Seismic capacity estimation of a reinforced concrete containment building considering bidirectional cyclic effect

Abstract

This study serves to estimate the seismic capacity of the reinforced concrete containment building considering its bidirectional cyclic effect and variations of energy. The implementation of the capacity estimation has been performed by extending two well-known methods: nonlinear static pushover and incremental dynamic analysis. The displacement and dissipated energy demands are obtained from the static pushover analysis considering bidirectional cyclic effect. In total, 18 bidirectional earthquake intensity parameters are developed to perform the incremental dynamic analysis for the reinforced concrete containment building. Results show that the bidirectional static pushover analysis tends to decrease the capacity of the reinforced concrete containment building in comparison with unidirectional static pushover analysis. The 5% damped first-mode geometric mean spectral acceleration strongly correlates with the maximum top displacement of the containment building. The comparison of the incremental dynamic analysis and static pushover curves is employed to determine the seismic capacity of the reinforced concrete containment building. It is concluded that bidirectional static pushover and incremental dynamic analysis studies can be used in performance evaluation and capacity estimation of reinforced concrete containment buildings under bidirectional earthquake excitations.

Keywords

bidirectional interaction capacity estimation containment building incremental dynamic analysis pushover

Introduction

Throughout the world, there are hundreds of nuclear power plants, the failure of which due to seismic activities could result in serious or even catastrophic consequences. The reinforced concrete containment (RCC) building preventing the radiation from leaking into the environment is one of the most important nuclear structures. The behavior of an RCC building in an earthquake and its structural losses are important issues in the seismic evaluation of nuclear power plants. Therefore, presented herein is only the RCC building, which also has a weak connection with other nuclear structures such as the auxiliary building, the fuel building, and the diesel generator building established by a nuclear power company.

Seismic analyses of RCC buildings have been investigated extensively in the last few decades, and growth in the processing power of computers has expedited this improvement. Varpasuo (1996) presented a finite element method to estimate the ultimate capacity of a nuclear containment building by performing the stress analysis of the containment structure. He numerically studied the nonlinear behavior of the containment building represented by a stick model and finally obtained the failure probabilities of the building. Based on comparative analyses, site-dependent ground motions and ground motions determined using Newmark’s spectra are employed to conduct the seismic analyses of nuclear structures by Cho and Joe (2005). Their results showed that seismic analysis based on the Newmark’s spectra might overestimate the seismic capacities of seismic facilities. Choi et al. (2008) investigated the effects of near-fault ground motions on seismic response of a CANDU type containment building. They used nonlinear pushover analysis and a three-dimensional element model to determine its structural capacity and a simple stick model to perform the incremental dynamic analysis (IDA). Using the simplified model of a sample conventional and a base-isolated nuclear containment building, Huang et al. (2010) evaluated the seismic performance of base-isolated safety-related nuclear structures. They found that the seismic capacity of the nuclear building could be largely improved by means of isolation systems. With the aid of a detailed three-dimensional nonlinear finite element model considering soil-structure interaction and basemat uplift behavior, seismic response analyses were conducted by Nakamura et al. (2010) increasing the input acceleration up to 3500 Gal, until the damage of the building reached the ultimate condition. However, these studies primarily considered seismic shaking in one principal horizontal direction of the structure though in reality, shaking occurs simultaneously in both horizontal principal directions. It should be noticed that bidirectional horizontal shaking considerations are specified in many design codes for common buildings and nuclear power plants. For example, the Uniform Building Code (UBC) (International Conference of Building Officials (ICBO), 1997) states that bidirectional orthogonal shaking effects must be considered if a column of a structure forms part of two or more intersecting lateral-force-resisting systems and that such columns requires to be designed using 100% of the prescribed design seismic forces in one direction plus 30% of the prescribed design seismic forces in the perpendicular direction. Alternatively, the effects of the two orthogonal directions may be combined based on the square-root-of-the-sum-of-squares (SRSS) rule. For seismic design criteria in nuclear facilities, ASCE 4-86 (1986) specifies either a 40% rule or SRSS rule. In Regulatory Guide 1.92 (US Nuclear Regulatory Commission, 2006) and ASCE/SEI 4-16 (2016), the combined response for all three spatial components shall be conducted by one of the following: the SRSS rule, the maximum response from the algebraically combined response history, or the 100-40-40 rule. Nevertheless, the above design codes are only available for linear response history analysis. The RCC under strong ground motions may deform significantly and enter the inelastic stage and therefore the simplified combination rule cannot be adopted for seismic capacity estimation of the RCC. Moreover, a large number of preceding literatures showed that the bidirectional cyclic effect should not be neglected in order not to overestimate the structural capacity (Lee and Hong, 2010; Lin et al., 2013; Rodrigues et al., 2013; Wang et al., 2015). Accordingly, this study attempts to provide more information on estimation of the seismic capacity of the RCC building under bidirectional earthquake excitations.

In the case of RCC buildings, because there is limited database of field and experimental observations of damage, the numerical methods for estimation of capacity are preferred. Usually, the capacities of a structure can be computed from its maximum response values under increasing applied loads. Two well-known methods including the static pushover (SPO) analysis and the IDA are extended for computing the structural capacities under bidirectional earthquake excitations. In an SPO analysis, a monotonically increasing lateral load is statically applied to the structure, and the response parameters are monitored until yielding and failure states of the structure are captured (Chopra and Goel, 2002; Faramarz and Mehdi, 2011). In this study, bidirectional cyclic pushover analysis for the RCC building is conducted. The significance of the practice is its consideration of inelastic bidirectional interaction and variations of energy caused by bidirectional load histories for a whole structure. In an IDA, which is, in fact, a dynamic pushover analysis, the intensity of ground shaking applied to the base of the structure is incrementally increased until limit states of the structure are captured and probable collapse mechanisms are determined (Vamvatsikos and Cornell, 2002). Rather than unidirectional IDA, bidirectional IDA is performed for the RCC building. For unidirectional ground motions, intensity measures (IMs) such as peak ground acceleration (PGA), peak ground velocity (PGV), peak ground displacement (PGD), 5% damped first-mode spectral acceleration (S_a(T₁,5%)), and cumulative absolute velocity (CAV) are always compared and one of them is then adopted to be the most suitable IM (Elenas and Meskouris, 2001; Lu et al., 2013). However, these IMs may not be used as bidirectional IMs because these IMs in any direction cannot represent the earthquake intensity for bidirectional ground motions. Therefore, 18 bidirectional earthquake intensity parameters are considered to perform the IDA for the RCC building. Some of these bidirectional ground motion definitions are extended from unidirectional IMs such as PGA, PGV, PGD, S_a(T₁,5%), and CAV by taking the arithmetic mean, the geometric mean and the max of IMs for bidirectional as-recorded components. Other bidirectional IMs such as GMRotD00, GMRotD50, and GMRotD100 are selected from Boore et al. (2006). These three definitions are determined as the 0th, 50th, and 100th percentile of the set of geometric means computed from the as-recorded orthogonal horizontal motions rotated through all possible non-redundant rotation angles for a given oscillator period. The advantage of these three parameters is because that they can reduce the variance of the ground-motion measure and produce less arbitrary approximations to components that have genuinely random orientation. Currently, although unidirectional IMs have been applied to evaluate seismic demands for three-dimensional buildings (Alembagheri and Ghaemian, 2013; Lagaros, 2010), the correlation between the above bidirectional IMs and demand measures (DMs) for the structure has received little attention. Moreover, a comprehensive and comparative analysis of all of these measures would facilitate the selection of IMs for bidirectional ground motions and reduce the variance of estimated capacities for the considered building.

This article aims to determine the seismic capacity of the RCC building under bidirectional earthquake excitations. To this end, a three-dimensional numerical model of the RCC building is first built using the Abaqus software (Abaqus, 2010). In order to determine the structural capacity, three different load paths are employed to perform the bidirectional cyclic pushover analysis. For comparison, unidirectional monotonic and cyclic pushover analyses are also conducted. To perform the IDA analysis, a suite of 12 appropriate bidirectional ground motions for the specified site is extracted from the Pacific Earthquake Engineering Research (PEER) center. Bidirectional IDA analyses are then carried out and correlations between IMs and DMs are ranked. Finally, seismic capacity of the RCC building is determined through the comparison between the IDA and SPO results.

Numerical modeling

Figure 1(a) presents a cutaway view of a typical RCC building to be analyzed in this study. The inside and outside diameters of the cylinder are 37.796 and 39.93 m, respectively. The height of the cylinder and the total height of the containment building are 43.83 and 63.26 m, respectively. The wall thickness of the dome is 0.762 m. The elastic modulus, Poisson’s ratio, density, tensile strength, and compressive strength of the containment concrete are 33,000 MPa, 0.2, 2400 kg/m³, 2.67 MPa, and 32.4 MPa, respectively. The yield strength of the steel is 350 MPa. The three-dimensional RCC model shown in Figure 1(b) is established using the Abaqus finite element software. As the thickness of the cylinder of the containment building is less than one-tenth of the radius of the cylinder, the cylinder is modeled using shell elements into which rebars are embedded. The maximum size of the shell element for the cylinder is 1.02 m × 1.0 m. The semi-elliptical dome is also modeled using shell elements and eight-node solid elements are applied to the base. The maximum size of the shell element for the dome and the solid element for the base is 1.02 m × 0.9 m and 2.3 m × 1.3 m × 1.5 m, respectively. A concrete damaged plasticity model developed by Lubliner et al. (1989) and modified by Lee and Fenves (1998) is employed here as the constitutive model to capture the seismic behavior of the containment building. This constitutive model, including its strain hardening and softening behavior, is widely used for seismic cracking and crushing analyses of concrete structures (Alembagheri and Ghaemian, 2013; Resta et al., 2013; Zhai et al., 2015, 2017; Zhang et al., 2013). In this model, the uniaxial strength functions are composed of plastic deformation and degradation of stiffness. Therefore, the tensile damage (dt) and the compressive damage (dc) are addressed by normal strain at any point as shown in Figure 2. The rebars are considered uniaxial materials in the arranged direction with a bilinear restoring force. The second gradient is set to be 1/100 of the initial stiffness, provided by a nuclear company. It should be noted that the rebars are completely bonded to the concrete in the following analyses. Using the Rayleigh method, the critical damping ratio is set to be the same value (5% according to an engineering report of the containment building) over the dominant periods of the structure (5.0 and 14.9 Hz) so that the mass- and stiffness-proportional damping coefficients are determined. As the containment building is supported on a rock soil site, therefore, the soil–structure interaction is neglected in this investigation.

Figure 1.

(a) Cutaway view and (b) finite element mesh generation of the RCC building.

Figure 2.

Assumed constitutive behavior of concrete in tension (a) and in compression (b).

SPO analysis of the RCC building

The SPO analysis plays an important part in providing the expected behavior of a structural model (Vamvatsikos and Fragiadakis, 2010). It can be realized by first exerting gravitational loads and then applying a lateral incremental load to the RCC building. In the past, Varpasuo (1996) and Cho et al. (2004) obtained the structural capacities for typical nuclear buildings using the monotonic SPO analysis and the detailed finite element model. However, the existing practice only focused on one-component seismic shaking and did not take into account the cyclic effect and neglected the variations of energy.

Application of the unidirectional lateral load

After exerting the weight loads, the lateral pushover load along the x-direction, as shown in Figure 3(a), is applied statically from zero with an increasing top displacement, until the top displacement reaches 200 mm. The uniform distributed load is selected for two reasons: first, the stiffness of the RCC is uniformly distributed along its height, leading to the soft story of the RCC always occurring at lower heights under lateral loads; moreover, compared to the inverted triangle and modal force distributions, the uniform force distribution results in larger drifts and plastic rotations in lower stories (Goel and Chopra, 2004), which practice makes the pushover results more conservative. The SPO curves for both force and energy forms are developed, as shown in Figure 3(b) and (c). Variation of model’s energies versus total lateral displacement in the SPO analysis has been developed according to the energy form of the equation of motion of the RCC as

E_{I} + E_{K} + E_{D} = E_{E}

(1)

E_{S} + E_{F} = E_{I}

(2)

in which $E_{E}$ is the work done by externally applied loads, $E_{I}$ is the internal energy, $E_{K}$ is the kinetic energy, $E_{D}$ is the viscous energy dissipated, $E_{S}$ is the recoverable elastic strain energy, and $E_{F}$ is the energy dissipated by inelastic processes and damage. Because the SPO analysis is essentially a static analysis, all of the external work is transformed to the internal energy and all of the internal energy is composed of recoverable strain energy and the dissipated energy in terms of the plastic deformations and damage. The damage contour results are presented in Figure 3(d) to (f). It is found that the tension softening (yielding state) is initiated in the parallel direction of the lateral load and at the bottom of the RCC, with the top displacement of 13.2 mm. Increasing the top displacement to 22.7 mm induces the first unit tensile damage (cracking initiation), which appears at the bottom of the structure. The increase in the lateral load is continued until the top displacement of 149 mm at which, by further increasing, an unstable crush is formed at the bottom of the RCC and rapidly propagated to the upside of the RCC, as shown in Figure 3(f). Thus, the RCC loses its stiffness and deforms drastically under the fixed lateral loads. This behavior is considered the ultimate (collapse) state. Between the top displacements of 22.7 and 149 mm, there is stable crack growth from the base to the top of the structure. Figure 3(b) and (c) also shows the pushover curves along with the obtained limit states.

Figure 3.

The SPO analysis for monotonic loading: (a) schematic diagram, (b) load–displacement curve, (c) energy–displacement curve, (d) softening initiation, (e) cracking initiation, and (f) crushing initiation.

Rather than the monotonic lateral load, the SPO is performed by applying the unidirectional cyclic lateral load, which can better monitor the behavior of the RCC under earthquake excitations. The lateral displacement step is developed with an increment of 12.5 mm on the basis of the results of the monotonic SPO analyses. The lateral load against top displacement in the cyclic and monotonic analyses has been plotted in Figure 4(b). As can be seen, the backbone curve for the cyclic loading and the pushover curve for the monotonic loading generally overlap. The dissipated energy due to the plastic deformations and damage for monotonic loading and cyclic loading has been compared in Figure 4(c). It is clear that the dissipated energy for cyclic loading is much greater than that for monotonic loading because the damage only occurs at specified regions for the RCC and cyclic manner significantly increases the dissipated energy due to the plastic deformations. The damage contour results are presented in Figure 4(d) to (f). Similar to its monotonic loading, it is found that the tension softening and cracking initiation occur at the top displacement of 13.2 and 22.7 mm, respectively. With the increase in the amplitude for top displacement, the damage of the RCC propagates and crushing initiation is induced at the base of the RCC with the top displacement of 110 mm, which is much lower than its monotonic loading. The obtained limit states are also presented in Figure 4(b) and (c).

Figure 4.

The SPO analysis for cyclic loading: (a) schematic diagram, (b) load–displacement curve, (c) energy–displacement curve, (d) softening initiation, (e) cracking initiation, and (f) crushing initiation.

Application of the bidirectional cyclic lateral load

Three different load histories including circular, rhombus, and quadrangular load paths are employed to conduct the SPO analysis for the RCC model. The evolution of the lateral displacement step is shown in Figure 5. The shear–displacement and energy–displacement curves appear in Figure 6. For the lower top displacement, the skeleton curves for three bidirectional load histories almost overlap with the unidirectional pushover curve, while for the larger top displacement, strength values are lower for all biaxial results compared with those for the corresponding uniaxial results. It can be found that the reduction in the specific strength of the RCC due to bidirectional cyclic effect is between 14.0% and 29.8%, as shown in Figure 6. Investigating the damage contour, softening initiation for the RCC under biaxial loading occurs for similar top displacement demands in comparison with the uniaxial loading. Increasing the top displacement demands, cracking initiation is induced with the total top displacement of 21.1, 21.5, and 20.99 mm for circular, rhombus and quadrangular load paths, respectively. As the damage continues, crushing initiation is also significantly affected by the biaxial loading, with the total top displacement of 98.88, 99.83, and 95.13 mm for circular, rhombus and quadrangular load paths, respectively. In summary, the yielding and ultimate response values for the RCC have been listed in Table 1. In this table, total means the top displacement is computed using the SRSS rule for two directions. In this study, the lowest and the most conservative top displacement demands for each damage state are chosen here for biaxial load paths, that is, 13.2 mm for softening initiation, 20.99 mm for cracking initiation, and 95.13 mm for crushing initiation, respectively. In addition, the dissipated energy for crushing initiation under different load paths is also given in Table 1, which is 22.26 MJ for unidirectional monotonic loading, 99 MJ for unidirectional cyclic loading, 89.6 MJ for circular loading, 90.7 MJ for rhombus loading, and 89.3 MJ for quadrangular loading. Because the unidirectional monotonic loading cannot reflect the reciprocating characteristic of the earthquake, the dissipated energy for this practice is not suitable to estimate the accumulated energy under earthquake excitations. Similar to the displacement results, the dissipated energy for bidirectional cyclic loading is slightly smaller than that for unidirectional cyclic loading.

Figure 5.

Schematic diagram and load path for bidirectional cyclic loading.

Figure 6.

The SPO analysis results considering bidirectional cyclic loading: (a) load-displacement curve for circular load path-X, (b) load-displacement curve for circular load path-Y, (c) energy-displacement curve for circular load path, (d) load-displacement curve for rhombus load path-X, (e) load-displacement curve for rhombus load path-Y, (f) energy-displacement curve for rhombus load path, (g) load-displacement curve for quadrangular load path-X, (h) load-displacement curve for quadrangular load path-Y, (i) energy-displacement curve for quadrangular load path.

Table 1.

Summary of the RCC model responses from SPO analyses.

Load case	Softening initiation	Cracking initiation	Crushing initiation
Unidirectional monotonic	13.2 mm	22.7 mm	149 mm; 22.26 MJ
Unidirectional cyclic	13.2 mm	22.7 mm	110 mm; 99 MJ
Circular load	13.2 mm (Total)	21.1 mm (Total)	98.88 mm (Total); 89.6 MJ
Rhombus load	13.2 mm (Total)	21.5 mm (Total)	99.83 mm (Total); 90.7 MJ
Quadrangular load	13.2 mm (Total)	20.99 mm (Total)	95.13 mm (Total); 89.3 MJ

RCC: reinforced concrete containment; SPO: static pushover.

IDA of the RCC building

The IDA is a parametric analysis that estimates more thoroughly structural performance under seismic loads (Vamvatsikos and Cornell, 2002). For unidirectional earthquake excitations, common examples of scalable IMs are the PGA, PGV, and S_a(T₁,5%). However, the earthquake always occurs in terms of two horizontal and one vertical ground motion records, and use of only unidirectional IMs seems not be appropriate to represent the real earthquake intensity. Therefore, in order to select the most representative bidirectional IM, a large number of bidirectional earthquake intensity parameters are developed to thoroughly investigate the correlation between different IMs and DMs. The complete list of bidirectional IMs is summarized in Table 2. A DM is a non-negative observable scalar that characterizes the state of the structural model to a prescribed seismic loading that can be deduced from the output of the corresponding nonlinear dynamic analysis. In general, the damage indexes are separated into cumulative and non-cumulative ones. The displacement demand belongs to the non-cumulative ones while those indexes that can consider the accumulation effect of seismic excitation to structural damages are called cumulative indexes (e.g. dissipated energy). It is generally recognized that the cumulative damage indexes can better capture the damage progress in the structure. Hence, in this work, the maximum top displacement and maximum dissipated energy are selected as the structural response parameters. The maximum top displacement is calculated as the maximum value of top displacement history by combining the bidirectional top displacement using the SRSS rule.

Table 2.

Summary of selected bidirectional IMs.

	Arithmetic mean (AM)	Max	Geometric mean (GM)
The PGA set	$(PGAx + PGAy) / 2$	$max {PGAx, PGAy}$	$\sqrt{PGAx \cdot PGAy}$
The PGV set	$(PGVx + PGVy) / 2$	$max {PGVx, PGVy}$	$\sqrt{PGVx \cdot PGVy}$
The PGD set	$(PGDx + PGDy) / 2$	$max {PGDx, PGDy}$	$\sqrt{PGDx \cdot PGDy}$
The S_a(T₁,5%) set	$(S ax (T 1, 5 %) + S ay (T 1, 5 %)) / 2$	$\max {S ax (T 1, 5 %), S ay (T 1, 5 %)}$	$\sqrt{S ax (T 1, 5 %) \cdot S ay (T 1, 5 %)}$
The CAV set	$(CAVx + CAVy) / 2$	$max {CAVx, CAVy}$	$\sqrt{CAVx \cdot CAVy}$
The GMRotD set	GMRotD00 (GM)	GMRotD50 (GM)	GMRotD100 (GM)

IM: intensity measure; PGA: peak ground acceleration; PGV: peak ground velocity; PGD: peak ground displacement; CAV: cumulative absolute velocity.

Selected earthquake ground motions

To perform bidirectional IDA of the RCC building, a suite of ground motion records including two horizontal components should be selected appropriately. According to the uniform risk spectrum (URS) for the specific site, 12 pairs of earthquake histories used for the analysis are generated for earthquake magnitude from 5.5 to 7, rupture distance (R_rup) from 0 to 30 km and V_s30 from 600 to 1500 m/s from the PEER Center. The complete list of these earthquakes appears in Table 3. Each pair of earthquake histories is slightly scaled to minimize the sum of the squared error between the target spectral values and the geometric mean of the spectral ordinates. Acceleration spectra for the resulting 12 pairs of ground motions are shown in Figure 7.

Table 3.

Detailed information of selected earthquake ground motions.

No.	Ground motion	Year	Station	Magnitude (Ms)	R _rup (m/s)	V _s30 (m/s)	Component
1	San Fernando	1971	Lake Hughes #12	6.61	19.3	602.1	21;291
2	Loma Prieta	1989	Gilroy-Gavilan Coll.	6.93	9.96	729.65	67;337
3	Loma Prieta	1989	San Jose-Santa Teresa Hills	6.93	14.69	671.77	225;315
4	Loma Prieta	1989	UCSC	6.93	18.51	713.59	0;90
5	Northridge-01	1994	LA 00	6.69	19.07	706.22	180;270
6	Northridge-01	1994	Santa Susana Ground	6.69	16.74	715.12	0;90
7	Kobe, Japan	1995	Nishi-Akashi	6.9	7.08	609	0;90
8	Chi-Chi, Taiwan-03	1999	TCU071	6.2	16.46	624.85	N;E
9	Tottori, Japan	2000	OKYH14	6.61	26.51	709.86	NS;EW
10	Niigata, Japan	2004	NIG023	6.63	25.82	654.76	EW;NS
11	L’Aquila, Italy	2009	L’Aquila-V. Aterno-Colle Grilli	6.3	6.81	685	E;N
12	Chuetsu-oki, Japan	2007	Joetsu Uragawaraku Kamabucchi	6.8	22.74	655.45	NS;EW

Figure 7.

The comparison of determined URS and spectral accelerations of all time histories.

IDA results and their properties

Each of the records has been scaled to multiple levels of S_aGM(T₁,5%) from 0g to 4.5g that have been arranged in 0.2g (sometimes 0.1g) steps. From the primary modal analysis of the RCC model, it is found that the period of the first mode of vibration of the system is T₁ = 0.2 s. After extraction of DMs, a set of discrete points are left for each record that resides in the IM–DM plane and lies on its IDA curve. By interpolating them, the entire IDA curve can be approximated without performing additional analyses. To do so, the spline interpolation is utilized. Then, the representative curves including the running mean, running median, mean plus one σ (84% fractile), and mean minus one σ (16% fractile) can be plotted.

The nonlinear structural model of the RCC after exerting the weight load is analyzed under the set of earthquake ground motions listed in Table 3, in multiple levels of intensity. Figure 8(a) to (c) shows the IDA curves for the maximum top displacement and the PGA set for individual records. The IDA curves are record and structural model dependent; thus, by subjecting the same structural model to different earthquakes, these curves often thoroughly differ from each other. Some curves in Figure 8(a) to (c) are normally softened or hardened, but the others twist and display successive segments of softening and hardening, and have non-monotonic function of DM in terms of IM. The hardening behavior, which has been reported in other literatures for IDA curves of buildings (Vamvatsikos and Cornell, 2002, 2004; Vamvatsikos and Fragiadakis, 2010), is because of variation in damage patterns resulted from more effects of the first cycles of the records that influence the responses with scaling and alter the distribution of damage through the structure. It is seen from Figure 8(d) that selection of the PGA_GM as IM, instead of PGA_AM or PGA_max, produces a lower dispersion (standard deviation divided by mean) over the full range of top displacements. The IDA curves for other IMs display the similar characteristics with the PGA set. Observing Figure 8(h), selection of the PGV_GM as IM, instead of PGV_AM or PGV_max, causes a lower dispersion over the full range of top displacements. For the PGD set, PGD_GM as IM is better than PGD_AM or PGD_max due to the lowest dispersion over the whole range of top displacements, as shown in Figure 8(l). For the S_a(T₁,5%) set, as shown in Figure 8(p), employing S_aGM(T₁,5%) as IM develops the lowest dispersion for the full range of top displacements in comparison with S_aAM(T₁,5%) and S_amax(T₁,5%). It is also seen from Figure 8(t) that the use of the CAV_GM as IM induces the smallest dispersion compared with CAV_AM and CAV_max. To conclude, the selection of geometric mean parameters can produce a better correlation between IM and DM in comparison with arithmetic mean and max parameters for the RCC model under bidirectional earthquake excitations. As shown in the GMRotD set, it presents the similar distribution between IM and DM with the S_a(T₁,5%) set. Further observing Figure 8(x), applying GMRotD50 as IM results in the lowest dispersion in comparison with GMRotD0 and GMRotD100. For the best selection of IM, the lowest dispersion for each set is selected and plotted in Figure 9. It can be seen from the figure that the lowest dispersion for all considered IM sets is caused by S_aGM(T₁,5%) and the dispersion for the full range of top displacements is between 0.11 and 0.36. The CAV_GM induces the largest dispersion for the mean top displacement of 50 mm, with the value of about 0.80 reached. In addition, the dispersion between GMRotD50 and S_aGM(T₁,5%) shows the similar tendency, and using S_aGM(T₁,5%) as IM is not much worse than GMRotD50.

Figure 8.

IDA results for different IM sets: (a) PGA_max-displacement, (b) PGA_AM-displacement, (c) PGA_GM-displacement, (d) displacement dispersion for PGA set, (e) PGV_max-displacement, (f) PGV_AM-displacement, (g) PGV_GM-displacement, (h) displacement dispersion for PGV set, (i) PGD_max-displacement, (j) PGA_AM-displacement, (k) PGD_GM-displacement, (l) displacement dispersion for PGD set, (m) S_amax-displacement, (n) S_aAM-displacement, (o) S_aGM-displacement, (p) displacement dispersion for S_a set, (q) CAV_max-displacement, (r) CAV_AM-displacement, (s) CAV_GM-displacement, (t) displacement dispersion for CAV set, (u) GMRotD50-displacement, (v) GMRotD_max-displacement, (w) GMRotD_min-displacement, (x) displacement dispersion for GMRotD set.

Figure 9.

The dispersion for all considered IM sets.

Because the IDA can be used for estimation of seismic capacity of structures, the S_aGM(T₁,5%) seems preferable to other bidirectional IMs for this structure as the dispersion of IDA curves is less at both low and high values of IM. Thus, the remaining IDA curves are plotted only for S_aGM(T₁,5%) as IM.

Figure 10 shows the IDA curves using S_aGM(T₁,5%) as IM for individual records and their mean, median, 16%, and 84% fractiles. The median and mean IDA curves show a good agreement. Because there is no infinite value for DMs even at high values of IM, the mean representative curve is preferable among the others. All IDA curves have been plotted until S_aGM(T₁,5%) = 4.5g for different earthquakes. The first-mode spectral acceleration of 4.5g generates extensive damages to the RCC in all of the analyses that are indeed unrealistic.

Figure 10.

IDA curves using S_aGM(T₁,5%) as IM.

Limit states and capacity estimation

An IDA illustrates the demands imposed upon the structure by each ground motion record scaled to multiple intensities; it is usable for the definition of capacity and limit states of the structural model. For example, both DM- and IM-based methods can be employed for the determination of limit states of the structural model from IDA analysis. In DM-based methods, a certain value (C_DM) is defined, and when the DM passes this value, limit state is exceeded. Such values of C_DM can be obtained from experiments or field investigations. Unlike other structures, a definite value for C_DM has not been defined for the RCC building. A C_DM based on the maximum top displacement that accounts for damage can be a possible choice. The DM-based rules have the advantage of simplicity and ease of implementation, especially for performance levels other than collapse; however, even a unique C_DM value may imply multiple limit-state points on an IDA curve (Vamvatsikos and Cornell, 2002). Nevertheless, comparison of the RCC’s responses at various states makes possible the definition of such C_DM values. For example, from IDA analysis, the initiation of softening at the bottom of the RCC occurs at the maximum top displacements of 11.763 to 19.829 mm with the average of 14.298 mm. This value is comparable with the SPO result, that is, 13.2 mm (Table 1), with the difference of 8.3%. Because the dynamic analysis is more realistic, the C_DM = 14.298 mm can be a critical value for the limit state of RCC’s softening initiation with corresponding IM of S_aGM(T₁,5%) ≈ 0.5g (Figure 10). Initiation of crack at the bottom of the RCC occurs at the maximum top displacement of 17.542–33.214 mm with the average of 21.629 mm. The corresponding extracted value from the SPO analysis is 20.99 mm (Table 1), which differs 3.04% with the IDA result. Thus, C_DM = 21.629 mm can be a good choice for cracking initiation of the RCC performance with corresponding IM of S_aGM(T₁,5%) ≈ 0.7g (Figure 10).

The dispersion of the response results at each level of IM for various IDA curves is shown in Figure 11. The low dispersion of the results at corresponding selected IMs, that is, 0.5g and 0.7g, assures the quality of the selected C_DMs.

Figure 11.

Dispersion of IDA curves used in limit-state determination.

The alternative IM-based method is primarily intended for better assessment of collapse capacity because it generates a single point on the IDA curve that divides it into two regions, non-collapse and collapse. Definition of a critical value on IM (C_IM) that signals collapse for all IDA curves is more difficult than on DM (Vamvatsikos and Cornell, 2002). For example, the Federal Emergency Management Agency’s (FEMA) 20% tangent slope approach (FEMA, 2000) is in effect an IM-based method; the last point on the curve with a tangent slope equal to 20% of the initial elastic slope is assumed to be the capacity point. Such a flattening that is a typical final state in building’s IDA curves is not observed in the RCC’s IDA curves. However, there is extensive damage imposed on the RCC model at high intensities; the maximum top displacement IDA curves follow closely the equal displacement rule, that is, the inelastic section generally approximately follows the elastic section. Thus, the FEMA method is not applicable here.

The common incremental loading nature of the IDA study and the SPO implies an investigation of the connection between their results (Vamvatsikos and Cornell, 2002). Hence, estimation of the model’s capacity is investigated by comparison of the IDA and SPO results. For reasonable comparison of the IDA and SPO curves, it is required to establish a one-to-one mapping between the S_aGM(T₁,5%) in an IDA curve and the lateral load (base shear) in an SPO curve. This purpose is achieved by dividing the base shear by the mass of the model and then adjusting it so that the elastic segments of the SPO curve and the mean IDA curve lie on each other. Here, the SPO curve in the direction of Y under quadrangular load path is selected as it is the most conservative capacity curve in this study. Now, the two curves are traceable in Figure 12.

Figure 12.

Comparison of mean IDA and nonlinear SPO curves.

The IM values of the IDA curves are almost higher than the SPO curves at the same DM. It is because of cyclic application of loads in the SPO analyses that causes concentration of damage at specific regions. But in IDA analyses, the damages are not scattered in specific regions and more elements would participate in resisting the applied load.

The ultimate value of maximum top displacement in the SPO analysis is 95.13 mm (Table 1) that corresponds to S_aGM(T₁,5%) ≈ 3.30g in the IDA curve (Figure 12). It can be said that C_IM = 3.3g can be used as ultimate state of the RCC performance. It is worth noting that the damage energy dissipation of the crushing initiation of the SPO analyses, that is, E_F = 89.3 MJ for quadrangular load path, is comparable with the corresponding value from the mean IDA curve at C_IM = 3.3g, that is, 93.6 MJ (Figure 10). In addition, the selected C_IM has low dispersion from Figure 11. The recognized limit states of the RCC based on the explanations of this and previous sections have been shown in Figure 13.

Figure 13.

Limit states of the RCC model on IDA curves.

The IDA curves and correspondingly the limit-state capacities display large record-to-record variability as evident in the previous sections. This observed dispersion is closely connected to the selected IM; for example, it is proven that PGA_GM is deficient relative to S_aGM(T₁,5%) in expressing limit-state capacities as it increases their dispersion, as shown in Figure 9. For unidirectional IDA, using spectral accelerations at other periods, instead of the first-mode period, has shown that it often does not decrease the dispersion especially for first-mode-dominated short-period structures (Shome and Cornell, 2004). For bidirectional IDA using S_aGM(T,5%) as IM, this issue is shown for the dispersion of IDA curves of the RCC model in Figure 14, for three limit states of the RCC model. The lowest dispersion belongs to the first mode with the approximate value for different limit states. The dispersion curves vary with the similar pattern and show that S_aGM(T₁,5%) is the best choice among other S_aGM(T,5%) or even combination of them and that the dispersion is mostly affected by the value of S_aGM(T₁,5%), not by elastic spectral shape.

Figure 14.

Dispersion of IDA curves for various limit states versus the period.

Comparison with capacity estimation under unidirectional earthquakes

In order to generate unidirectional IDA curves, 24 ground motion records selected from Table 3 are applied to the RCC building. Each of the records has been scaled to multiple levels of S_a(T₁,5%) from 0g to 5.0g that have been arranged in 0.2g steps. After extraction of DMs and using the spline interpolation method, the entire IDA curve can be obtained under unidirectional earthquakes, as shown in Figure 15(a). By making a comparison between the SPO curve and the IDA curve under unidirectional earthquake excitations, the seismic capacity is obtained, as shown in Figure 15(b). It is found that the seismic capacity is represented using the S_a(T₁,5%) as IM and is 0.5g, 0.84g, and 4.3g for softening initiation, cracking initiation, and crushing initiation, respectively, which is about 1 times, 1.2 times, and 1.3 times larger than that under bidirectional earthquakes.

Figure 15.

(a) The IDA curves and (b) limit states of the RCC model under unidirectional earthquakes.

Conclusion

By extending the concepts of unidirectional SPO and IDA into the field of RCC buildings, structural performance, capacity, and limit states of such structures are investigated for bidirectional earthquake excitations. The detailed three-dimensional finite element model for the RCC is utilized. The damage-plastic model is considered for the nonlinear concrete behavior.

For SPO analysis, the structural capacities including the displacement and energy demands for the RCC building are obtained from three bidirectional loadings. For larger top displacements, the reduction in the strength of the RCC due to biaxial loading can reach 29.8%. For the ultimate state, the dissipated energy due to bidirectional cyclic loading is slightly lower than that for unidirectional cyclic loading. On the basis of the results of bidirectional SPO analyses, the lowest and most conservative seismic demand for each damage state is determined.

For IDA, 18 bidirectional earthquake intensity parameters are developed to perform the IDA for the RCC building. It is observed that geometric mean parameters as IM are much better than the corresponding arithmetic mean and max parameters, and S_aGM(T₁,5%) is the best choice for IM among the other selected IMs and geometric mean spectral acceleration at other periods (S_aGM(T,5%)). Two limit states (softening and cracking initiation) are determined based on the DM, and collapse limit state is determined based on the IM. Investigating the dispersion of the IDA results assures the quality of the defined limit states.

The comparison between the seismic capacity under unidirectional and bidirectional earthquakes shows that the seismic capacity of the RCC due to unidirectional earthquakes would be overestimated by 30% than that under bidirectional earthquakes.

It is concluded that bidirectional SPO and IDA studies can be used in performance evaluation and capacity estimation of RCC buildings for bidirectional earthquake excitations. The results of these methods well correlate with each other, especially for determination of the limit states of the RCC model.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This investigation is supported by the National Key R&D Program of China (2017YFC1500602), the National Natural Science Foundation of China (No. 51878130).These supports are greatly appreciated.

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