’ _u), yield load P _y(P’ _y), yield displacement Δ _y(Δ' _y), peak displacement Δ _m(Δ' _m), and descending section stiffness K _d(K’ _d).

(1) Numerical analysis regression obtains the expression of the initial section stiffness B _s of the control section in the span of the composite beam as shown in Eq. 16.

Where, λ is the ratio of shear span to depth, B _ss is the sectional stiffness of H-shaped steel, and B _prc is the sectional stiffness of reinforced concrete.

The Initial stage stiffness expression of components (Ji et al., 2021a) is shown in Eq. 17. K e = 24 B s / b 3 L 2 − 4 b 2 (17)

Where, L is the span of the beam, and b is the concentration of the distance from the loading point to the bearing.

(2) The peak load is shown in Eq. 18.

Where, D is the distance between two loading points, P _u is the ultimate load when subjected to upward loads; P _u ' is the ultimate load when subjected to downward loads.

(3) The yield load is shown in Eq. 19.

Where P _y is yield load when subjected to upward load; P _y ' is Yield loads when subjected to downward loads.

(4) Yield displacement is shown in Eq. 20.

Where Δ _y is P _y corresponding displacement, Δ _y ' is P _y ' corresponding displacement.

(5) Peak displacement is shown in Eq. 21.

Where Δ _m is peak displacement when subjected to upward loads, Δ _m ' is peak displacement when subjected to downward loads.

(6) Falling section stiffness is shown in Eq. 22.

When the bearing capacity drops to 85% of the ultimate bearing capacity, the specimen reaches failure. K d = 0.85 P u − P u Δ y − Δ m K d ′ = 0.85 P u ′ − P u ′ Δ y ′ − Δ m ′ (22)

Where Δ _y is 0.85P _u Corresponding displacement, Δ _y ' is 0.85 P _u ' Corresponding displacement.

Regression gets Eq. 23. K d = 0.123 + 0.018 λ − 2 + 0.095 λ − 4 K e − 0.37 K d ′ = 0.08 + 0.42 λ − 2.5 − 0.05 λ − 3 − 0.25 λ − 5.5 K e − 0.25 (23)

(7) Verification of trilinear skeleton curve model.

The comparisons of the simulated and calculated skeleton curves are shown in Figure 14 (2). It can be found that the skeleton curves obtained by equation are in good agreement with the numerical simulation results, which can provide a reference for engineering design.

The restoring force model (RFM) of PSRHSCB is established by statistical regression, as shown in Figure 15 (1), where the numbers: 0, 1, 2, … , express the walking routes of the model in the process of loading and unloading in forward-backward directions (Ji et al., 2022). The hysteretic curves of 13 specimens were observed, and it was found that the forward and reverse loading paths approximately passed through a “fixed point”. The statistical analysis of the experimental data showed that the fixed point was the intersection of the loading path and the elastic section in the positive and reverse directions, and the ordinates were 0.65 P_u and-0.65 P_u, respectively.

The hysteretic rules are expressed as follows:

(1) Loading and unloading rules of elastic stage

Before the restoring force reaches P _y, the loading is carried out along the skeleton curve. During unloading, the initial stiffness is taken into account, regardless of stiffness degradation and residual deformation.

(2) Loading and unloading rules of elastoplastic stage

When the restoring force exceeds the horizontal yielding load P _y in the forward or backward direction, but is smaller than the peak load P _u, loading occurs along the skeleton curve, whereas unloading conforms to Eq. 24, and the unloading stiffness (K _un) is developed through regression analysis. K u n = 3.2117 Δ u n Δ y − 0.2192 K e (24)

Where Δ _y is the yielding displacement, Δ _un is the displacement at the unloading point, and K _e is the initial stiffness.

(3) Loading-unloading rules at the descending segment stage

When the restoring force exceeds P _u in the forward or backward direction, K _d in descending segment stage of the skeleton curve is taken as the loading stiffness and unloading stiffness is still calculated according to Eq. 24.

(4) Backward loading and forward reloading rules

In the case of backward loading after forward unloading, when the maximum displacement in the backward direction does not exceed the yield displacement, the walking route is from the point of p = 0.65P _u directly to the yielding point in the backward direction. When the maximum displacement in the backward direction exceeds the yielding displacement, the walking route is from the point of p = 0.65P _u to the maximum displacement point in the backward direction. For forward reloading after backward unloading, the walking route is from the point of p = 0.65P _u to the maximum point in the forward direction.

The comparisons between the hysteretic curves obtained by finite element simulation (FES) and RFM are shown in Figure 15 (2), and it can be seen that both of them are in good agreement, therefore, the restoring force model of this kind of composite beam is reasonable.