The influence of coupled horizontal–vertical ground excitations on the collapse margins of modern RCMRFs
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Abstract
With the increasing interest in vertical ground motions, the focus of this study is to investigate the effect of concurrent horizontal–vertical excitations on the seismic response and collapse fragilities of RC buildings designed according to modern seismic codes and located near active faults. It must be stressed that only mid to highrise buildings are of significant concern in the context of this research. The considered structures are categorized as intermediate and special RCMRFs and have been remodeled using distributed and lumped plasticity computational approaches in nonlinear simulation platforms, so that the utilized NL models can simulate all possible modes of deterioration. For better comparison, not only was the combined vertical and horizontal motion applied, but also a single horizontal component was considered for direct evaluation of the effect of the vertical ground motions (VGMs). At the member level, axial force variation and shear failure as the most critical brittle failure mechanisms were studied, while on the global level, adjusted collapse margin ratios (ACMRs) and mean annual frequency of collapse (λ _{Collapse}) using a new vectorvalued intensity measure were investigated. Findings from the study indicate that VGMs have significant effects on both local and global structural performance and cannot be neglected.
Keywords
Vertical excitation Vector IM Adjusted collapse margin ratios (ACMRs) Mean annual frequency (MAF) of collapse Seismic fragility Nonlinear (NL) modelsList of symbols
 d_{max,i}
Current deformation that defines the end of the reload cycle for deformation demand
 F_{i}^{+} and F_{i}^{−}
Deteriorated yield strength after and before excursion i, respectively
 F_{ref}^{+/−}
Intersection of the vertical axis with the projection of the postcapping branch
 F_{y}
Yield strength
 k_{1} and k_{u}
Constants specifying the lower and upper bounds in the vector IM
 K_{0}
Element stiffness at Δ_{cr}
 K_{1}
Element stiffness at Δ_{y}
 K_{2}
Element stiffness at Δ_{m}
 K_{deg}
Degrading slope of the shear spring based on the limitstate material
 K_{e}
Elastic (initial) stiffness of the element
 K_{rel}
Reloading stiffness of the element
 K_{s}
Slope of the hardening branch
 \(K_{ \text{deg} }^{\text{t}}\)
Degrading slope for the total response in OpenSees model
 K_{u,i} and K_{u,i−1}
Deteriorated unloading stiffness after and before excursion i, respectively
 K_{unloading}
Unloading slope of the rotational spring in the OpenSees model
 T_{1}
Dominant period of vibration for a specific structure
 T_{h}
Horizontal period of vibration for a specific structure
 T_{low} and T_{upp}
Lower and the upper periods of the elastic spectrum
 T_{v}
Vertical period of vibration for a specific structure
 V_{cr}
Shear force corresponding to displacement which causes concrete cracking
 V_{m}
Shear force corresponding to the maximum displacement
 V_{y}
Shear force corresponding to the displacement which causes steel yielding
 β_{c}, β_{DIM} and β_{M}
Uncertainties in capacity, demand and modeling
 β_{TOT}
Total uncertainty
 δ_{Ci}
Cap deformation at the ith cycle
 δ_{t,i}^{+/−}
Target displacement for each loading direction at the ith cycle
 Δ
Total deformation
 Δ_{cr}
Deformation at cracking
 Δ_{f}
Flexural deformation
 Δ_{m}
Maximum displacement
 Δ_{s}
Shear deformation
 Δ_{y}
Deformation at yielding
 Φ(.)
Standard normal cumulative distribution function
 ρ_{t}
Transverse reinforcement ratio in beams and columns
 λ_{Collapse}
Mean annual frequency of collapse
 λ_{IM} (x)
Mean annual frequency of the ground motion intensity exceeding x
 μ_{T}
Periodbased ductility
 χ_{c} and χ_{DIM}
Natural logarithm of the median capacity and demand of the structural system
Introduction
From a historical point of view, the horizontal component amplitude of ground motions normally plays a dominant role compared to the vertical counterpart. However, acceleration records from the (1989) Loma Prieta earthquake and the (1994) Northridge earthquake in the USA, the (1995) HyogokenKobe earthquake in Japan, (2003) Bam earthquake in Iran, and the (2011) Christchurch earthquake in New Zealand, among others showed that the magnitudes of the vertical component can be as large as, or exceed, the horizontal component. The report from Elnashai et al. (1995) also highlighted cases of brittle failure induced by direct compression, or by reduction in shear strength and ductility due to variation in axial forces arising from the vertical motion in the (1994) Northridge earthquake. In such situations, most existing code specifications assume that the ratio of vertical component of the ground motion to that of the horizontal component (V/H) varies from 1/2 to 2/3, which must be considered unconservative and needs to be investigated.

The attenuation rate for vertical ground motion is much higher than that of the horizontal ground motion. This rate increase in the farfield areas. Thus, structures built in the nearfault regions experience higher vertical excitations.

Vertical ground motion includes more highfrequency content than horizontal ground motion. The difference increases with the decrease in the soil stiffness.
It should also be noted that the higher values of V/H ratio do not necessarily imply more energy content on the desired structure. The reason is that the two components may not coincide in time to cause strong interaction effects.
Besides these, many of the current seismic design codes and damage estimation tools do not include the effect of vertical ground motions on the seismic response of structures and especially columns. However, the observed damage on the columns (diagonal shear cracks) during historical seismic events such as the 1994 Northridge earthquake and the 1995 Kobe earthquake was partly attributed to the effect of vertical motions (Broderick et al. 1994; Elnashai et al. 1995). Field and analytical evidence by Papazoglou and Elnashai (1996) indicated that strong vertical earthquakes can cause a significant fluctuation in the axial force in columns, resulting in a reduction in their shear capacity and compression failure of some of the columns. During the 1995 Kobe earthquake in Japan, the RC structures exhibited very high amplifications of the vertical component of more than two times. The main reasons were the low damping mechanism in the vertical direction and the absence of supplement seismic energy dissipating systems in this direction. On the other hand, because of the high stiffness in the vertical direction, a quasiresonant response was observed in these structures. Highfrequency pulses from vertical motion were recognized as the other reason for such a phenomena (AIJ 1995).
Iyengar and Shinozuka (1972) investigated the effect of selfweight and vertical accelerations on the behavior of tall structures. The structures have been idealized as cantilevers and the ground motion as a random process. The main conclusion from their study was that inclusion of selfweight simultaneously with the vertical ground acceleration can increase or decrease the global peak response. These fluctuations in structural response had been considerable in most cases. On the local level, beams were identified as the most critical elements and the effect of vertical ground motion on them had been pronounced. Iyengar and Sahia (1977) investigated the effect of vertical ground motion on the response of cantilever structures using the mode superposition method; their main conclusion is that the consideration of the vertical component is essential in analyzing towers. Anderson and Bertero (1977) used numerical methods to evaluate the inelastic response of a tenstory unbraced steel frame subjected to a horizontal component of earthquake and to combinations of this component with the vertical one; they deduced the following points; the inclusion of the vertical motion on one hand does not increase the displacements, but on the other increases the girder ductility requirement by 50 % and induces plastic deformations in columns. Mostaghel (1974) and Ahmadi (1978, 1980) studied the effect of vertical motion on columns and tall buildings which have been idealized as cantilevers, using the mathematical theory of stability of Liapunov. Their main conclusion was that, in the inelastic region, if the maximum applied earthquake loading would be less than the Euler buckling load, it is guaranteed that the column will remain stable irrespective to the type of earthquake loading, and the inclusion of vertical ground excitation can be neglected. But this is unlikely to be the case for reinforced concrete columns because of the crushing of concrete in compression and the buckling of the yielded reinforcement.
Munshi and Ghosh (1998) investigated the seismic performance of a 12story RC building under a combination of horizontal and vertical ground motions. This analysis showed a slight increase in the maximum deformation when the vertical ground motion was included. The formation patterns revealed that vertical accelerations induced a slightly different hinge formation pattern and hinge rotation magnitude, and the response of the frame–wall system did not show sensitivity to the vertical acceleration in this case. Antoniou (1997) studied the effect of vertical accelerations on RC buildings by analyzing a eightstory reinforced concrete building designed for high ductility class in Euro code (EC8) with a design acceleration of 0.3 g. This analysis showed that the vertical ground motion can increase the compressive forces by 100 % or even more and lead to the development of tensile forces in columns. These fluctuations in axial forces can result in shear failure in these elements.
Kim et al. (2011) studied the effect of various peak V/H ground acceleration ratios and the time lag between the arrival of the peak horizontal and vertical accelerations on the inelastic vibration period and column response for infrastructures. It was observed that the inclusion of vertical motions notably influenced the inelastic response vibration periods and considerably increased or decreased the lateral displacement. It was also noticed that the arrival time had a minimal effect on the axial force variation and shear demand.
None of the previous studies have investigated the codeconforming RCMRFs utilizing fragility curves and reliability methods. As the seismic vulnerability assessment of highrise structures is a complex task, it is important to consider that both the lower and higher structural modes might be excited, because of the wide range of frequency content of the applied earthquake loads. On the other hand, the imposed displacements to these structures can be very significant, since the fundamental period of many highrise structures are within the period range of 1–5 s, which corresponds to the peak displacement spectra of the standard earthquakes. To this end, in the current study, both distributed fiberbased and lumped plasticity approaches and various modes of collapse are considered in the simulation process. To show the significance of vertical ground excitations and to get the most accurate results, a new definition for V/H ratio and an optimum intensity measure are proposed. Various mass distributions are considered in the eigenvalue analysis to determine the most accurate and computationally efficient structural model.
Seismic fragility curves as the main component of the current study can be derived using various approaches: observational, experimental, analytical and hybrid techniques to quantify damage and estimate monetary losses (Calvi et al. 2006). While the observational method is the most realistic and rational one, as the entire inventory is taken into consideration it is usually difficult to be utilized because no or insufficient observationalbased date are available from the past events. The experimental method is not a feasible option in many cases, because of its cost and the time needed, since a wide range of structures should be tested. In the current study, the third approach based on extensive nonlinear analytical simulations is adopted. This option is the most feasible and possible methodology which can be used in many cases.
The main objective of this study is to calculate the collapse margins and mean annual frequency of collapse as the performance metrics employing displacementbased fragility curves for multiple limit states from concrete cracking to structural collapse in the nearfault areas. The collapse of structures is determined on the basis of the global failure mechanism of the structural system rather than the failure of a structural element. To achieve this goal, numerical models that capture the axial–shear–flexural behavior of the columns are created in nonlinear seismic simulation platforms, ZeusNL and OpenSees (Elnashai et al. 2004; McKenna 2014).
Selection and characterization of input ground motions
Input ground motions used for the nonlinear response history analyses (NLRHA)
No.  Earthquake name  Date  Station name  Moment magnitude, M _{w}  Epicentral distance, (km)  (PGA)_{H}, g  (PGA)_{V}, g 

1  Wenchuan, China  2008  Wenchuanwolong  7.90  19.54  0.77  0.96 
2  ChiChi, Taiwan  1999  TCU078  7.62  4.96  0.38  0.17 
3  ChiChi, Taiwan  1999  TCU089  7.62  14.16  0.75  0.34 
4  ChiChi, Taiwan  1999  TCU079  7.62  15.42  0.59  0.42 
5  Kocaeli, Turkey  1999  Izmit  7.51  5.31  0.19  0.14 
6  Kocaeli, Turkey  1999  Yarimca  7.51  19.30  0.29  0.24 
7  Tabas, Iran  1978  Dayhook  7.35  20.63  0.33  0.19 
8  Landers, USA  1992  Joshua Tree  7.28  13.67  0.27  0.18 
9  Landers, USA  1992  Morongo Valley Fire Station  7.28  21.34  0.19  0.16 
10  Duzce, Turkey  1999  Duzce  7.14  1.61  0.43  0.35 
11  Duzce, Turkey  1999  Lamont 1058  7.14  24.05  0.68  0.19 
12  Duzce, Turkey  1999  IRIGM 487  7.14  24.31  1.00  0.33 
13  Golbaft, Iran  1981  Golbaft  7.00  13.00  0.28  0.24 
14  Darfield, New Zealand  2010  GDLC  7.00  4.42  0.73  1.25 
15  Darfield, New Zealand  2010  HORC  7.00  10.91  0.47  0.81 
16  Loma Prieta, USA  1989  Corralitos  6.93  7.17  0.50  0.46 
17  Loma Prieta, USA  1989  BRAN  6.93  18.46  0.59  0.90 
18  Loma Prieta, USA  1989  Capitola  6.93  20.35  0.44  0.14 
19  Kobe, Japan  1995  NishiAkashi  6.90  8.70  0.47  0.39 
20  Kobe, Japan  1995  IWTH26  6.90  13.12  0.67  0.28 
21  Kobe, Japan  1995  Takatori  6.90  19.25  0.32  0.57 
22  Nahanni, Canada  1985  Site 2  6.76  6.52  0.40  0.67 
23  Nahanni, Canada  1985  Site 1  6.76  6.80  1.16  2.28 
24  Northridge, USA  1994  Rinaldi Receiving  6.69  5.41  1.64  1.05 
25  Northridge, USA  1994  ArletaNordhoff Fire Sta  6.69  8.48  0.75  0.32 
26  Northridge, USA  1994  LA Dam  6.69  20.36  1.39  1.23 
27  Niigata, Japan  2004  NIG019  6.63  4.36  1.26  0.80 
28  Niigata, Japan  2004  NIG020  6.63  21.52  1.48  0.57 
29  Bam, Iran  2003  Bam  6.60  12.59  0.74  0.97 
30  Zarand, Iran  2005  Zarand  6.40  16.00  0.31  0.30 
31  Imp. Valley, USA  1979  Bonds Corner  6.53  6.19  0.69  0.53 
32  Imp. Valley, USA  1979  Calexico Fire Station  6.53  19.44  0.31  0.25 
33  Imp. Valley, US  1979  Chihuahua  6.53  24.82  0.17  0.21 
34  Christchurch, New Zealand  2011  Heathcote Valley Primary School  6.20  1.11  1.39  2.18 
35  Christchurch, New Zealand  2011  LPCC  6.20  4.89  0.65  1.90 
36  Morgan Hill, USA  1984  Halls Valley  6.19  16.67  0.35  0.21 
37  Morgan Hill, USA  1984  Zack Brothers Ranch  6.19  24.55  0.94  0.39 
38  Talesh, Iran  1978  Talesh  6.00  15.00  0.23  0.13 
39  Parkfield, USA  2004  ParkfieldStone Corral 1E  6.00  7.17  0.72  0.33 
40  Parkfield, USA  2004  ParkfieldStone Corral 2E  6.00  9.28  0.83  0.72 
Representative set of structures
Four RCMRFs ranging from 7 to 20 stories are selected to represent medium and highrise buildings. The frames are designed and detailed according to ACI building code ACI318 (2011) and ASCE 7 (2010) provisions. Two categories of RCMRFs, special and intermediate, are used in the current study. The ordinary MRFs, because of their low level of ductility during an earthquake, are not considered here. The special MRF employs the strong column weak beam (SCWB) concept and specifies elaborate detailing of joints. Thus, the SMRF is expected to form the sway mechanism and possesses a high degree of ductility. On the other hand, the intermediate MRF has enough strength as well as reasonable ductility and can be used throughout most of the seismicprone areas. 7 and 12story buildings are designed as intermediate MRFs, while 15 and 20story buildings are designed as special MRFs. The behavior factors (R) are considered as 5 and 8, respectively (ASCE 7 2010).
Frame element sizes and reinforcements details
Member specifications  Reference structures and stories range  

20S4B  15S4B  12S3B  7S3B  
1–5  6–10  11–15  16–20  1–5  6–10  11–15  1–4  5–8  9–12  1–4  5–7  
Beam  
b (cm)  60  50  45  35  55  45  35  50  40  35  45  35 
h (cm)  90  70  60  45  75  60  45  60  50  45  50  45 
ρ _{l} (%)  2.2  2.2  1.8  1.8  2.0  1.9  1.7  2.0  2.0  1.9  2.2  2.0 
ρ _{t} (%)  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0 
Exterior column  
b (cm)  100  80  65  50  85  65  50  65  55  45  50  40 
h (cm)  100  80  65  50  85  65  50  65  55  45  50  40 
ρ _{l} (%)  2.1  1.9  1.9  1.8  2.0  1.8  1.8  2.1  1.8  1.7  1.9  1.8 
ρ _{t} (%)  2.0^{a} 1.4  1.4  1.4  1.4  2.0^{a} 1.4  1.4  1.4  2.0^{a} 1.4  1.4  1.4  2.0^{a} 1.4  1.4 
Interior column  
b (cm)  120  90  75  60  95  75  55  75  60  50  60  45 
h (cm)  120  90  75  60  95  75  55  75  60  50  60  45 
ρ _{l} (%)  1.8  1.7  1.5  1.5  1.8  1.5  1.5  1.7  1.7  1.6  1.9  1.7 
ρ _{t} (%)  2.0^{a} 1.6  1.6  1.6  1.6  2.0^{a} 1.6  1.6  1.6  2.0^{a} 1.6  1.6  1.6  2.0^{a} 1.6  1.6 
Structural modeling approaches
In general, most of the current models provided by other researchers cannot be utilized to predict the accurate behavior of the RC elements in the presence of vertical excitations. The main reason is that such models do not account for important response features, such as the interaction between shear, flexure and axial forces. One of the main failure modes of RC columns due to vertical excitations as mentioned previously is the shear failure in these members. To include this type of failure in the analytical models, some modification should be incorporated in the modeling approaches. To this end, two nonlinear simulation platforms; ZEUSNL (Elnashai et al. 2004) and OpenSees (McKenna 2014), are used to simulate the seismic response of the reference structures. Fixedbase models are used in the analysis stage; as a result, soil–structure foundation interaction is neglected. A leaning column to account for the PΔ effect from loads on the gravity system is also considered in both platforms.
Watanabe and Ichinose (1992), Aschheim and Moehle (1992), Priestley et al. (1994) and Sezen (2002) have shown that the shear strength in RC elements decays with increased plastic deformations. Hence, the dashed line provided in Fig. 8 cannot be realistic and accurate, if the column yields in flexure close to its estimated shear strength. The main deficiency of the hysteretic uniaxial model is that it determines the point of shear failure based only on the column shear, while it should be determined based on both force and deformation.
To resolve this problem, the shear load versus deformation model for the shear spring was developed using the existing OpenSees limitstate material model and shear limit curve developed by Elwood (2004). As shown in Fig. 8, the shear limit curve is activated and shear failure is initiated once the column shear demand exceeds the column shear capacity. In this case, the limitstate material model simulates and captures the RC column response to detect the possibility of shear failure. To this end, the shear limit curve is defined according to both the column shear and the total displacement or drift ratio (Fig. 8). In case the columns are vulnerable to shear failure after flexural yielding, then the drift capacity model proposed by Elwood and Moehle (2005) can be utilized to define the accurate limit curve. Other shear failure criterion such as plastic rotation at the two ends of the column is also proposed by LeBorgne (2012).
Basic strength degradation This mode of deterioration is defined in a way to show a reduction in the yield strength. This mode also includes the strain hardening slope degradation. Based on the formulation, the values for positive and negative directions are defined independently (Fig. 9a).
Postcapping strength degradation It is defined by translating the postcapping branch toward the origin; however, the branch slope will keep constant and it will move inward to reduce the reference strength. The process will be done each time the X axis is crossed (Fig. 9b).
Unloading stiffness degradation This mode of deterioration indicates the reduction in both positive and negative unloading stiffness. The unloading stiffness (K _{ u,i }) in each cycle depends on the unloading stiffness in the previous excursion (K _{ u,i−1}) (Fig. 9c).
Accelerated reloading stiffness degradation This mode of deterioration escalates the target displacement according to the loading trend in both positive and negative directions (Fig. 9d).
Eigenvalue analysis
Vertical and horizontal periods of the case studies considering various mass models
Reference structures  Mass model  Horizontal period (s), T _{h}  Vertical period (s), T _{v} 

7 Story  LMass 1  0.79  0.36 
LMass 2  0.79  0.17  
LMass 3  0.77  0.13  
LMass 4  0.76  0.12  
Distributed mass  0.74  0.11  
12 Story  LMass 1  1.38  0.41 
LMass 2  1.35  0.22  
LMass 3  1.35  0.18  
LMass 4  1.34  0.17  
Distributed mass  1.32  0.15  
15 Story  LMass 1  1.81  0.53 
LMass 2  1.81  0.29  
LMass 3  1.79  0.22  
LMass 4  1.78  0.19  
Distributed mass  1.78  0.18  
20 Story  LMass 1  2.53  0.67 
LMass 2  2.51  0.38  
LMass 3  2.50  0.29  
LMass 4  2.49  0.27  
Distributed mass  2.45  0.24 
Compared to LMass 1, LMass 2 is much more accurate. However, there are clearly differences in the vertical periods. LMass 3 and LMass 4 show both significantly similar periods in the horizontal and vertical directions. Even though LMass 3 has a coarse lumpedmass approach compared to the distributed mass model, the eigenvalues of this model imply very similar tendencies to the exact solution. As a result, because of the small differences in LMass 3 compared to LMass 4 results, LMass 3 is the most simplified lumpedmass model to cover realistic vertical motion with minimum computational effort and is implemented in the NLRHA of the studied structures to provide fragility curves.
V/H ratio (conventional and proposed approaches)
Performance criteria
In this study, interstory drift ratio (IDR) is considered as the engineering demand parameter (EDP). This is particularly a suitable choice for RCMRFs, since it relates the global response of the structure to joint rotations where most of the inelastic behavior in the momentresisting frames is concentrated.
Performance levels according to FEMA356 and NLpushover analysis in terms of IDR (%)
Reference structures  Immediate occupancy (IO)  Life safety (LS)  Collapse prevention (CP)  

FEMA356  Pushover  FEMA356  Pushover  FEMA356  Pushover  
7S3B  1  0.87  2  2.03  4  3.42 
12S3B  1  0.93  2  2.27  4  3.78 
15S4B  1  1.12  2  2.29  4  4.08 
20S4B  1  1.27  2  2.38  4  4.25 
Besides the limit states mentioned above, the collapse limit state can be defined on the basis of a different approach. The onset of ‘collapse’ for a ground motion record is identified as the point where maximum IDR response increases ‘drastically’ when the spectral acceleration of the record is increased by a ‘small’ amount (Vamvatsikos and Cornell 2002). To this end, in the next stages, incremental dynamic analysis (IDA) is performed on the reference MRFs and the collapse is defined as the point of dynamic instability when IDR increases without bounds for a small increase in the ground motion intensity for each structure individually.
Proposing an optimum intensity measure for fragility analysis
The ground motions are characterized by intensity measures (IMs), and their choice plays a crucial role in the seismic fragility estimation. An optimal IM is the one that has good efficiency, sufficiency, practicality, hazard computability and predictability among other characteristics (Mackie and Stojadinovic 2001; Luco and Cornell 2007; Giovenale et al. 2004; Padgett et al. 2008). Efficiency means the ability to accurately predict the response of a structure subjected to earthquakes (i.e., small dispersion of structural response subjected to earthquake ground motions for a given IM). A sufficient IM is defined as one that renders structural responses subjected to earthquake ground motions for a given IM conditionally, independent of other ground motion properties (i.e., no other ground motion information is needed to characterize the structural response). Previous studies have shown that PGA is not an accurate and ideal IM for evaluating the geotechnical phenomenon, as it cannot consider the ground motion duration (Kramer et al. 2008). It is found that the fragility curves based on vectorvalued IM are better able to represent the damage potential of earthquake (Baker 2015). Thus, an optimized vectorvalued intensity measure, which includes the geometric mean of spectral accelerations over a range of period, is considered in the current study. The parameters in the vectorvalued intensity measure should be chosen to convey the most possible information between the ground motion hazard and the structural response stages of analysis. This requires identifying parameters that most affect the structure under consideration.
As can be seen from Fig. 14, S _{a}(T _{1}) can be an efficient predictor for structures with short period, but not for the structures with medium to large periods. On the other hand, the vector IM can predict the results in an efficient way for both short and large period structures; however, the results have more standard deviations compared to the proposed IM. In all cases, IM (New) has been more efficient and sufficient, as it is always associated with small values of dispersion. The maximum dispersion values are 20, 31 and 50 % for IM (New), IM (Vector) and S _{a}(T _{1}), respectively.
Probabilistic demand models
NLincremental dynamic analyses and seismic fragility estimate
In the fragility estimation process, a suitable analysis procedure should be implemented. Based on the recommendations provided by Vamvatsikos and Cornell (2002) and FEMA P695 (2009), incremental dynamic analysis (IDA) is recognized as one of the best and most common procedure, where a suite of earthquake records is scaled repeatedly to find the intensity measure in which the structural collapse will occur. In this study to isolate the effect of each component, horizontal and vertical ground motions were applied separately and simultaneously.
Using the IDA approach, information about variability in ground motions can be directly incorporated into the collapse performance assessment. However, this process only captures the recordtorecord variability and does not account for how well the nonlinear simulation model represents the collapse performance of the reference structures; hence, model uncertainties should also be accounted in the collapse simulation, which will be discussed later.
Effect of vertical excitations on the structural responses (local level)
Seismic performance evaluation of RC structures subjected to (medium to strong) multicomponent ground motions has some complexities. One of these difficulties is the increment in axial force variation in RC columns, which can be superimposed in the overturning forces. Since there is a direct relation between flexural, shear and axial forces, the fluctuation in axial force increases the possibility of shear failure. This is mainly due to the significant variation in strength and stiffness in the columns caused by vertical excitation. NLRHA was performed using the selected natural horizontal ground motions applied with and without the vertical component.
Averaged ratio of vertical seismic force to gravity load force for all the structures
Reference structure  V/H ratio  Horizontal only  Combined (H + V)  Increment due to vertical excitations (%) 

7S3B  1.0  1.53  1.92  25.49 
1.5  1.62  2.45  51.23  
2.0  1.28  2.67  108.59  
12S3B  1.0  1.92  2.18  13.54 
1.5  1.68  1.92  14.29  
2.0  2.15  2.24  4.19  
15S4B  1.0  1.78  2.10  17.98 
1.5  1.93  2.19  13.47  
2.0  1.67  2.78  66.47  
20S4B  1.0  1.78  2.27  27.53 
1.5  1.95  2.34  20.00  
2.0  2.52  2.95  17.06 
Seismic fragility estimation (global level)
For the case of earthquake excitations, a closedform solution is not usually available, since there are a large number of random variables and different probability density functions associated with these events. To resolve this problem, the reliability of the structures under these complex phenomena can be represented using a probabilistic methodology incorporating fragility curves. These curves define the probability of exceeding a specific damage state subjected to a hazard by a suitable IM.
Based on Figs. 18 and 19, when the vertical component is coupled with the horizontal excitations, the fragility results can be changed extensively. The maximum increase in the structure’s fragility appeared in the CP and Collapse damage modes. Figure 20 shows the developed fragility surface based on a vectorvalued IM and can be visualized as fragility curves by projecting the surface onto the planes. These figures demonstrate the wide variation between fragility curves based on scalarvalued intensity measure.
It can be seen in Fig. 20 that there is a discrepancy of up to 30 % between the curves calculated for various HM values. The main advantage of these fragility surfaces is that the variability of structural fragility due to a second parameter can be accounted for in contrast to when fragility curves are used. This means that which records should be used depends on the seismic hazard at the site when scalarvalued IM [S _{a}(T _{1})] is used to evaluate the structural fragility.
Ignoring the effect of HM will bias the final results. For example, if the seismic hazard disaggregation suggests that extreme motions are associated with records having a mean value of HP of about 0.75, but records are selected with a mean value of about 0.50, then the S_{a}(T _{1})based result will underestimate the seismic fragility. In other words, the evaluation of structural fragility by means of vectorvalued IMs reduces the complexity of record selection procedure based on the seismic environment (i.e., magnitude, distance, site conditions, etc.).
Collapse performance evaluation
Prior to the development of incremental dynamic analysis (IDA) and its use in the FEMA P695 methodology (2009), accurate modeling of buildings near collapse, in the negative postpeak response range, was not a high priority of research. In recent FEMA guidelines (e.g., FEMA P440A 2009; FEMA P695 2009), sideway collapse (where collapse is defined based on unrestrained lateral deformations with an increase in ground motion intensity) is typically assumed to be the governing collapse mechanism.
Comparing the intermediate and special MRFs located in the same hazard region, Fig. 21 indicates that the 20 and 15story buildings have better collapse behavior compared to the 7 and 12story buildings. The reason for this behavior is due to the higher ductility level in SMRFs.
Summary of final collapse margins and comparison to acceptance criteria
Reference structure  SSF  S _{a}(T _{1})_{@MCE}  Loading type  S _{a}(T _{1})_{@50%Col.}  CMR  ACMR  Acceptance ACMR  Performance 

7S3B  1.31  1.24  H  1.31  1.06  1.38  2.28  Fail 
H + V  0.88  0.71  0.93  2.28  Fail  
12S3B  1.35  0.73  H  1.25  1.71  2.31  2.28  Marginal pass 
H + V  0.81  1.11  1.50  2.28  Fail  
15S4B  1.61  0.50  H  1.85  3.70  5.96  2.28  Pass 
H + V  1.43  2.86  4.60  2.28  Pass  
20S4B  1.61  0.37  H  1.91  5.16  8.31  2.28  Pass 
H + V  1.76  4.76  7.66  2.28  Pass 
Focusing on the four reference structures, Table 6 shows that SMRFs have acceptable ACMR, while a disturbing trend becomes evident for the IMRF buildings. The results show that in terms of coupled horizontal–vertical excitation, both 7S3B and 12S3B buildings have unacceptable ACMR, and surprisingly 7S3B would also fail for the horizontal excitation case while the 12story frame passed the criteria marginally. As a result, the IMRFs do not attain the collapse performance required by FEMA P695 methodology, and additional design requirement adjustments would be needed to improve the overall performance. It means that even the codeconforming design structures with acceptable level of ductility can be vulnerable to seismic excitations.
Comparison of the calculated ACMRs in terms of (H) and (H + V) excitations demonstrates that generally in all models, the safety margin against collapse reduces by including the vertical component of earthquake and the reductions are very remarkable and pronounced. Fortunately, the SMRF models could pass the collapse performance criteria for both types of excitation. It shows that their elements, being controlled by many detailing and capacity design requirements of the building code, limit possible failure modes.
Mean annual frequency (MAF) of collapse
Based on the extracted results in Fig. 23, the λ _{Collapse(max)} in the intermediate MRFs are 2.1 × 10^{−5} and 1.86 × 10^{−4} collapse/year for the horizontal and combined H + V, respectively. However, these values are relatively smaller for the special MRFs under both types of excitations. The maximum values for the SMRFs are 2.3 × 10^{−6} and 4.5 × 10^{−5} collapse/year for the horizontal and combined H + V, respectively. It is worth mentioning that these results correspond to collapse return periods of 2475 years. Based on MAF calculation, SMRFs are in a higher confidence bounds of safety compared to the intermediate moment frames, while the effect of vertical ground motion is very significant for both groups and cannot be neglected.
Conclusions and recommendations
In this study, the effect of vertical excitations on the seismic performance of intermediate and special RCMRFs has been evaluated. It is important to emphasize that the main objective of this study was not to quantify numerical design values. The objective was rather to focus on the importance or otherwise of including vertical ground motion in design of RC buildings and its impact on the member and the structure levels. Hence, a large number of NL dynamic analyses were performed using fiberbased and concentrated plasticity approaches. The computational models were utilized in ZEUSNL and OpenSees platforms to compare the results. The VGM was shown to be significant and should be included in the analysis when the proposed structure is located within 25 km of a seismic source.

The vertical component of an earthquake tends to concentrate all its energy content in a narrow band, unlike the horizontal counterpart. This energy concentration can be very destructive for the (mid and high)rise RCMRFs with vertical periods in the range of vertical components periods. Extracted ACMR values and fragility curves proved that the intermediate RCMRFs are very vulnerable and need major revision in their design stage, while the SMRFs can well resist both horizontal and vertical seismic excitations.

Based on the frequency content of the vertical ground motion, it can be concluded that structural failure modes from past earthquakes might be attributed to underestimating the effect of vertical acceleration in the design procedures, and there is an urgent need for the adoption of more realistic vertical spectra in future version of seismic design codes.

Although the effect of axial force on shear capacity of the structural elements is an accepted fact and is proved in the current study, current seismic codes do not have a consensus on this effect, and different code equations might lead to different shear capacity estimations. Both the ACI318 and ASCE41 equations captured the shear strength degradation due to axial force. ASCE equation predictions could be considered as accurate, because the strength reduction caused by ductility demand could be more significant than that by tension.

As the V/H ratio increases, more fluctuations can be observed in the columns axial force. This phenomenon leads to a significant reduction in the shear capacity in the range of (15–30) %. This reduction in shear capacity of the vertical members increases the potential for shear failure.

Geometric nonlinearities, in terms of the deformed configuration of the system, do not come into play in either IO or LS damaged states of the system. However, PΔ effects due to higher interstory drifts of combined H + V excitations do influence the response of the building in the region near collapse and must be taken into account.

A new vector IM is proposed in the current study to predict accurate fragility results. One of the main advantages of the proposed IM is its hazard compatibility, in which a GM prediction model can be easily developed for the second parameter (HM), implementing the existing attenuation models with an arbitrary set of periods. In case a probabilistic seismic hazard analysis (PSHA) would be required, the calculations can be performed using Ln (HM) as IM in the same way as any single spectral acceleration value.

The results presented in this study indicate that coupling horizontal and vertical ground excitations increases the ductility demand. Therefore, it is highly suggested that the conventional response spectrum (RS) be replaced with a multicomponent RS in the next version of seismic design codes. Doing this will consider the effect of vertical ground excitations on the enhanced seismic demand and will provide a better understanding to structural designers.
Taking into account the above observations, the authors would like to recommend for the next version of seismic codes to make sure that the structures locate within 25 km from the active faults be designed to the combined effect of horizontal and vertical ground motions.
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