# Numerical Simulation Analysis of the Seismic Performance of Precast Bridge Piers

Authors: Jia Wang; Fangyuan Li
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Precast technology is increasingly favoured by bridge engineers, and the seismic performance of precast bridge piers urgently needs to be addressed. Precast piers have a complex load-bearing mechanism, and their design method has not been clearly determined. Fibre plastic hinge theory can satisfactorily analyse the seismic performance of precast piers. In this study, the seismic performance of precast piers is analysed through numerical simulation of pier columns based on the pseudostatic testing of a set of existing precast piers and combined with fibre plastic hinge theory. The results show the following: With reasonable constructional measures, precast piers connected by grouted sleeves indeed have seismic performance similar to that of cast-in-place components. Finite-element programs can be employed to simulate the mechanical properties of steel rebars before yielding well but not those after yielding or under fatigue. When fibre plastic hinges are used for finite-element analysis, if the fibre cross-section is reasonably divided, the length of the plastic hinge has little influence on the results; otherwise, too densely distributed plastic hinges may lead to nonconvergence of the results in a later working condition. The plastic hinge method recommended in this paper, especially the sectional fibre-cross-section division method, can obtain the plastic hinge performance state, it can help save the test cost and make up for the shortage of test data.


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## Body

## 1. Introduction

Compared with traditional cast-in-place bridge construction, precast technology can reduce the impact on existing traffic, improve the safety of the construction process, improve project quality through shop standardized prefabrication and curing, improve the constructability at the bridge site, substantially reduce construction costs, and reduce environmental pollution and therefore are garnering increasing attention around the world.Precast components, an alternative to “cast-in-place” construction, are usually used in concrete structures. The construction of precast concrete bridges began in the United Kingdom during World War II [1]. In 1943, the British Ministry of War Transport manufactured a batch of concrete standard section girders for the London Midland and Scottish Railway as well as the London and North Eastern Railway. In 1973, the Prestressed Concrete Association was established in the United Kingdom, marking the beginning of the era of the standardization of precast concrete bridges [2].Precast prestressed concrete was introduced in the United States in 1949 and quickly became the most-used material for bridge construction. The first prestressed concrete bridge in the United States came into being in 1949 and was designed by Belgian engineer Gustave Magnel. As of 1990, most of the bridges in the United States had been in service for nearly 40 years, and many required upgrading, and the increasing traffic made traffic diversion due to old bridge retrofitting a major problem. To solve this problem, the Federal Highway Administration (FHWA) of the United States and the American Association of State Highway and Transportation Officials have vigorously promoted the shop prefabrication and site assembly of bridges as a construction technology to reduce the disruption of site construction to road traffic [3, 4].Precast technology was introduced later in China than in Europe and America. Completed in 1956, the Xinyi River Bridge on the Lianyungang–Lanzhou Railway Line was the first precast concrete bridge in China. The development of precast concrete bridges in the highway system occurred at almost the same time as that in the railway system. The first precast concrete highway bridge, which spans the Yaba River on the Beijing-Zhoukoudian Highway, was also built in 1956 [5]. The successful construction of this bridge promoted a series of standardized designs of precast concrete simply supported girder bridges in the highway system.The FHWA divides bridge precast technology into five parts, namely, that for bridge decks, girders, piers, abutments, and other components [2]. Usually, precast technology is also divided into that for substructures and superstructures. Precast superstructures have been widely studied and applied in various countries because of the light weight of their segments and thus the ease of lifting and handling during construction. In contrast, precast substructures are difficult to handle and install due to the heavy weight of their segments, and its development is limited to some extent due to the presence of immature construction equipment that fails to fulfil the role of precast substructures as the support of precast superstructures.Precast piers as part of the substructure of a bridge find only limited application compared with the mature use of precast superstructures. In Europe and the United States, the engineering application of precast substructures began in the 1960s, with that in the construction of the Victory Bridge in New Jersey, United States, as an example. In the mainland of China, precast segmental bridge piers were initially applied in the Shanghai Donghai Bridge and later to the projects of the Hangzhou Bay Bridge, Hong Kong-Zhuhai-Macau Bridge, and Shanghai Jiamin Viaduct, successively. The piers of the substructure in the expressway reconstruction project of the Xiangfu Road in Changsha and in the reconstruction project of the city access section of the Chengdu-Pengzhou viaduct in Chengdu are currently being constructed using precast segmental technology. At present, precast segmental piers are primarily used in nonseismic and low-seismic fortification areas, with restricted application in moderate- and high-intensity regions mainly due to the uncertainty of the seismic performance of prestressed segmental concrete piers.Precast piers with different types of connections exhibit different seismic performance, and the corresponding seismic theory may also vary. The connection types for precast piers can be broadly classified into sleeve connections, grouted metal corrugated duct connections, prestressed connections, hybrid connections, socket connections, and pocket connections [6–8]. There are two main concepts in the design and study of the seismic performance of precast pier columns. First, the connections are designed to make the precast pier columns and the monolithic cast-in-place pier columns have similar seismic indices to ensure that the existing seismic design methods remain applicable. Second, new connections are developed, and methods for the seismic design of precast pier columns with the new connections are studied since the existing seismic design methods are not fully applicable. Numerical simulation methods include the concentrated plastic hinge method [9], fibre model method [9, 10], and three-dimensional (3D) solid finite-element method [11]. According to the findings from previous tests, the seismic design methods for precast piers with sleeve connections, grouted metal corrugated duct connections, and socket and pocket connections are generally similar to those of monolithic cast-in-place pier columns; previous theoretical studies on the seismic performance of precast pier columns with hybrid connections have mainly focused on finite-element simulation methods, including the fibre model and the solid model analysis methods. Different precast segmental pier types and joint types have different load-bearing mechanisms. However, analytical methods have been little studied, and the design methods have still not been clearly established.At present, precast segmental piers are primarily used in nonseismic and low-seismic fortification areas, with restricted application in moderate and high-intensity regions mainly due to the uncertainty of the seismic performance of prestressed segmental concrete piers.It has been pointed out that the assembly of precast substructures mainly relies on different types of connections, including sleeve connections, grouted metal corrugated duct connections, prestressed connections, hybrid connections, socket connections, and pocket connections to realize the connections between precast segmental piers, between precast piers and bent caps, and between precast piers and pier caps. A literature survey reveals that the bridge engineering community lacks a clear and consistent understanding of the ductility, damage process, bending capacity, shear failure mechanism, collapse failure mechanism, and nonlinear mechanical behaviour at the joints of precast piers constructed with different types of connections. The seismic performance of precast segmental piers under different force conditions is affected by a variety of factors. For example, the different types of joint connections and the changes in the arrangement of longitudinal rebars and prestressing tendons can influence the seismic performance of piers. However, corresponding provisions are missing from the seismic code in China.Currently, the development of precast piers is relatively immature, and there are few practical engineering applications. Researchers have conducted a large number of tests to prove that precast piers with sleeve connections, grouted metal corrugated duct connections, and socket and pocket connections have seismic performance similar to that of cast-in-place piers [12–14] and hence can meet the corresponding seismic design requirements with reasonable constructional measures. A prestressed connection has a low hysteretic energy dissipation capacity and a small residual displacement and can be used for precast substructures in low-intensity areas. A hybrid connection exhibits a high hysteretic energy dissipation capacity and a small residual displacement and thus has broad application prospects; however, its development is limited due to its complex structural construction and low construction speed.


## 2. Seismic Tests of Precast Piers as a reference

As mentioned earlier, the precast technology for substructures is not as mature as that for superstructures, nor is the corresponding design method as complete. Therefore, it is necessary to combine the test data with the numerical simulation results to determine the method and possibility of the seismic design of precast piers. In this study, a small three-span continuous box girder bridge in Section 6 of the Shanghai S6 Highway project designed and studied by the Shanghai Urban Construction Design and Research Institute and Tongji University, respectively, is taken as an engineering example, based on which the seismic performance of the precast columns of this project is investigated through numerical simulation in combination with the relevant pseudostatic test data [15, 16].In this project, the seismic performance (e.g., hysteresis performance, ductility, energy dissipation, and minimum residual plastic deformation) of precast segmental piers with grouted sleeve connections as a constructional measure was investigated through large-scale pseudostatic tests and compared with that of conventional cast-in-place concrete piers. Details about some of the tested specimens used as control data for numerical analysis in this study are shown in Table1.Table 1
Design of the test specimens.
Specimen no.Specimen typePurpose of test1Specimen of cast-in-place reinforced concrete single-column pierA benchmark for the seismic performance of precast segmental columns2Specimen of precast segmental single-column pier connected by large-diameter steel rebars and grouted sleeves (with steel rebars connecting the sleeve pre-embedded in the pier cap)For the investigation of the seismic performance and damage mechanism of precast segmental columns with this constructional measureA set of 1:3-scale models were designed and fabricated considering the characteristics of the structural components in the actual project. The cast-in-place specimen was made of the same materials as those for the piers, that is, C40 concrete and HRB335 (Grade II) steel rebars. The grouted sleeve connections with large-diameter rebars were made of C40 concrete and HRB400 steel rebars. The following dimensions were used consistently for the test specimens: 530 × 500 × 3200 mm for the column, 900 × 500 × 400 mm for the loading end, and 1600 × 1600 × 600 mm for the base. The structural form of the specimen is shown in Figure1.Figure 1
Schematic diagram of specimens.The reinforcement design for the column of specimen#1 is shown in Figure 2(a). The 12-mm HRB335 hot-rolled ribbed steel bars were used as the longitudinal reinforcement, and the 6-mm hot-rolled plain steel bars were used as stirrups and tie bars. The clear concrete cover of the stirrups had a thickness of 15 mm.Figure 2
Column reinforcement design.
(a)(b)(c)The column reinforcement design of specimen#2 is shown in Figure 2(b). The 20-mm HRB400 hot-rolled ribbed steel bars were used as the longitudinal reinforcement, the 8-mm hot-rolled plain steel bars were used as constructional reinforcement, and the 8-mm and 6-mm hot-rolled plain steel bars were used as stirrups and tie bars, respectively, with a 39-mm-thick clear concrete cover of the stirrups. The longitudinal reinforcement passed through the joint and was connected by a grouted sleeve. The design of the cross-sectional reinforcement at the location of the grouted sleeve is shown in Figure 2(c), where 8-mm hot-rolled plain steel bars were used as stirrups and tie bars and the clear concrete cover of the stirrups had a thickness of 15 mm.The axial compression ratio of the pier column has a very large influence on the seismic performance of the pier and needs to be accurately simulated. Considering the difference in the self-weight of the model and the scale ratio of the axial force, the vertical load of the specimen was finally determined to be 56.0 t.The specimen was loaded to failure under both constant axial compression and uniaxial horizontal cyclic loading. In the loading protocol, a displacement-controlled loading was applied with a loading frequency of 0.01 Hz and a sampling frequency of 5 Hz. When the maximum displacement was reached at each level of the cyclic load, the load was held to observe and mark the damage conditions. The loading was terminated when the strength of the specimen decreased to 80% of its maximum strength. The specimen loading protocol is shown in Figure3.Figure 3
Specimen loading protocol.The following four items were measured in the test: (1) The curvature distribution in the plastic hinge zone; (2) the strains in the rebars, stirrups, and prestressing tendons in the plastic hinge zone; (3) the displacements at key locations of the pier; and (4) the horizontal and vertical loads. To better describe the test results of the seismic performance of each specimen, five component level-based performance levels are defined, as shown in Table2.Table 2
Definition of component level-based performance levels.
Performance levelPerformance objectiveQualitative description of the performance levelQuantitative description of the performance levelICrackingHairline cracks appearCracks are almost invisibleIIYieldingThe first longitudinal rebar theoretically yieldsCrack width <1 mmIIIPlastic hinge beginning to formNonlinear deformation occurs; the concrete cover starts to spall; inclined cracks developCrack width of 1–2 mm; height of the concrete spalling zone >1/10 of the height of the cross-sectionIVPlastic hinge completely formedLarge cracks are formed; the concrete in local damaged regions spalls completelyCrack width >2 mm; inclined cracks develop within 2/3 of the height of the cross-section from the pier cap; height of the concrete spalling zone >1/2 of the height of the cross-sectionVStrength degradationLongitudinal rebars buckle; stirrups fracture; core concrete crushedCrack width of core concrete >2 mm; expansion of core concrete >5%The seismic performance of the component in the test was evaluated using the rebar strainεs, drift ratio DR, displacement ductility factor of the component μΔ, residual deformation index RDI, equivalent viscous damping ratio ξeq, and normalized equivalent stiffness nk. εs was measured by strain gauges; DR is defined as the ratio of the horizontal control displacement of the specimen at each loading level to the effective loading height of the specimen; and μΔ is defined as the ratio of the lateral displacement at a given loading level to the idealized yield displacement, as illustrated in Figure 4 [17].Figure 4
Illustration of the calculation ofμΔ.The RDI refers to the unrecoverable plastic deformation of the component after unloading.ξeq is an index that characterizes the energy dissipation capacity of the structure. The larger the value of ξeq is, the more energy the structure dissipates under seismic action, and hence, the safer the structure is. nk is defined as the ratio of the equivalent stiffness Keff to the initial stiffness K0, where Keff is the slope of the equivalent linear elastic system, that is, the ratio of the horizontal force to the lateral displacement of the specimen at a given loading level. The calculation of the RDI, ξeq, and nk is illustrated in Figure 5 [17].Figure 5
Illustration for the calculation of the RDI,ξeq, and nk.The test results are as follows. According to the RDI, specimen#1 had the smallest residual deformation after unloading and had an RDI of 4.5 at performance level V (which corresponds to the performance objective of strength degradation and is qualitatively described as longitudinal reinforcement buckling, stirrup fracture, and core concrete crushing and quantitatively described as core concrete crack width >2 mm and core concrete expansion >5%). The precast column specimens #2 and #3 had RDI values of 6.3 and 5.5, respectively, at performance level V. Comparison of the μΔ values revealed that the displacement ductility levels of specimens #2 and #3 were higher than that of specimen #1. The detailed test procedure and results are available elsewhere [15].


## 3. Numerical Simulation Analysis of the Seismic Performance of Precast Pier Columns

Numerical simulation methods mainly include the concentrated plastic hinge method, fibre model method, and 3D solid finite-element method. The fibre model method is used in this study [18, 19].Based on the aforementioned seismic tests and test results of the piers, the low-cycle reverse loading tests on specimens nos. 1 and 2 are simulated in SAP2000, and then the numerical simulation results are compared with the test results to evaluate their reliability to provide the basis for the corresponding parameter analysis [20].The elastic analysis model of the test specimen is established first using the fibre PMM hinge model in SAP2000, and then a plastic hinge with a certain length is assigned to the elastic element to build the fibre element model. The frame fibre hinge element discretizes the cross-section into a number of fibres, and each fibre has a position, an attached area, and a stress-strain relationship. The fibre hinge model can simulate the performance of the specimen before and after the onset of yielding and can simulate the strength degradation behaviour of the specimen caused by concrete cracking, rebar yielding, and rebar strain hardening [21, 22].


### 3.1. Definition of Fibre Plastic Hinges

The behaviour of the specimen before and after yielding can be simulated using discrete plastic hinges, which are generally defined as axial hinges for trusses, flexural hinges, and shear hinges in the principal direction for beams and PMM-related hinges for columns. The types of plastic hinges provided by SAP2000 include uncoupled hinges, interaction PMM hinges, and fibre PMM hinges. In this study, the fibre PMM hinge is used to simulate the nonlinear mechanical behaviour of the specimen. The fibre PMM hinge is defined and specified as follows [23, 24]:(1) The stress-strain relationship of the material is defined, including the constitutive relations of the concrete cover, the core concrete, and the rebar. To ensure that the numerical simulation results match the test results as much as possible, the measured values of the yield stress, elastic modulus, ultimate stress, and ultimate strain in the stress-strain relationship of the material are used.There are roughly three types of constitutive relationship models for rebars: the ideal elastic-plastic model, the elastic-hardening model, and the elastic-plastic hardening model. In particular, the elastic-plastic hardening model can be divided into linear hardening and parabolic hardening types considering different hardening segments, as shown in Figure6.Figure 6
Calculation models for the stress-strain relationships of rebar.The curvilinear elastic-plastic hardening model is used in this study as the calculation model for the stress-strain relationship of the rebar, as described in the following equation:(1)fs=Es⋅ε,ε<εy,fy,ε<εsh,fu−fu−fy⋅εsu−εεsu−εsh2,ε<εsu,where ε is the rebar strain; fy is the rebar stress; Es is the elastic modulus of the rebar; εy is the yield strain of the rebar; εsh is the initial strain of the hardening segment of the rebar; and εsu is the ultimate strain of the rebar.The concrete models are divided into the concrete cover model and the core concrete model, and the Mander model is used to describe the uniaxial stress-strain relationship of the concrete [25, 26], as shown in Figure 7.Figure 7
Concrete stress-strain relationship.The stress-strain relationship of the core concrete based on the Mander model is expressed in equations (2)–(11).(2)fc=0,ε>0,ε⋅Ec,ε<0,fcc′⋅x⋅rr−1+xr,ε<εcu,(3)x=εεcc,(4)r=EcEc−Esec,(5)Esec=fcc′εcc,(6)εcc=0.002×1+5⋅fcc′fc′−1,(7)fcc′=fc′2.2541+7.94fl′fc′−2fl′fc′−1.254,(8)εcu=0.004+1.4ρsfyhεsufcc′,where ε is the concrete strain; fc is the concrete stress; Ec is the elastic modulus of concrete; εcu is the ultimate compressive strain of concrete; εcc is the peak strain of concrete; fc′ is the standard value of the 28-day compressive strength of unconfined concrete; fcc′ is the peak stress of the core concrete; ρs is the volumetric reinforcement ratio of the stirrups; fyh is the yield stress of the stirrups; and εsu is the ultimate strain of the stirrups, which is set to 0.09 in this study.fl′ is the effective transverse confining stress of the concrete. For the concrete cover, fl′=0, fcc′=fc′. For the concrete in the core region, fl′ is calculated using the following equation:(9)fl′=12keρsfyh,where ke is the effective transverse confinement coefficient of concrete, calculated as the ratio of the area of the effective confined concrete (taken at the middle section between adjacent stirrups, where the confined area is the smallest) to the area of the core region of the confined concrete.For the rectangular cross-section, the stirrup ratioρs may differ in the x and y directions. Hence, the effective transverse confining stresses of the concrete in the x and y directions, respectively, are as follows:(10)flx′=keρxfyh,(11)fly′=keρyfyh,Then, the ratio of the peak stress of the core concretefcc′ to the 28-day compressive strength of the unconfined concrete fc′ is determined from Figure 8, which in turn determines the strength of the core concrete.Figure 8
Curves showing the relationship between the strength ratio and effective confining stress ratio of confined concrete with a rectangular cross section.(2) A custom deformation-controlled plastic hinge is created, and its length is specified. The length of the plastic hinge influences, to some extent, the accuracy and convergence of the calculation results and thus needs to be carefully selected. At present, there is no simple method for selecting the length of the plastic hinge; however, it can be determined based on test results.(3) The cross-section of the pier is divided into multiple fibre elements, and the parameters (area, coordinates, and material) of each fibre element are determined.The cross-section of specimen#1 is used as an example to illustrate the method for fibre cross-section division, as shown in Figure 9. The rebars are treated as independent fibres with the actual locations of the rebars as the fibre coordinates. The concrete fibres are divided into concrete cover fibres and core concrete fibres, with the area centroid of each fibre element as the fibre coordinates.Figure 9
Division of the fibre cross-section of specimen#1.Similarly, the fibre cross-section of specimen#2is divided using the aforementioned approach. Different from the case of specimen #1, the concrete cover of specimen #2 is divided into two layers of fibre elements, as shown in Figure 10.Figure 10
Division of the fibre cross-section of specimen #2.(4) The fibre plastic hinge is assigned to the potential plastic zone of the element. The plastic hinge can be specified at the midpoint of the plastic zone height.


### 3.2. Numerical Model of Specimens

#### 3.2.1. Cross-Sectional Parameters of Specimens

The cross-sectional design of specimen#1 is shown in Figure 2(a). The 12-mm HRB335 hot-rolled ribbed steel bars are used as the longitudinal reinforcement, and 8-mm and 6-mm hot-rolled plain steel bars are used as stirrups and tie bars, respectively, with a 15-mm concrete cover of the stirrups. The cross-sectional design of specimen #2 is shown in Figure 2(b). The 20-mm HRB400 hot-rolled ribbed steel bars are used as the longitudinal reinforcement, and 8-mm and 6-mm hot-rolled plain bars are used as the stirrups and tie bars, respectively, with a 39-mm concrete cover of the stirrups. The longitudinal reinforcement passes through the joint and is connected by a grouted sleeve.


#### 3.2.2. Determination of the Height of the Potential Plastic Zone of the Specimens

According to article 8.1.1 of the Guidelines for Seismic Design of Highway Bridges, the length of the potential plastic zone of the pier columnLpr satisfies Lpr>h or hm, where h is the height of the cross-section and hm is the range on the pier column with a bending moment exceeding 80% of the maximum bending moment.Therefore,Lpr>0.53mm, Lpr>hm=0.2×3.4=0.68m.Based on the specimen damage states and the collected rebar strains during the test, the plastic zones of the specimens are determined as follows:(1)
According to the test observations, the height of the failure zone of specimen#1 was approximately 1/2 H, that is, 265 mm. The measured rebar strains showed that the rebars at a height of 425 mm had just yielded. Therefore, the height of the potential plastic hinge zone was greater than 0.425 m.(2)
According to the test observations, the height of the failure zone of specimen#2 was approximately 15 cm. The measured rebar strains indicated that, at a height of approximately 550 mm, some rebars had just theoretically yielded and some others were about to yield. Therefore, the height of the potential plastic zone was greater than 0.55 m.In summary, the height of the potential plastic zone of the specimen model is set to 1 m in the numerical simulation.


#### 3.2.3. Specimen Meshing

To meet the expected accuracy requirements, the model should be discretized with a minimum of eight elements. For specimens#1 and #2, the loading centre is 3.4 m aforementioned the bottom of the column, and the height of the potential plastic region is 1 m. Outside the potential plastic region, each specimen is discretized into 12 elastic elements.To investigate the influence of the plastic hinge length in the potential plastic hinge region, different numbers of plastic hinges are considered in the model, with each plastic hinge simulated by a single element. Three relative plastic hinge lengths (0.1, 0.5, and 1) are studied. The schematic diagram of the SAP2000 model (Figure11) of the specimen is as follows:Figure 11
Finite-element analysis model of the specimen.


### 3.3. Results and Discussion of the Cyclic Loading Analysis

The calculation results of the cyclic loading of specimens#1 and #2 are analysed as followed:The calculation results when the plastic hinge in the potential plastic hinge region has a relative length of 1.0 are shown in Figure12.Figure 12
Hysteresis curves of specimens#1 and #2 ①.The seismic performance of specimens#1 and #2 is calculated according to the parameter indices in Section 3, as shown in Tables 3 and 4.Table 3
Parameter indices of specimen#1 ①.
LevelDR (%)RDIξeq (%)nkI0.44———II0.740.045.40.75III2.41.122.40.28IV3.53.528.40.19V4.75.029.30.14Table 4
Parameter indices of specimen#2 ①.
LevelDR (%)RDIξeq (%)nkI0.44———II0.590.040.30.93III2.40.9512.40.30IV3.51.7122.90.14V4.72.7315.10.16The parameter indices of specimens#1 and #2 are found to differ significantly.As the grouted sleeve cannot be simulated in SAP2000, the difference between the two is naturally regarded as the result of different rebar arrangements. The two specimens have similar stirrup ratios and longitudinal reinforcement ratios; the difference is that specimen#1 has a larger number of small-diameter longitudinal rebars, while specimen #2 has a small number of large-diameter longitudinal rebars. The results show that with the same reinforcement ratio and stirrup ratio, the smaller the diameter of the longitudinal rebars is, the higher the energy dissipation capacity of the pier, but the larger the residual deformation. Comparison of the test results and numerical simulation results of specimen #1 and #2 finds that SAP2000 is capable of satisfactorily simulating the mechanical behaviour of the specimen before it reaches the peak strength while after the peak strength, the SAP2000 simulation results differ substantially from the test results. First, SAP2000 fails to simulate the fatigue damage to longitudinal reinforcement after multiple cyclic loadings; as a result, during the loading process, the longitudinal rebars are always in the hardening stage, and the response of the base increases continuously. Therefore, SAP2000 does not have the capability of simulating the strength degradation behaviour of the specimen. Second, due to the presence of grouted sleeve, the longitudinal rebars of specimen #2 are not continuous and hence the specimen might experience slight relative slip during the test, thereby causing the larger base response obtained by SAP2000. Finally, only one element is used in the potential plastic hinge region, and hence, there may not be sufficient calculation accuracy, resulting in a larger error in the calculated peak base response.The calculation results when the plastic hinge in the potential plastic hinge region has a relative length of 0.5 are shown in Figure13.Figure 13
Hysteresis curves of specimens#1 and #2 ②.The calculation results of the parameter indices are shown in Tables5 and 6.Table 5
Parameter indices of specimen#1 ②.
LevelDR (%)RDIξeq (%)nkI0.44———II0.740.043.20.71III2.41.822.30.26IV3.53.425.60.18V4.74.928.40.14Table 6
Parameter indices of specimen#2 ②.
LevelDrift (%)RDIξeq (%)nkI0.44———II0.590.040.420.91III2.40.9513.10.30IV3.51.8616.00.19V4.72.7616.60.14Comparing the parameter indices of the two shows that the smaller the diameter of the rebars is, the higher the energy dissipation capacity and the larger the residual deformation.The results of specimens#1 and #2 do not change much compared with those obtained when the plastic hinge has a length of 1 in the potential plastic hinge region, indicating that given the division of the fibre cross-section, merely changing the plastic hinge length does not necessarily have a large influence on the results.The hysteresis curves of the two specimens when the plastic hinge in the potential plastic hinge region has a relative length of 0.1 are shown in Figure14.Figure 14
Hysteresis curves of specimens#1 and #2 ③.The calculation results of the parameter indices are shown in Tables7 and 8.Table 7
Parameter indices of specimen#1 ③.
LevelDR (%)RDIξeq (%)nkI0.44———II0.740.043.20.71III2.42.123.50.27IV3.53.627.60.19V4.75.129.90.14Table 8
Parameter indices of specimen#2 ③.
LevelDR (%)RDIξeq (%)nkI0.44———II0.590.020.560.91III2.40.9613.60.29IV3.51.8616.40.18V4.7———Comparison of the hysteresis curves of specimen#1 in three different cases finds that the plastic hinge length has almost no influence on the hysteresis curve if the fibre cross-section is divided reasonably.The results of specimen#2 in the three cases differ, and nonconvergence may even occur, possibly because the fibre cross-section division method used for specimen #1 is unreasonable when used for specimen #2. For example, the rebars in specimen #2 have a larger diameter than those in specimen #1, while the concrete fibre areas of the two specimens are similar to each other. As a result, the stiffness matrix becomes singular for the calculation under a certain working condition, which may cause the results to not converge. This also further proves that the division of the fibre cross-section has an enormous influence on the convergence of the results when analysing the seismic resistance of bridge piers with fibre plastic hinges.The stresses of the two models under the last loading condition are analysed, and the stress results of the two models are shown in Figure15. For example, when the fibre plastic hinge length is 1, it can be shown from the figure that the maximum principal stress of concrete is concentrated near the bottom of the pier, which corresponds to the cracks of the specimens in the previous test being concentrated at the bottom of the pier. Meanwhile, it can be seen that the final principal stress of specimen 1# is nearly 8 MPa larger than that of specimen 2#, indicating that although the stress distribution of the two specimens is roughly the same, the size of the final concrete failure area is completely different. As the steel bar used in specimen 2# is larger in diameter and less in quantity, its energy dissipation capacity is lower, so the stress of concrete in the final loading condition is also smaller.Figure 15
Stress of specimen1# and 2# under the final loading condition; (a) specimen 1# and (b) specimen 2#.
(a)(b)


## 3.1. Definition of Fibre Plastic Hinges

The behaviour of the specimen before and after yielding can be simulated using discrete plastic hinges, which are generally defined as axial hinges for trusses, flexural hinges, and shear hinges in the principal direction for beams and PMM-related hinges for columns. The types of plastic hinges provided by SAP2000 include uncoupled hinges, interaction PMM hinges, and fibre PMM hinges. In this study, the fibre PMM hinge is used to simulate the nonlinear mechanical behaviour of the specimen. The fibre PMM hinge is defined and specified as follows [23, 24]:(1) The stress-strain relationship of the material is defined, including the constitutive relations of the concrete cover, the core concrete, and the rebar. To ensure that the numerical simulation results match the test results as much as possible, the measured values of the yield stress, elastic modulus, ultimate stress, and ultimate strain in the stress-strain relationship of the material are used.There are roughly three types of constitutive relationship models for rebars: the ideal elastic-plastic model, the elastic-hardening model, and the elastic-plastic hardening model. In particular, the elastic-plastic hardening model can be divided into linear hardening and parabolic hardening types considering different hardening segments, as shown in Figure6.Figure 6
Calculation models for the stress-strain relationships of rebar.The curvilinear elastic-plastic hardening model is used in this study as the calculation model for the stress-strain relationship of the rebar, as described in the following equation:(1)fs=Es⋅ε,ε<εy,fy,ε<εsh,fu−fu−fy⋅εsu−εεsu−εsh2,ε<εsu,where ε is the rebar strain; fy is the rebar stress; Es is the elastic modulus of the rebar; εy is the yield strain of the rebar; εsh is the initial strain of the hardening segment of the rebar; and εsu is the ultimate strain of the rebar.The concrete models are divided into the concrete cover model and the core concrete model, and the Mander model is used to describe the uniaxial stress-strain relationship of the concrete [25, 26], as shown in Figure 7.Figure 7
Concrete stress-strain relationship.The stress-strain relationship of the core concrete based on the Mander model is expressed in equations (2)–(11).(2)fc=0,ε>0,ε⋅Ec,ε<0,fcc′⋅x⋅rr−1+xr,ε<εcu,(3)x=εεcc,(4)r=EcEc−Esec,(5)Esec=fcc′εcc,(6)εcc=0.002×1+5⋅fcc′fc′−1,(7)fcc′=fc′2.2541+7.94fl′fc′−2fl′fc′−1.254,(8)εcu=0.004+1.4ρsfyhεsufcc′,where ε is the concrete strain; fc is the concrete stress; Ec is the elastic modulus of concrete; εcu is the ultimate compressive strain of concrete; εcc is the peak strain of concrete; fc′ is the standard value of the 28-day compressive strength of unconfined concrete; fcc′ is the peak stress of the core concrete; ρs is the volumetric reinforcement ratio of the stirrups; fyh is the yield stress of the stirrups; and εsu is the ultimate strain of the stirrups, which is set to 0.09 in this study.fl′ is the effective transverse confining stress of the concrete. For the concrete cover, fl′=0, fcc′=fc′. For the concrete in the core region, fl′ is calculated using the following equation:(9)fl′=12keρsfyh,where ke is the effective transverse confinement coefficient of concrete, calculated as the ratio of the area of the effective confined concrete (taken at the middle section between adjacent stirrups, where the confined area is the smallest) to the area of the core region of the confined concrete.For the rectangular cross-section, the stirrup ratioρs may differ in the x and y directions. Hence, the effective transverse confining stresses of the concrete in the x and y directions, respectively, are as follows:(10)flx′=keρxfyh,(11)fly′=keρyfyh,Then, the ratio of the peak stress of the core concretefcc′ to the 28-day compressive strength of the unconfined concrete fc′ is determined from Figure 8, which in turn determines the strength of the core concrete.Figure 8
Curves showing the relationship between the strength ratio and effective confining stress ratio of confined concrete with a rectangular cross section.(2) A custom deformation-controlled plastic hinge is created, and its length is specified. The length of the plastic hinge influences, to some extent, the accuracy and convergence of the calculation results and thus needs to be carefully selected. At present, there is no simple method for selecting the length of the plastic hinge; however, it can be determined based on test results.(3) The cross-section of the pier is divided into multiple fibre elements, and the parameters (area, coordinates, and material) of each fibre element are determined.The cross-section of specimen#1 is used as an example to illustrate the method for fibre cross-section division, as shown in Figure 9. The rebars are treated as independent fibres with the actual locations of the rebars as the fibre coordinates. The concrete fibres are divided into concrete cover fibres and core concrete fibres, with the area centroid of each fibre element as the fibre coordinates.Figure 9
Division of the fibre cross-section of specimen#1.Similarly, the fibre cross-section of specimen#2is divided using the aforementioned approach. Different from the case of specimen #1, the concrete cover of specimen #2 is divided into two layers of fibre elements, as shown in Figure 10.Figure 10
Division of the fibre cross-section of specimen #2.(4) The fibre plastic hinge is assigned to the potential plastic zone of the element. The plastic hinge can be specified at the midpoint of the plastic zone height.


## 3.2. Numerical Model of Specimens

### 3.2.1. Cross-Sectional Parameters of Specimens

The cross-sectional design of specimen#1 is shown in Figure 2(a). The 12-mm HRB335 hot-rolled ribbed steel bars are used as the longitudinal reinforcement, and 8-mm and 6-mm hot-rolled plain steel bars are used as stirrups and tie bars, respectively, with a 15-mm concrete cover of the stirrups. The cross-sectional design of specimen #2 is shown in Figure 2(b). The 20-mm HRB400 hot-rolled ribbed steel bars are used as the longitudinal reinforcement, and 8-mm and 6-mm hot-rolled plain bars are used as the stirrups and tie bars, respectively, with a 39-mm concrete cover of the stirrups. The longitudinal reinforcement passes through the joint and is connected by a grouted sleeve.


### 3.2.2. Determination of the Height of the Potential Plastic Zone of the Specimens

According to article 8.1.1 of the Guidelines for Seismic Design of Highway Bridges, the length of the potential plastic zone of the pier columnLpr satisfies Lpr>h or hm, where h is the height of the cross-section and hm is the range on the pier column with a bending moment exceeding 80% of the maximum bending moment.Therefore,Lpr>0.53mm, Lpr>hm=0.2×3.4=0.68m.Based on the specimen damage states and the collected rebar strains during the test, the plastic zones of the specimens are determined as follows:(1)
According to the test observations, the height of the failure zone of specimen#1 was approximately 1/2 H, that is, 265 mm. The measured rebar strains showed that the rebars at a height of 425 mm had just yielded. Therefore, the height of the potential plastic hinge zone was greater than 0.425 m.(2)
According to the test observations, the height of the failure zone of specimen#2 was approximately 15 cm. The measured rebar strains indicated that, at a height of approximately 550 mm, some rebars had just theoretically yielded and some others were about to yield. Therefore, the height of the potential plastic zone was greater than 0.55 m.In summary, the height of the potential plastic zone of the specimen model is set to 1 m in the numerical simulation.


### 3.2.3. Specimen Meshing

To meet the expected accuracy requirements, the model should be discretized with a minimum of eight elements. For specimens#1 and #2, the loading centre is 3.4 m aforementioned the bottom of the column, and the height of the potential plastic region is 1 m. Outside the potential plastic region, each specimen is discretized into 12 elastic elements.To investigate the influence of the plastic hinge length in the potential plastic hinge region, different numbers of plastic hinges are considered in the model, with each plastic hinge simulated by a single element. Three relative plastic hinge lengths (0.1, 0.5, and 1) are studied. The schematic diagram of the SAP2000 model (Figure11) of the specimen is as follows:Figure 11
Finite-element analysis model of the specimen.


## 3.2.1. Cross-Sectional Parameters of Specimens

The cross-sectional design of specimen#1 is shown in Figure 2(a). The 12-mm HRB335 hot-rolled ribbed steel bars are used as the longitudinal reinforcement, and 8-mm and 6-mm hot-rolled plain steel bars are used as stirrups and tie bars, respectively, with a 15-mm concrete cover of the stirrups. The cross-sectional design of specimen #2 is shown in Figure 2(b). The 20-mm HRB400 hot-rolled ribbed steel bars are used as the longitudinal reinforcement, and 8-mm and 6-mm hot-rolled plain bars are used as the stirrups and tie bars, respectively, with a 39-mm concrete cover of the stirrups. The longitudinal reinforcement passes through the joint and is connected by a grouted sleeve.


## 3.2.2. Determination of the Height of the Potential Plastic Zone of the Specimens

According to article 8.1.1 of the Guidelines for Seismic Design of Highway Bridges, the length of the potential plastic zone of the pier columnLpr satisfies Lpr>h or hm, where h is the height of the cross-section and hm is the range on the pier column with a bending moment exceeding 80% of the maximum bending moment.Therefore,Lpr>0.53mm, Lpr>hm=0.2×3.4=0.68m.Based on the specimen damage states and the collected rebar strains during the test, the plastic zones of the specimens are determined as follows:(1)
According to the test observations, the height of the failure zone of specimen#1 was approximately 1/2 H, that is, 265 mm. The measured rebar strains showed that the rebars at a height of 425 mm had just yielded. Therefore, the height of the potential plastic hinge zone was greater than 0.425 m.(2)
According to the test observations, the height of the failure zone of specimen#2 was approximately 15 cm. The measured rebar strains indicated that, at a height of approximately 550 mm, some rebars had just theoretically yielded and some others were about to yield. Therefore, the height of the potential plastic zone was greater than 0.55 m.In summary, the height of the potential plastic zone of the specimen model is set to 1 m in the numerical simulation.


## 3.2.3. Specimen Meshing

To meet the expected accuracy requirements, the model should be discretized with a minimum of eight elements. For specimens#1 and #2, the loading centre is 3.4 m aforementioned the bottom of the column, and the height of the potential plastic region is 1 m. Outside the potential plastic region, each specimen is discretized into 12 elastic elements.To investigate the influence of the plastic hinge length in the potential plastic hinge region, different numbers of plastic hinges are considered in the model, with each plastic hinge simulated by a single element. Three relative plastic hinge lengths (0.1, 0.5, and 1) are studied. The schematic diagram of the SAP2000 model (Figure11) of the specimen is as follows:Figure 11
Finite-element analysis model of the specimen.


## 3.3. Results and Discussion of the Cyclic Loading Analysis

The calculation results of the cyclic loading of specimens#1 and #2 are analysed as followed:The calculation results when the plastic hinge in the potential plastic hinge region has a relative length of 1.0 are shown in Figure12.Figure 12
Hysteresis curves of specimens#1 and #2 ①.The seismic performance of specimens#1 and #2 is calculated according to the parameter indices in Section 3, as shown in Tables 3 and 4.Table 3
Parameter indices of specimen#1 ①.
LevelDR (%)RDIξeq (%)nkI0.44———II0.740.045.40.75III2.41.122.40.28IV3.53.528.40.19V4.75.029.30.14Table 4
Parameter indices of specimen#2 ①.
LevelDR (%)RDIξeq (%)nkI0.44———II0.590.040.30.93III2.40.9512.40.30IV3.51.7122.90.14V4.72.7315.10.16The parameter indices of specimens#1 and #2 are found to differ significantly.As the grouted sleeve cannot be simulated in SAP2000, the difference between the two is naturally regarded as the result of different rebar arrangements. The two specimens have similar stirrup ratios and longitudinal reinforcement ratios; the difference is that specimen#1 has a larger number of small-diameter longitudinal rebars, while specimen #2 has a small number of large-diameter longitudinal rebars. The results show that with the same reinforcement ratio and stirrup ratio, the smaller the diameter of the longitudinal rebars is, the higher the energy dissipation capacity of the pier, but the larger the residual deformation. Comparison of the test results and numerical simulation results of specimen #1 and #2 finds that SAP2000 is capable of satisfactorily simulating the mechanical behaviour of the specimen before it reaches the peak strength while after the peak strength, the SAP2000 simulation results differ substantially from the test results. First, SAP2000 fails to simulate the fatigue damage to longitudinal reinforcement after multiple cyclic loadings; as a result, during the loading process, the longitudinal rebars are always in the hardening stage, and the response of the base increases continuously. Therefore, SAP2000 does not have the capability of simulating the strength degradation behaviour of the specimen. Second, due to the presence of grouted sleeve, the longitudinal rebars of specimen #2 are not continuous and hence the specimen might experience slight relative slip during the test, thereby causing the larger base response obtained by SAP2000. Finally, only one element is used in the potential plastic hinge region, and hence, there may not be sufficient calculation accuracy, resulting in a larger error in the calculated peak base response.The calculation results when the plastic hinge in the potential plastic hinge region has a relative length of 0.5 are shown in Figure13.Figure 13
Hysteresis curves of specimens#1 and #2 ②.The calculation results of the parameter indices are shown in Tables5 and 6.Table 5
Parameter indices of specimen#1 ②.
LevelDR (%)RDIξeq (%)nkI0.44———II0.740.043.20.71III2.41.822.30.26IV3.53.425.60.18V4.74.928.40.14Table 6
Parameter indices of specimen#2 ②.
LevelDrift (%)RDIξeq (%)nkI0.44———II0.590.040.420.91III2.40.9513.10.30IV3.51.8616.00.19V4.72.7616.60.14Comparing the parameter indices of the two shows that the smaller the diameter of the rebars is, the higher the energy dissipation capacity and the larger the residual deformation.The results of specimens#1 and #2 do not change much compared with those obtained when the plastic hinge has a length of 1 in the potential plastic hinge region, indicating that given the division of the fibre cross-section, merely changing the plastic hinge length does not necessarily have a large influence on the results.The hysteresis curves of the two specimens when the plastic hinge in the potential plastic hinge region has a relative length of 0.1 are shown in Figure14.Figure 14
Hysteresis curves of specimens#1 and #2 ③.The calculation results of the parameter indices are shown in Tables7 and 8.Table 7
Parameter indices of specimen#1 ③.
LevelDR (%)RDIξeq (%)nkI0.44———II0.740.043.20.71III2.42.123.50.27IV3.53.627.60.19V4.75.129.90.14Table 8
Parameter indices of specimen#2 ③.
LevelDR (%)RDIξeq (%)nkI0.44———II0.590.020.560.91III2.40.9613.60.29IV3.51.8616.40.18V4.7———Comparison of the hysteresis curves of specimen#1 in three different cases finds that the plastic hinge length has almost no influence on the hysteresis curve if the fibre cross-section is divided reasonably.The results of specimen#2 in the three cases differ, and nonconvergence may even occur, possibly because the fibre cross-section division method used for specimen #1 is unreasonable when used for specimen #2. For example, the rebars in specimen #2 have a larger diameter than those in specimen #1, while the concrete fibre areas of the two specimens are similar to each other. As a result, the stiffness matrix becomes singular for the calculation under a certain working condition, which may cause the results to not converge. This also further proves that the division of the fibre cross-section has an enormous influence on the convergence of the results when analysing the seismic resistance of bridge piers with fibre plastic hinges.The stresses of the two models under the last loading condition are analysed, and the stress results of the two models are shown in Figure15. For example, when the fibre plastic hinge length is 1, it can be shown from the figure that the maximum principal stress of concrete is concentrated near the bottom of the pier, which corresponds to the cracks of the specimens in the previous test being concentrated at the bottom of the pier. Meanwhile, it can be seen that the final principal stress of specimen 1# is nearly 8 MPa larger than that of specimen 2#, indicating that although the stress distribution of the two specimens is roughly the same, the size of the final concrete failure area is completely different. As the steel bar used in specimen 2# is larger in diameter and less in quantity, its energy dissipation capacity is lower, so the stress of concrete in the final loading condition is also smaller.Figure 15
Stress of specimen1# and 2# under the final loading condition; (a) specimen 1# and (b) specimen 2#.
(a)(b)


## 4. Conclusion

This study focuses on the seismic performance of precast piers. The test results of the seismic behaviour of precast piers are compared with numerical simulation results by SAP2000, and the following conclusions are reached:(1)
Both the test results and numerical simulation results demonstrate that precast piers connected by grouted sleeves indeed have seismic performance similar to that of cast-in-place components under reasonable constructional measures.(2)
SAP2000 is unable to quantitatively simulate the properties of the grouted sleeve of the precast component, and hence, the obtained results actually reflect the influence of the difference in rebar arrangement. However, the results show that the use of a large number of small-diameter steel rebars in the concrete cross-section improves the energy dissipation capacity of the pier but also leads to large residual deformation due to the susceptibility of small-diameter steel rebars to yielding.(3)
SAP2000 can simulate the mechanical properties of steel rebars before yielding well but cannot simulate those of rebars after yielding or under fatigue, resulting in hysteresis curves slightly different from those derived from the test.(4)
When the fibre plastic hinge is used for finite-element analysis, the division of the fibre cross-section has a very large impact on the results. If the fibre cross-section is reasonably divided, the length of the plastic hinge has little influence on the results; otherwise, too densely distributed plastic hinges may lead to nonconvergence of the results in a later working condition.(5)
Final-element method is used in this study to simulate the seismic performance of precast piers. However, the influences of various aspects such as the selection of the plastic hinge length and the method for fibre cross-section division on the convergence of results are not investigated, awaiting much research effort in the future.



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*Source: 6337791-2022-05-06.xml*