Comparison of ductility reduction factors for MDOF flexure-type and shear-type 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.

Fig. 1Comparison of design response spectra and measured response spectra

2.2. MDOF systems

It’s widely acknowledged that the flexure-shear 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 shear-type frames and pure flexure-type walls, respectively. While the $\lambda$ takes value between 0 and $\infty$, structure is defined as frame shear wall structure predominated by composite flexure-shear 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, multi-mass 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 shear-type spring modelis used to simulate the shear-type 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].

3.1. 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 flexure-type system and shear-type 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 shear-type and flexure-type 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 flexure-type structure than that in shear-type structure. The modal period before and after loading in shear-type structure is smaller than that in flexure-type structure.

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 flexure-type structure is larger than that in shear-type 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 shear-type structure is greatly larger than that in flexure-type structure. It can be explained that the flexure-type structure deforms lager in upper stories while the shear-type 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 inter-story displacement ductility

3.2. Comparison results for same initial periods

The flexure-type systems and shear-type 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 flexure-type structure is still larger than that of shear-type 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 flexure-type 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 flexure-type structure is larger than that of shear-type structure. It also demonstrates that the shear-type 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 inter-story displacement ductility

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 flexure-type and shear-type structures with 5-, 10-, 15- and 20-stories, 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 flexure-type structure increases with the increment of target displacement ductility, which has the same trend with SDOF systems and shear-type 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 shear-type and flexure-type 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 flexure-type structure is observed to be larger than that for the shear-type structure. As shown in Table 3, the $R$ for the flexure-type structure is about 10 % larger than the $R$ for the shear-type 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 shear-type system to flexure-type 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 20-story 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 high-ductility concrete frame structure as 4.5, while the behavior factor for high-ductility 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 over-strength, and etc.; and (2) the comparison on $R$ value between flexure-type and shear-type 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.