Abstract
This research investigates the difference of seismic ductility reduction factors between flexuretype and sheartype multidegreefreedom (MDOF) systems. Multimass column cantilever systems are employed to simulate flexuretype shearwall structure, while multimass series spring connection systems are used to simulate sheartype frame structure. Four earthquake records in hard soil site are employed to conduct nonlinear dynamic time history analysis. The effects of storey displacement ductility and vibration period on the seismic ductility reduction factors for flexuretype structure are studied, and the ductility reduction factors between flexuretype and sheartype structures are compared. The results show that the ductility reduction factors for flexuretype structure are about 40 % larger than those for flexuretype structure. Meanwhile, storey displacement ductility and storey number are important factors on seismic ductility reduction factors.
1. Introduction
The seismic ductility reduction factor, denoted as $R$, is not only an important indicator in strengthbased seismic design, but also a key factor to determine the inelastic response spectra in performancebased seismic design theory. Seismic ductility reduction factors are defined as the ratios of the minimum bearing capacity for structures keeping perfect elastic mechanical behavior to those keeping given ductility. In respect to the development and application in international seismic design codes, Zhai conducted a comprehensive review on seismic ductility reduction factors [12]. Previous researches on seismic ductility reduction factors mainly focused on singledegreeoffreedom (SDOF) system, such as classic Newmark rule, Nessar and Miranda models [15]. In recent years, investigation aspects have been extended to MDOF systems. For instance, Miranda and Zhai studied those factors based on concrete frame structures, SantaAna based on steel frames, and Moghaddamand Zhou based on sheartype MDOF systems [68]. All of these researches have gained deep and new insights into the seismic ductility reduction factor [810].
However, the current studies on the seismic ductility reduction factor are mainly based on structure models predominated by sheartype deformation. Few researches report on this factor using flexure or flexuresheartype models. It needs to be further studied regarding whether there are apparent effects on the seismic ductility reduction factor for different flexureshear stiffness ratios. In order to deeply understand the differences between various models, multimass column cantilever systems and multimass series spring connection systems are employed to simulate flexuretype shearwall structure and sheartype frame structure, respectively. The effects of flexureshear ratio on the seismic response in terms of top displacement, story maximum displacement ductility demand and base shear are studied. In addition, the displacement ductility demand and the seismic ductility reduction factor are compared between flexuretype and sheartype MDOF systems.
2. Ground motion records and MDOF systems
2.1. Ground motion records
Four sets of earthquake record are selected and matched to moderate hard soil condition (360$<{V}_{s}<$800 m/s, EC8) employing SeismoMatch software to eliminate the influence of sites [11, 12]. The details are presented in Table 1. The comparisons of the matched acceleration response spectra and the design response spectra are shown in Figure 1. The peak value of the response spectra ranges from 0.2 to 0.5 times period.
Table 1Ground motion records used in analysis
Number  Earthquake event  Date  Station  $PGA$ / g  $\mathrm{\Delta}t$ / s  Duration / s 
Q1  Kobe  1995.01  NishiAkashi NIS090  0.486  0.01  40 
Q2  Imperial Valley  1940.04  El Centro Site Irrigation District N180  0.319  0.02  30 
Q3  Loma Prieta  1989.10  SaratogaAloha Ave N360  0.504  0.02  40 
Q4  Northridge  1994.01  NewhallLa County Fire Station N360  0.589  0.02  30 
Fig. 1Comparison of design response spectra and measured response spectra
2.2. MDOF systems
It’s widely acknowledged that the flexureshear stiffness eigenvalue $\lambda $ has great influence on the mechanic and deformation behavior for structures subjected to lateral loads with even or inverse triangle distribution along height. $\lambda $ equals to 0 and $\infty $ for pure sheartype frames and pure flexuretype walls, respectively. While the $\lambda $ takes value between 0 and $\infty $, structure is defined as frame shear wall structure predominated by composite flexureshear deformation.
The design response spectrum is defined by the European specifications using the shear wave velocity to define different types of venues, where 360$<{V}_{s}<$800 m/s (EC8), and this it is widely considered as a standard method all over the world. The reason of recording is to ensure the consistency of site conditions since different site conditions would lead to a greater variation of seismic response. Figure 1 illustrates the site of this study is similar to site conditions in such a comparison, and in the comparison under conditions other venues.
The simplified column cantilever model, namely Sandwich beam, is widely used to model the shear walls in analyzing the global seismic behavior of tall buildings [13]. Choprahas validated the effectiveness of this simplified model [14]. In this research, multimass column cantilever system is employed to simulate the shear walls predominated by parabolic curve flexure deformation under static lateral loads with even distribution along height. The hysteretic curve of this system is based on the modified Clough hysteretic model. The stiffness degradation factor is set to be 0.001, and the shear deformation of the column is not considered. In comparison, a MDOF sheartype spring model^{}is used to simulate the sheartype frame structures, also employing the modified Clough hysteretic model. The $P\mathrm{\Delta}$ effects of all the systems are not considered, and the first and second damping ratio of the structures is assigned to be 0.05. In this study, systems are modeled by structural static/dynamic analysis program CANNY2010 to conduct dynamic time history analysis, as shown in Figure 2 [15, 16].
3. The influence of flexureshear stiffness ratio
For simplification, 8storey models are designed as shown in Figure 2, where the storey height and the mass of each floor are equal in height. Simple harmonic sinusoidal excitations are inputted as presented in Figure 3 to conduct nonlinear dynamic time history analysis. The seismic responses in terms of vibration mode, top displacement and storey displacement ductility have been compared.
Fig. 2Deformation mode for systems and multimass model
(a) Lateral distribution model
(b) MDOF column system
(c) MDOF sheartype system
Fig. 3Sample harmonic sinusoidal excitation
3.1. Comparison results for different initial periods
The story lateral strength and stiffness along height are firstly set to be the same both in the column cantilever flexuretype system and sheartype spring system. The 1st to 5th modal periods before and after loading are shown in Table 2, which shows that the modal periods differ greatly from each other between sheartype and flexuretype structure. The differences in mode periods are attributed to different story shear strength and different lateral deformation mechanism. The first vibration period is lager in flexuretype structure than that in sheartype structure. The modal period before and after loading in sheartype structure is smaller than that in flexuretype structure.
Table 2Comparison of mode periods
Mode  Sheartype structure  Flexuretype structure  
Before loading  After loading  Before loading  After loading  
1  0.7975  1.0850  1.0316  1.3514 
2  0.2546  0.2911  0.3478  0.3849 
3  0.1519  0.1870  0.2135  0.2356 
4  0.1137  0.1382  0.1579  0.1840 
5  0.0929  0.1126  0.1288  0.1575 
The top displacement demand (TDD) of two different systems with different modal periods, as presented in Table 2, is compared in Figure 4. It is demonstrated that the TDD in flexuretype structure is larger than that in sheartype structure due to different periods and deformation mechanism. In contrast, it is observed from Figure 5 that the maximum story displacement ductility demand (MSDDD) in sheartype structure is greatly larger than that in flexuretype structure. It can be explained that the flexuretype structure deforms lager in upper stories while the sheartype structure deforms lager in lower stories when subjected to external excitations. Thus, two different systems show the contrast variation trends in the displacement and ductility demand.
Fig. 4Comparison of top displacement of two structural systems with different periods
Fig. 5Comparison of maximum interstory displacement ductility
3.2. Comparison results for same initial periods
The flexuretype systems and sheartype systems with a same initial period of 0.7 seconds are also studied by changing the stiffness of the corresponding system described in section 2.1. In other word, the lateral story stiffness in these two systems is different while their initial periods and the lateral story strength are kept the same with each other.
The TDD and MSDDD of two different systems with same period are shown in Figure 6 and Figure 7, respectively. Figure 6 indicates that the TDD of flexuretype structure is still larger than that of sheartype structure, in which there is a same variation trend with Figure 4. In addition, the TDD of both systems becomes smaller due to relatively small initial vibration periods. As shown in Figure 7, the MSDDD of flexuretype structure increases in comparison to that in Figure 5. This increment reduces the difference in MSDDD between the two systems. However, Figures 7 and 5 indicate the same trend that the MSDDD of flexuretype structure is larger than that of sheartype structure. It also demonstrates that the sheartype structure shows a better displacement ductility behavior.
In order to compare the maximum base shear demand of the two different systems, namely the seismic ductility reduction factors, the storey strength of these two systems are changed to be identical and the target ductility are set to be the same level. The following section will discuss these differences deeply by analyzing various parameters in terms of storey numbers, displacement ductility targets and initial periods.
Fig. 6Comparison of top displacement of two structural systems with same period
Fig. 7Comparison of maximum interstory displacement ductility
4. The seismic ductility reduction factor
4.1. Analysis method
For MDOF systems, the seismic ductility reduction factor is defined as:
where ${R}_{\mu}$ is the seismic ductility reduction factor; $V\left({\mu}_{i}=1\right)$ and $V\left({\mu}_{i}={\mu}_{i}^{t}\right)$ are the maximum base shear demand for MDOF systems under same earthquake intensity to keep complete elasticity and that to reach the ductility target of ${\mu}_{i}^{t}$, respectively.
For simplicity, a flowchart is presented in Figure 8 to better understand the method principle and steps. A total of 768 nonlinear time history analysis are conducted accounting for the following permutations: 4 systems for both flexuretype and sheartype structures with 5, 10, 15 and 20stories, respectively; 4 vibration periods ranging from 0.04 to 0.1 times the story number; 6 target ductility levels of 1, 2, 3, 4, 5, 6. It should be noted that the target ductility level in analysis is the maximum story displacement ductility factor rather than the target displacement ductility for global system.
This flowchart not directing structural design process, but rather how to get the ductility demand for computational analysis of the reduction factor, in fact, are reflected in the paper. As for each seismic record, there are mainly three aspects to analysis. First, adjust the model structure of the lateral stiffness of two structures to ensure the fundamental vibration period of the same type; second, adjust the lateral strength of the model structure ensures two types of structure ductility level of the basic the same; third, levels are calculated based on different ductility reduction factor and comparisons between the two types of structural models.
Fig. 8Flowchart for ductility reduction factors calculation
4.2. Influence factor analysis
The averaged values of the ductility reduction factor $R$ are shown in Table 3 (see Appendix). As can be seen from Figure 9, the $R$ for flexuretype structure increases with the increment of target displacement ductility, which has the same trend with SDOF systems and sheartype structures. With the increment of the basic vibration period, there is no regular variation trend in $R$. The $R$ shows a decline trends with increasing story numbers (also means increasing vibration periods to some extent). However, for an identical vibration period, the $R$ is generally observed to be small for both sheartype and flexuretype structure with higher height or more stories, especially for higher ductility level cases. It indicates that the story number has great influence on the ductility reduction factors, as shown in Figure 10.
4.3. Comparisons between two structural systems
For an identical story number, ductility level or basic vibration period, the $R$ for the flexuretype structure is observed to be larger than that for the sheartype structure. As shown in Table 3, the $R$ for the flexuretype structure is about 10 % larger than the $R$ for the sheartype structure under the conditions of a ductility level ranging from 2 to 6 and different basic vibration periods. There is no regular trend in $R$ ratio of sheartype system to flexuretype system with the increment of ductility and vibration period. The mean results of $R$ ratio are 1.387, 1.458, 1.372, and 1.392 for 5, 10, 15, and 20story structures, respectively. The total mean value of $R$ ratio for the above four structures is 1.402.
Fig. 9Effect of ductility factors on ductility reduction factors
Fig. 10Effect of storey number on ductility reduction factors
According to several international seismic design codes, the seismic ductility reduction factor $R$ (namely the response modification factor or behavior factor) of the shear wall structure is generally smaller than those of the frame structures. For instance, EC8 defined the behavior factor for highductility concrete frame structure as 4.5, while the behavior factor for highductility concrete shear wall structure is 4.0. It seems that the results of this study are contradicted with these definitions. However, through profound analysis, it demonstrates that the two factors are not completely identical indexes, and the reasons include following two points: (1) the $R$ studied in this paper are completely based on the ductility dissipation energy mechanism, while the $R$ defined in seismic codes is based on composite mechanism of structural ductility, system damping, redundancy, structural overstrength, and etc.; and (2) the comparison on $R$ value between flexuretype and sheartype structure is assumed to be an ideal case with same storey number, ductility or vibration periods. In practical engineering, when the number of storey is equal, the ductility level and the basic vibration period of the shear wall structure are normally smaller than those of the frame structure (see Figure 5 and Figure 7). The difference in ductility level will induce great differences in seismic demand. However, the seismic design codes do not define different $R$ values accounting for the differences in storey number, ductility level and vibration period, and just adopt several values to consider all the differences together. Therefore, the seismic ductility reduction factor defined in seismic codes for the shear wall structure is generally smaller than those for the frame structure.
5. Conclusions
Seismic ductility reduction factors for flexuretype and sheartype structures are compared based on simplified MDOF systems and nonlinear dynamic time history analysis. The following conclusions can be drawn:
• The seismic response in terms of structural displacement ductility, base shear, and top displacement in flexuretype structure are different from those in sheartype systems, due to different deformation mechanism and lateral deformation modes.
• The structural storey ductility level and storey number have significant influence on ductility reduction factors of MDOF systems, while the basic vibration period of a structure does not show obvious influence on this factor.
• When parameters of structures, such as storey number, displacement ductility level and the basic vibration period, are the same, the ductility reduction factors of flexuretype structure are obviously larger than those of sheartype structure.
• The ratios of ductility reduction factors between the flexuretype and sheartype structures vary with the vibration period and the ductility level, but showing no regular change trends. In the present study, the ductility reduction factors for flexuretype structure are generally about 40 % larger than those for sheartype structures.
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