# Seismic Behavior Factors of RC Staggered Wall Buildings

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## Abstract

In this study seismic performance of reinforced concrete staggered wall system structures were investigated and their behavior factors such as overstrength factors, ductility factors, and the response modification factors were evaluated from the overstrength and ductility factors. To this end, 5, 9, 15, and 25-story staggered wall system (SWS) structures were designed and were analyzed by nonlinear static and dynamic analyses to obtain their nonlinear force–displacement relationships. The response modification factors were computed based on the overstrength and the ductility capacities obtained from capacity envelopes. The analysis results showed that the 5- and 9-story SWS structures failed due to yielding of columns and walls located in the lower stories, whereas in the 15- and 25-story structures plastic hinges were more widely distributed throughout the stories. The computed response modification factors increased as the number of stories decreased, and the mean value turned out to be larger than the value specified in the design code.

### Keywords

staggered wall systems seismic design overstrength factors ductility factors response factors## 1 Introduction

Reinforce concrete (RC) shear walls are key elements to resist both gravity and lateral loads in building structures, and the seismic performance and analysis modeling of RC shear wall structures have been widely investigated by many researchers (e.g., Wallace 2012). Recently an alternative building structure system, a staggered wall system, has drawn attention due mainly to its capability to provide wider open space. The staggered-wall system consists of a series of storey-high RC walls spanning the total width between two rows of exterior columns and arranged in a staggered pattern on adjacent column lines. With the columns only on the exterior of the building, a full width of column-free area can be created. Compared with traditional shear wall structures, the structures with vertical walls placed at alternate levels have advantage for their enhanced spatial flexibility. Currently Korean government provides various incentives for apartment buildings designed with increased spatial flexibility. In this regard the apartment buildings with vertical walls placed at alternate levels have advantage for their enhanced spatial flexibility. Such a structural system has already been widely applied in steel residential buildings, which is typically called a staggered truss system.

The system was first proposed by Fintel (1968), who found out that the staggered wall systems are very competitive with the conventional form of construction and are more economical. Mee et al. (1975) carried out shaking table tests of 1/15 scaled models for the staggered wall systems and found that the consistent mass analysis gave reasonable estimation of dynamic behavior of the system. Kim and Jun (2011) evaluated the seismic performance of partially staggered wall apartment buildings using non-linear static and dynamic analysis, and compared the results with those of conventional shear wall apartment buildings. Lee and Kim (2013) investigated the seismic performance of six and 12-story staggered wall structures with a middle corridor based on the FEMA P695 procedure. It was found that the collapse margin ratios of the model structures obtained from incremental dynamic analyses turned out to be larger than the limit states specified in the FEMA P695. Kim and Han (2013) investigated the sensitivity of design variables to the seismic response of staggered wall structures. It was observed that when the earthquake intensity is relatively small, the yield stress of rebars and the concrete strength in the link beams are important factors as well as inherent damping ratio. As the intensity of seismic load increased, the strength of columns became another important factor. Lee and Kim (2013) derived empirical formulas for fundamental natural period of reinforced concrete staggered wall structures. They found that the natural periods of the staggered wall structures are similar to those of the shear wall structures having the same overall configuration.

The staggered wall systems, however, have not been considered as one of the basic seismic-force-resisting systems in most design codes due mainly to the vertical discontinuity of the main structural elements. ASCE 7 (2010) requires that lateral systems that are not listed as the basic seismic-force-resisting systems shall be permitted if analytical and test data are submitted to demonstrate the lateral force resistance and energy dissipation capacity. The American Institute of Steel Construction (AISC) Design Guide 14 (AISC 2002) recommends the response modification factor of 3.0 for seismic design of staggered truss system buildings; however none is specified for reinforced concrete staggered wall systems.

Seismic behavior factors including the response modification factors are essential for seismic design of structures. The factors are provided for typical structure systems in most design codes. However for non-typical structures the determination of the behavior factors is an important issue (e.g. Tomaževič and Weiss 2010, Skalomenos et al. 2015). In this study the behavior factors such as overstrength factors, ductility factors, and the response modification factors of reinforced concrete staggered wall system (SWS) structures were evaluated following the procedure recommended in the ATC 19 (1995). To this end, 5, 9, 15, and 25-story SWS structural models were designed and were analyzed by nonlinear static and dynamic analyses to obtain their force–displacement relationship up to failure. The response modification factors were computed based on the overstrength and the ductility capacities obtained from the capacity envelopes.

## 2 Analysis Model Structures

### 2.1 Configuration of Staggered Wall System Analysis Models

Naming plan for analysis model structures.

Length of wall (m) | Seismic region | Number of story | Name |
---|---|---|---|

6 | Low seismic region | 5 | 5F_6 m low |

9 | 9F_6 m low | ||

15 | 15F_6 m low | ||

25 | 25F_6 m low | ||

Medium seismic region | 5 | 5F_6 m medium | |

9 | 9F_6 m medium | ||

15 | 15F_6 m medium | ||

25 | 25F_6 m medium | ||

9 | Low seismic region | 5 | 5F_9 m low |

9 | 9F_9 m low | ||

15 | 15F_9 m low | ||

25 | 25F_9 m low | ||

Medium seismic region | 5 | 5F_9 m medium | |

9 | 9F_9 m medium | ||

15 | 15F_9 m medium | ||

25 | 25F_9 m medium |

### 2.2 Structural Design of Analysis Model Structures

^{2}including the weight of the structure itself and immovable fixtures, and live loads of 1.92 kN/m

^{2}was used assuming that the structure was used as residential buildings. The staggered wall structures, as well as the staggered truss structures, have not been included in seismic load resisting systems due mainly to the fact that the lateral load resisting system, the staggered walls, is not vertically continuous. In addition, as the staggered walls act like story-high deep beams, the structures are similar to typical weak column-strong beam systems. Therefore in this study response modification factor of 3.0 was used in the structural design of the staggered wall systems, which is generally used for the structures to be designed without consideration of seismic detailing. Table 2 shows the seismic coefficients used for evaluation of design seismic load following the ASCE 7-13 specification, where the parameters S

_{s}and S

_{1}represent the maximum considered earthquake (MCE) spectral response acceleration parameters at short period and at 1 s period, respectively, and the parameters S

_{DS}and S

_{D1}represent the design spectral response acceleration at short and at a period of 1 s, respectively. To consider the effect of design seismic load levels, the model structures were designed with two different levels of seismic loads corresponding to low and medium seismic regions. The design seismic loads for the structures located in the low and the medium seismic regions were determined based on the assumption that the structures were located in the class B site (rock) and C site (very dense soil or soft rock), respectively. The design spectrum for low seismic region was constructed using the design spectral response acceleration parameters, S

_{DS}and S

_{D1}, of 0.31 and 0.13, respectively. The design spectrum for medium seismic region was constructed using the acceleration parameters of 0.57 and 0.20, respectively. At zero natural period (T = 0), the spectral response accelerations for the Low and the Medium earthquakes are 0.12 and 0.23, respectively. The ultimate strength of concrete is 24 MPa and the tensile yield stress of re-bars is 400 MPa. The thickness of the staggered walls is 20 cm throughout the stories. The thickness of the floor slabs is 21 cm which is the minimum thickness required for wall-type apartment buildings in Korea to prevent transmission of excessive noise and vibration through the floors. The thickness of the staggered walls is 20 cm throughout the stories. The rebar placements in the columns and staggered walls are presented in Table 3. The reinforcements of the columns followed the seismic detailing of ordinary moment resisting frames specified in the ACI 318 (2014). The staggered walls were designed as deep beams, for which only minimum reinforcement of D10@400 was needed both horizontal and vertical directions due to their large depth. Table 4 shows the fundamental natural periods of the analysis model structures along the transverse direction where the staggered walls are located, where it can be observed that the natural periods of the structures designed for medium seismic load are slightly shorter than those of the structures designed for low seismic load.

Seismic coefficients used for evaluation of design seismic load.

Seismic load level | Low | Medium |
---|---|---|

Maximum considered earthquake | ||

S | 0.46 | 0.80 |

S | 0.19 | 0.30 |

Site class | B | C |

Design earthquake | ||

S | 0.31 | 0.57 |

S | 0.13 | 0.20 |

S | 0.12 | 0.23 |

R-factor | 3 |

Rebar details in the analysis model structure.

(a) Beams | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Bar-placement detail (beam) | ||||||||||

Type | Low seismic region | Medium seismic region | ||||||||

Dimension (mm) | Rebar | Dimension (mm) | Rebar | |||||||

H | B | Upper | Lower | H | B | Upper | Lower | |||

5F | G1 | 1–3F | 350 | 220 | 4-D22 | 2-D22 | 400 | 250 | 5-D25 | 3-D25 |

4–5 F | 350 | 220 | 4-D19 | 2-D19 | 350 | 220 | 4-D22 | 2-D22 | ||

GR | 5 F | 400 | 250 | 3-D19 | 3-D19 | 350 | 220 | 4-D19 | 3-D19 | |

9F | G1 | 1–3 F | 350 | 220 | 4-D22 | 2-D22 | 400 | 250 | 5-D25 | 4-D25 |

4–6 F | 350 | 220 | 4-D22 | 2-D22 | 400 | 250 | 5-D25 | 4-D25 | ||

7–9 F | 350 | 220 | 4-D19 | 2-D19 | 400 | 250 | 5-D22 | 3-D22 | ||

GR | 9 F | 400 | 250 | 2-D22 | 3-D22 | 350 | 220 | 4-D19 | 3-D19 | |

15F | G1 | 1-3 F | 350 | 220 | 4-D25 | 2-D25 | 450 | 300 | 6-D25 | 5-D25 |

4–6 F | 350 | 220 | 4-D25 | 2-D25 | 450 | 300 | 6-D25 | 5-D25 | ||

7–9 F | 350 | 220 | 5-D22 | 2-D22 | 450 | 300 | 5-D25 | 4-D25 | ||

10–12 F | 350 | 220 | 5-D19 | 2-D19 | 400 | 250 | 5-D25 | 3-D25 | ||

13–15 F | 350 | 220 | 4-D19 | 2-D19 | 400 | 250 | 5-D22 | 3-D22 | ||

GR | 15 F | 400 | 250 | 2-D22 | 3-D22 | 350 | 220 | 4-D19 | 3-D19 | |

25F | G1 | 1–3 F | 400 | 250 | 4-D25 | 2-D25 | 500 | 320 | 6-D25 | 5-D25 |

4–6 F | 400 | 250 | 4-D25 | 2-D25 | 500 | 320 | 6-D25 | 5-D25 | ||

7–9 F | 400 | 250 | 4-D25 | 2-D25 | 500 | 320 | 7-D25 | 5-D25 | ||

10–12 F | 400 | 250 | 4-D25 | 2-D25 | 450 | 300 | 6-D25 | 5-D25 | ||

13–15 F | 400 | 250 | 4-D25 | 2-D25 | 450 | 300 | 6-D25 | 5-D25 | ||

16–18 F | 350 | 220 | 4-D22 | 2-D22 | 400 | 250 | 6-D25 | 4-D25 | ||

19–21 F | 350 | 220 | 5-D19 | 2-D19 | 400 | 250 | 5-D25 | 3-D25 | ||

22–25 F | 350 | 220 | 4-D19 | 2-D19 | 350 | 220 | 4-D25 | 2-D25 | ||

GR | 25 F | 400 | 250 | 3-D19 | 3-D19 | 350 | 220 | 4-D19 | 3-D19 |

(b) Columns | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Bar-placement detail (column) | |||||||||||||

Type | Low seismic region | Medium seismic region | |||||||||||

C1 | C2 | C3 | C1 | C2 | C3 | ||||||||

BH | Rebar | BH | Rebar | BH | Rebar | BH | Rebar | BH | Rebar | BH | Rebar | ||

5F | 1–3 F | 400 | 8-D22 | 600 | 16-D22 | 400 | 12-D22 | 600 | 16-D22 | 400 | 8-D22 | 600 | 12-D22 |

4–5 F | 400 | 8-D22 | 550 | 8-D22 | 400 | 12-D22 | 550 | 8-D22 | 400 | 6-D22 | 550 | 16-D22 | |

9F | 1–3 F | 400 | 12-D22 | 650 | 16-D22 | 500 | 16-D22 | 650 | 12-D25 | 500 | 12-D22 | 650 | 16-D22 |

4–6 F | 400 | 6-D19 | 600 | 14-D19 | 450 | 8-D19 | 600 | 10-D22 | 450 | 6-D22 | 600 | 10-D22 | |

7–9 F | 400 | 8-D22 | 550 | 6-D25 | 400 | 12-D22 | 550 | 6-D25 | 400 | 6-D19 | 550 | 8-D25 | |

15F | 1–3 F | 550 | 6-D25 | 750 | 12-D25 | 600 | 8-D25 | 650 | 12-D25 | 600 | 12-D25 | 750 | 16-D25 |

4–6 F | 500 | 6-D25 | 700 | 10-D25 | 550 | 6-D25 | 600 | 10-D22 | 550 | 16-D19 | 700 | 10-D25 | |

7–9 F | 450 | 4-D25 | 650 | 10-D25 | 550 | 6-D25 | 550 | 6-D25 | 500 | 16-D16 | 650 | 10-D25 | |

10–12 F | 420 | 4-D25 | 600 | 10-D22 | 500 | 6-D25 | 500 | 6-D25 | 500 | 16-D16 | 600 | 8-D25 | |

13–15 F | 420 | 8-D25 | 550 | 6-D25 | 500 | 8-D25 | 500 | 6-D25 | 450 | 4-D25 | 550 | 6-D25 | |

25F | 1–3 F | 700 | 10-D25 | 750 | 20-D29 | 800 | 14-D25 | 600 | 20-D25 | 800 | 16-D25 | 750 | 20-D29 |

4–6 F | 650 | 10-D25 | 700 | 20-D25 | 750 | 12-D25 | 550 | 12-D25 | 750 | 16-D25 | 700 | 18-D29 | |

7–9 F | 600 | 8-D25 | 650 | 20-D25 | 700 | 10-D25 | 500 | 10-D25 | 700 | 10-D25 | 650 | 16-D29 | |

10–12 F | 550 | 6-D25 | 600 | 16-D25 | 650 | 10-D25 | 450 | 10-D25 | 650 | 10-D25 | 600 | 16-D29 | |

13–15 F | 500 | 6-D25 | 550 | 10-D25 | 600 | 10-D25 | 400 | 8-D25 | 600 | 10-D25 | 550 | 12-D29 | |

16–18 F | 450 | 4-D25 | 500 | 10-D25 | 550 | 6-D25 | 400 | 8-D25 | 550 | 6-D25 | 500 | 8-D29 | |

19–21 F | 450 | 4-D25 | 450 | 12-D25 | 500 | 6-D25 | 400 | 10-D25 | 500 | 6-D25 | 450 | 12-D29 | |

22–25 F | 400 | 8-D25 | 400 | 10-D25 | 450 | 10-D25 | 400 | 8-D25 | 450 | 4-D25 | 400 | 8-D29 |

Fundamental natural periods of model structures along the transverse direction.

Wall length (m) | Seismic load level | Story | Period (s) |
---|---|---|---|

6 | Low | 5 | 0.15 |

9 | 0.43 | ||

15 | 1.09 | ||

25 | 2.37 | ||

Medium | 5 | 0.14 | |

9 | 0.36 | ||

15 | 0.88 | ||

25 | 2.04 | ||

9 | Low | 5 | 0.13 |

9 | 0.32 | ||

15 | 0.69 | ||

25 | 1.47 | ||

Medium | 5 | 0.11 | |

9 | 0.26 | ||

15 | 0.59 | ||

25 | 1.36 |

### 2.3 Modeling for Analysis

*E*is the elastic modulus of steel rebars. The expected ultimate strengths of the concrete and steel were taken to be 1.5 and 1.25 times the nominal strengths based on the recommendation of the FEMA-356 (FEMA 2000). As the model structures were designed without considering seismic detailing, the confinement effect of concrete was neglected in the stress–strain relationship. The columns and walls were modeled by the multi-axial spring model with fiber elements as shown in Fig. 4. The axial/bending deformation was simulated by elongation or contraction of each fiber element. The in-plane shear force is resisted by the spring

*W*and the out-of-plane shear is resisted by the springs

*C1*and

*C2*, respectively. The symbols

*I*

_{1}and

*A*

_{s1}denote the moment of inertia and rebar cross-sectional area of the element

*C1*, and

*I*

_{2}and

*A*

_{s2}denote the moment of inertia and rebar cross-sectional area of the element

*C2*, respectively. The symbols

*I*and

*A*

_{s}denote the moment of inertia and rebar cross-sectional area of the element

*W.*The symbols

*d*and \( \theta \) denote the displacement and the rotation at a joint, respectively. The hysteretic behavior of the shear springs was idealized by the origin-oriented model based on the tri-linear hysteresis curve as described in Fig. 5, which can consider the decrease in gradient of loading as the loading cycle and the deformation increase. It is assumed that the cross section of the shear walls remains plane when an in-plane wall deformation occurs. Following the plane section remain plane assumption, strain of the fiber element in the cross section is proportional to the distance from the neutral axis. The stress of each slice is calculated using the stress–strain relation from the strain of each fiber slice, and the bending moment is calculated by summing the moments to the center of the cross section. Figure 6 shows the modified Clough model used to simulate the bending deformation of the elements (Clough and Johnston 1966). The model is composed of bi-linear lines and may represent the degradation of stiffness after yielding. Even though the simplified origin-oriented hysteresis model may not be quite accurate for predicting shear response of the wall element, especially under high shear stresses, it was employed in this study for the following reasons: (i) the staggered walls act more like deep beams rather than shear walls, and (ii) for design level earthquakes, inelastic deformations are concentrated mostly in columns and most staggered walls remain elastic. At the first yield point,

*f*

_{e}, the post-yield stiffness was set to be 16 % of the initial stiffness, and after the final yield point,

*f*

_{y}, the stiffness was reduced to 0.1 % of the initial stiffness. The slabs were considered as rigid diaphragm.

## 3 Seismic Performance Evaluation

To evaluate the nonlinear behavior of the model structures subjected to seismic load, pushover analyses were carried out along the transverse direction by applying incremental lateral load proportional to the fundamental mode of vibration. To define the failure limit state of the model structures, the following two approaches were followed: first, the structure was assumed to have reached a limit state when the inter-story drift reached 1.5 % of the story height as recommended by most seismic design codes such as the ASCE 7 (2010). Second, a structural failure was defined when formation of plastic hinges leaded to failure mechanism. The model structures were assumed to have failed when either of the two limit states occurred.

*S*

_{a}at the fundamental period of the structure becomes 0.1 g; (2) Carry out nonlinear dynamic analysis and estimate the maximum inter-story drift and base shear of the structure; (3) Increase

*S*

_{a}by 0.1 g and carry out the same analysis procedure. Figure 15 compares the base shear-roof displacement relationships of the model structures obtained by incremental dynamic analyses and nonlinear static pushover analyses. Except for the slight discrepancy in the results of the 15- and 25-story model structures, the base shear-roof displacement curves obtained from IDA and pushover analyses generally coincide well with each other. In the linear elastic deformation stage the two results are almost identical. After yielding slight difference is observed between the two results; however the difference is not significant.

## 4 Behavior Factors of the Model Structures

*R*

_{o}is the overstrength factor to account for the observation that the maximum lateral strength of a structure generally exceeds its design strength. Similar procedure was applied to evaluate the seismic design factors for reinforced concrete moment frames (AlHamaydeh et al. 2011), reinforced masonry structures (Shedid et al. 2011), and steel moment resisting frames with buckling restrained braces (Abdollahzadeh et al. 2012). The FEMA (2000) specified three components of overstrength factors in Table C5.2.7-1: design overstrength, material overstrength, and system overstrength.

*R*

_{μ}is a ductility factor which is a measure of the global nonlinear response of a structure, and

*R*

_{γ}is a redundancy factor to quantify the improved reliability of seismic framing systems constructed with multiple lines of strength. In this study the redundancy factor was assumed to be 1.0 based on the fact that there are more than four seismic load-resisting frames along the transverse direction. Then the response modification factor is determined as the product of the overstrength factor and the ductility factor. From the base-shear versus roof displacement relationships, the overstrength factor and the ductility factor are obtained as follows (ATC-19 1995):

*V*

_{d}is the design base shear,

*V*

_{e}is the maximum seismic demand for elastic response, and

*V*

_{y}is the base shear corresponding to the yield point, which can be obtained from the capacity curves. To find out the yield point, straight lines are drawn on the pushover curve as depicted in Fig. 16 in such a way that the area under the original curve is equal to that of the idealized one as recommended in the FEMA-356 (2000). In this study the ductility factor was obtained using the system ductility ratio \( \mu \) as proposed by Newmark and Hall (1982)

*T*is the fundamental natural period of the structure and the ductility ratio was obtained by dividing the roof displacement at failure with the displacement at yield. Equation (3) is plotted in Fig. 17.

## 5 Conclusions

One of the main obstacles to be overcome for application of staggered wall systems is to ensure the seismic safety of the systems and to provide valid seismic design coefficients. In this study seismic performance and the behavior factors such as overstrength factors, ductility factors, and the response modification factors of reinforced concrete SWS structures were evaluated. The analysis results showed that the behavior factors obtained by pushover analysis and incremental dynamic analysis turned out to be similar to each other. The overstrength factors of the structures designed with medium-level seismic load turned out to be smaller than those of the structures designed with low-level seismic load. This is due mainly to the fact that the participation of gravity load is more significant in the design of the latter system. The structures with 9 m-long staggered walls showed higher overstrength than the structures with 6 m-long walls. The ductility factors were relatively uniform regardless of the height of the model structures and the length of the staggered walls with average value of 2.34. The response modification factors obtained by multiplying overstrength factor and ductility factor decreased as the number of stories increased. Except for the structure with 6 m-long staggered walls designed for medium-level seismic load, the response modification factors turned out to be higher than 3.0 which was used for evaluation of design seismic load. The magnitude of the response modification factors were contributed mainly from large overstrength rather than from large deformability. The response modification factors of the structures designed for low-level seismic load were higher than those of the structures designed for higher seismic load. Based on the analysis results, it is concluded that the RC SWS structures generally have adequate strength and ductility capacities to resist design seismic load. As the response modification factor of the model structures analyzed in this study ranged from 3.5 to 8, the current response modification factor of 3.0 seems to be in the conservative side and a little higher value of 3.5 or 4.0 may be more appropriate value for seismic design of staggered wall structures.

It was also observed that the maximum strength of the model structures did not increase proportionally to the design base shear, even though the design base shear increased as the number of story increased. This is due to the fact that as the number of story increased the damage was concentrated in the lower few stories. Even though no story failure mechanism was observed until maximum inter-story drift of 1.5 % was reached in all model structures, it would be necessary to delay the occurrence of story failure mechanism by reinforcing lower story columns to increase seismic-load resisting capacity of the structures. Also the adoption of seismic joint details specified in the ACI code will help increase the ductility of the system.

Finally it needs to be stated that, as the seismic performance of the staggered wall structures has not been validated by proper tests, further experimental research is still required for accurate evaluation of the seismic load resisting capacity of the staggered wall structures. Also the use of more accurate nonlinear concrete model will help enhance the validity of this study

## Notes

### Acknowledgments

This research was supported by a Grant (13AUDP-B066083-01) from Architecture & Urban Development Research Program funded by Ministry of Land, Infrastructure and Transport of Korean government.

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