#### Abstract

The seismic performance of reinforced concrete members under earthquake excitation is different from that of whole structures; collapse mechanism may occur because of severe damage to individual members, even if the structural damage is not significant. Therefore, the potential seismic damage of each member should be investigated specifically apart from that of overall structure. In this study, a global damage model based on component classification is proposed to analyze the structural damage evolution rule and failure mechanism; then, the computed damage is compared with the experimental phenomena of three 1/3-scale models of three-storey, three-bay reinforced concrete frame structures under low-reversed cyclic loading. In addition, a probabilistic approach is finally adopted to quantify the seismic performance of RC frame structures based on the proposed global damage model. Results indicate that the structures with lower vertical axial force and beam-to-column linear stiffness ratio still maintain a certain load-bearing capacity even when the interstorey drift angle exceeds the elastoplastic limit value and the cumulative damage of structures is mainly concentrated on the beam ends and column bottoms of the first floor at final collapse. Moreover, the structural failure probability at different performance levels would increase significantly if reinforced concrete frame structures suffer ground motions higher than the design fortification intensity, even up to eight times.

#### 1. Introduction

Structural collapse refers to the loss of capacity to resist gravity loads and dynamic instability in a side-way mode when subjected to seismic excitation, which is usually caused by the deterioration in stiffness and strength of components and -Δ effects. Protection has been the major target of seismic design as collapse being the main reason for casualties and property losses; thus, reasonable provisions and constructional measures have been given in current building codes and standards to alleviate seismic damage and prevent structural collapse, but frame structures still suffer severe damage even designed according to modern building codes strictly [1]. The aforementioned problems have been attributed to the lack of perception in structural damage evolution rule and failure mechanism and then defining the collapse as an acceptable storey drift or a limit value of individual component deformation, but this assumption could not reflect the fact that the capacity of global structure to resist deformation is significantly greater than that of individual members. The main goal of this study is to present a methodology for evaluating the collapse state of deteriorating reinforced concrete frame structures and then studying the failure mechanism.

Researches on collapse assessment have been developed on several respects during last decades. The experiment on the seismic performance of frame structures is the most direct method to study the failure rule, and a large number of experiments have been carried out [2–10]. For instance, Zaharia et al. tested two full-scale 3-storey RC structures with flat slab under pseudodynamic loading and concluded that the deformations mainly concentrated in the slab-column connections and column bases; Bousias et al. tested two 2-storey RC structures with one bay in each direction under earthquake loading. The results indicated that structural damage modified its fundamental frequency; Yavari et al. carried out a shaking table test on a two-storey, two-bay frame to evaluate the effect of axial load and joint confinement reinforcement on the seismic performance of RC frames. The previous researches mostly focused on the global performance of frame structures; although they may not be significantly affected by some alterations in earthquake characteristics, the performance of individual components may change dramatically. Hence, it is necessary to specifically assess the seismic performance of columns, beams, and beam-to-column joints in addition to the global response of structures. This could be achieved by carrying out a multilevel damage assessment of each individual component.

Experimental studies also indicate that the hysteretic behavior of the structure is mainly dependent on the parameters that affect the characteristics of deformation and energy dissipation. Thus, the development of smooth hysteretic degrading model [11, 12] was performed to replace the bilinear elastic-plastic hysteresis model, which was widely used for their advantage of simplicity. Analytical investigation based on numerical simulation is an effective supplement to assess structural collapse [13–16]. For example, Haselton et al. studied the influences of axial compression ratio, strong column weak-beam ratio, and -Δ effect on the collapse risk of special moment-frame (SMF) buildings based on nonlinear dynamic analyses. The results showed that the plastic deformation capacity of columns and -Δ effect were the most important factors affecting the interstorey displacement. Shi et al. evaluated the collapse-resistant capacities and safety margins against collapse of multistorey reinforced concrete frames with different seismic fortification levels based on incremental dynamic analysis (IDA). The influences of axial compression ratio and failure mode on the collapse-resistant capacity of RC frames were discussed by taking the loss of vertical bearing capacity as the evaluation index.

Damage indices could quantify the damage degree of structures and provide a theoretical basis for postearthquake repairing schemes. At present, it is generally agreed that global damage indices defined as weighted average component level indices referring to the damage degree of a single member are more precise than those defined in terms of global property variation before and after earthquake [17–19], which could provide reasonable evaluation of the overall damage level at the premise that the damage distributes evenly. However, the global damage models defined as weighted individual component indices in previous researches tended to give more weight to the members at lower stories [20–24] but failed to take the negative influence of different types of damaged components on the structural seismic performance into account. The effect of localized column failure on structural collapse is more obvious than that of beam failure or other components; thus, the definition of weighted coefficients should consider the differences particularly.

In this study, three 1/3-scale models of three-storey, three-bay reinforced concrete frame structures are tested under low-reversed cyclic loading, which could synthetically reflect the force characteristics of components in sidespan and midspan at the bottom floor, middle floor, and top floor. Based on experimental results, the damage distribution of individual components is investigated and the damage quantitative deduction of overall structures is accomplished through a proposed global damage model. This model considers the different influences of component types on the deterioration of overall seismic performance and divides structural components into different types, expressed as component classification. In addition, a probabilistic assessment relied on nonlinear dynamic simulation is conducted to exhibit the further application of the global damage model in predicting the failure probability of reinforced concrete frame structures.

#### 2. Specimens and Test Setup

##### 2.1. Specimen Design

The prototype structure was a typical RC moment-resisting frame structure located in an earthquake-prone region with seismic fortification intensity 8, site soil class II, and design group 1. The criteria of strong-column weak-beam were adopted in the structural design according to the Chinese Code for Seismic Design of Buildings [25], to ensure that the test frames would fail in the flexural mode under combined lateral displacement and axial compression load. Thus, the cross-sectional areas of frame columns were designed to a larger size (600 × 600 mm), while those of beams were 300 × 600 mm. The component sizes in the test specimens were defined according to the geometric reduced scale of prototype ones, as 200 × 200 mm for columns and 100 × 200 mm for beams, respectively.

To investigate the ultimate elastoplastic interstorey drift angle for preventing building collapse under strong earthquake, the test specimens with excellent ductile deformation were designed through reasonable construction measures. The specimens were optimized by properly raising the longitudinal reinforcement ratio of frame columns, in order to facilitate the formation of structural “beam-hinge” failure pattern under low-reversed cyclic loading. The sum of ultimate flexural capacity of columns framing into joints should be larger than that of beams in the same plane, and the overstrength factors of exterior joints at the first floor and second floor were 3.23 and 2.95, respectively, while those of interior joints were 2.69 and 2.53, meeting the conclusions in previous researches that the strong-column weak-beam ratio ranging from 2.0 to 3.0 is beneficial to the formation of the “beam-hinge” mechanism [14, 26–30]. The overstrength factors of joints at the third floor were ignored for the reason that the interference effect of the loading device on the strength of beams was inevitable.

For the model design of overall structures, the specimens are mostly scaled models due to the limitations of testing equipment and manufacturing cost. Hence, the lower three layers of a single plane frame in the central axis were selected as the prototype substructure, and three 1/3-scale models of three-storey, three-bay RC frames were constructed. The scaled models could accurately reflect the seismic behavior of prototypes, such as the failure pattern, the appearance order of plastic hinges, the ultimate bearing capacity, and the ultimate deformation capacity, with the method of keeping the reinforcement ratio and material strength constant before and after scaling. The similarity relation of mechanical behavior during the cracking process between models and prototypes was difficult to fulfill because the influence factors such as steel diameter, reinforcement ratio, and cover thickness as well as relevant variables could not be scaled completely according to geometric similarity [31], but this shortcoming could be improved through the method adopted above. The details of Specimen KJ-1 are presented as an illustration in Figure 1.

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In order to study the influence of axial compression ratio on the ultimate deformability and seismic performance of structures, the axial compression loads applied on the top of exterior columns and interior columns in Specimen KJ-1 were 262 kN and 330 kN, respectively, determined based on the experimental axial compression ratio converted from the prototype structural design axial compression ratio 0.45 with a ratio of 1 : 1.68 [32], while those in Specimen KJ-2 and Specimen KJ-3 were 421 kN and 566 kN, which were corresponding to the limiting value of axial compression ratio for Aseismic Grade II. To study the influence of beam-to-column stiffness ratio, the height of the first storey in Specimen KJ-1 and Specimen KJ-2 was 1.1 m for both, while that in Specimen KJ-3 was 1.5 m.

The commercial concrete and steel used for the test models were C40 and HRB400, respectively. According to the results of material sampling tests, little variation was found in mechanical properties among the specimens. The average compressive strength of 150 mm concrete cubes was measured as 30.5 MPa after curing at an ambient temperature for 28 days. 6 bars, 8 bars, and 10 bars were used as the longitudinal reinforcements in columns and beams; the actual yield and ultimate strength of longitudinal reinforcement with a diameter of 6 mm were 471.2 MPa and 606.2 MPa, respectively, while those with the diameter of 8 mm were 548.9 MPa and 640.2 MPa and those with the diameter of 10 mm were 539.2 MPa and 593.7 MPa. 4 low-carbon steel wires were used as the stirrups in both columns and beams, and the actual ultimate strength of steel wires was 678.6 MPa.

##### 2.2. Test Setup and Loading Procedure

The model frames were tested under constant vertical loads and lateral low-reversed cyclic loads in the Earthquake Engineering Laboratory of Chang’an University. The test setup and instrumentation are illustrated in Figure 2. Two rigid steel beams were used to brace the specimens in the out-of-plane direction, with frictionless rollers at the top beam level to allow the free in-plane motion of frames. The lateral cyclic loads were applied to the top beam by using a horizontal MTS electrohydraulic servosystem; the force was transmitted by using four high-strength threaded rods attaching the actuator and connecting two sides of each column with sleeves and fixtures. The actuator was arranged to move along the guild rail freely so that the P-Δ effect could be considered with great precision. Constant vertical loads were applied to the top of steel distributive girders by two 300 t electrohydraulic jacks with small sliding plates. The axial loads were obtained from 1/1.68 design axial compression ratio, measured strength of concrete material, and scaled member sizes and transferred to each column by using two 100 t manual-hydraulic jacks (located in the interior columns) and two 50 t manual-hydraulic jacks (located in the exterior columns) with pressure sensors for each to facilitate real-time monitoring of vertical loads. The vertical force was compensated by the manual-hydraulic jacks to ensure the loads remained constant during the whole loading process. All the hydraulic jacks could move with the structural deformation during the loading history.

To examine the allowable values of interstorey drift angle corresponding to different performance levels specified in standards (1/550 for operational level, 1/250 for slight damage level, 1/120 for medium damage level, 1/50 for serious damage level, and 1/25 for collapse level, respectively), through the comparison with the experimental failure phenomena, this paper makes some improvements to the loading protocol based on the basic principles in specification for earthquake-resistant buildings [31]: the lateral displacement varied from 0 mm to 18 mm, which is the structural yielding displacement obtained from the finite element simulation ahead of conducting the experiment, with an interval of 3 mm (the corresponding drift ratio is 1/1100) and one loading cycle for each displacement amplitude, to catch the yielding feature points; then, some damage phenomenon occurred and the test specimens went into the plastic stage and, subsequently, three full loading cycles were applied for each displacement amplitude with an increment of 9 mm, which was the half of structural yielding displacement (1/2 × 18 mm = 9 mm), to approach the allowable values for the five performance levels as far as possible. Until the roof drift angle reached 1/25, the test frame structures were supposed to collapse. In this experiment, the roof drift angle was defined as *D* = Δ/*H*, where *H* was the total height of structure and Δ was the lateral roof displacement. The lateral loading, as presented in Figure 3, was applied to the centerline of top-floor beams in the form of a displacement-control mode.

Two kinds of measurement methods were used in the experiment. One was the traditional data acquisition instrument such as resistance strain gauges and linear variable differential transformers (LVDTs) to obtain the steel strains, force, and displacement during the testing process, as illustrated in Figure 4. Strain gauges were installed on the longitudinal and transverse steel bars at the sections of 100 mm from component ends and transverse steel bars at the middle position of each beam-column joint to monitor the section strains. LVDTs were used to record the deformation of members and the displacement at each floor. Two wide-ranging LVDTs were placed at each floor level to measure the lateral displacement, and one LVDT was arranged at the basebeam level to monitor the potential sliding. LVDTs with a lower range were placed vertically at the ends of beams in each floor to obtain the beam-to-column relative rotation, and two LVDTs were placed diagonally to one joint to measure the joint shear response. Another measurement technique named digital image correlation (DIC), which is an emerging noncontact optical technique for measuring displacement and strain [33], was also used on the south side of test frames, as presented in Figure 5. All the concrete surfaces of test frames were made speckled pattern artificially with an approximate diameter of 4∼8 mm, and five high-resolution cameras were used to capture the undeformed image before loading and the deformed images at every loading stage. An open source software, Ncorr-V1.2 [34], was introduced to analyze the acquired digital images and obtain the local deformation of structural components.

#### 3. Damage Observation

Specimens were considered to fail when the roof drift angle reached 1/26.2, and then, the loading continued until severe damage occurred. For an accurate description, the parts of test frames are named as shown in Figure 6, where columns are assigned by the axis and the storey number and beams are assigned by the axis on both sides and the storey number. Based on the limit value of interstorey drift angle at different performance levels, the test frames were assumed to go through five periods, i.e., operational, slight damage, medium damage, serious damage, and collapse, respectively.

Minor flexural cracks first occurred at the beam ends, with a maximum width of 0.04 mm at the roof drift ratio of 0.09%. There were no visible cracks on columns and joints. As the roof drift ratio increased to 0.18%, hairlike horizontal cracks occurred at the bottom of first-storey and second-storey columns in Specimen KJ-1. The cracks at the beam ends of Specimen KJ-2 continued to increase and extended toward the midposition, but the number of cracks remained relatively low. In Specimen KJ-3, the cracks formed at the midspan of beams partly and increased in length and number, with the width within the range of 0.06∼0.12 mm (operational level).

For all the three specimens, the number and width of cracks at the beam ends increased substantially as the roof drift ratio rose to 0.27%. The length extended to 5∼10 cm and the width was 0.08∼0.24 mm, and a few penetrating cracks formed at the bottom of beam ends. New cracks appeared at the bottom of first-storey columns in Specimens KJ-2 and KJ-3, but no cracks occurred at the joints in the same cycle (slight damage level).

When the roof drift ratio reached 0.82%, the number of penetrating cracks at the beam ends of the three specimens increased drastically and those at the midspan developed with the width of 0.12∼0.44 mm. Concrete peeling initiated at the beam-column interface of the second storey in Specimens KJ-1 and KJ-2. Hairlike horizontal cracks aligning with the top of beams were detected at the joints, and a small amount of penetrating cracks were observed at the bottom of first-storey columns in Specimen KJ-2. The development of cracks at the beam ends of Specimen KJ-3 was lower than that of Specimen KJ-1 and KJ-2, but the cracks at the bottom of first-storey columns developed significantly with numerous penetrating cracks on both the east and west sides (medium damage level).

The concrete at the beam ends of the three specimens spalled to different degrees at the roof drift ratio of 1.91%. Minor concrete crushing occurred at the left side of Beam-AB2 in Specimens KJ-1 and KJ-2, which caused the exposure of longitudinal reinforcements. The damage degree of the column bottoms in Specimen KJ-2 was more severe than that of Specimen KJ-1, with the phenomena that massive penetrating cracks formed at the bottom of first-storey columns and concrete peeled at Column-A1 and Column-D1. The concrete at Beam-AB1, Beam-AB2, Column-B1, and Column-C1 of Specimen KJ-3 was crushed and peeled; meanwhile, horizontal cracks aligning with the top surface of beams appeared at Joint J-3 (serious damage level).

In Specimens KJ-1 and KJ-2, buckling of the naked longitudinal reinforcements at the left side of Beam-AB1 and Beam-AB2 occurred when the roof drift ratio reached 2.73% and massive concrete flaked away at the other beams, inducing the exposure of longitudinal reinforcements. Besides, Specimen KJ-2 showed a larger extent of concrete spalling at the bottom of first-storey columns, and the longitudinal reinforcements and stirrups inside could be observed clearly. The damage degree at the beam ends of Specimen KJ-3 was lighter than that of Specimens KJ-1 and KJ-2; although the longitudinal reinforcements were exposed, there was no buckling (Beam-CD2 right side, Beam-AB1 left side, and Beam-BC1 left side). The concrete at the bottom of Column-A1, Column-B1, and Column-C1 was crushed as severely as that of Specimen KJ-2. The naked longitudinal reinforcements at the beam ends (Beam-AB2 left side, Beam-AB1 left side, and Beam-BC1 left side) of Specimen KJ-1 fractured at the roof drift ratio of 3.00%. The longitudinal reinforcements and stirrups at the column bottoms were exposed without buckling, and the joints remained intact as the roof drift ratio rose almost as high as 4.09%. The test loading was terminated at this displacement amplitude to ensure the experimental safety of Specimen KJ-1. The bending degree of the buckling longitudinal reinforcements at the beam ends (Beam-CD2 right side, Beam-AB2 left side, and Beam-BC1 right side) of Specimens KJ-2 and KJ-3 increased as the displacement amplitude increased, but there was no fracture at the end of loading. The concrete at the bottom of Column-B1 and Column-C1 in Specimen KJ-2 was crushed to a large scale; the longitudinal reinforcements and stirrups with large deformation ruptured at the roof drift ratio of 3.61%. Specimen KJ-2 collapsed due to the severe loss of vertical bearing capacity. The damage degree of the column bottoms in Specimen KJ-3 was lighter than that in Specimen KJ-2, and the concrete at the bottom of Column-B1 and Column-C1 was crushed, and the longitudinal reinforcements and stirrups bended without rupturing. Loading was halted immediately owing to the sudden drop in the structural vertical bearing ability to ensure safety. Figures 7 and 8 illustrated the damage characteristics and the force distribution of the specimens at the end of loading, respectively (collapse level).

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In general, the experimental phenomena of the three test frames were almost identical at the small-loading stage (operational, slight damage, and medium damage), in spite of the difference in design parameters. Cracks at beam ends were observed obviously, while those at the column bottoms appeared laggingly; i.e., only slight horizontal cracks were detected in the joint areas. At the large-loading stage (Serious Damage and Collapse), all the three test frames exhibited different failure process and failure pattern. From the perspective of failure phenomena, the influence of cycle number increase on the structural damage was slightly greater than that of displacement amplitude increase. The force condition of columns and beams alternated between tension and compression under multiple positive and negative loads, especially in the position of beam-column junction and the connection of ground beam-column end at the first floor. The serious failure phenomena such as the bending and fracture of steel bars or massive peeling of concrete occurred due to uncoordinated deformation caused by the stiffness difference of members. While the P-Δ effect highlighted when the structures approached the collapse level, then the influence of displacement amplitude increase on the structural failure became more serious relatively. From the perspective of plastic hinge development, the test specimens presented the similar damage characteristics with actual seismic failure of frame structures: The plastic hinges in the columns formed mostly subsequent to those at the beam ends though the growth of column hinges was faster than that of the beam hinges. When the specimens approached the ultimate state of the collapse level, the plastic hinges at the bottom of first-storey columns fully developed and the -Δ effect highlighted at the same time and, subsequently, the structures collapsed due to the sudden loss of vertical bearing capacity, which arose from the massive concrete crushing at first-storey column bottoms. The failure characteristics indicated that the structures still possessed a certain vertical bearing capacity and were far from reaching the limit state of collapse; even the interstorey drift angle exceeded 1/50, which is specified as the elasto-plastic limit value in the Chinese Code for Design of Concrete Structures [35].

#### 4. Proposed Seismic Damage Model

##### 4.1. Component Level

Computational expressions taking internal force parameters or deformation parameters as variables are widely used in the evaluation of component damage; numerous damage models specific to the component-level have been established by both local and international scholars to reflect the influence of earthquake excitation on component failure [18, 36–45]. The Mehanny–Deierlein damage model is selected as the quantitative expression for component level in this article, owing to the easy acquisition of local deformation data gauged by the digital image correlation method. The effect of the loading path on component failure is taken into consideration in this model, accompanied with stable computational convergence [46]. The formula is shown as follows:where *θ*_{p∣currentPHC} is the inelastic component deformation referring to any half cycle whose amplitude exceeds that of previous cycles; *θ*_{p∣FHC,j} is the inelastic component deformation referring to all the subsequent cycles of smaller amplitude; *θ*_{pu} is the associated capacity under monotonic loading; and *α*, *β*, and *γ* are the calibration coefficients, and the values are *α* = 1, *β* = 1.5, *γ* = 6 for reinforced concrete members.

##### 4.2. Storey Level

Quantifying component damage is to evaluate the damage degree of overall structure eventually, so it is necessary to establish a combination mode with simple calculation for component damage. Storey damage is usually used as a transition to deliver component damage to structural collapse. In this paper, the components in RC frame structures are divided into two types and the corresponding storey-weighted coefficients are defined as the concept of damage indices [47]:where , are the storey damage indices for beam and column members in the *i*-th floor, respectively; *m* and *n* are the number of beam and column members in the *i*-th floor, respectively; and are the damage indices of individual beam and column members, respectively.

##### 4.3. Structure Level

The storey damage indices for different types of components should be summed up in order to evaluate the seismic performance of the overall structure. Structure-weighted coefficients, referring to the damage severity and the relative position of storeys, are defined to establish the relation between storey damage and structural damage:where *ζ*_{i} is the total weighted coefficient; *ζ*_{Di} is the weighted coefficient referring to the damage degree of the storey; *ζ*_{Fi} is the weighted coefficient referring to the relative position of the storey; and *N* is the entire floor of building structure.

Structural components are divided into three types in the Chinese Technical Specification for Concrete Structure of Tall Buildings [48], namely, “key components,” “normal vertical components,” and “energy dissipation components,” to highlight the importance of different component types to structural stability. According to the detailed description in code, the “key components” and “normal vertical components” are referred to as “vertical members” in this article and the global damage model is defined as follows:where *D*_{total} is the global damage indices for overall structure; *λ*_{h} and are the importance coefficients for “vertical members” and “horizontal members,” respectively.

Component damage could be accumulated into structural damage through abovementioned the calculation theory; however, defining the importance coefficients for different types of components becomes the key to solving the problem. The definition of importance factors should satisfy the following requirements: (1) The weighted coefficient of “vertical members” should be larger than that of “horizontal members,” for the reason that the failure of vertical bearing members under earthquake excitation would have a catastrophic effect on structural collapse. (2) The global damage model based on component classification should maintain a good consistency with the damage model defined as global property parameter variation [49], in respect of reflecting the damage degree. The correlation between damage indices and damage states is shown in Table 1. (3) The damage indices should be larger than 1.00 once the structure collapses.

In compliance with the principles above, the importance coefficients for different types of components are given based on the statistical results of a series of elastoplastic time-history analysis. The initial combination values are set as *λ*_{h} = 1.00 and , and the combination is supposed to be reasonable if most evaluation results of these two damage models (global damage model based on component classification and damage model based on structural energy dissipation capability) are in good agreement. On the contrary, the combination values vary with an interval of 0.25, and the operation mentioned above repeats. After several trial calculations, the importance coefficients are valued as *λ*_{h} = 0.50 and tentatively, which guarantees the consistency of evaluation results in most working conditions. The proposed global damage model reflects the inherent relation between local component damage and global structure collapse. The cumulative damage and loading path are represented as the ratio of cross-sectional rotation angle during the loading process to the maximum rotation capacity; meanwhile, the different influence of element types, damage degree, and relative position of storeys on structural deterioration is also taken into consideration to explain the physical behavior reasonably.

##### 4.4. Model Verification

The global damage model *D*_{total} based on component classification is verified by comparing it with the final softening model *D*_{T} [50] and the stiffness damage model *D*_{k} [51] in the form of evaluating the damage degree of RC test frames presentedin the following sections. The results are shown in Table 2.

It is concluded that the evaluation results of the three damage models *D*_{total}, *D*_{T}, and *D*_{k} are almost identical in the aspect of determining the damage degree although the values differ owing to separate performance level division. All the values based on *D*_{total} are larger than 1.00 at structural failure, which indicates that the global damage model based on component classification could assess the structural damage degree accurately and reflect the collapse failure logically.

#### 5. Damage Assessment and Failure Analysis

##### 5.1. Component Damage Evaluation

The Mehanny–Deierlein damage model is expressed as the ratio of maximum deformation during the loading process to the inelastic deformation capacity. The deformation capacity refers to the disparity between the ultimate rotation and the yield rotation of member sections. The formulas differ for component types: Columns:where and are the yield curvature and the ultimate curvature of column, respectively; *l*_{p} is the length of plastic hinge; *ε*_{y} and *f*_{y} are the yield strain and the yield strength of longitudinal reinforcement, respectively; *d* is the diameter of longitudinal reinforcement; *h* is the section height in calculation direction; *ε*_{cu} is the ultimate strain of concrete; *n* is the axial compression ratio; is the volumetric percentage of stirrup; and *f*_{yh} is the yield strength of stirrup. Beams:where is the yield curvature of beam; *l*_{s} is the shear-span length; *h*_{0} is the effective height of section; *h*_{1} is the distance between the centroid of compression reinforcement and the edge of concrete; *f*_{c} is the axial compressive strength of concrete; *f*_{y} and are the yield strength of compression reinforcement and tensile reinforcement, respectively; *E*_{s} is the elastic modulus of steel; *α*_{st} is the steel type coefficient, valued as 1.25 for hot-rolled bar; *α*_{cy} is the load type coefficient, valued as 1 and 0.6 for static loading and cyclic load, respectively; *ρ* and *ρ*^{′} are the reinforcement ratios of compression reinforcement and tensile reinforcement, respectively; *α* is the restriction coefficient of stirrup; *ρ*_{d} is the reinforcement ratio of web bars.

The inelastic deformation capacity of structural columns and beams is shown in Table 3. The damage distribution of three RC test frame structures at different performance levels is obtained by substituting DIC data into equations (1) to (3), as presented in Figure 9. It is found that the seismic damage initiated at the local position of structural columns and beams in the three test frame structures at the operational performance level, with the damage indices of columns varying in the range 0∼0.10 and that of beams generally exceeding 0.10. The number and width of cracks at the beam ends were larger than those at the column ends obviously in actual test phenomena; the damage degree of the three test frame structures is different at the slight damage level and medium damage level:

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The damage at the beam ends of Specimens KJ-1 and KJ-2 increases faster than that at the column ends, with the damage indices of columns being about 0.20, while that of beams exceeds 0.40. In comparison with Specimens KJ-1 and KJ-2, the damage degree of beams in Specimen KJ-3 develops slowly, but the damage to columns develops faster, indicating that the structural failure path deteriorates owing to the “vertical components” breaking anterior to the “horizontal components” gradually. From the serious damage level to the point of collapse, the damage degree of beam ends is larger than that of column ends all the time in Specimen KJ-1, with all the damage indices of beams exceeding 1.00 at the end of loading. On the contrary, the damage indices of beams in Specimens KJ-2 and KJ-3 are lower than the limit value, while the damage indices of columns exceed 1.00; that is to say, these two test frame structures no longer have the ability to withstand the vertical bearing capacity and have reached the limit state of collapse. It is also noteworthy that the damage degree of Specimen KJ-3 is lighter than that of Specimen KJ-2 in the end because of the larger beam-to-column stiffness ratio.

It could be observed from the damage distribution of the three RC test frame structures that the damage indices of bottom members are generally larger than that of the upper ones. To illustrate with Specimen KJ-2, the average damage indices of columns in the first floor, second floor, and third-floor are 0.97, 0.62, and 0.15, respectively, and those of beams are 0.95, 0.86, and 0.74 at the final collapse, which indicates that the structural cumulative damage proceeds from bottom storey to top storey under seismic excitation. Besides, the damage degree of beam ends is more serious than that of column ends in the specimen with smaller vertical load; however, the cumulative damage of column ends increases rapidly in the structure with larger beam-to-column linear stiffness.

##### 5.2. Storey Damage Evaluation

The damage distribution along the floors is obtained by substituting the damage indices of components into equations (4) to (5), as shown in Figure 10.

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It is found that the storey damage of beams is evenly distributed, with test results showing that the damage indices of the first storey in Specimen KJ-1 are about 13.91% and 39.13% larger than that of the second storey and third storey, respectively (for Specimen KJ-2: 12.74% and 25.49%; for Specimen KJ-3: 17.71% and 35.42%), at final collapse. The damage degree of columns increases slowly with lower displacement amplitude, but increases rapidly as the amplitude and cycle times rise; the damage is mainly concentrated on the bottom floor of structures with test results showing that the damage indices of the first storey in Specimen KJ-1 are about 34.73% and 69.47% larger than that of the second storey and third storey, respectively (for Specimen KJ-2: 38.46% and 74.36%; for Specimen KJ-3: 32.43% and 73.87%), at final collapse, reflecting the fact that the structural collapse arose from the local accumulation of damage at the bottom of columns.

##### 5.3. Structure Damage Evaluation

The damage evolution curves of overall structures are obtained by substituting the storey damage indices into eequations (6) to (9), as shown in Figure 11.

It is found that the damage evolution curves of the three test frame structures show a similar trend: the slope is flat when the displacement loading is small and gradually becomes steeper as the amplitude increases. This phenomenon indicates that the beams acting as energy-consuming components have positive effects on delaying the progression of damage in the overall structure at the small-loading stage, while the columns gradually replace the beams and begin to consume the seismic energy, as seriously damaged beams lose their functions with the displacement amplitude and cycle times increasing. The damaged columns subsequently induce a sharp decline in the structural energy dissipation capacity and, then, aggravate the degradation of seismic performance at the large-loading stage.

The members, including beams and columns, consume the seismic energy through the method of changing their deformation: Large deformation occurred to dissipate energy when the displacement amplitude increased; obvious failure phenomena were found once the deformation exceeded a certain limiting value. The strength and stiffness of members degraded after several repeated deformations when suffering cyclic loading at the same amplitude, and then, the energy dissipation capacity declined sharply and serious failure phenomena occurred along with the next larger displacement amplitude.

Comparing the damage evolution curves of the three specimens, it is found that the curve of Specimen KJ-1 is flatter than that of Specimens KJ-2 and KJ-3. The damage indices of Specimen KJ-1 for the final collapse is the smallest, which indicates that the full development of “beam hinge” is beneficial to postpone the collapse process of integral structure and mitigate the damage degree at failure. The curves of Specimen KJ-2 and KJ-3 are both steeper, with few differences between them: there are several feature points presented in the form of slope variation in the Specimen KJ-2 curve, which is related to the alternation of “beam hinge failure” and “column hinge failure.” While the slope of the Specimen KJ-3 curve is almost unchanged during the loading process, for the reason that the plastic hinges at the beam ends are not fully formed, and the columns consume a considerable part of seismic energy due to the larger beam-to-column linear stiffness ratio. The damage indices of Specimen KJ-2 are a little larger than those of KJ-3 at final collapse, indicating that the influence of axial compression ratio increasing on the degradation of seismic performance is greater than that of the beam-to-column linear stiffness ratio.

In conclusion, the global damage model based on component classification could reflect the structural damage evolution rule and the different influence of component types on the seismic performance of overall structure.

#### 6. Seismic Vulnerability Assessment

Over the past decade, efforts to develop performance-based seismic evaluation have progressed and resulted in guideline documents such as ASCE/SEI 41-06 (2006) and FEMA 356 (2000) [52]. FEMA-356 (2000) proposes a probabilistic description of the performance level on establishing a quantitative evaluation of damage states, which is accomplished by nonlinear elasto-plastic methods. As numerous studies have shown, ground motion characteristics is the largest source of uncertainty in structural seismic response; hence, three ground motion parameters that affect the structural response are considered. Response parameters, such as plastic rotation, are also interpreted into a damage form (*D*) through the global damage model presented in the previous section, and the probabilistic approach resulting in the fragility surfaces that characterize the conditional probability of predicted demand and performance state is carried out.

##### 6.1. Nonlinear Simulation Model

In this section, probabilistic seismic demand analysis is performed using Abaqus/Explicit finite element program to study the seismic vulnerability of archetype RC frame structure. The moment frame is designed to resist both gravity loads and lateral loads, satisfying all the seismic criteria regarding strength and deformation limits according to the codes.

Compared with the implicit method, it is not necessary to form or invert stiffness matrices in the explicit dynamic analysis and the displacements are calculated at the beginning of each time increment, so it is efficient for highly nonlinear problems such as structural dynamic collapse. The fiber-beam element B31 in the Abaqus element library is selected to simulate the beams and columns of a structure, and the rectangular cross-sectional shape is defined for beam property. This type of element is suitable for exhibiting the nonlinear materials where large deformation and rotation occur. At each time increment, the stress over cross section is integrated numerically to follow the development of individual elastoplastic response. The section is divided into multifiber bundles with the uniaxial stress-strain relationship of concrete material imparted on each fiber, and steel reinforcements are inserted into each element by definition at appropriate depth of cross section, using the keyword REBAR to ensure computational convergence, as shown in Figure 12.

The material property is simulated using the subroutine PQ-Fiber [53] through the converter program UMAT. UConcrete02 is used for the concrete to consider the confined effect of stirrups in this paper because the hoops cannot be defined directly in fiber-beam element. It is an isotropic elastoplastic concrete material with the character of confined concrete, which defines the modified Kent-Park model [54, 55] as compression constitutive relation and the bilinear model with a softening segment as tension constitutive relation, as shown in Figure 13. Eight parameters are defined in this model: axial compressive strength, corresponding compressive strain, ultimate compressive strength, ultimate compressive strain, the ratio of unloading stiffness to initial elasticity modulus, axial tensile strength, tensile softening modulus, and yield strain of steel bars in the section. All the values of these parameters are related to the measured strength of the concrete material, as described in Section 2. Usteel02 is used for steel to consider the Bauschinger effect caused by stiffness degradation and the bending capacity attenuation caused by cumulative damage. It is the improved form of the proposed maximum point-oriented bilinear model [56], as shown in Figure 14. Four parameters are used in this material: elasticity modulus, yield strength, the ratio of postyield stiffness to elasticity modulus, and ultimate plastic deformation rate. Similar with adopted values in UConcrete02, these parameters are obtained from measured values and default values. Besides the material nonlinearity in beams and columns, the option of geometric nonlinearity in the Step Module of Abaqus is selected in simulating structural collapse due to the large deformation (-Δ) effects experienced during the strong earthquake. The axial loads of beams are negligible but imposed at column tops in each floor as concentrated forces.

Three analysis steps, including initial step for fixed constraints of column bottoms at the first floor, Step-1 for axial loads applying, and Step-2 for seismic records inputting, are set to complete the applied loads in order to ensure the boundary condition and loading sequence of the numerical model being consistent with the experimental specimens. The initial time and the minimum time step are set as 0.001 s and 10^{−5}, respectively. The seismic loading is defined as the acceleration field acting on the whole structure, with the consideration of fixed support in the foundation. The numerical results are compared with the experimental results of Specimen KJ-1 to verify the reliability of simulation method, as shown in Figure 15.

It can be seen that the analytical hysteresis loop lies reasonably close to the experimental one, but the energy dissipation capacity of simulated frame structure is slightly smaller than that of experimental specimen.

The major reason for the difference may attribute to the method of defining the embedded steel bars; that is to say, the keyword “rebar” treats concrete material and steel material as a whole; thus, the energy dissipation ability of steel could not be made full use. However, in actual experiment, the bond slip in the interface between concrete and steel bars occurs under low-reversed cycle loading; thus, the tensile and energy dissipation properties of steel bars could be utilized sufficiently. After comprehensive consideration, the numerical model for the experimental prototype structure (a multilayer RC office building) is established based on the simulation method above.

The selection of seismic intensity parameters has a greater impact on the accuracy of seismic vulnerability analysis results. Considering the complexity of structural failure mechanism under earthquake action, a single seismic intensity parameter is not sufficient to reflect the influence of earthquake characteristics on the structural failure probability. Therefore, three seismic intensity parameters, i.e., PGD (in relation to the amplitude characteristics), *M* (in relation to the spectrum characteristics), and SA_{1} (commonly used in time-history analysis), are selected in this section to carry out the seismic reliability evaluation of RC frame structures. The time-history analysis is accomplished by using a set of 50 ground motions from PEER Ground Motion Database to reduce the biases in structural response. These 50 ground motions involve a wide range of magnitudes (5.0–7.6), epicentral distances (0.6 km–222.4 km), site classification (types A–D), and spectral range.

##### 6.2. Collapse Resistance Assessment

Fragility surface represents the conditional probability that structural response parameter *D* reaches or exceeds the structural resistance *C* at a certain performance level when the intensity parameters *IM*_{1} and *IM*_{2} take different values. The formula could be expressed as follows:

Assuming that the parameters *C* and *D* are independent random variables obeying log-normal distribution, then the failure probability of the structure at a specific performance level could be expressed as follows:where Φ(*x*) is the normal distribution function, and the value could be obtained by consulting relevant tables, and *σ* is the standard deviation.

The structural aseismic capacity at different performance levels (characterized by global damage indices based on component classification) are shown in Table 1. The linear regression equation between two seismic intensity parameters (*IM*_{1}, *IM*_{2}) and the global damage indices *D*_{total} (Table 4) is setup as follows:where *a*, *b*, *c*, *d*, *e*, and *f* are the regression coefficients.

It could be seen in Figure 16 that the failure probability that RC structural seismic response exceeding different damage levels is affected by PGD and SA_{1} simultaneously; furthermore, SA_{1} has a greater influence on the structural failure probability at the operational level and slight damage level. The fragility surface also indicates that the probability of structural damage exceeding the serious damage level is 72.80% based on equations (23) and (25).

**(a)**

**(b)**

**(c)**

**(d)**

In the fragility surface composed of SA_{1} and *M*, as shown in Figure 17, the failure probability of RC structural seismic response exceeding different damage levels is mainly affected by SA_{1}; however, *M* has little impact on the structural failure probability. The fragility surface indicates that the probability of structural damage exceeding the serious damage level is 70.59% based on equations (24) and (26).

**(a)**

**(b)**

**(c)**

**(d)**

The failure probability of RC frame structures under 8-degree and 9-degree design fortification intensity is also investigated in this section.

Based on the relevant parameters of selected 50 ground motions, the PGD is set as 1 m, 3 m, 6 m, and 10 m, respectively, and the *M* is set as 5, 6, 7, and 8, respectively. The value of SA_{1} is taken as the standard proposal, i.e., 102.71 cm/s^{2} for frequent earthquake, 269.62 cm/s^{2} for basic fortification, and 577.76 cm/s^{2} for rare earthquake, respectively, corresponding to 8-degree fortification intensity, and 205.42 cm/s^{2} for frequent earthquake, 513.56 cm/s^{2} for basic fortification, and 898.74 cm/s^{2} for rare earthquake, respectively, corresponding to 9-degree fortification intensity. The structural failure probability under different performance levels is obtained via equations (25) and (26), and the calculation results are shown in Tables 5∼6.

It could be seen in Table 5 that the failure probability of RC frame structure subjected to frequent earthquake is 0.1%–2.5% for operational level, 0%–1.0% for slight damage level, 0%–0.6% for medium damage level, and 0%–0.2% for serious damage level corresponding to 8-degree fortification intensity when PGD is assumed to be 100∼1000 cm; the failure probability is 33.5%–88.5%, 6.9%–63.8%, 4.6%–26.9%, and 2.0%–16.0%, respectively, for RC frame structure subjected to rare earthquake at the same fortification intensity. The failure probability of RC frame structure subjected to frequent earthquake is 4%–24.6% for operational level, 1.6%–14.3% for slight damage level, 1%–10.3% for medium damage level, and 0.3%–5% for serious damage level corresponding to 9-degree fortification intensity when PGD is assumed to be 100∼1000 cm.

The failure probability is 71.1%–97.3%, 31.8%–75.2%, 18.4%–47.6%, and 3.9%–34.3%, respectively, for RC frame structure subjected to rare earthquake at the same fortification intensity.

It could be seen in Table 6 that the failure probability of RC frame structure subjected to frequent earthquake is 2.4%–4.0% for operational level, 1.1%–2.0% for slight damage level, 0.7%–1.4% for medium damage level, and 0.3%–0.6% for serious damage level corresponding to 8-degree fortification intensity when *M* is assumed to be 5∼8; the failure probability is 46.1%–69.2%, 17.3%–30.0%, 9.1%–13.5%, and 5.1%–8.0%, respectively, for RC frame structure subjected to rare earthquake at the same fortification intensity. The failure probability of RC frame structure subjected to frequent earthquake is 9.6%–14.1% for operational level, 5.3%–8.4% for slight damage level, 3.8%–6.2% for medium damage level, and 1.9%–3.3% for serious damage level corresponding to 9-degree fortification intensity when *M* is assumed to be 5∼8; the failure probability is 75.8%–96.0%, 18.5%–75.4%, 20.6%–34.6%, and 8.7%–23.0%, respectively, for RC frame structure subjected to rare earthquake at the same fortification intensity.

Based on the information presented above, it is found that the failure probability of RC frame structure under different performance levels would increase significantly once the structure is subjected to earthquake excitation higher than the initial design fortification intensity and the increasing amplitude could rise as much as eight times.

#### 7. Conclusion

An experimental study was conducted on three 1/3-scale models of three-storey, three-bay reinforced concrete frame structures under low-reversed cyclic loading, and a proposed global damage model based on component classification was established to investigate the damage evolution rule by discussing the relationship between local damage and structural collapse. Further application of the global damage model was introduced in terms of assessing the failure probability of RC frame structure at different damage levels; the following conclusions can be drawn from this study:(1)The structural damage develops from the bottom storey to top storey under earthquake excitation, and the damage distribution is mainly concentrated on the beam ends and column bottoms of the first floor at final collapse. The anticipated beam-hinged failure mechanism is easier to achieve in the structures with lower vertical axial force, and the structures still maintain a certain load-bearing capacity even when the interstorey drift angle exceeds the elastoplastic limit value. The damage degree of columns increases rapidly in structures with larger beam-to-column linear stiffness, which is harmful to integral ductility and deformation capacity.(2)The damage distribution along the floors varies directly with component types. The damage of beams is distributed along the floor evenly, but that of columns is concentrated. The full development of plastic hinges at the beam ends has positive effect on delaying the damage growth of the integral structure and mitigating the damage degree; besides, the influence of the axial compression ratio increasing on the seismic performance degradation is greater than that of the beam-to-column linear stiffness ratio increasing(3)The formula ln(*D*_{total}) = *a* + bln(IM_{1}) + *c*(ln(IM_{1}))^{2} + dln(IM_{2}) + *e*(ln(IM_{2}))^{2} + *f* ln(IM_{1}) ln(IM_{2}) could be used to reveal the relationship between the two seismic intensity parameters and global damage model based on component classification. From this equation, fragility surfaces with different intensity parameter combinations are drawn, showing that the probability of structural damage exceeding the serious damage level ranges from 70.59% to 72.80%;(4)The maximum failure probabilities of structural damage exceeding operational level, slight damage level, medium damage level, and serious damage level are 4.0%, 2.0%, 1.4%, and 0.6%, respectively, when RC frame structure suffers the frequent earthquake at 8-degree fortification intensity; failure probabilities are 88.5%, 63.8%, 26.9%, and 16.0%, respectively, when the structure suffers the rare earthquake at the same fortification intensity. The failure probabilities would increase significantly once the structure encountering ground motions higher than the initial design fortification intensity and the increasing amplitude reaches as much as eight times.

#### Data Availability

The data given in Table 2 and Figure 9 used to support the findings of this study were related to the original data of experiments funded by the National Natural Science Foundation of China and so cannot be made freely available concerning legal restrictions. Requests for access to these data should be made to the leader of this academic project (email address: [email protected]). The data in Tables 6–9 and Figures 16 and 17 used to support the findings of this study were based on the ground motions selected from PEER Ground Motion Database (website: ngawest2.berkeley.edu) and calculated by commercial software SeismoSignal, Abaqus, and Matlab. The results are presented in the article. The other data used to support the findings of this study were obtained through equations presented in article, and the calculation methods were included within the paper.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This research was partially supported by the National Natural Science Foundation of China (Grant no. 51578077) and the International Science and Technology Cooperation Project of Shaanxi Province (Grant no. 2016KW-056).