3.5 Dynamic Amplification Factor
3.5.0 General
Nonlinear dynamic analyses carried out in the present study require huge computational effort and time, which is not always possible to realize for analyzing a large number of building models. However, under certain circumstances nonlinear assessment becomes necessary, for example when sufficient amount of statistical data is required for fragility assessment. In such a case, nonlinear static procedures can be performed and the resulting displacement demand (performance point obtained using CSM) can be modified by a suitable factor to obtain the nonlinear dynamic displacement demand. Such a factor is dynamic amplification factor (DAF). DAF is a dimensionless number, which describes how many times the seismic response obtained by static nonlinear analysis, should be multiplied in order to get the response under dynamic loads. A DAF higher than 1.0 implies dynamic response is higher, whereas, a DAF of less than 1.0 implies static response is higher. In order to capture the nonlinear static deformation response, capacity spectrum method is utilized. The CSM is carried out for different hazard levels in the form of response spectra. These response spectra are developed for the ground motions and samples of ground motions used in the nonlinear dynamic analyses of the frames.
3.5.1 Dynamic Amplification Factor for the Frames
The dynamic amplification factor is determined in the study as the ratio of the dynamic response to the static response of the building frames for a given seismic demand. The dynamic responses are obtained using the results of nonlinear time-history analysis conducted separately for two different cases: demand variation, and capacity variation. In order to capture the variation in the seismic demand, nonlinear dynamic analyses are carried out considering different ground motions (Table 3.5) and its samples with increasing seismic intensity as discussed in Section 3.4.4. The peak seismic responses are obtained for each seismic intensity and the median response is obtained.
Corresponding nonlinear static analysis followed by CSM is carried out considering the same ground motions in order to obtain the median estimate. The DAF is then obtained as the ratio of the median responses from static and dynamic analyses at each seismic intensity measure level. The DAF in demand obtained for all the frame configurations are shown in Fig. 3.24(a) with respect to PGA.
The variation in capacity is related to the variation of uncertain parameters in geometry (sectional details), nonlinear material data and loading details related to the frame. Nonlinear time-history analyses are carried out for a number of sample frames modeled using the uncertain parameters using 1940 El Centro ground motion. Details regarding the variation of the uncertain data are provided in Chapter 5 (sensitivity analysis). The corresponding static responses (displacement demands) are obtained using the capacity spectrum method for the same ground motion as in NLTHA. Similar to demand variation, DAFs are obtained for the median responses at each level of the intensity measure (PGA). Fig. 3.24(b) shows the variation in DAF obtained for the three frames with respect to PGA by a variation in the parameters related to the capacity of the frames.
(a)
(b)
(c)
Figure 3.24 Dynamic Amplification Factors (DAF) obtained for the three frame configurations (Bare frame, OGS frame, and FI frame) with respect to: (a) demand, (c) capacity, and (c) combined response.
0.0 0.5 1.0 1.5 2.0
0.1 0.4 0.7 1.0
DAF (demand)
PGA (g) BF
0.0 0.5 1.0 1.5
0.1 0.4 0.7 1.0
DAF (demand)
PGA (g) OGS
0.0 0.5 1.0 1.5 2.0
0.1 0.4 0.7 1.0
DAF (demand)
PGA (g) FI
0.0 0.5 1.0 1.5 2.0
0.1 0.4 0.7 1.0
DAF (capacity)
PGA (g) BF
0.0 0.5 1.0 1.5 2.0
0.1 0.4 0.7 1.0
DAF (capacity)
PGA (g) OGS
0.0 0.5 1.0 1.5 2.0
0.1 0.4 0.7 1.0
DAF (capacity)
PGA (g) FI
0.0 1.0 2.0 3.0
0.1 0.4 0.7 1.0
DAF
PGA (g) BF
0.0 1.0 2.0 3.0
0.1 0.4 0.7 1.0
DAF
PGA (g) OGS
0.0 1.0 2.0 3.0
0.1 0.4 0.7 1.0
DAF
PGA (g) FI
3.5 Dynamic Amplification Factor
It is observed that the dynamic amplification factor values obtained for demand (seismic hazard) variation is nearly constant for different PGA range for all the three frame configurations. However, when a variation in the capacity of the frames is considered, DAF is high for lower values of PGA in bare and OGS frames, and it decreases with increasing PGA. In case of FI frame, the DAF is very low as compared to the other two frames and is more or less constant for all ranges of PGA. Combining the DAF for demand and capacity variation (Fig. 3.24(c) and Fig. 3.25), it is observed for all the ranges of PGA that the DAF is highest and greater than 1.0 for bare frame and lowest (< 1.0) for FI frame. This seems logical, since in bare frames dynamic response is higher because of the flexibility of the frame. In FI frames, mostly the DAF is less than 1.0, i.e., the static response is higher because of the higher stiffness of the FI frame. In OGS frame, static and dynamic responses are more or less similar (and the ratio is approximately 1.0) because, in addition to the ground storey flexibility, the OGS frames also have concentrated mass acting on all the upper floors that gives a balancing effect on the dynamic to static response.
Figure 3.25 Combined DAF over a range of PGA for the three frames.
In addition to the continuous variation of dynamic amplification factor with respect to PGA, the values are averaged over a certain range of PGA values for simple application of the obtained DAF along with the demand model (Eq. (3.10)) as given in Table 3.8. An overall combined average for dynamic amplification factors is also provided over a PGA range of 0.1g to 1.0g. The combined average of the dynamic amplification factor is in the order, BF > OGS > FI, reflecting the structural type and behavior of the considered frames.
0.0 0.5 1.0 1.5 2.0 2.5
0.1 0.3 0.5 0.7 0.9
DAF
PGA (g)
BF OGS
FI
Table 3.8. Average Dynamic Amplification Factors obtained for the three frames for different PGA ranges.
PGA range DAF for Bare frame DAF for OGS frame DAF for FI frame Demand Capacity Demand Capacity Demand Capacity
< 0.3g 1.21 1.7 0.94 1.28 0.74 0.39
0.3-0.65g 1.46 1.5 0.95 1.28 0.81 0.38
0.65-1.0g 1.39 0.81 1.02 0.65 0.81 0.36
Combined
Average 1.72 0.99 0.30
3.5.2 Application of Dynamic Amplification Factor
The nonlinear time-history analysis is the most desired method to predict the force and deformation demands in various components of the structure. However, the use of time-history analysis is limited because the dynamic response is very sensitive to the modelling and ground motion characteristics. It requires proper modelling of the cyclic load-deformation characteristics and careful consideration of the material degradation characteristics of all the important components. The computation time, time required for generation of input parameters, and interpreting the voluminous output of the analyses, make the use of the time-history analysis difficult for seismic performance evaluation.
Nonlinear static analysis, on the other hand, is essentially a static analysis, in which the static loads are applied in an incremental fashion monotonically until the ultimate state of the structure, i.e., the structural failure defined by particular damage state or failure of certain structural members, is attained. The nonlinear material data are provided in the form of envelope or backbone curves obtained from cyclic tests or analysis. Thus, computationally the nonlinear static analysis are less demanding. In this regard, the DAF can be efficiently utilized which when multiplied by static response gives the dynamic response of the structure concerned.
The dynamic amplification factor obtained in the current study can be utilized for similar RC frame configurations to obtain the equivalent dynamic displacement response from static displacement response. The DAF for the frames are obtained over a range of PGA, implying a wide range of seismic hazard intensity can be considered for performance assessment of the framed buildings. The DAF provided would aid in seismic performance assessment more accurately by converting the static response to dynamic response, which is more realistic.