Application of precast reinforced concrete fuse units as energy dissipation units in various self-centering precast bridge pier systems

Model description

Based on the experimental tests of Wang et al. (2008), Zhu and Guo (2017), and Han et al. (2019), the FE models of a precast single-column bridge pier system, a precast double-column bridge pier system, and a precast wall system were developed. Table 1 shows a summary of the properties of the materials used for each sample. In the experimental test of Wang et al. (2008), two large-scale precast columns (P1, and P2) were tested. The P1 column is composed of nine hollow-core precast RC segments, precast RC cap-beam, and precast RC footing. Eight unbonded post-tensioned tendons were used to connect the segments with the cap beam and the footing. The outer cross-section size of each segment was 1200 × 1800 mm², and the size of the hollow core was 600 × 1200 mm². At the RC footing, the unbonded tendons took the shape of a U and were anchored to the cap beam. In each segment, transverse and longitudinal reinforcement were provided by steel bars with a diameter of 13 mm and 22 mm, respectively. The longitudinal reinforcement did not extend to the other segments. As a result, the only reinforcement passing through the column height was the unbonded tendons. The vertical gravity load and prestressing force were equal to 4000 kN. The same number of segments with the same longitudinal and transverse reinforcement was used for specimen P2 as for specimen P1. Compared to specimen P1, specimen P2 also contains the same number of tendons with the same distribution in the cross-section, but the tendons were bonded to the adjacent components. A hybrid precast system was used for this specimen, utilising 36 mm diameter high-strength ED steel bars that were run continuously through the segments. These ED steel bars were secured using custom T-headed threaded couplers embedded in the foundation. Half of these bars (four in total) extended through grouted steel corrugated ducts to the upper section of segment S2, where they were anchored with custom steel plates and nuts. The remaining four bars continued to the top of segment S5, employing the same anchoring technique.

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Table 1.

Properties of the materials.

Material	Items	P1	P2	TRB-N	TRB-B	EHW1
Concrete	Max Strength (MPa)	37	40	38.5	38.5	35.38
	Elastic modulus (GPa)	28.589	29.725	29.163	29.163	27.956
	Poisson’s ratio	0.2
ED bars	Diameter	–	Φ 13	–	Φ20	Φ10
	Elastic modulus (GPa)	–	200	–	200	200
	Yield strength (MPa)	–	1080	–	291	430
	Ultimate strength (MPa)	–	1230	–	440	557
	Poisson’s ratio	–	0.3	–	0.3	0.3
Transverse bars	Diameter	Φ 13		Φ8	Φ8	Φ10
	Elastic modulus (GPa)	200
	Yield strength (MPa)	372		425	425	451
	Ultimate strength (MPa)	559		520	520	618
	Poisson’s ratio	0.3
Longitudinal bars	Diameter	Φ 22		Φ32	Φ32	Φ16
	Elastic modulus (GPa)	200		200	200	200
	Yield strength (MPa)	480		455	455	438
	Ultimate strength (MPa)	637		630	630	620
	Poisson’s ratio	0.3
	Elastic modulus (GPa)	200		200	200	200
Tendons	Yield strength (MPa)	1792		1670	1670	1650
	Ultimate strength (MPa)	1888		1860	1860	1860
	Poisson’s ratio	0.3

In the experiment of Zhu and Guo (2017), three precast hybrid wall systems were carried out. One precast wall specimen denoted as EHW1 was used herein to evaluate the response of the SC-PC system with the PC-ED system. The EHW1 specimen consisted of an RC cap-beam, RC wall, and RC footing. The concrete wall was assembled with the other components using an unbonded post-tensioned tendon passing through the centerline of the wall cross-section. The cross-section of the precast RC wall was rectangular, the size was 200 × 1700 mm, and the height was 3460 mm. Eight 16 mm diameter vertical ED bars were used on the boundary elements on both sides of the wall, and ten 10 mm diameter ED bars were used in the middle of the wall. Two hundred mm of the ED bars were debonded from the top surface of the footing. The characteristic strength of concrete was 35.38 MPa. The vertical load of the precast wall was 946 kN (representing gravity load), and the average post-tensioned force was 296.8 kN. Since a typical SC wall specimen was not included in the study by Zhu and Guo (2017), the authors adopted the specimen (EHW1) without ED bars as a typical SC precast wall model, which is denoted as Typical EHW1 (TY- EHW1) in the following sections.

Han et al. (2019) conducted an experimental study on three specimens of SC double-column bridge piers equipped with two replaceable types of ED devices. Only two specimens were used in this study. Each specimen consisted of three main prefabricated members: RC footing, RC columns, and RC cap-beam. Eight unbonded post-tensioned tendons were set in each specimen to assemble two columns, the footing, and cap-beam; each column contained four tendons. The initial tensile force of each tendon was 105 kN. In addition, to simulate the vertical dead load, an 866.4 kN axial load was applied on the top of each column. The horizontal distance between the centerlines of the two columns was 3300 mm, and the vertical distance between the bottom and the top rocking surfaces of the column was 2250 mm. The column cross-section size was 540 × 400 mm, and the size was reduced to 380 × 240 mm at the bottom side and encased in a 16mm-thickness steel jacket to install the external replaceable ED devices. One of the two specimens was a typical SC specimen denoted as TRB-N: no ED devices were used. The second specimen (TRB-B) was identical to the (TRB-N) specimen, but it contained external replaceable ED mild steel bars (12 mild steel bars). For more details regarding the ED system, refer to the study by Han et al. (2019). The compressive characteristic strength of the concrete material was 38.5 MPa.

Modeling strategy

To accurately simulate the behavior of concrete components within the developed models, the study employed an eight-node, three-dimensional solid brick element (C3D8R), as illustrated in Figure 4. The components modeled using this element type include RC footings, precast columns, precast cap beams, and precast walls. These elements were selected due to their relevance in the structural analysis of bridge systems under seismic loading.OPEN IN VIEWER

For the material definition of concrete within the ABAQUS software environment, the Concrete Damage Plasticity (CDP) model was utilized. The CDP model was chosen because it is widely recognized for its capability to represent the nonlinear behavior of concrete, particularly under lateral cyclic loading conditions. This model accounts for key phenomena such as the progressive damage of concrete under tension and compression, the degradation of stiffness under repeated loading, and the effect of cyclic softening. To define the constitutive relationships for concrete, the study relied on the concrete softening damage plasticity model proposed by Feng et al., (2018a). Feng’s model enhances the standard CDP approach by incorporating the effects of compression-softening and the degradation of concrete under cyclic loading into the calculation of stress-strain relationships. This advanced model has demonstrated its applicability across various RC structural configurations, including RC walls, beams, and beam-column connections (Feng et al., 2018b; Feng et al., 2018c), making it particularly suitable for the present study’s objectives. Figure 5 illustrates the CDP input data for the concrete material used in the precast column segments of the P1 specimen.OPEN IN VIEWER

Key parameters that need to be specified include the concrete compressive strength (f’ _c ), and the modulus of elasticity (E _c ). The ultimate compressive strength of concrete for each modeled component was taken from the corresponding experimental descriptions, with a peak strain value of 0.003 assumed for all concrete elements at their ultimate compressive strength. The modulus of elasticity of concrete was estimated using empirical formulas recommended by the American Concrete Institute (ACI 318-19, 2019), which are commonly used in structural engineering (E _c = 4700 f ′ c ). The calibration of the CDP model parameters was performed following recommendations from prior research and ABAQUS documentation. Specifically, the plasticity parameters were set as follows: dilation angle (ψ) of 30°, flow potential eccentricity (ϵ) of 0.1, ratio of initial biaxial to uniaxial compressive strength (σb0/σc0) of 1.16, the shape factor of the yield surface (Kc) of 0.667, and the viscosity parameter (μ) of 0.005 (Dawood et al., 2012; Li et al., 2017). These values were selected to ensure that the simulation results align closely with the observed experimental behavior of concrete structures under similar loading conditions.

In the developed models, the truss element (T3D2) was employed to simulate all steel reinforcing components, including the main steel bars, longitudinal and transverse mild steel bars, and steel tendons. The use of T3D2 elements, which are ideal for modeling slender members under axial load, ensures that the behavior of these reinforcing components is accurately captured under various loading conditions. For the steel jackets and steel connections, C3D8R elements were utilized. The C3D8R elements are well-suited for capturing the complex stress distribution and potential plastic deformation within these components. This approach allows for a detailed representation of the interactions between the steel jackets, steel connections, and the surrounding concrete elements. A specific modeling technique was applied to the steel tendons to accurately represent the behavior of unbonded tendons observed in experimental tests. The ends of the steel tendons were embedded within the surrounding solid elements, which ensures a fixed connection at these points. In contrast, the middle portions of the tendons were left unbonded, allowing them to move freely and simulate the real-world behavior of unbonded tendons under loading conditions.

The longitudinal and transverse reinforcement, as well as the main reinforcement within the models, were embedded within the concrete members. This embedding technique ensures that the steel reinforcement and the concrete elements work together as a composite system, reflecting the actual structural behavior observed in experiments. To define the material behavior of the various reinforcing elements, a bilinear stress-strain relationship was adopted. This approach was selected based on the material characteristics of each reinforcing component, providing a balance between accuracy and computational efficiency. The bilinear model effectively captures the initial elastic behavior and the subsequent yielding of the steel, which is critical for accurately simulating the response of the reinforced concrete structures under loading.

In the simulation of the developed models, a surface-to-surface contact algorithm was employed to accurately represent the interaction between prefabricated components. This algorithm was crucial for capturing the opening-and-closing behavior at these interfaces, which is a key aspect of the structural response under loading conditions. To prevent the unintended penetration of prefabricated elements during the activation of the rocking mechanism, the interface of the rocking joints was modeled using a “hard contact” definition. This approach ensures that the prefabricated elements do not overlap or penetrate each other when subjected to rocking motions, thus accurately simulating the physical constraints observed in real-world applications. Additionally, a friction coefficient of 0.5 was applied to model the tangential interaction between the contact surfaces. This coefficient was chosen based on previous studies (Dawood et al., 2012; Li et al., 2017) and reflects the typical frictional resistance encountered between concrete and steel surfaces in such systems.

The boundary conditions for the models were defined to replicate the conditions observed in the experimental tests. Specifically, the bottom surface of the concrete footing was fully restrained, preventing any movement in the X, Y, and Z directions. This restriction simulates the fixed support conditions typically found at the base of structural components. The loading scenario in the numerical analysis of each model was carefully designed to replicate the loading conditions applied in the experimental tests. The analysis was conducted in two distinct steps. In the first step, the initial stress feature in ABAQUS was utilized to apply pre-stress to the tendons, simulating the initial tensioning process observed in experiments. Concurrently, a distributed load was applied to simulate the vertical axial load, matching the axial load conditions from the experimental setup. In the second step, displacement control was employed to simulate the lateral drift ratios used in the experimental tests. This approach allowed the numerical models to closely follow the same loading paths as the physical specimens, ensuring that the simulated response was directly comparable to the experimental results.

To model the proposed PC-ED components, two distinct forms were considered: PRC and HP. Both components were simulated using C3D8R elements to accurately represent the concrete material, with the CDP model applied for the material definition of the concrete. Truss elements (T3D2) were utilized to represent the embedded longitudinal and transverse mild steel reinforcement in both the PRC and HP PC-ED units. The bottom surface of the PRC PC-ED component was connected to the main SC system using a Tie connection. This type of connection ensures that the two components move together as a single unit, reflecting the integrated behavior intended in the structural design. In contrast, the bottom of the HP PC-ED component was connected to the main SC system using a surface-to-surface contact algorithm, allowing for rocking behavior. The interaction between the head-side faces of the PC-ED component and adjacent precast components or the corresponding inner surface of the steel connections attached to the main SC system was modeled using a surface-to-surface contact algorithm, with hard contact defined at the interface to prevent penetration during loading. A friction coefficient of 0.5 was applied to simulate the tangential interaction between contact surfaces, ensuring realistic behavior during the rocking mechanism. The unbonded steel tendons were incorporated in the HP PC-ED component by embedding their ends in the head of the HP PC-ED unit and the RC footing of the main SC system, allowing for free movement in the middle.

Validation of FEM results

In this section, the FE outputs (damage mode and cyclic load-displacement curves) of the models are compared with the available experimental test results to validate the accuracy of the numerical FE results.

Failure mode

Figure 8 shows the stresses generated in the connections (steel connection and footing socket gap filling) of the proposed system at a maximum displacement of 4%. The connections of the double-column PBP system were taken as a standard for the other models. The maximum stress on the steel connection is 91 N/mm². This value represents 33% of the yield stress of the used steel. No deformations or dents appeared on the surfaces of the connections, which indicates that the function of the connection is fully performed to transfer lateral loads to the PC-ED units accompanying the main system. Figure 8(a) shows the maximum stress value occurring on the bolts that fix the steel connection to the main SC system. At the maximum lateral displacement of 4%, the stresses on the bolts did not exceed 210 N/mm². This stress demonstrates that the bolts were able to efficiently secure the steel connection without any risk of failure during lateral displacements. The maximum tensile stresses and compressive stresses of the grout were 2.91 N/mm² and 33.4 N/mm², respectively. There was no movement or sliding between the grout elements and the PC-ED units during the lateral displacements. From these results, we can conclude that the socket connection is an appropriate choice in the proposed system to install the PC-ED units at the base of the main SC system.OPEN IN VIEWER

Figure 6 illustrates the damage levels of the proposed single-column, double-column, and wall-type precast bridge piers, each incorporating the two proposed energy dissipation mechanisms (PRC and HP) and designed with a 1% SC coefficient, λsc. These damage levels are compared to those of typical SC precast bridge pier systems without the proposed PC-ED units. Specifically, the specimens P1-PRC-1.0 and P1-HP-1.0 represent the general damage levels for the proposed single-column PBPs, TRB-PRC-1.0 and TRB-HP-1.0 for the double-column PBPs, and EHW1-PRC-1.0 and EHW1-HP-1.0 for the wall-type PBPs.

As shown in Figure 6, the proposed double-column and wall-type PBPs did not exhibit stress concentrations in the main piers. The behavior of these components is limited to the rocking mechanism through the unbonded tendons. Instead, the stresses were localized within the proposed damping element, a phenomenon attributed to the sliding capability of the damping element head at the contact interface. In the single-column PBPs, the inclusion of the proposed internal damping element led to an improvement in stress levels within the concrete segments, characterized by a reduction in stress and a more uniform distribution along the height of the pier due to the redistribution of gaps between segments. This redistribution effectively reduces stress concentrations and mitigates damage to the concrete segments. Overall, the proposed PC-ED units play a crucial role in alleviating stress levels in the precast bridge piers, rather than exacerbating them, particularly under transverse loading conditions.

Figure 9 shows the damage mode of the two types of PC-ED elements used in all investigated cases, which are determined by the plastic strain in the concrete material under the influence of different SC coefficients, λsc of 1.33, 1.0, and 0.67. Figure 9(a) and (b) show the damage mode of the PC-ED units used with the single-column PBP system and the double-column PBP system at a drift of 4%, while Figure 9(c) shows the damage mode of the PC-ED units used with the wall PBP system at a drift of 2.5%. As can be seen in Figure 9, the damage mode is flexural when the PRC units are used with the main SC system, as a plastic hinge forms at the bottom of the PRC units. The flexural behavior becomes more pronounced when the SC coefficient increases from 0.67 to 1.33. This increase in flexural cracks results from a reduction in the reinforcement ratio of the PC-ED elements from 6% to 2%.OPEN IN VIEWER

The same figure (Figure 9) shows the failure mode of the non-emulative HP units used with all the proposed PBP systems. As shown in the figure, since the behavior of these units depends mainly on a rocking mechanism, the failure mode differs from that which occurred in the PRC units. In HP units, the damage is limited to the crushing of concrete in the toes of the unit, with the distribution of tensile cracks along the lower part of the unit. As the SC coefficient, λ_sc, decreases, the high tensile extend upward. This is due to the increase in the unit lateral strength capacity and the corresponding strain of the elements incorporating the mild steel bars.

Compared to the PC-ED units used with the single-column PBP system and the double-column PBP system, Figure 9(c) illustrates that the PC-ED units used with the wall PBP system have a small damage index. The damage to these units does not exceed the concrete cover for the PRC units and is limited tension cracks at the bottom of the HP units. This is due to the lateral drift ratio of these units being only 2.5 % compared to 4 % for other PC-ED units. These results confirm the aforementioned claim that the traditional PRC units can withstand a lateral displacement of 2-3 %, with recoverable damage. All HP units were able to withstand 4 % drift without significant damage or excessive strains that could affect the unit’s functionality after seismic actions.

Hysteretic response of the proposed SC system with PC-ED system

Figures 10–12 show the hysteresis curves of all proposed SC PBP systems with PRC units (emulative system) and HP units (non-emulative system). The proposed system generally shows an improvement in the initial stiffness, yield point of the curve, load-carrying capacity, and area of the loading cycle compared to the hysteresis response of the typical main SC PBP systems of P1, TRB-N and EHW1. This improvement in lateral loading results from the fact that the lateral resistance of the PC-ED unit is activated in parallel with the main SC systems. Ultimately, all hysteresis curves of the proposed PBP systems exhibit a flag shape compared to the hysteresis curve of the typical main SC PBP systems of P1, TRB-N and EHW1 specimens. A plastic hinge is formed at the bottom of the PRC units, or the yielding of the steel bars at the rocking joints results in a flag shape. As the SC coefficient decreases, the flag-shaped area increases, as shown in Figures 10–12.OPEN IN VIEWER

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Figure 11.

Hysteretic curves of the proposed double-column PBP system with a SC coffetient of (a) λ_sc = 1.33; (b) λ_sc = 1.0; and (c) λ_sc = 0.67.

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Figure 12.

Hysteretic curves of the proposed wall PBP system with a SC coffetient of (a) λ_sc = 1.33; (b) λ_sc = 1.0; and (c) λ_sc = 0.67.

The hysteresis curves of the proposed single-column PBP system with the two types of PC-ED units (PRC unit and HP unit) at a drift ratio of 4% are shown in Figure 10(a)–(c). The SC coefficients of 1.33, 1.0, and 0.67 are the only variables between the specimens. It can be seen that the initial stiffness of the hysteresis curve of the proposed single-column PBP system is 28%, 35%, and 42% higher than that of the P1 specimen when the PRC units are used with the main system, respectively. The initial stiffness of the hysteretic curve increases by 23%, 31%, and 36% when the HP units are used with the main system. The shared stiffness of the PC-ED units causes this increase. Moreover, the yield load of the hysteretic curves of specimens P1-PRC-1.33, P1-PRC-1.0, and P1-PRC-0.67 shifts upward by 33%, 57%, and 71% of the yield load of the proposed single-column PBP system, respectively, and by 30%, 51%, and 62% for specimens P1-HP-1.33, P1-HP-1.0, and P1-HP-0.67, respectively. The cause of this increase is the lateral resistance of the PC-ED unit during yielding. Also, compared with specimen P1, the load-carrying capacity of the proposed single-column PBP is increased by 20%, 39%, and 56% by using PRC elements and by 31%, 48% and 61% by using HP elements with SC coefficients of 1.33, 1.0, and 0.67, respectively. For comparison, the calculated results of the proposed analytical model with the 3D FEM results of the proposed single-column PBP system with the PC-ED unit are also shown in Figure 10 with solid lines. As can be seen in the figure, the proposed analytical model successfully predicts the main characteristic points of the lateral response of the proposed system with good accuracy compared with the numerical FE results. It is worth noting that the yield strength values predicted by the analytical model are lower than those observed in the FE modeling results. The primary reason for this discrepancy can be attributed to the redistribution of gaps between the concrete segments of the main column, influenced by the presence of the internal PC-ED unit. This PC-ED unit delays the opening between the concrete segments, resulting in a higher yield point in the curves of the proposed columns.

The hysteresis curves of the proposed double-column PBP system with different SC coefficients of 1.33, 1.0, and 0.67 are shown in Figure 11(a)–(c). As can be seen from the figure, the yield load of the hysteretic curve of the proposed double-column PBP system is 466 MPa, 647 MPa, and 671 MPa when the PRC units are used with the main system with SC coefficients of 1.33, 1.0, and 0.67, respectively. The yield load of the hysteretic curve is 434 MPa, 517 MPa, and 600 MPa when the HP units are used with the main system with SC coefficients of 1.33, 1.0, and 0.67, respectively. Compared to the typical TRB-N specimen (275 MPa yield load), all the proposed double-column PBP specimens show a higher yield load of the hysteresis curve. This improvement is due to the additional lateral resistance of the two PC-ED units during yielding. It is worth noting that using the HP unit with the main SC system gives a lower yield load than using the PRC units with the main SC system. This lower yield load is due to the opening at the rocking joints of the HP units through the post-tensioned tendons, resulting in a reduction in the yield load of the entire system. Compared with the lateral load-carrying capacity of 385 MPa for the typical TRB-N specimen at lateral displacement of 80 mm, the load-carrying capacity of the proposed double-column PBP is 540 MPa, 642 MPa, and 724 MPa using PRC elements and 592 MPa, 706 MPa, and 804 MPa using HP elements with SC coefficients of 1.33, 1.0, and 0.67, respectively, at the same lateral displacement of 80 mm. This increase in the lateral load-carrying capacity of the proposed double-column PBP specimens is due to the additional lateral load resistance of the two PC-EDs. On the other hand, the proposed SC system in combination with HP units shows an increase in the maximum lateral load capacity compared to the use of PRC units. This increase is due to the increased post-tensioning force in the tendons during lateral displacement.

The hysteresis curves of the proposed wall PBP system with the two types of PC-ED units (PRC unit and HP unit) at a drift ratio of 2.5% are shown in Figure 12(a)–(c). The SC coefficients of 1.33, 1.0, and 0.67 are the only variable between the specimens. In general, the proposed wall PBP system shows an improvement in the initial stiffness, yield point of the curve, load carrying capacity, and area of the loading cycle compared to the hysteretic behavior of the typical main SC PBP systems of EHW1. Although the yield load increased from 395 MPa to 555 MPa when PRC units were used and from 412 MPa to 585 MPa when HP units were used, when the SC coefficient decreased from 1.33 to 0.67, no significant differences in maximum lateral resistance occurred when both units were used. This is due to the lateral displacement ratio, which only reached 2.5%. In other words, the forces in the tendons were not sufficiently activated to cause a significant difference in lateral resistance, as was the case with the single-column PBP system and the double-column PBP system.

Energy dissipation of the proposed SC system with PC-ED system

Figure 13 shows the equivalent viscous damping ratio of all the simulated cases of the proposed single-column PBP system, double-column PBP system, and wall PBP system with different SC coefficients, (λ_sc) of 1.33, 1.0, and 0.67. Elmenshawi’s approach was used to calculate the equivalent viscous damping ratio (Elmenshawi and Brown, 2010). As shown in Figure 13, the proposed system with PC-ED elements displays an equivalent viscous damping ratio in the range of 9% to 27%.OPEN IN VIEWER

Figure 13(a) shows the equivalent viscous damping ratio of the proposed single-column PBP system with the two forms of PC-ED units. Compared with the P1 specimen, all the proposed single-column PBP specimens display a higher equivalent viscous damping ratio. The use of a PRC unit with the main SC system results in equivalent viscous damping ratios of 12%, 15%, and 17% for columns with reinforcement ratios of 1.33, 1.0, and 0.67, respectively. In contrast, the use of an HP unit with the main SC system yields equivalent viscous damping ratios of 9%, 12%, and 13% for columns with reinforcement ratios of 1.33, 1.0, and 0.67, respectively. The increase in the equivalent viscous damping ratio is due to the formation of the flag-shaped hysteretic response of the proposed single-column PBP system. It is worth noting that all simulated PBP columns that used PRC units showed higher values of the equivalent viscous damping ratio. This is due to the contribution of concrete in dissipating energy with the yield steel bars, which leads to an increase in the ED capacity of the entire system.

Figure 13(b) shows the equivalent viscous damping ratio of the proposed double-column PBP system with the two forms of PC-ED units. As shown in Figure 13(b) the used of emulative PC-ED units (PRC units) in the main SC double-column PBP models, the equivalent viscous damping ratios with SC coefficient, λ_sc, of 1.33, 1.0, and 0.67 are 13%, 17%, and 19%, respectively. With the same SC coefficient, λ_sc, of 1.33, 1.0, and 0.67, the equivalent viscous damping ratios of the SC double-column PBP models with the non-emulative PC-ED units (HP units) are 16%, 21%, and 25%, respectively. Compared with the proposed SC system with non-emulative PC-ED units, all cases with emulative PC-ED units show higher values in terms of the equivalent viscous damping ratio. This increase is due to an enlarged hysteric curve.

Figure 13(c) shows the equivalent viscous damping ratio of the proposed SC wall PBP system under different SC coefficient (λ_sc) of 1.33; 1.0; and 0.67. As shown in Figure 13(c), the equivalent viscous damping ratio of the proposed system with PC-ED units is between 9% and 18%. When the SC coefficient (λ_sc) are 1.33, 1.0, and 0.67 the proposed SC precast wall models with emulative PC-ED units show equivalent viscous damping ratio of 12%, 16%, and 18%, respectively. At the same SC coefficient (λ_sc), the equivalent viscous damping ratios of the wall PBP models with the non-emulative PC-ED units are 9%, 12%, and 14%, respectively. The difference between the equivalent viscous damping ratio of the two types of PC-ED units is due to the smaller hysteretic loop area of the models with non-emulative PC-ED units compared to the models with emulative PC-ED units as shown in Figure 13(c).

According to Seo and Sause (2005), SC systems with an equivalent viscous damping ratio from 12.5% to 25% produced ductility demands comparable to conventional systems under seismic loading. The adequate ED capacity of the SC system is based on the ratio of the post-yield stiffness to initial stiffness. In other words, the SC system with higher ratio of the post-yield stiffness to initial stiffness requires less ED capacity. As shown in Figures 10–12, the hysteretic curves of the SC system with non-emulative PC-ED unit have higher post-yield stiffness than that of SC systems with emulative PC-ED unit. This increase in post-yield stiffness is due to the hardening of post-tensioned tendons used with non-emulative PC-ED units. Therefore, although the equivalent viscous damping ratios of SC systems with non-emulative PC-ED units are reduced, the higher post-yield stiffness can improve the damping capacity of these systems under the seismic loads.

Residual displacements of the proposed SC system with PC-ED system

Residual deformation is one of the most important criteria for measuring how well RC bridges perform after an earthquake. Figure 14 shows the relationship between the residual deformation ratio of the proposed PBP systems with SC coefficient ratios of 1.33, 1.0, and 0.67. Figure 14(a) shows the maximum residual drift ratio of the proposed single-column PBP system and double-column PBP system at a drift ratio of 4%. Figure 14(b) shows the maximum residual drift ratio of the proposed wall PBP system at a drift ratio of 2.5%. As can be seen in Figure 14, the residual drift ratio ranged from 0.04% to 1.7% for all simulated cases, depending on the type of PC-RD unit (PRC unit, HP unit) and the SC coefficient ratio. In general, a decrease in the SC coefficient ratio leads to an increase in residual displacement for all samples analyzed. In other words, an increase in the reinforcement ratio in the PC-RD units leads to an increase in the residual displacement. This is due to the yielding of a larger proportion of the steel reinforcement used in the damping elements, which in turn affects the residual displacement of the entire system. At a constant reinforcement ratio for the PC-ED unit, all systems containing PRC units have a higher residual displacement ratio than the systems using HP units. This is due to the force in the tendons contributing to the HP units returning to their original vertical position and thus reducing the overall residual displacement of the system. Comparing the residual displacement results in Figure 14(b) with those in Figure 14(a), one can clearly see a decrease in the residual displacement values for all the wall PBP specimens compared to the other specimens (single-column PBP and the double-column PBP). This is due to the fact that the wall PBP samples were subjected to a lateral displacement ratio of only 2.5 %, in contrast to the single-column PBP and the double-column PBP, which were subjected to a lateral displacement ratio of 4%.OPEN IN VIEWER

It’s worth noting that the double-column PBP system with the PRC or HP unit exhibits a more pronounced residual drift compared to the single-column and wall PBP systems. In this study, the authors aimed to maintain a consistent cross-sectional area ratio of 40% for for the PC-ED unit across the different PBP systems. However, due to practical considerations during the design dimension selection phase, the actual ratios slightly deviated from the target, with cross-sectional area ratios for the PC-ED unit in the single-column, double-column, and wall PBP systems being 38%, 42%, and 37%, respectively. The larger cross-sectional area of the PC-ED unit in the double-column PBP system (42%) results in a higher quantity of reinforcement compared to the other systems. This increased reinforcement, coupled with the slightly larger cross-sectional area, enhances the damping contribution in the double-column system. Consequently, as the SC coefficients decrease, the double-column PBP system exhibits a more significant residual drift due to its increased stiffness and greater sensitivity to stiffness degradation. This effect is less pronounced in the single-column and wall PBP systems, which have slightly smaller cross-sectional areas, and therefore lower reinforcement levels and damping contributions.

It is clear that the SC coefficient (λsc) plays an important role in defining the SC capability of the proposed system. Based on the recommendation of the Japanese Code, (2002) and the Pacific Earthquake Engineering Research Center (PEER), the residual drift ratio of the bridge piers should not exceed 1% to ensure the serviceability of the bridges after earthquakes. As can be seen in Figure 14, the use of an SC coefficient (λsc) equal to or higher than 1.0 ensures that the proposed SC PBP with the emulative PRC units has excellent SC capability with a residual drift ratio of ≤ 0.5%. When the SC coefficient λsc is equal to 0.67, the proposed system starts to lose its SC capacity gradually, as shown in Figure 14(a). When the non-emulative PC-ED units (HP units) are used with the main SC PBP specimens and the SC coefficient λsc is equal to or higher than 0.67, the proposed system can guarantee the SC capability with a residual drift ratio ≤ 0.65%.

Resilient-based-design of the proposed SC system with PC-ED system

According to the required resilience and recoverability of modern structures (Wu et al., 2009; Fahmy et al., 2010a), the performance of the proposed system can be divided into three states: Resilience, recoverability, and safe-exit. These states vary depending on the resisting mechanism of ED unit (PRC unit or HP unit). In the state of resilience, care should be taken in the design to ensure that the prestressed system does not yield and that the main SC precast system is not locally damaged. In addition, the residual deformation is controlled so that the original function of the structure is restored immediately after the earthquake.

According to the previous test results (Fahmy et al., 2010b), the conventional emulative RC column can withstand a lateral displacement of 2% to 3%, with recoverable damage and a residual deformation of less than 1%. On the other hand, as shown by the damage mode (see Figure 9), hysteresis curves (see Figures 10–12), and residual displacement (see Figure 14), the proposed SC PBP systems with the emulative PRC system can ensure the resilience response up to a drift ratio of 2.5%: at a drift of 2.5%, the residual displacement is less than 1%, the lateral strength does not decrease and the PC-ED elements showes limited damage localized at the concrete cover.

On the other hand, when utilizing the non-emulative HP unit (SC system), the opening at the rocking interface can reduce the overall damage to the HP units. Furthermore, the tendons can limit the residual drift of the entire system. This type of ED units (HP unit) can ensure a resilient response up to a higher drift ratio (equal to or greater than 5% (Ou et al., 2010). Since both the main system and the ED system (HP units) are characterized as SC systems, the combined system can guarantee a resilience response up to or exceeding 5% lateral drift, as indicated in Figures 10–12. The superior lateral performance of this integrated system makes the proposed system preferable for application in high earthquake areas, as it can absorb strong earthquake shocks and restore its original function immediately after earthquake without any significant repair up to a 4% drift ratio.

The state of recoverability of the proposed system can be divided into two zones. In the first zone, the residual drift is limited to 1% for the PBP system. In this case, the structure can quickly recover its original function, but some quick repair work is required. According to all the simulated cases with different SC coefftient ratios, except for one specimen of the double-column PBP system (TRB-PRC-0.67 specimen), all the proposed SC PBP system with the emulative PRC units can provide recoverable performance up to 4% lateral drift or higher with quick repair work for the PRC units. For the double-column PBP specimen TRB-PRC-0.67, it can be classified as in the second zone of recoverability, in which the damage to the PC-ED components may be significant, and the residual deformation may exceed 1% but should be less than 1.75%. In this case, extensive repair is necessary to reuse the structure function. In addition, replacing the severely damaged PC-ED elements can help the post-tensioning system bring the structure back to the vertical position and then new damping elements should be placed, or damaged units should be repaired if possible, as shown in Figure 11(c).

If the structure exceeds the recoverability state criteria, the structural performance will be shifted to the last safe-exit state, in which the post-tensioned system of the structure may yield, and most of the compressed areas will be excessively damaged with residual deformation less than 1.75%, resulting in a reduction in structural resistance. In this case, the structure can be safely dismantled.