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Seismic collapse performance of special moment steel frames with torsional irregularities
Engineering Structures 141 (2017) 482–494
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Seismic collapse performance of special moment steel frames with torsional irregularities Sang Whan Han ⇑, Tae-O Kim, Dong Hwi Kim, Seong-Jin Baek Department of Architectural Engineering, Hanyang University, Seoul 133-791, Republic of Korea
a r t i c l e
i n f o
Article history: Received 24 February 2016 Revised 16 March 2017 Accepted 20 March 2017
Keywords: Torsional irregularity Inelastic response Ground motion Collapse performance Story drift Nonlinear response history analysis
a b s t r a c t Many building structures exhibit torsional irregularity, which is a type of horizontal irregularity. It is difficult to estimate the inelastic response of torsionally irregular structures subjected to earthquake ground motions using numerical analyses because torsionally irregular structures experience both lateral displacement and floor rotation. This study evaluates the collapse performance of multi-story model structures with various degrees of torsional irregularity via nonlinear response history analyses. This study also proposes a procedure for computing design story drift demands, which allows torsionally irregular structures to have uniform collapse risk irrespective of the degree of torsional irregularity. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Torsional irregularities are a type of horizontal irregularity (Table 12.3-1, ASCE 7-10 [1]) that are induced by an asymmetric distribution of mass, stiffness and (or) strength. Since torsionally irregular structures experience both lateral displacement and floor rotation during an earthquake, they sustain greater member forces and drifts compared to regular structures. Without careful design consideration, torsionally irregular structures may be more vulnerable to earthquakes than regular structures. To reduce the vulnerability of torsionally irregular structures, current seismic codes specify more stringent requirements for torsionally irregular structures than regular structures; in other words, torsionally irregular structures must be designed to satisfy the design requirements for regular structures as well as additional requirements for torsionally irregular structures. Many studies have been conducted to investigate the effect of torsional irregularity on the seismic response of single-story torsionally irregular structures designed according to seismic design provisions (Chopra and Goel [2]; Tso and Wong [3]; Humar and Kumar [4]; Dutta and Das [5]; Aziminejad and Moghadam [6]; Herrera and Soberon [7]). Although it is convenient to draw insight about the effect of torsional irregularity by analyzing simple single-story structures, there are limitations when it comes to ⇑ Corresponding author. E-mail address: [email protected] (S.W. Han). http://dx.doi.org/10.1016/j.engstruct.2017.03.045 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.
extrapolate the results from single-story structures to multi-story torsionally irregular structures (Stathopoulos and Anagnostopoulos [8]). Recently, due to the development of high-performance computers and software, the seismic behavior of multi-story structures with torsional irregularity has been investigated by conducting nonlinear response analyses (Jeong and Elnashai [9]; Stathopoulos and Anagnostopoulos [10]; Reyes and Quintero [11]). De-la-Colina [12] investigated the seismic behavior of multi-story torsionally irregular structures, and proposed design recommendations to control the ductility demands. DeBock et al. [13] investigated the seismic collapse performance of multi-story reinforced concrete frame structures using sophisticated plastic models that accounted for cyclic and in-cyclic deterioration in order to evaluate the design recommendations related to accidental torsion. The objective of this study is to evaluate the seismic collapse performance of multi-story structures with various degrees of torsional irregularity, which are designed according to current seismic design provisions. For this purpose, three- and nine-story model structures with various degrees of torsional irregularity are designed according to ASCE 7-10 [1], AISC 341-10 [14] and AISC 360-10 [15], in which special moment steel frames are used as seismic force resisting systems. Incremental dynamic analyses are conducted for the model structures with repeated nonlinear response history analyses using a three-dimensional inelastic analytical model. Based on the analysis results, this study evaluates the collapse performance of the model structures, and proposes a
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procedure for computing the design story drift demand in an attempt to allow torsionally irregular structures to have uniform collapse risk. In order to monitor the change in member sections of torsionally irregular structures according to the design stages, this study also estimates the total steel weight of the moment frames in model structures at each design stage. 2. Summary of design process for torsionally irregular structures In ASCE 7-10 (Table 12.3-1), when the maximum story drift at one end of the structure is more than 1.2 times the average story drift at both ends of the structure, the structure is classified as being torsionally irregular. When the drift ratio is greater than 1.4, the structure is considered to have extreme torsional irregularity. When a structure has torsional irregularity, the structure should be designed to satisfy the design requirements for regular structures as well as additional requirements for torsionally irregular structures, as specified in ASCE 7-10 [1]. The seismic design process for torsionally irregular structures is similar to that of regular structures. In this study, the design process for torsionally irregular structures is separated into three stages: (1) strength design stage, (2) drift (stiffness) checking stage, and (3) stability (P Deffect) checking stage. Fig. 1 illustrates the schematic flow of the seismic design process for torsionally irregular structures that is used in this study. The additional requirements specified in ASCE 7-10 are: (1) increases in forces due to irregularities for SDC D, E and
Stage 1: Strength design
Stage 2: Drift check
F (12.3.3.4), (2) 3-D modeling for structures having torsional irregularity type 1a, 1b, 4, or 5 of Table 12.3-1 in ASCE 7-10 (12.7.3), (3) amplification of accidental torsion for structures assigned to SDC C through F (12.8.4.3), (6) analysis procedure selection for torsionally irregular structures (12.6), and (7) story drift calculation using the largest difference of the deflections of vertically aligned points at the top and bottom of the story along any edges of the structure (12.8.6). The numbers in parentheses are the section numbers of ASCE 7-10 [1] for each of the additional requirements. 2.1. Strength design In the strength design stage, seismic design forces are calculated to determine the member section. Three different analysis procedures are permitted in ASCE-7-10 [1]: (1) equivalent lateral force analysis, (2) modal response spectrum analysis, and (3) response history analysis. A three-dimensional analytical model is required for the analysis of torsionally irregular structures assigned to all levels of SDCs, with the exception of SDC A. For torsionally irregular structures assigned to SDCs D, E, and F, the equivalent lateral force analysis procedure is not permitted. In this study, a modal response spectrum analysis method that has no limitations according to the level of SDCs is used.For torsionally irregular structures, both inherent and accidental torsions should be properly accounted for when calculating the member forces and drifts. The inherent torsion results from an asymmetric distribution of mass, stiffness, and strength, whereas accidental torsion is induced by uncertainties in the distribution of mass, stiffness, and strength. The accidental torsional moment is calculated using
Determine seismic design forces
Modal spectral response analysis
Calculate member force demands and select the member sections
If the structure is torsional irregular, amplify accidental torsional moment
Calculate story drift Δ (If torsional irregular, Δ shall be computed as the largest difference of the deflections) Change section
Δ < Δ allowable
NG
OK Calculate stability coefficient, θ
θ < θ max Stage 3: Stability check
Change section NG
OK
θ < 0.1
NG
Amplify member force demands and amplify story drift demands
Check the member sections OK OK
Complete the design
Detailing Requirements (AISC 341, 358) Fig. 1. Seismic design process for torsion-irregular structures.
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the assumed displacement of the center of mass in each direction from its actual location by a distance equal to 5% of the dimension of the structure that is perpendicular to the direction of the applied forces. The accidental torsional moment for torsion-irregular structures assigned to SDCs C, D, E, and F should be amplified using a torsional amplification factor (Ax ), which can be calculated using Eq. (1).
Ax ¼
dmax 1:2davg
2 ð1Þ
where dmax is the maximum displacement at level x, and davg is the average displacement at the extreme points of the structure at level x (Fig. 2). Using proper analysis procedures, the member forces are calculated by considering both the inherent and accidental torsion, and the adequate member sections are selected.
3. Seismic design of model structures
2.2. Drift check After determining the proper member sections that are capable of resisting member forces, the design story drift of a structure should be checked. According to Section 12.8.6 of ASCE 7-10, the design story drift (D) can be calculated as the difference of the deflections (dCM ) at the center of mass at the top and bottom of the story under consideration except for structures of SDCs C, D, E and F having horizontal irregularity Type 1a or 1b. For these structures, the design story drift is calculated as the largest difference of deflections (dmax ) along any of the edges of the structure at top and bottom of the story under consideration, which is the largest difference in deflection observed at any column line. As shown in Fig. 2, dmax is larger than dCM . The design story drift (D) should not exceed the allowable story drift that is specified in Table 12.12-1 of ASCE 7-10, which varies according to the structure type and risk category. Otherwise, the process of selecting member sections is repeated until the drift requirement is satisfied. 2.3. Stability check To incorporate the P D effect, the stability coefficient (h) is calculated using Eq. (2):
h¼
P x DI e V x hx C d
ð2Þ
where P x is the total vertical design load, V x is the seismic shear force in story x, hx is the story height, Ie is the importance factor, and C d is the deflection amplification coefficient that is specified
δA
δB
δ CM
δ avg =
in Table 12.2-1 of ASCE 7-10. The stability coefficient h should be less than hmax , as specified by Eq. (12.8-17) in ASCE 7-10 [1]. When h is greater than 0.1, P D effects should be accounted for by increasing the member forces and the design story drifts based on rational analyses. Alternatively, it is permitted to increase the seismic response demand by an amplification factor: 1=ð1 hÞ. In this study, the latter method is used to account for the P D effect. Member sections, which were determined from previous design stages, should be checked to determine whether the members are capable of sustaining the increased member forces and design story drifts. The process of selecting member sections should be repeated until the structure is capable of resisting the increased member forces and drifts.
δA + δB 2
⎡ δ ⎤ Ax = ⎢ max ⎥ ⎣⎢1.2δ avg ⎦⎥
Fig. 2. Floor displacements in torsion-irregular structures.
3.1. Model structures Office buildings are designed according to ASCE7-10 [1], which are considered as model structures. Fig. 3 shows the floor plans and elevations of the model structures. To sustain seismic loads, two steel special moment frames (SMF) are placed in each orthogonal direction as shown in Fig. 3b (marked by thicker solid lines). For steel SMFs, the response modification factor (R), deflection amplification factor (C d ), overstrength factor (Xo ), and importance factor (Ie ) are 8, 5.5, 3, and 1, respectively. In Fig. 3b, the dotted lines denote gravity frames. Among the configuration types of moment frames shown in Fig. 3b, Type 1 is the floor plan for regular structures, in which moment frames are symmetrically placed along the perimeter of the structure. In order to impart different degrees of torsional irregularity to a structure, the west side moment frame (W-frame) is placed at five different locations as shown in Fig 3b. As the W-frame approaches the E-frame, the torsional irregularity becomes more severe. Thus, this study considers the six different floor plans shown in Fig. 3b. It is noted that to place the W- frame inside a building rather than along the perimeter of the building is neither practical nor economical. Nevertheless, the different locations of the W-frame are considered because the objective of this study is to investigate effect of different degrees of torsional irregularity on the seismic performance of buildings. As shown in Fig. 3b, the overall floor plan does not vary according to the location of the W-frame. Because heavy cast-in-place concrete rigid diaphragms are used in model structures, the total weight of a building is much larger than that of the W-frame in the building. The weight of the W-frame for three and nine story buildings is less than 2.3% of the total weight of these building. Therefore, the center of mass was assumed to locate at the center of plan irrespective of the location of the W-frame. Variability in mass center is not considered in the present study. To investigate the effect of building height on the collapse performance of torsionally irregular structures, this study considers three- and nine-story building structures. All building structures have five bays in each direction (Fig. 3) where the story height and span length are 4 m and 6 m, respectively. All structures are symmetric about the east-west direction (x-axis), but are asymmetric about the north-south direction (y-axis), with the exception of Type 1 (regular) structures.The model structures are assumed to be located at a site with design spectral response accelerations of 1.0 g and 0.6 g at a short period (SDS ) and a 1-s period (SD1 ), respectively; thus, according to ASCE 7-10 [1], the buildings are assigned to SDC D. The dead and live loads on the floors are 4.60 kPa and 2.39 kPa, respectively, whereas the corresponding loads on the roof are 4.12 kPa and 0.96 kPa, respectively [16].
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Fig. 3. Model structures.
Member force and story drift demands are calculated using a modal response spectrum analysis, which can be used for torsionally irregular structures regardless of SDC levels as described in ASCE 7-10 [1]. The commercial software MIDAS-Gen [17] is used for the analysis and design. Member sections are determined according to AISC 360-10 [15]; the strong column and weak beam requirements specified in E3 of AISC 341-10 [14] are also considered. According to ASCE 7-10 [1], three- and nine-story building structures with irregularity types 4, 5, and 6 (Fig. 3b) are classified as torsionally irregular structures. Alternatively, structures with irregularity types 1, 2, and 3 are classified as regular structures. Thus, torsionally irregular structures (3-TP4-O, 3-TP5-O, 3-TP6-O, 9-TP4-O, 9-TP5-O, and 9-TP6-O) are designed according to the requirements for regular structures with additional requirements for torsionally irregular structures. It is noted that 3-TP4-O stands for a three-story structure (3) of irregularity Type 4 (TP4) that is designed according to the requirements for torsionally irregular structures (O). For nine-story structures [i.e., 9-TP5-O and 9-TP6-O with severe torsion-irregularity (types 5 and 6)], member
sections cannot be found from the list of W-shaped sections provided in AISC 360-10 [15]. This indicates that it may not be economical and feasible to construct buildings having nine stories or higher if they have severe torsional irregularity. Thus, these structures are excluded from the list of model structures. In Appendix B, the member sections for the 3-TP-5O building are summarized at individual design stages. For comparison purposes, this study also considers torsionally irregular structures (3-TP4-X, 3-TP5-X, 3-TP6-X, 9-TP4-X, and 3-TP4-X) having the same member sections with regular structures (3-TP1 and 9-TP1), thus without taking into account the additional requirements for torsional irregular structures. Therefore, these structures do not satisfy the building code requirements, and they may not be safe. Tables 1 and 2 summarize member sections used for beams and columns in Type 1- and Type 5-three story model structures (3-TP1 and 3-TP5-O), respectively. Each model structure is designed repeatedly to find the optimal member sections considering torsion due to the change in the location of the W-frame.
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Table 1 Member sections for Type 1 three story building (3-TP1). Story number (Ns)
N and S frames
E and W frames
Column
1 2 3
Beam
Exterior
Interior
W14x48 W14x48 W14x48
W14x68 W14x68 W14x38
Column
W14x38 W14x38 W14x38
Beam
Exterior
Interior
W14x74 W14x74 W14x53
W14x74 W14x74 W14x53
W14x48 W14x48 W12x45
Table 2 Member sections for Type 5 three story building (3-TP5-O). Ns
N and S frames
E frame
Column
1 2 3
Beam
Exterior
Interior
W14x 68 W14x 68 W14x 68
W24x 94 W24x 94 W14x 68
W24x 55 W24x 55 W21x 68
W frame
Column
Beam
Exterior
Interior
W14x 74 W14x 74 W14x 53
W14x 74 W14x 74 W14x 53
It is observed that member sections in the W-frame of the torsionally irregular structure (3-TP5-O) are significantly larger than those of corresponding regular structure (3-TP1). With increasing the degree of torsional irregularity from Type 1 to Type 5, member sections for N- and S-frames increase moderately. To resist the enlarged forces due to torsion, it is less effective to increase member sections of the E-frame compared to increasing member sections in W-, N-, and S-frames. In this study, the same member sections are used for the E- frame of torsionally irregular structures as those for the E-frame of corresponding regular structures. Table 3 summarizes the natural periods of the first three modes. Centerline dimensions are used for calculating the natural periods. Gravity columns are not included in period calculation and all columns in the first story are rigidly fixed to the ground. The present study investigates the deformed shapes of the first three modes of model structures and the mass participations of two translation components and one rotation component for these modes. The results are not included in this paper. For torsionally regular structures (3TP-1 and 9TP-1), the first mode includes translation in the y-direction exclusively, whereas the second mode includes only translation in the x-direction. The third mode
W14x 48 W14x 48 W12x 45
Column
Beam
Exterior
Interior
W24x 162 W24x 162 W24x 131
W24x 162 W24x 162 W24x 131
W27x 102 W27x 94 W24x 76
includes only rotation. However, unlike the regular structures, the first and third modes of torsionally irregular structures contain both translational component along y-axis and rotation component about z-axis, whereas the 2nd mode of these structures contains only translation component along x-axis. 3.2. Total weight of moment frames at three different design stages The total steel weight of the moment frames in the model structures is estimated at three different seismic design stages: (Stage 1) strength design stage: member sections are determined to resist the member force demands; (Stage 2) drift (stiffness) checking stage: design story drift demands should be smaller than the allowable story drift, and member sections can be changed if necessary; and (Stage 3) stability checking stage: member forces and design story drifts are amplified to account for the P D effects, and member sections are changed if necessary.Fig. 4a presents the total steel weight (W) of the moment frames in three-story model structures that are calculated at each design stage. At Stage 1 (strength checking), the weight for all structures is nearly the same regardless of the degree of torsional irregularity. For three-
Table 3 Natural Periods of the model structures. Number of stories
3
Plan type
Type 1 Type 2 Type 3 Type 4 Type 5 Type 6
9
Type 1 Type 2 Type 3 Type 4
ID
3-TP1 3-TP2-X 3-TP2-O 3-TP3-X 3-TP3-O 3-TP4-X 3-TP4-O 3-TP5-X 3-TP5-O 3-TP6-X 3-TP6-O 9-TP1 9-TP2-X 9-TP2-O 9-TP3-X 9-TP3-O 9-TP4-X 9-TP4-O
Periods, T (s) 1st mode
2nd mode
3rd mode
1.39 1.41 1.40 1.50 1.44 1.65 1.14 1.83 1.19 2.34 1.09 2.61 2.71 2.62 2.92 2.73 3.26 2.40
1.27 1.27 1.27 1.27 1.22 1.27 1.09 1.27 1.17 1.28 0.79 2.51 2.63 2.49 2.63 2.45 2.64 2.32
0.95 0.87 0.87 0.89 0.86 0.87 1.05 0.82 1.12 0.89 0.34 1.56 1.73 1.73 1.77 1.99 1.71 1.71
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3500
2500 3-TP1
9-TP1
(a) 3-story 3000
3-TP3-O
Steel weights (kN)
Steel weights (kN)
2000
3-TP4-O
1500
(b) 9-story
9-TP2-O
3-TP2-O
3-TP5-O 3-TP6-O
1000
9-TP3-O 9-TP4-O
2500
2000
1500
500
1000
0 1
2
3
Design stages
1
2
3
Design stages
Fig. 4. Total steel weights of the moment frames in structures for each design stage.
story regular structures (3-TP1, 3-TP2-O, and 3-TP3-O), the weight is nearly constant irrespective of the design stages. However, for nine-story regular structures (9-TP1, 9-TP-2-O, and 9-TP-3-O), the weight at Stage 2 is significantly larger than the weight at Stage 1; the weight at Stage 3 is also larger than the weight at Stage 2. This indicates that nine-story structures are affected by drifts and the P D effect more than three-story structures. For torsionally irregular structures (3-TP4-O, 3-TP5-O, 3-TP6-O, and 9-TP4-O), the weight increases sharply from Stage 1 to Stage 2. It is noted that when drifts are checked for torsionally irregular structures, dmax is used instead of dCM , as shown in Fig. 2; this requires larger member sections. For three-story torsionally irregular structures, the increase in the weight is small between Stage 2 and Stage 3; alternatively, for nine-story torsionally irregular structures, the increase in the weight is significant. This observations confirms that nine-story torsionally irregular structures are influenced by the P D effect more than three-story torsionally irregular structures. At Stages 2 and 3, the weights for torsionally irregular structures are much larger than those for corresponding regular structures. For example, the weights of 3-TP6-O and 9-TP4-O are 3.9 and 1.5 times larger than those of 3-TP1 and 9-TP1, respectively. As shown in Fig. 4, as the degree of torsional irregularity increases, larger steel weights are required.
4. Nonlinear response history analyses of torsionally irregular structures 4.1. Analytical model Nonlinear response history analyses (RHA) with a threedimensional analytical model are conducted to evaluate the seismic collapse performance of the model structures. For analyses, Opensees (Mazzoni et al. [18]) software is used. Fig. 5a presents the three-dimensional frame model used in the present study. Moment frames are denoted by thick solid lines. Gravity columns are modeled as rigid elements with pins at their ends, which account for the P D effects due to gravity loads. Slabs are modeled to behave as rigid diaphragms by restraining in-plane slab deformation. Fig. 5b presents the analytical model for the beam-column connections in moment frames. Columns are modeled using fiber elements (Fig. 5b-1), which effectively simulate interaction between axial force and moment. The behavior of the fiber elements is represented by a bilinear model with a post elastic slope of 3%. Five integration points are used, equally spaced along the column (Fig. 5b). Neuenhofer and Filippou [19] reported that four to six integration points are typically sufficient to represent the spread of plasticity along an element.
Although fiber section elements may not incorporate cyclic deterioration in strength and stiffness, they can efficiently capture distributed plasticity in members. Because most plastic hinges are developed in beams rather than columns in special moment frames owing to strong column-weak beam requirements, cyclic deterioration may not be a serious problem for columns in SMFs. At each integration point, the member section is divided into 160 fibers (64 fibers for each flange and 32 fibers for web). It is noted that the 108 fibers offers an almost perfect match of the exact solution [20]. Beams are modeled to behave in an elastic range (Fig. 5b-2). Inelastic beam behavior is represented by inelastic rotational springs placed at the ends of each beam (Fig. 5b-3), where M y and M c are the yield and maximum moment strengths, respectively, while hp is the plastic rotation capacity, which can be calculated using the equations proposed by Lignos and Krawinkler [21]. Cyclic deterioration in strength and stiffness is not considered in this study.
4.2. Nonlinear response history analysis (RHA) of torsionally irregular structures To investigate the effect of torsional irregularity on the seismic demands of torsionally irregular structures, this study conducts nonlinear RHA under an orthogonal pair of horizontal ground motions: these motions were recorded at the Beverly Hills station with an epicentral distance of 13 km during the Northridge earthquake. Their peak ground accelerations (PGAs) are 0.42 g and 0.52 g, respectively. Fig. 6 shows the distribution of plastic hinges in the N- and S-frames in the three-story model structures (3-TP1, 3-TP5-X, and 3-TP5-O) that are obtained from nonlinear RHA. As mentioned earlier, 3-TP1 is a three-story regular structure, 3-TP5-X is a torsion-irregular structure with the same member sections as 3-TP1, and 3-TP5-O is also a torsion-irregular structure that is designed according to the requirements for torsionally irregular structures that are specified in ASCE 7-10 [1]. As shown in Fig. 6, the plastic hinges developed in the N- and S-frames of the three model structures are denoted by solid circles. For the regular structure (3-TP1), the distribution and number of plastic hinges developed in the N- and S-frames are almost identical; this result is expected. However, for 3-TP5-X, the distribution of plastic hinges in the N- and S-frames is significantly asymmetric (Fig. 6b). The numbers of plastic hinges in the N- and S-frames are 36 and 10, respectively. Unlike 3-TP5-X, the N- and S-frames of 3-TP5-O have an almost symmetric distribution of plastic hinges. The numbers of plastic hinges detected in the N- and S-frames of 3-TP5-X are 31 and 32, respectively.
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M Mc My
M e
p
(b-3) Rotational spring
M (b-1) Column element (b-2) Beam element (a) Frame model
(b) Connection model Fig. 5. Analytical model.
N
N
N
0.42g
W
0.52g
E
W
S
E
S
W
E
S
N-frame (35)*The number of plastic hinges
N-frame (36)
N-frame (31)
S-frame (36)
S-frame (10)
S-frame (32)
(a) 3-TP1
(b) 3-TP5-X
(c) 3-TP5-O
Fig. 6. Distribution of plastic hinges.
To investigate the cause of the difference in plastic hinge distributions in the N- and S-frames, the displaced roof positions are plotted with thin solid lines at each time step of the nonlinear RHA, as shown in Fig. 7. The original roof position is also plotted with a dotted line as a reference. In Fig. 7, the thick solid lines denote the displaced roof positions when the N- and S-frame displacements are the largest.The ratios of the maximum displacement of the N-frame (DmaxN ) to the maximum displacement of the S-frame (DmaxS ) are 1.02, 2.22, and 1.3 for 3-TP1, 3-TP5-X, and 3-TP5-O, respectively. As expected, the ratio (DmaxN /DmaxS ) for the regular structure is the smallest and the ratio for 3-TP5-X is the largest.This study also evaluates the effect of torsional irregularity on the collapse intensities. For this purpose, incremental dynamic analyses (IDA) are conducted (Vamvatsikos and Cornell [22]; Moon et al. [23]). Collapse is defined as the state of global dynamic instability in one or more stories of the structural system.
In this study, the collapse intensity is defined as a 5% damped pseudo spectral acceleration at the fundamental period [Sa ðT 1 Þ], which is the ordinate of IDA curves. The approximate period (C u T a ) is used as the fundamental period, which is calculated according to ASCE 7-10 [1] (section 12.8.2); for the three-story structures, C u T a is 0.74 s. The approximate period of the three-story regular structure (3-TP1) is shorter than the fundamental period (=1.39 s) obtained from eigenvalue analyses. For torsionally irregular structures having the same member sections, the period difference between approximate and eigenvalue periods is more significant with an increase in torsional irregularity. As shown in Table 3, these structures have a longer period as the level of torsional irregularity increases. However, for torsionally irregular structures properly designed according to seismic design provisions with additional requirements for torsionally irregular structures, the fundamental
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N-frame
N-frame
S-frame
S-frame
S-frame
(a) 3-TP1
(b) 3-TP5-X
(c) 3-TP5-O
N-frame
y x
Fig. 7. Displaced roof positions for three-story model structures.
period obtained from eigenvalue analyses generally decreases as the level of torsional irregularity. The eigenvalue periods of these structures are also shorter than that of the corresponding torsionally irregular structures having the same member sections as regular structures. Therefore, the period difference for these structures is not as significant as torsional irregular structures having the same member sections as 3-TP1. Similar observation was made for 9-story torsional irregular structures.To determine the bi-directional acceleration spectrum, the acceleration responses of an elastic single-degree-of-freedom system with a period of C u T a are calculated subjected to individual ground motions. The vector sum of the bi-directional acceleration responses is computed for each time step and the maximum is defined as the value of the acceleration response spectrum [24]. The abscissa of the IDA curve is represented by an engineering demand parameter (EDP). In this study, the maximum story drift ratio (hmax ) is used as the EDP. In the IDA curve, the global dynamic instability is identified by a characteristic flattening of each IDA, referred to as the flatline, where the seismic demand increases greatly with only a slight increase in the ground motion intensity. To conduct IDA, the same pair of ground motions recorded at the Beverly Hills station during the Northridge earthquake, is used (Fig. 8). The collapse intensities of 3-TP1 and 3-TP5-X are 1.8 g and 0.8 g, respectively. This indicates that the collapse intensities of torsionally irregular structures with the same member sections
as regular structures are significantly smaller than those of corresponding regular structures. The collapse intensity of 3-TP5-O is 3.6 g, which is twice the collapse intensity of 3-TP1. The larger collapse intensity of 3-TP5-O is due to the heavy section sizes that are selected in accordance with the stringent requirements for torsionally irregular structures (ASCE 7-10). 5. Seismic collapse performance evaluation This study evaluates the seismic collapse performance of torsionally irregular structures. Current seismic codes are intended to provide buildings designed according to code requirements with the capability to have a low probability of collapse if subjected to very rare, intense ground motions (referred to as maximum considered earthquake (MCE) ground motion) [1]. FEMA P-695 [25] provides a methodology for quantifying the system performance and response parameters in seismic design. The two performance objectives used in FEMA P-695 [25] are: (1) The probability of collapse for maximum considered earthquake (MCE) ground motions is 10% or less, on average, across a performance group containing model frames with a specific seismic force resisting system with different configurations, and (2) the probability of collapse is 20% or less for an individual model frame.The probability of collapse under MCE ground motions [PðCollapsejSMT Þ] is calculated using Eq. (3).
Fig. 8. IDA curves and collapse intensities.
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PðCollapsejSMT Þ ¼ U
lnðSMT Þ lnð^SCT SSFÞ bTOT
! ð3Þ
where UðxÞ is the cumulative distribution function of the standard normal variate x;SMT is the MCE 5% damped-spectral response acceleration at the fundamental period (T) of the structure, ^SCT is the median value of the 5% damped spectral response acceleration at T for a collapse level earthquake, SSF is the spectral shape factor accounting for the spectral shape of rare ground motions, and adjust the collapse probability, UðxÞ, while bTOT is the total system collapse uncertainty calculated using Eq. (4).
bTOT ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2RTR þ b2DR þ b2TD þ b2MDL
ð4Þ
where bRTR ; bDR ; bTD and bMDL are the record-to-record uncertainty, design requirement-related uncertainty, test data-related uncertainty and modeling uncertainty, respectively. These values range from 0.1 to 0.5, and are determined from tables in pages 3–8, 3– 20, and 5–23 of FEMA P-695 [25]. The step-by-step procedure used to calculate the probability of collapse is described in Appendix A. The three- and nine-story structures are assigned to SDC D. For these structures, SMT values are 1.15 g and 0.51 g, respectively. Table 4 summarizes the probabilities of collapse for model structures, which are calculated using the procedure listed in Appendix A. Fig. 9 presents the effect of the degree of torsional irregularity on the probability of collapse in which the abscissa is the degree of torsional irregularity (Fig. 3) and the ordinate is the probability of collapse. Twenty-two pairs of ground motions, provided in FEMA P695 [25], are used as input ground motions. Fig. 9 includes the probabilities of collapse for torsionally irregular structures designed with and without the additional requirements for torsional irregularity. For three-story torsionally irregular structures with the same member sections as those of regular structures, the probability of collapse becomes larger as the degree of torsional irregularity increases. The collapse probabilities of 3-TP2-X and 3-TP3-X are higher than 0.1, which is a limiting value for performance group, whereas exceed those of 3-TP4-X, 3-TP5-X, and 3-TP6-X exceed 0.2, which is the limiting value for individual structures [25]. In contrast, for three-story torsionally irregular structures designed with the additional requirements, the probability of collapse becomes smaller as the degree of torsional irregularity
increases. The collapse probabilities of 3-TP2-O and 3-TP3-O are slightly higher than 0.1 (<0.2), whereas those of 3-TP4-O, 3-TP5-O, and 3-TP6-O (=0.0441, 0.0123, and 0.0062) are much lower than 0.1. For nine-story torsionally irregular structures, similar observations are also made. For torsionally irregular structures with the same member sections as those of the regular structures, the probability of collapse becomes higher as the degree of torsional irregularity increases; however, this increment is not as significant as that of the three-story torsionally irregular structures. For the nine-story torsionally irregular structures that are designed according to the additional requirements for torsion, the probability of collapse becomes smaller as the degree of irregularity increases. It is noted that 9-TP4-O exhibited the smallest collapse probability (=0.0752). Among eight torsionally irregular model structures designed according to additional requirements for torsion specified in ASCE 7-10, four structures have the probability of collapse lower than 0.1, which are 3-TP4-O, 3-TP5-O, 3-TP6-O, and 9-TP4-O. Therefore, it is observed that as the degree of torsional irregularity increases, the collapse probability of torsionally irregular structures that are designed according to the additional requirements becomes smaller. This phenomenon is attributed to the heavy member sections selected for torsionally irregular structures, which serve to fulfill the drift requirement. 6. Proposed procedure for computing the design story drift for torsionally irregular structures As shown in Fig. 9, the probability of collapse for codecompliant torsionally irregular structures is much smaller than that of regular structures. The probability of collapse also decreases as the degree of torsional irregularity increases. This is attributed to the large member sections of the torsionally irregular structures. In order to design torsionally irregular structures that have a similar probability of collapse to regular structures, irrespective of the degree of torsional irregularity, this study proposes a procedure to determine the design story drift demands that are required in the drift checking stage (Fig. 2). As shown in Fig. 4, the sharpest increase in the total steel weight was observed at the drift checking stage. In the proposed procedure, the drift demand of torsionally irregular structures is estimated by calculating the difference between the displacements at the mass centers of two adjacent
Table 4 Probability of collapse for the model structures. Number of stories
Model structures
SMT
lT
^ SCT
SSF
PðCollapsejSMT Þ
3
3-TP1 3-TP2-X 3-TP3-X 3-TP4-X 3-TP5-X 3-TP6-X 3-TP2-O 3-TP3-O 3-TP4-O 3-TP5-O 3-TP6-O 3-TP4-OM 3-TP5-OM 3-TP6-OM 9-TP3-X 9-TP4-X 9-TP2-O 9-TP3-O 9-TP4-O 9-TP4-OM
1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 0.51 0.51 0.51 0.51 0.51 0.51
5.64 5.75 5.14 4.57 3.75 9.44 5.63 6.49 6.60 6.15 5.36 8.49 2.77 4.39 3.14 3.82 3.99 4.09 4.61 4.59
1.70 1.65 1.61 1.30 1.30 0.85 1.65 1.70 2.40 3.60 4.38 1.70 1.90 2.86 0.59 0.55 0.64 0.70 0.93 0.82
1.33 1.33 1.31 1.29 1.26 1.14 1.34 1.36 1.36 1.35 1.32 1.42 1.21 1.29 1.23 1.26 1.27 1.27 1.29 1.29
0.12 0.12 0.14 0.22 0.23 0.36 0.12 0.11 0.04 0.01 0.01 0.14 0.13 0.06 0.23 0.25 0.19 0.15 0.08 0.10
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491
Fig. 9. Collapse probabilities of the model structures.
floors, as opposed to calculating the maximum drift values of the vertical members in the story, which is the same procedure for regular structures. Four torsionally irregular structures (3-TP4-OM, 3-TP5-OM 3, 3TP6-OM, and 9-TP4-OM) are re-designed with the proposed procedure for drift calculations. Nonlinear response history analyses are conducted for structures that are subjected to the same pair of ground motions that were used in the previous section. The probability of collapse for these structures calculated according to FEMA P-695 is summarized in Table 4. Fig. 10 shows the distribution of plastic hinges in the N- and Sframes of the newly designed three-story structure (3-TP5-OM) with torsional irregularity Type 5. The numbers of plastic hinges in the N-and S-frames of 3-TP5-OM are 30 and 22, respectively (Fig. 10b). Thus, the difference between the numbers of plastic hinges in the N-and S-frames of 3-TP5-OM is 12. It is noted that, for 3-TP5-X and 3-TP-5-O, the differences are 26 and 1, respectively (Fig. 6). This indicates that the contribution of torsional
responses in 3-TP5-OM is more significant than that in 3-TP5-O, but less than that in 3-TP5-X. IDA is also conducted for 3-TP5-OM under the same pair of ground motions (Fig. 10c). The collapse intensity is 1.9 g, which is less than that of 3-TP5-O (=3.6 g), but close to that of the regular frame 3-TP1 (=1.7 g). This study also calculated the probabilities of collapse for 3TP4-OM, 3-TP5-OM, 3-TP6-OM and 9-TP4-OM using the 22 pairs of ground motions provided in FEMA P695 [25], which were also used previously. Fig. 11 shows the probability of collapse. In this figure, ‘ASCE 7’ indicates the torsionally irregular structures designed according to ASCE 7-10, whereas ‘MC’ denotes the torsionally irregular structures designed according to ASCE 7-10 [1] with drift demands calculated using the proposed procedure. The collapse probability of ‘MC’ structures is more uniform than that of the ‘ASCE 7’ structures. Also, the collapse probability of ‘MC’ structures is larger than that of the ‘ASCE 7’ structures but is similar with that of regular structures.
Fig. 10. Results of nonlinear RHA for 3-TP5-OM.
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Fig. 11. Probability of collapse according to the degree of torsional irregularity.
Fig. 12. Total steel weights required for the model structures.
The collapse probabilities of three-story ‘MC’ structures (3-TP4OM, 3-TP5-OM, and 3-TP6-OM) with different degrees of torsional irregularity are 0.1372, 0.1307, and 0.0549, respectively. Alternatively, the collapse probabilities of three-story ‘ASCE 70 structures (3-TP4-O, 3-TP5-O, and 3-TP6-O) are 0.0441, 0.0123, and 0.0062, respectively. This indicates that ‘MC’ structures have probabilities of collapse that are closer to those of regular structures than ‘ASCE 70 structures. It is also noted that ‘MC’ structures produced more uniform probabilities of collapse according to the degree of torsional irregularity than ‘ASCE 70 structures. Similar observations are also made for the nine-story structures (Fig. 11b). This study also compared the total steel weight of the structures as shown in Fig. 12. The total steel weight of ‘MC’ structures, using the proposed design methods is less than that of ‘ASCE7’ structures. It is noted that there may be other design solutions that would bring the ‘MC’ buildings to the desired collapse level.
(2)
(3)
7. Conclusions
(4)
In order to evaluate the seismic collapse performance of torsionally irregular structures, three- and nine-story steel special moment frames in model structures with various degrees of torsional irregularity were designed according to ASCE 7-10 [1]. Nonlinear response history analyses were conducted. The conclusions and recommendations of this study are as follows:
(5)
(1) When the degree of torsional irregularity was increased, larger member sections were required. At the strength design
stage (Stage 1), the total steel weight of the moment frames in torsionally irregular structures was similar to that of regular structures. However, at the drift checking stage (Stage 2), the weight of torsionally irregular structures was much larger than that of regular structures. The increase in the total steel weight for three-story torsionally irregular structures was negligible between Stage 2 and Stage 3; however, the increase for nine- story torsionally irregular structures was significant between Stage 2 and Stage 3 due to the large P D effects. It is observed that for regular structures, the distribution of plastic hinges was almost symmetric in the N- and Sframes. Torsionally irregular structures designed according to ASCE 7-10 [1] also exhibited a symmetric distribution of plastic hinges in the N- and S-frames; this was due to the large member sections that were selected to satisfy the additional requirements for torsionally irregular structures, as specified in ASCE 7-10 [1]. For torsionally irregular structures designed without considering the additional requirements, the probability of collapse is larger as the degree of torsional irregularity increases. However, torsionally irregular structures that were designed while also considering the additional requirements generally exhibited smaller probabilities of collapse compared to regular structures. The probabilities of collapse of torsionally irregular structures, designed with drift demands calculated using the proposed method, is similar to those of regular structures regardless of the degree of torsional irregularity.
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(6) The results of this study is obtained using only torsionally irregular structures with steel SMF frames. Therefore, caution should be made for buildings having different seismic force resisting systems such as braced frames and shear walls.
Acknowledgements Authors would like to acknowledge the financial supports provided by the National Research Foundation of Korea (No. 2014R1A2A1A11049488). The valuable comments offered by anonymous reviewers are greatly appreciated.
where V max is the maximum base shear resistance, V d is the design base shear, du is the roof drift displacement at the point of 20% strength loss (=0.8 V max ), dyeff is the effective yield roof displacement, C o relates the displacement of the single-degree-of-freedom system at the fundamental mode of a frame to the roof displacement of the frame, g is the gravity constant, T n is the fundamental period (defined as C u T a , specified in ASCE 7-10) [1], and T 1 is the fundamental period of the model frame computed via eigenvalue analysis. (3) Calculate the spectral shape factor (SSF),that accounts for that accounts for the spectral shape of rare ground motions as depicted in Fig. A1(c).
SSF ¼ exp½b1 ðeo ðTÞ eðTÞrecord
ðA4Þ
eo ðTÞrecord ¼ 0:6ð1:5 TÞ
ðA5Þ
b1 ¼ 0:14ðlT 1Þ0:42
ðA6Þ
Appendix A A step-by-step procedure for calculating the probability of collapse according to FEMA P-695 [23]: (1) Idealize the model frame using a proper analytical model. (2) Conduct nonlinear static analysis to determine the overstrength factor (X) and period-based ductility (lT ). This is done by using Eqs. (A1) and (A2) with Eq. (A3), respectively [Fig. A1(a)].
X¼
V max Vd
(4) Calibrate the probability of collapse by applying the SSF.
! lnðSMT Þ lnð^SCT SSFÞ PðCollapsejSMT Þ ¼ U bTOT
ðA7Þ
ðA1Þ Appendix B
lT
du ¼ dyeff
dy;eff ¼ C 0
ðA2Þ V max h g i ðmaxðT n ; T 1 ÞÞ2 W 4p2
ðA3Þ
The following table includes information about member sections for the Type 5-three story structure at three different design stages. Tables B-1, B-2 and B-3.
Fig. A1. Procedure for calculating the SSF and calculating the probability of collapse.
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Table B-1 Member sections used for the Type 5-three story structure at strength design stage. Level
N-and S-frames
E-frame
Colum
Girder
Exterior
Interior
1
W14x48
W14x68
W14x38
2
W14x48
W14x68
W14x38
3
W14x38
W14x38
W12x35
W-frame
Colum
Girder
Exterior
Interior
W14x 74 W14x 74 W14x 53
W14x 74 W14x 74 W14x 53
W14x 48 W14x 48 W12x 45
Colum
Girder
Exterior
Interior
W14x 74 W14x 74 W14x 68
W14x 74 W14x 74 W14x 68
W14x 48 W14x 48 W12x 45
Table B-2 Member sections used for the Type 5-three story structure at drift checking stage. Level
N-and S-frames
E-frame
Colum
Girder
Exterior
Interior
1
W14x68
W24x94
W24x55
2
W14x68
W24x94
W24x55
3
W14x68
W14x68
W21x68
W-frame
Colum Exterior
Interior
W14x 74 W14x 74 W14x 53
W14x 74 W14x 74 W14x 53
Girder
Colum
Girder
Exterior
Interior
W14x 48 W14x 48 W12x 45
W24x 162 W24x 162 W24x 131
W24x 162 W24x 162 W24x 131
W27x 102 W27x 94 W24x 76
Table B-3 Member sections used for the Type 5-three story structure at stability checking stage. Level
N-and S-frames
E-frame
Colum
Girder
Exterior
Interior
1
W14x68
W24x94
W24x55
2
W14x68
W24x94
W24x55
3
W14x68
W14x68
W21x68
W-frame
Colum
Girder
Exterior
Interior
W14x 74 W14x 74 W14x 53
W14x 74 W14x 74 W14x 53
References [1] ASCE 7-10. Minimum design loads for buildings and other structures. USA: American Society of Civil Engineers, Structural Engineering Institute; 2010. [2] Chopra AK, Goel RK. Evaluation of torsional provisions in seismic codes. J Struct Eng 1991;117(12):3762–82. [3] Tso WK, Wong CM. An evaluation of the New Zealand code torsional provision. Bull New Zealand Natl Soc Earthquake Eng 1993;26(2):194–207. [4] Humar JL, Kumar P. Effect of orthogonal inplane structural elements on inelastic torsional response. Earthq Eng Struct Dyn 1999;28:1071–97. [5] Dutta SC, Das PK. Inelastic seismic response of code-designed reinforced concrete asymmetric buildings with strength degradation. Eng Struct 2002;24 (10):1295–314. [6] Aziminejad A, Moghadam AS. Performance of asymmetric single story buildings based on different configuration of center of mass, rigidity and resistance.. In: Proceedings of the 4th European workshop on the seismic behaviour of irregular and complex structures. [7] Herrera RG, Soberón CG. Influence of plan irregularity of buildings. In: The 14th World Conference on Earthquake Engineering. [8] Stathopoulos KG, Anagnostopoulos SA. Inelastic earthquake response of singlestory asymmetric buildings: an assessment of simplified shear-beam models. Earthq Eng Struct Dyn 2003;32(12):1813–31. [9] Jeong SH, Elnashai AS. Analytical assessment of an irregular RC frame for fullscale 3D pseudo-dynamic testing part I: Analytical model verification. J Earthq Eng 2005;09(01):95–128. [10] Stathopoulos KG, Anagnostopoulos SA. Accidental design eccentricity: is it important for the inelastic response of buildings to strong earthquakes. Soil Dyn Earthq Eng 2010;30(9):782–97. [11] Reyes JC, Quintero OA. Modal pushover-based scaling of earthquake records for nonlinear analysis of single-story unsymmetric-plan buildings. Earthq Eng Struct Dyn 2014;43(7):1005–21. [12] De-la-Colina J. Assessment of design recommendations for torsionally unbalanced multistory buildings. Earthq Spectra 2003;19(1):47–66.
W14x 48 W14x 48 W12x 45
Colum
Girder
Exterior
Interior
W24x 162 W24x 162 W24x 131
W24x 162 W24x 162 W24x 131
W27x 102 W27x 94 W24x 76
[13] DeBock DJ, Liel AB, Haselton CB, Hooper JD, Henige RA. Importance of seismic design accidental torsion requirement for building collapse capacity. Earthq Eng Struct Dyn 2014;43(6):831–50. [14] American Institute of Steel Construction (AISC). Seismic provisions for structural steel buildings (ANSI/AISC 341-10). Chicago (IL): American Institute of Steel Construction; 2010. [15] American Institute of Steel Construction. AISC): Specification for Structural Steel Buildings (ANSI/AISC 360–10. Chicago (IL): American Institute of Steel, Construction; 2010. [16] Gupta A, Krawinkler H. Seismic demands for performance evaluation of steel moment resisting frame structures, Report No. 132. Blume Earthquake Engineering Center, Department of Civil and Environmental Engineering, Stanford University; 1999. [17] MIDAS IT. Midas/GEN User’s Manual (Version 6.3.2), MIDAS IT; 2004. [18] Mazonni S, McKenna F, Scott MH, Fenves GL. OpenSees Command Language Manual 2006.. [19] Neuenhofer A, Filippou FC. Evaluation of nonlinear frame finite-element models. J Earthq Eng, ASCE 1997;123(7):958–66. [20] Kostic SM, Filippou DC. Section discretization of fiber beam-column elements for cyclic inelastic response. J Struct Eng, ASCE 2002;138(5):592–601. [21] Lignos DG, Krawinkler H. Deterioration modeling of steel components in support of collapse prediction of steel moment frames under earthquake loading. J Struct Eng ASCE 2011;137(11):1291–302. [22] Vamvatsikos D, Cornell CA. Incremental dynamic analysis. Earthq Eng Struct Dyn 2002;31(3):491–514. [23] Moon KH, Han SW, Lee TS, Seok SW. Approximate MPA-based method for performing incremental dynamic analysis. Nonlinear Dyn 2012;67 (4):2865–88. [24] Wieser JD, Pekhan G, Zaghi AE, Itani AM, Maragakis EM. MCEER-12-0008. Assessment of floor accelerations in yielding buildings. MCEER Technical Report. University at Buffalo – the State University of New York; 2012. [25] FEMA. Quantification of building seismic performance factors. FEMA P695. Federal Emergency Management Agency, Washington, DC; June 2009.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Seismic collapse performance of special moment steel frames with torsional irregularities Sang Whan Han ⇑, Tae-O Kim, Dong Hwi Kim, Seong-Jin Baek Department of Architectural Engineering, Hanyang University, Seoul 133-791, Republic of Korea
a r t i c l e
i n f o
Article history: Received 24 February 2016 Revised 16 March 2017 Accepted 20 March 2017
Keywords: Torsional irregularity Inelastic response Ground motion Collapse performance Story drift Nonlinear response history analysis
a b s t r a c t Many building structures exhibit torsional irregularity, which is a type of horizontal irregularity. It is difficult to estimate the inelastic response of torsionally irregular structures subjected to earthquake ground motions using numerical analyses because torsionally irregular structures experience both lateral displacement and floor rotation. This study evaluates the collapse performance of multi-story model structures with various degrees of torsional irregularity via nonlinear response history analyses. This study also proposes a procedure for computing design story drift demands, which allows torsionally irregular structures to have uniform collapse risk irrespective of the degree of torsional irregularity. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Torsional irregularities are a type of horizontal irregularity (Table 12.3-1, ASCE 7-10 [1]) that are induced by an asymmetric distribution of mass, stiffness and (or) strength. Since torsionally irregular structures experience both lateral displacement and floor rotation during an earthquake, they sustain greater member forces and drifts compared to regular structures. Without careful design consideration, torsionally irregular structures may be more vulnerable to earthquakes than regular structures. To reduce the vulnerability of torsionally irregular structures, current seismic codes specify more stringent requirements for torsionally irregular structures than regular structures; in other words, torsionally irregular structures must be designed to satisfy the design requirements for regular structures as well as additional requirements for torsionally irregular structures. Many studies have been conducted to investigate the effect of torsional irregularity on the seismic response of single-story torsionally irregular structures designed according to seismic design provisions (Chopra and Goel [2]; Tso and Wong [3]; Humar and Kumar [4]; Dutta and Das [5]; Aziminejad and Moghadam [6]; Herrera and Soberon [7]). Although it is convenient to draw insight about the effect of torsional irregularity by analyzing simple single-story structures, there are limitations when it comes to ⇑ Corresponding author. E-mail address: [email protected] (S.W. Han). http://dx.doi.org/10.1016/j.engstruct.2017.03.045 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.
extrapolate the results from single-story structures to multi-story torsionally irregular structures (Stathopoulos and Anagnostopoulos [8]). Recently, due to the development of high-performance computers and software, the seismic behavior of multi-story structures with torsional irregularity has been investigated by conducting nonlinear response analyses (Jeong and Elnashai [9]; Stathopoulos and Anagnostopoulos [10]; Reyes and Quintero [11]). De-la-Colina [12] investigated the seismic behavior of multi-story torsionally irregular structures, and proposed design recommendations to control the ductility demands. DeBock et al. [13] investigated the seismic collapse performance of multi-story reinforced concrete frame structures using sophisticated plastic models that accounted for cyclic and in-cyclic deterioration in order to evaluate the design recommendations related to accidental torsion. The objective of this study is to evaluate the seismic collapse performance of multi-story structures with various degrees of torsional irregularity, which are designed according to current seismic design provisions. For this purpose, three- and nine-story model structures with various degrees of torsional irregularity are designed according to ASCE 7-10 [1], AISC 341-10 [14] and AISC 360-10 [15], in which special moment steel frames are used as seismic force resisting systems. Incremental dynamic analyses are conducted for the model structures with repeated nonlinear response history analyses using a three-dimensional inelastic analytical model. Based on the analysis results, this study evaluates the collapse performance of the model structures, and proposes a
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procedure for computing the design story drift demand in an attempt to allow torsionally irregular structures to have uniform collapse risk. In order to monitor the change in member sections of torsionally irregular structures according to the design stages, this study also estimates the total steel weight of the moment frames in model structures at each design stage. 2. Summary of design process for torsionally irregular structures In ASCE 7-10 (Table 12.3-1), when the maximum story drift at one end of the structure is more than 1.2 times the average story drift at both ends of the structure, the structure is classified as being torsionally irregular. When the drift ratio is greater than 1.4, the structure is considered to have extreme torsional irregularity. When a structure has torsional irregularity, the structure should be designed to satisfy the design requirements for regular structures as well as additional requirements for torsionally irregular structures, as specified in ASCE 7-10 [1]. The seismic design process for torsionally irregular structures is similar to that of regular structures. In this study, the design process for torsionally irregular structures is separated into three stages: (1) strength design stage, (2) drift (stiffness) checking stage, and (3) stability (P Deffect) checking stage. Fig. 1 illustrates the schematic flow of the seismic design process for torsionally irregular structures that is used in this study. The additional requirements specified in ASCE 7-10 are: (1) increases in forces due to irregularities for SDC D, E and
Stage 1: Strength design
Stage 2: Drift check
F (12.3.3.4), (2) 3-D modeling for structures having torsional irregularity type 1a, 1b, 4, or 5 of Table 12.3-1 in ASCE 7-10 (12.7.3), (3) amplification of accidental torsion for structures assigned to SDC C through F (12.8.4.3), (6) analysis procedure selection for torsionally irregular structures (12.6), and (7) story drift calculation using the largest difference of the deflections of vertically aligned points at the top and bottom of the story along any edges of the structure (12.8.6). The numbers in parentheses are the section numbers of ASCE 7-10 [1] for each of the additional requirements. 2.1. Strength design In the strength design stage, seismic design forces are calculated to determine the member section. Three different analysis procedures are permitted in ASCE-7-10 [1]: (1) equivalent lateral force analysis, (2) modal response spectrum analysis, and (3) response history analysis. A three-dimensional analytical model is required for the analysis of torsionally irregular structures assigned to all levels of SDCs, with the exception of SDC A. For torsionally irregular structures assigned to SDCs D, E, and F, the equivalent lateral force analysis procedure is not permitted. In this study, a modal response spectrum analysis method that has no limitations according to the level of SDCs is used.For torsionally irregular structures, both inherent and accidental torsions should be properly accounted for when calculating the member forces and drifts. The inherent torsion results from an asymmetric distribution of mass, stiffness, and strength, whereas accidental torsion is induced by uncertainties in the distribution of mass, stiffness, and strength. The accidental torsional moment is calculated using
Determine seismic design forces
Modal spectral response analysis
Calculate member force demands and select the member sections
If the structure is torsional irregular, amplify accidental torsional moment
Calculate story drift Δ (If torsional irregular, Δ shall be computed as the largest difference of the deflections) Change section
Δ < Δ allowable
NG
OK Calculate stability coefficient, θ
θ < θ max Stage 3: Stability check
Change section NG
OK
θ < 0.1
NG
Amplify member force demands and amplify story drift demands
Check the member sections OK OK
Complete the design
Detailing Requirements (AISC 341, 358) Fig. 1. Seismic design process for torsion-irregular structures.
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the assumed displacement of the center of mass in each direction from its actual location by a distance equal to 5% of the dimension of the structure that is perpendicular to the direction of the applied forces. The accidental torsional moment for torsion-irregular structures assigned to SDCs C, D, E, and F should be amplified using a torsional amplification factor (Ax ), which can be calculated using Eq. (1).
Ax ¼
dmax 1:2davg
2 ð1Þ
where dmax is the maximum displacement at level x, and davg is the average displacement at the extreme points of the structure at level x (Fig. 2). Using proper analysis procedures, the member forces are calculated by considering both the inherent and accidental torsion, and the adequate member sections are selected.
3. Seismic design of model structures
2.2. Drift check After determining the proper member sections that are capable of resisting member forces, the design story drift of a structure should be checked. According to Section 12.8.6 of ASCE 7-10, the design story drift (D) can be calculated as the difference of the deflections (dCM ) at the center of mass at the top and bottom of the story under consideration except for structures of SDCs C, D, E and F having horizontal irregularity Type 1a or 1b. For these structures, the design story drift is calculated as the largest difference of deflections (dmax ) along any of the edges of the structure at top and bottom of the story under consideration, which is the largest difference in deflection observed at any column line. As shown in Fig. 2, dmax is larger than dCM . The design story drift (D) should not exceed the allowable story drift that is specified in Table 12.12-1 of ASCE 7-10, which varies according to the structure type and risk category. Otherwise, the process of selecting member sections is repeated until the drift requirement is satisfied. 2.3. Stability check To incorporate the P D effect, the stability coefficient (h) is calculated using Eq. (2):
h¼
P x DI e V x hx C d
ð2Þ
where P x is the total vertical design load, V x is the seismic shear force in story x, hx is the story height, Ie is the importance factor, and C d is the deflection amplification coefficient that is specified
δA
δB
δ CM
δ avg =
in Table 12.2-1 of ASCE 7-10. The stability coefficient h should be less than hmax , as specified by Eq. (12.8-17) in ASCE 7-10 [1]. When h is greater than 0.1, P D effects should be accounted for by increasing the member forces and the design story drifts based on rational analyses. Alternatively, it is permitted to increase the seismic response demand by an amplification factor: 1=ð1 hÞ. In this study, the latter method is used to account for the P D effect. Member sections, which were determined from previous design stages, should be checked to determine whether the members are capable of sustaining the increased member forces and design story drifts. The process of selecting member sections should be repeated until the structure is capable of resisting the increased member forces and drifts.
δA + δB 2
⎡ δ ⎤ Ax = ⎢ max ⎥ ⎣⎢1.2δ avg ⎦⎥
Fig. 2. Floor displacements in torsion-irregular structures.
3.1. Model structures Office buildings are designed according to ASCE7-10 [1], which are considered as model structures. Fig. 3 shows the floor plans and elevations of the model structures. To sustain seismic loads, two steel special moment frames (SMF) are placed in each orthogonal direction as shown in Fig. 3b (marked by thicker solid lines). For steel SMFs, the response modification factor (R), deflection amplification factor (C d ), overstrength factor (Xo ), and importance factor (Ie ) are 8, 5.5, 3, and 1, respectively. In Fig. 3b, the dotted lines denote gravity frames. Among the configuration types of moment frames shown in Fig. 3b, Type 1 is the floor plan for regular structures, in which moment frames are symmetrically placed along the perimeter of the structure. In order to impart different degrees of torsional irregularity to a structure, the west side moment frame (W-frame) is placed at five different locations as shown in Fig 3b. As the W-frame approaches the E-frame, the torsional irregularity becomes more severe. Thus, this study considers the six different floor plans shown in Fig. 3b. It is noted that to place the W- frame inside a building rather than along the perimeter of the building is neither practical nor economical. Nevertheless, the different locations of the W-frame are considered because the objective of this study is to investigate effect of different degrees of torsional irregularity on the seismic performance of buildings. As shown in Fig. 3b, the overall floor plan does not vary according to the location of the W-frame. Because heavy cast-in-place concrete rigid diaphragms are used in model structures, the total weight of a building is much larger than that of the W-frame in the building. The weight of the W-frame for three and nine story buildings is less than 2.3% of the total weight of these building. Therefore, the center of mass was assumed to locate at the center of plan irrespective of the location of the W-frame. Variability in mass center is not considered in the present study. To investigate the effect of building height on the collapse performance of torsionally irregular structures, this study considers three- and nine-story building structures. All building structures have five bays in each direction (Fig. 3) where the story height and span length are 4 m and 6 m, respectively. All structures are symmetric about the east-west direction (x-axis), but are asymmetric about the north-south direction (y-axis), with the exception of Type 1 (regular) structures.The model structures are assumed to be located at a site with design spectral response accelerations of 1.0 g and 0.6 g at a short period (SDS ) and a 1-s period (SD1 ), respectively; thus, according to ASCE 7-10 [1], the buildings are assigned to SDC D. The dead and live loads on the floors are 4.60 kPa and 2.39 kPa, respectively, whereas the corresponding loads on the roof are 4.12 kPa and 0.96 kPa, respectively [16].
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Fig. 3. Model structures.
Member force and story drift demands are calculated using a modal response spectrum analysis, which can be used for torsionally irregular structures regardless of SDC levels as described in ASCE 7-10 [1]. The commercial software MIDAS-Gen [17] is used for the analysis and design. Member sections are determined according to AISC 360-10 [15]; the strong column and weak beam requirements specified in E3 of AISC 341-10 [14] are also considered. According to ASCE 7-10 [1], three- and nine-story building structures with irregularity types 4, 5, and 6 (Fig. 3b) are classified as torsionally irregular structures. Alternatively, structures with irregularity types 1, 2, and 3 are classified as regular structures. Thus, torsionally irregular structures (3-TP4-O, 3-TP5-O, 3-TP6-O, 9-TP4-O, 9-TP5-O, and 9-TP6-O) are designed according to the requirements for regular structures with additional requirements for torsionally irregular structures. It is noted that 3-TP4-O stands for a three-story structure (3) of irregularity Type 4 (TP4) that is designed according to the requirements for torsionally irregular structures (O). For nine-story structures [i.e., 9-TP5-O and 9-TP6-O with severe torsion-irregularity (types 5 and 6)], member
sections cannot be found from the list of W-shaped sections provided in AISC 360-10 [15]. This indicates that it may not be economical and feasible to construct buildings having nine stories or higher if they have severe torsional irregularity. Thus, these structures are excluded from the list of model structures. In Appendix B, the member sections for the 3-TP-5O building are summarized at individual design stages. For comparison purposes, this study also considers torsionally irregular structures (3-TP4-X, 3-TP5-X, 3-TP6-X, 9-TP4-X, and 3-TP4-X) having the same member sections with regular structures (3-TP1 and 9-TP1), thus without taking into account the additional requirements for torsional irregular structures. Therefore, these structures do not satisfy the building code requirements, and they may not be safe. Tables 1 and 2 summarize member sections used for beams and columns in Type 1- and Type 5-three story model structures (3-TP1 and 3-TP5-O), respectively. Each model structure is designed repeatedly to find the optimal member sections considering torsion due to the change in the location of the W-frame.
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Table 1 Member sections for Type 1 three story building (3-TP1). Story number (Ns)
N and S frames
E and W frames
Column
1 2 3
Beam
Exterior
Interior
W14x48 W14x48 W14x48
W14x68 W14x68 W14x38
Column
W14x38 W14x38 W14x38
Beam
Exterior
Interior
W14x74 W14x74 W14x53
W14x74 W14x74 W14x53
W14x48 W14x48 W12x45
Table 2 Member sections for Type 5 three story building (3-TP5-O). Ns
N and S frames
E frame
Column
1 2 3
Beam
Exterior
Interior
W14x 68 W14x 68 W14x 68
W24x 94 W24x 94 W14x 68
W24x 55 W24x 55 W21x 68
W frame
Column
Beam
Exterior
Interior
W14x 74 W14x 74 W14x 53
W14x 74 W14x 74 W14x 53
It is observed that member sections in the W-frame of the torsionally irregular structure (3-TP5-O) are significantly larger than those of corresponding regular structure (3-TP1). With increasing the degree of torsional irregularity from Type 1 to Type 5, member sections for N- and S-frames increase moderately. To resist the enlarged forces due to torsion, it is less effective to increase member sections of the E-frame compared to increasing member sections in W-, N-, and S-frames. In this study, the same member sections are used for the E- frame of torsionally irregular structures as those for the E-frame of corresponding regular structures. Table 3 summarizes the natural periods of the first three modes. Centerline dimensions are used for calculating the natural periods. Gravity columns are not included in period calculation and all columns in the first story are rigidly fixed to the ground. The present study investigates the deformed shapes of the first three modes of model structures and the mass participations of two translation components and one rotation component for these modes. The results are not included in this paper. For torsionally regular structures (3TP-1 and 9TP-1), the first mode includes translation in the y-direction exclusively, whereas the second mode includes only translation in the x-direction. The third mode
W14x 48 W14x 48 W12x 45
Column
Beam
Exterior
Interior
W24x 162 W24x 162 W24x 131
W24x 162 W24x 162 W24x 131
W27x 102 W27x 94 W24x 76
includes only rotation. However, unlike the regular structures, the first and third modes of torsionally irregular structures contain both translational component along y-axis and rotation component about z-axis, whereas the 2nd mode of these structures contains only translation component along x-axis. 3.2. Total weight of moment frames at three different design stages The total steel weight of the moment frames in the model structures is estimated at three different seismic design stages: (Stage 1) strength design stage: member sections are determined to resist the member force demands; (Stage 2) drift (stiffness) checking stage: design story drift demands should be smaller than the allowable story drift, and member sections can be changed if necessary; and (Stage 3) stability checking stage: member forces and design story drifts are amplified to account for the P D effects, and member sections are changed if necessary.Fig. 4a presents the total steel weight (W) of the moment frames in three-story model structures that are calculated at each design stage. At Stage 1 (strength checking), the weight for all structures is nearly the same regardless of the degree of torsional irregularity. For three-
Table 3 Natural Periods of the model structures. Number of stories
3
Plan type
Type 1 Type 2 Type 3 Type 4 Type 5 Type 6
9
Type 1 Type 2 Type 3 Type 4
ID
3-TP1 3-TP2-X 3-TP2-O 3-TP3-X 3-TP3-O 3-TP4-X 3-TP4-O 3-TP5-X 3-TP5-O 3-TP6-X 3-TP6-O 9-TP1 9-TP2-X 9-TP2-O 9-TP3-X 9-TP3-O 9-TP4-X 9-TP4-O
Periods, T (s) 1st mode
2nd mode
3rd mode
1.39 1.41 1.40 1.50 1.44 1.65 1.14 1.83 1.19 2.34 1.09 2.61 2.71 2.62 2.92 2.73 3.26 2.40
1.27 1.27 1.27 1.27 1.22 1.27 1.09 1.27 1.17 1.28 0.79 2.51 2.63 2.49 2.63 2.45 2.64 2.32
0.95 0.87 0.87 0.89 0.86 0.87 1.05 0.82 1.12 0.89 0.34 1.56 1.73 1.73 1.77 1.99 1.71 1.71
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3500
2500 3-TP1
9-TP1
(a) 3-story 3000
3-TP3-O
Steel weights (kN)
Steel weights (kN)
2000
3-TP4-O
1500
(b) 9-story
9-TP2-O
3-TP2-O
3-TP5-O 3-TP6-O
1000
9-TP3-O 9-TP4-O
2500
2000
1500
500
1000
0 1
2
3
Design stages
1
2
3
Design stages
Fig. 4. Total steel weights of the moment frames in structures for each design stage.
story regular structures (3-TP1, 3-TP2-O, and 3-TP3-O), the weight is nearly constant irrespective of the design stages. However, for nine-story regular structures (9-TP1, 9-TP-2-O, and 9-TP-3-O), the weight at Stage 2 is significantly larger than the weight at Stage 1; the weight at Stage 3 is also larger than the weight at Stage 2. This indicates that nine-story structures are affected by drifts and the P D effect more than three-story structures. For torsionally irregular structures (3-TP4-O, 3-TP5-O, 3-TP6-O, and 9-TP4-O), the weight increases sharply from Stage 1 to Stage 2. It is noted that when drifts are checked for torsionally irregular structures, dmax is used instead of dCM , as shown in Fig. 2; this requires larger member sections. For three-story torsionally irregular structures, the increase in the weight is small between Stage 2 and Stage 3; alternatively, for nine-story torsionally irregular structures, the increase in the weight is significant. This observations confirms that nine-story torsionally irregular structures are influenced by the P D effect more than three-story torsionally irregular structures. At Stages 2 and 3, the weights for torsionally irregular structures are much larger than those for corresponding regular structures. For example, the weights of 3-TP6-O and 9-TP4-O are 3.9 and 1.5 times larger than those of 3-TP1 and 9-TP1, respectively. As shown in Fig. 4, as the degree of torsional irregularity increases, larger steel weights are required.
4. Nonlinear response history analyses of torsionally irregular structures 4.1. Analytical model Nonlinear response history analyses (RHA) with a threedimensional analytical model are conducted to evaluate the seismic collapse performance of the model structures. For analyses, Opensees (Mazzoni et al. [18]) software is used. Fig. 5a presents the three-dimensional frame model used in the present study. Moment frames are denoted by thick solid lines. Gravity columns are modeled as rigid elements with pins at their ends, which account for the P D effects due to gravity loads. Slabs are modeled to behave as rigid diaphragms by restraining in-plane slab deformation. Fig. 5b presents the analytical model for the beam-column connections in moment frames. Columns are modeled using fiber elements (Fig. 5b-1), which effectively simulate interaction between axial force and moment. The behavior of the fiber elements is represented by a bilinear model with a post elastic slope of 3%. Five integration points are used, equally spaced along the column (Fig. 5b). Neuenhofer and Filippou [19] reported that four to six integration points are typically sufficient to represent the spread of plasticity along an element.
Although fiber section elements may not incorporate cyclic deterioration in strength and stiffness, they can efficiently capture distributed plasticity in members. Because most plastic hinges are developed in beams rather than columns in special moment frames owing to strong column-weak beam requirements, cyclic deterioration may not be a serious problem for columns in SMFs. At each integration point, the member section is divided into 160 fibers (64 fibers for each flange and 32 fibers for web). It is noted that the 108 fibers offers an almost perfect match of the exact solution [20]. Beams are modeled to behave in an elastic range (Fig. 5b-2). Inelastic beam behavior is represented by inelastic rotational springs placed at the ends of each beam (Fig. 5b-3), where M y and M c are the yield and maximum moment strengths, respectively, while hp is the plastic rotation capacity, which can be calculated using the equations proposed by Lignos and Krawinkler [21]. Cyclic deterioration in strength and stiffness is not considered in this study.
4.2. Nonlinear response history analysis (RHA) of torsionally irregular structures To investigate the effect of torsional irregularity on the seismic demands of torsionally irregular structures, this study conducts nonlinear RHA under an orthogonal pair of horizontal ground motions: these motions were recorded at the Beverly Hills station with an epicentral distance of 13 km during the Northridge earthquake. Their peak ground accelerations (PGAs) are 0.42 g and 0.52 g, respectively. Fig. 6 shows the distribution of plastic hinges in the N- and S-frames in the three-story model structures (3-TP1, 3-TP5-X, and 3-TP5-O) that are obtained from nonlinear RHA. As mentioned earlier, 3-TP1 is a three-story regular structure, 3-TP5-X is a torsion-irregular structure with the same member sections as 3-TP1, and 3-TP5-O is also a torsion-irregular structure that is designed according to the requirements for torsionally irregular structures that are specified in ASCE 7-10 [1]. As shown in Fig. 6, the plastic hinges developed in the N- and S-frames of the three model structures are denoted by solid circles. For the regular structure (3-TP1), the distribution and number of plastic hinges developed in the N- and S-frames are almost identical; this result is expected. However, for 3-TP5-X, the distribution of plastic hinges in the N- and S-frames is significantly asymmetric (Fig. 6b). The numbers of plastic hinges in the N- and S-frames are 36 and 10, respectively. Unlike 3-TP5-X, the N- and S-frames of 3-TP5-O have an almost symmetric distribution of plastic hinges. The numbers of plastic hinges detected in the N- and S-frames of 3-TP5-X are 31 and 32, respectively.
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M Mc My
M e
p
(b-3) Rotational spring
M (b-1) Column element (b-2) Beam element (a) Frame model
(b) Connection model Fig. 5. Analytical model.
N
N
N
0.42g
W
0.52g
E
W
S
E
S
W
E
S
N-frame (35)*The number of plastic hinges
N-frame (36)
N-frame (31)
S-frame (36)
S-frame (10)
S-frame (32)
(a) 3-TP1
(b) 3-TP5-X
(c) 3-TP5-O
Fig. 6. Distribution of plastic hinges.
To investigate the cause of the difference in plastic hinge distributions in the N- and S-frames, the displaced roof positions are plotted with thin solid lines at each time step of the nonlinear RHA, as shown in Fig. 7. The original roof position is also plotted with a dotted line as a reference. In Fig. 7, the thick solid lines denote the displaced roof positions when the N- and S-frame displacements are the largest.The ratios of the maximum displacement of the N-frame (DmaxN ) to the maximum displacement of the S-frame (DmaxS ) are 1.02, 2.22, and 1.3 for 3-TP1, 3-TP5-X, and 3-TP5-O, respectively. As expected, the ratio (DmaxN /DmaxS ) for the regular structure is the smallest and the ratio for 3-TP5-X is the largest.This study also evaluates the effect of torsional irregularity on the collapse intensities. For this purpose, incremental dynamic analyses (IDA) are conducted (Vamvatsikos and Cornell [22]; Moon et al. [23]). Collapse is defined as the state of global dynamic instability in one or more stories of the structural system.
In this study, the collapse intensity is defined as a 5% damped pseudo spectral acceleration at the fundamental period [Sa ðT 1 Þ], which is the ordinate of IDA curves. The approximate period (C u T a ) is used as the fundamental period, which is calculated according to ASCE 7-10 [1] (section 12.8.2); for the three-story structures, C u T a is 0.74 s. The approximate period of the three-story regular structure (3-TP1) is shorter than the fundamental period (=1.39 s) obtained from eigenvalue analyses. For torsionally irregular structures having the same member sections, the period difference between approximate and eigenvalue periods is more significant with an increase in torsional irregularity. As shown in Table 3, these structures have a longer period as the level of torsional irregularity increases. However, for torsionally irregular structures properly designed according to seismic design provisions with additional requirements for torsionally irregular structures, the fundamental
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N-frame
N-frame
S-frame
S-frame
S-frame
(a) 3-TP1
(b) 3-TP5-X
(c) 3-TP5-O
N-frame
y x
Fig. 7. Displaced roof positions for three-story model structures.
period obtained from eigenvalue analyses generally decreases as the level of torsional irregularity. The eigenvalue periods of these structures are also shorter than that of the corresponding torsionally irregular structures having the same member sections as regular structures. Therefore, the period difference for these structures is not as significant as torsional irregular structures having the same member sections as 3-TP1. Similar observation was made for 9-story torsional irregular structures.To determine the bi-directional acceleration spectrum, the acceleration responses of an elastic single-degree-of-freedom system with a period of C u T a are calculated subjected to individual ground motions. The vector sum of the bi-directional acceleration responses is computed for each time step and the maximum is defined as the value of the acceleration response spectrum [24]. The abscissa of the IDA curve is represented by an engineering demand parameter (EDP). In this study, the maximum story drift ratio (hmax ) is used as the EDP. In the IDA curve, the global dynamic instability is identified by a characteristic flattening of each IDA, referred to as the flatline, where the seismic demand increases greatly with only a slight increase in the ground motion intensity. To conduct IDA, the same pair of ground motions recorded at the Beverly Hills station during the Northridge earthquake, is used (Fig. 8). The collapse intensities of 3-TP1 and 3-TP5-X are 1.8 g and 0.8 g, respectively. This indicates that the collapse intensities of torsionally irregular structures with the same member sections
as regular structures are significantly smaller than those of corresponding regular structures. The collapse intensity of 3-TP5-O is 3.6 g, which is twice the collapse intensity of 3-TP1. The larger collapse intensity of 3-TP5-O is due to the heavy section sizes that are selected in accordance with the stringent requirements for torsionally irregular structures (ASCE 7-10). 5. Seismic collapse performance evaluation This study evaluates the seismic collapse performance of torsionally irregular structures. Current seismic codes are intended to provide buildings designed according to code requirements with the capability to have a low probability of collapse if subjected to very rare, intense ground motions (referred to as maximum considered earthquake (MCE) ground motion) [1]. FEMA P-695 [25] provides a methodology for quantifying the system performance and response parameters in seismic design. The two performance objectives used in FEMA P-695 [25] are: (1) The probability of collapse for maximum considered earthquake (MCE) ground motions is 10% or less, on average, across a performance group containing model frames with a specific seismic force resisting system with different configurations, and (2) the probability of collapse is 20% or less for an individual model frame.The probability of collapse under MCE ground motions [PðCollapsejSMT Þ] is calculated using Eq. (3).
Fig. 8. IDA curves and collapse intensities.
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PðCollapsejSMT Þ ¼ U
lnðSMT Þ lnð^SCT SSFÞ bTOT
! ð3Þ
where UðxÞ is the cumulative distribution function of the standard normal variate x;SMT is the MCE 5% damped-spectral response acceleration at the fundamental period (T) of the structure, ^SCT is the median value of the 5% damped spectral response acceleration at T for a collapse level earthquake, SSF is the spectral shape factor accounting for the spectral shape of rare ground motions, and adjust the collapse probability, UðxÞ, while bTOT is the total system collapse uncertainty calculated using Eq. (4).
bTOT ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2RTR þ b2DR þ b2TD þ b2MDL
ð4Þ
where bRTR ; bDR ; bTD and bMDL are the record-to-record uncertainty, design requirement-related uncertainty, test data-related uncertainty and modeling uncertainty, respectively. These values range from 0.1 to 0.5, and are determined from tables in pages 3–8, 3– 20, and 5–23 of FEMA P-695 [25]. The step-by-step procedure used to calculate the probability of collapse is described in Appendix A. The three- and nine-story structures are assigned to SDC D. For these structures, SMT values are 1.15 g and 0.51 g, respectively. Table 4 summarizes the probabilities of collapse for model structures, which are calculated using the procedure listed in Appendix A. Fig. 9 presents the effect of the degree of torsional irregularity on the probability of collapse in which the abscissa is the degree of torsional irregularity (Fig. 3) and the ordinate is the probability of collapse. Twenty-two pairs of ground motions, provided in FEMA P695 [25], are used as input ground motions. Fig. 9 includes the probabilities of collapse for torsionally irregular structures designed with and without the additional requirements for torsional irregularity. For three-story torsionally irregular structures with the same member sections as those of regular structures, the probability of collapse becomes larger as the degree of torsional irregularity increases. The collapse probabilities of 3-TP2-X and 3-TP3-X are higher than 0.1, which is a limiting value for performance group, whereas exceed those of 3-TP4-X, 3-TP5-X, and 3-TP6-X exceed 0.2, which is the limiting value for individual structures [25]. In contrast, for three-story torsionally irregular structures designed with the additional requirements, the probability of collapse becomes smaller as the degree of torsional irregularity
increases. The collapse probabilities of 3-TP2-O and 3-TP3-O are slightly higher than 0.1 (<0.2), whereas those of 3-TP4-O, 3-TP5-O, and 3-TP6-O (=0.0441, 0.0123, and 0.0062) are much lower than 0.1. For nine-story torsionally irregular structures, similar observations are also made. For torsionally irregular structures with the same member sections as those of the regular structures, the probability of collapse becomes higher as the degree of torsional irregularity increases; however, this increment is not as significant as that of the three-story torsionally irregular structures. For the nine-story torsionally irregular structures that are designed according to the additional requirements for torsion, the probability of collapse becomes smaller as the degree of irregularity increases. It is noted that 9-TP4-O exhibited the smallest collapse probability (=0.0752). Among eight torsionally irregular model structures designed according to additional requirements for torsion specified in ASCE 7-10, four structures have the probability of collapse lower than 0.1, which are 3-TP4-O, 3-TP5-O, 3-TP6-O, and 9-TP4-O. Therefore, it is observed that as the degree of torsional irregularity increases, the collapse probability of torsionally irregular structures that are designed according to the additional requirements becomes smaller. This phenomenon is attributed to the heavy member sections selected for torsionally irregular structures, which serve to fulfill the drift requirement. 6. Proposed procedure for computing the design story drift for torsionally irregular structures As shown in Fig. 9, the probability of collapse for codecompliant torsionally irregular structures is much smaller than that of regular structures. The probability of collapse also decreases as the degree of torsional irregularity increases. This is attributed to the large member sections of the torsionally irregular structures. In order to design torsionally irregular structures that have a similar probability of collapse to regular structures, irrespective of the degree of torsional irregularity, this study proposes a procedure to determine the design story drift demands that are required in the drift checking stage (Fig. 2). As shown in Fig. 4, the sharpest increase in the total steel weight was observed at the drift checking stage. In the proposed procedure, the drift demand of torsionally irregular structures is estimated by calculating the difference between the displacements at the mass centers of two adjacent
Table 4 Probability of collapse for the model structures. Number of stories
Model structures
SMT
lT
^ SCT
SSF
PðCollapsejSMT Þ
3
3-TP1 3-TP2-X 3-TP3-X 3-TP4-X 3-TP5-X 3-TP6-X 3-TP2-O 3-TP3-O 3-TP4-O 3-TP5-O 3-TP6-O 3-TP4-OM 3-TP5-OM 3-TP6-OM 9-TP3-X 9-TP4-X 9-TP2-O 9-TP3-O 9-TP4-O 9-TP4-OM
1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 1.14 0.51 0.51 0.51 0.51 0.51 0.51
5.64 5.75 5.14 4.57 3.75 9.44 5.63 6.49 6.60 6.15 5.36 8.49 2.77 4.39 3.14 3.82 3.99 4.09 4.61 4.59
1.70 1.65 1.61 1.30 1.30 0.85 1.65 1.70 2.40 3.60 4.38 1.70 1.90 2.86 0.59 0.55 0.64 0.70 0.93 0.82
1.33 1.33 1.31 1.29 1.26 1.14 1.34 1.36 1.36 1.35 1.32 1.42 1.21 1.29 1.23 1.26 1.27 1.27 1.29 1.29
0.12 0.12 0.14 0.22 0.23 0.36 0.12 0.11 0.04 0.01 0.01 0.14 0.13 0.06 0.23 0.25 0.19 0.15 0.08 0.10
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Fig. 9. Collapse probabilities of the model structures.
floors, as opposed to calculating the maximum drift values of the vertical members in the story, which is the same procedure for regular structures. Four torsionally irregular structures (3-TP4-OM, 3-TP5-OM 3, 3TP6-OM, and 9-TP4-OM) are re-designed with the proposed procedure for drift calculations. Nonlinear response history analyses are conducted for structures that are subjected to the same pair of ground motions that were used in the previous section. The probability of collapse for these structures calculated according to FEMA P-695 is summarized in Table 4. Fig. 10 shows the distribution of plastic hinges in the N- and Sframes of the newly designed three-story structure (3-TP5-OM) with torsional irregularity Type 5. The numbers of plastic hinges in the N-and S-frames of 3-TP5-OM are 30 and 22, respectively (Fig. 10b). Thus, the difference between the numbers of plastic hinges in the N-and S-frames of 3-TP5-OM is 12. It is noted that, for 3-TP5-X and 3-TP-5-O, the differences are 26 and 1, respectively (Fig. 6). This indicates that the contribution of torsional
responses in 3-TP5-OM is more significant than that in 3-TP5-O, but less than that in 3-TP5-X. IDA is also conducted for 3-TP5-OM under the same pair of ground motions (Fig. 10c). The collapse intensity is 1.9 g, which is less than that of 3-TP5-O (=3.6 g), but close to that of the regular frame 3-TP1 (=1.7 g). This study also calculated the probabilities of collapse for 3TP4-OM, 3-TP5-OM, 3-TP6-OM and 9-TP4-OM using the 22 pairs of ground motions provided in FEMA P695 [25], which were also used previously. Fig. 11 shows the probability of collapse. In this figure, ‘ASCE 7’ indicates the torsionally irregular structures designed according to ASCE 7-10, whereas ‘MC’ denotes the torsionally irregular structures designed according to ASCE 7-10 [1] with drift demands calculated using the proposed procedure. The collapse probability of ‘MC’ structures is more uniform than that of the ‘ASCE 7’ structures. Also, the collapse probability of ‘MC’ structures is larger than that of the ‘ASCE 7’ structures but is similar with that of regular structures.
Fig. 10. Results of nonlinear RHA for 3-TP5-OM.
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Fig. 11. Probability of collapse according to the degree of torsional irregularity.
Fig. 12. Total steel weights required for the model structures.
The collapse probabilities of three-story ‘MC’ structures (3-TP4OM, 3-TP5-OM, and 3-TP6-OM) with different degrees of torsional irregularity are 0.1372, 0.1307, and 0.0549, respectively. Alternatively, the collapse probabilities of three-story ‘ASCE 70 structures (3-TP4-O, 3-TP5-O, and 3-TP6-O) are 0.0441, 0.0123, and 0.0062, respectively. This indicates that ‘MC’ structures have probabilities of collapse that are closer to those of regular structures than ‘ASCE 70 structures. It is also noted that ‘MC’ structures produced more uniform probabilities of collapse according to the degree of torsional irregularity than ‘ASCE 70 structures. Similar observations are also made for the nine-story structures (Fig. 11b). This study also compared the total steel weight of the structures as shown in Fig. 12. The total steel weight of ‘MC’ structures, using the proposed design methods is less than that of ‘ASCE7’ structures. It is noted that there may be other design solutions that would bring the ‘MC’ buildings to the desired collapse level.
(2)
(3)
7. Conclusions
(4)
In order to evaluate the seismic collapse performance of torsionally irregular structures, three- and nine-story steel special moment frames in model structures with various degrees of torsional irregularity were designed according to ASCE 7-10 [1]. Nonlinear response history analyses were conducted. The conclusions and recommendations of this study are as follows:
(5)
(1) When the degree of torsional irregularity was increased, larger member sections were required. At the strength design
stage (Stage 1), the total steel weight of the moment frames in torsionally irregular structures was similar to that of regular structures. However, at the drift checking stage (Stage 2), the weight of torsionally irregular structures was much larger than that of regular structures. The increase in the total steel weight for three-story torsionally irregular structures was negligible between Stage 2 and Stage 3; however, the increase for nine- story torsionally irregular structures was significant between Stage 2 and Stage 3 due to the large P D effects. It is observed that for regular structures, the distribution of plastic hinges was almost symmetric in the N- and Sframes. Torsionally irregular structures designed according to ASCE 7-10 [1] also exhibited a symmetric distribution of plastic hinges in the N- and S-frames; this was due to the large member sections that were selected to satisfy the additional requirements for torsionally irregular structures, as specified in ASCE 7-10 [1]. For torsionally irregular structures designed without considering the additional requirements, the probability of collapse is larger as the degree of torsional irregularity increases. However, torsionally irregular structures that were designed while also considering the additional requirements generally exhibited smaller probabilities of collapse compared to regular structures. The probabilities of collapse of torsionally irregular structures, designed with drift demands calculated using the proposed method, is similar to those of regular structures regardless of the degree of torsional irregularity.
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(6) The results of this study is obtained using only torsionally irregular structures with steel SMF frames. Therefore, caution should be made for buildings having different seismic force resisting systems such as braced frames and shear walls.
Acknowledgements Authors would like to acknowledge the financial supports provided by the National Research Foundation of Korea (No. 2014R1A2A1A11049488). The valuable comments offered by anonymous reviewers are greatly appreciated.
where V max is the maximum base shear resistance, V d is the design base shear, du is the roof drift displacement at the point of 20% strength loss (=0.8 V max ), dyeff is the effective yield roof displacement, C o relates the displacement of the single-degree-of-freedom system at the fundamental mode of a frame to the roof displacement of the frame, g is the gravity constant, T n is the fundamental period (defined as C u T a , specified in ASCE 7-10) [1], and T 1 is the fundamental period of the model frame computed via eigenvalue analysis. (3) Calculate the spectral shape factor (SSF),that accounts for that accounts for the spectral shape of rare ground motions as depicted in Fig. A1(c).
SSF ¼ exp½b1 ðeo ðTÞ eðTÞrecord
ðA4Þ
eo ðTÞrecord ¼ 0:6ð1:5 TÞ
ðA5Þ
b1 ¼ 0:14ðlT 1Þ0:42
ðA6Þ
Appendix A A step-by-step procedure for calculating the probability of collapse according to FEMA P-695 [23]: (1) Idealize the model frame using a proper analytical model. (2) Conduct nonlinear static analysis to determine the overstrength factor (X) and period-based ductility (lT ). This is done by using Eqs. (A1) and (A2) with Eq. (A3), respectively [Fig. A1(a)].
X¼
V max Vd
(4) Calibrate the probability of collapse by applying the SSF.
! lnðSMT Þ lnð^SCT SSFÞ PðCollapsejSMT Þ ¼ U bTOT
ðA7Þ
ðA1Þ Appendix B
lT
du ¼ dyeff
dy;eff ¼ C 0
ðA2Þ V max h g i ðmaxðT n ; T 1 ÞÞ2 W 4p2
ðA3Þ
The following table includes information about member sections for the Type 5-three story structure at three different design stages. Tables B-1, B-2 and B-3.
Fig. A1. Procedure for calculating the SSF and calculating the probability of collapse.
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Table B-1 Member sections used for the Type 5-three story structure at strength design stage. Level
N-and S-frames
E-frame
Colum
Girder
Exterior
Interior
1
W14x48
W14x68
W14x38
2
W14x48
W14x68
W14x38
3
W14x38
W14x38
W12x35
W-frame
Colum
Girder
Exterior
Interior
W14x 74 W14x 74 W14x 53
W14x 74 W14x 74 W14x 53
W14x 48 W14x 48 W12x 45
Colum
Girder
Exterior
Interior
W14x 74 W14x 74 W14x 68
W14x 74 W14x 74 W14x 68
W14x 48 W14x 48 W12x 45
Table B-2 Member sections used for the Type 5-three story structure at drift checking stage. Level
N-and S-frames
E-frame
Colum
Girder
Exterior
Interior
1
W14x68
W24x94
W24x55
2
W14x68
W24x94
W24x55
3
W14x68
W14x68
W21x68
W-frame
Colum Exterior
Interior
W14x 74 W14x 74 W14x 53
W14x 74 W14x 74 W14x 53
Girder
Colum
Girder
Exterior
Interior
W14x 48 W14x 48 W12x 45
W24x 162 W24x 162 W24x 131
W24x 162 W24x 162 W24x 131
W27x 102 W27x 94 W24x 76
Table B-3 Member sections used for the Type 5-three story structure at stability checking stage. Level
N-and S-frames
E-frame
Colum
Girder
Exterior
Interior
1
W14x68
W24x94
W24x55
2
W14x68
W24x94
W24x55
3
W14x68
W14x68
W21x68
W-frame
Colum
Girder
Exterior
Interior
W14x 74 W14x 74 W14x 53
W14x 74 W14x 74 W14x 53
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W14x 48 W14x 48 W12x 45
Colum
Girder
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Interior
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W24x 162 W24x 162 W24x 131
W27x 102 W27x 94 W24x 76
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