Energy-based Adaptive Modal Pushover Analysis Using an Innovative Combination Method

B. Farhadi Islamic Azad University, Chaloos branch, Mazandaran, Iran M.A. Shayanfar Iran University of Science and Technology, Tehran, Iran K. Shakeri University of Mohaghegh Ardabili, Ardabil, Iran

ABSTRACT: In recent years, lots of efforts are devoted to develop the new pushover procedures to assess the seismic demands of the structures. But most of these procedures do not have enough accuracy in comparison to the nonlinear time history analysis because of using the non-adaptive load patterns and quadratic modal combination methods. In this paper a new energy based adaptive modal pushover analysis is proposed which is using an innovative weighted direct algebraic addition method to combine the effects of the higher modes. In the proposed procedure, the capacity spectrum is established based on the energy concept and also adaptive load pattern is implemented to consider the effects of the changes in the structural vibration characteristics. This method is applied to two steel moment resistant frames under 7 near field earthquake records to investigate the accuracy of the predicted responses. Results show the success of this method. Keywords: seismic demand, pushover, adaptive, energy, weighted modal combination. 1. INTRODUCTION During the last decade, the nonlinear static procedure (NSP) is developed widely in the engineering practice to estimate the seismic demands of the structures. The conventional pushover analyses implement uniform, triangular or first mode inertia forces as the lateral load pattern which is kept invariant during the inelastic phase (FEMA-356 2000; ATC-40 1996). In these procedures the structure is pushed with a monotonous increasing predefined load pattern until the target displacement achieved or the structure collapsed. These methods estimate seismic demands of the low-rise structures more accurate than their linear counterpart. Not including higher modes effects, especially in the high-rise buildings is one of the main drawbacks of the conventional procedures. Using the invariant load pattern is the other obstacle of the conventional methods. Recently some advanced pushover analyses proposed to overcome these drawbacks. (Chopra and Goel 2002; Jan et al. 2004; Kunnath 2004; Antoniou and Pinho 2004; Kalkan and Kunnath 2006; Shakeri et al. 2010). These procedures could be categorized into two main approaches: 1. Single-run procedures. 2. Multi-run procedures. In the single-run modal pushover analyses, the structure is pushed with the combined modal forces (Jan et al. 2004; Kunnath 2004). The applied load vectors reflect the contributions of the instantaneous higher modes and their interaction in the inelastic phase. In the adaptive forms of these methods, at each step, the load pattern is defined by combining the instantaneous modal loads (Antoniou and Pinho 2004; Shakeri et al. 2010). The progresses changes in the modal properties of the structure during the analysis are considered in these adaptive procedures. The multi-run modal pushover procedures are developed based on the elastic modal decomposition concepts (Moghadam 2002; Chopra and Goel 2002;). In the modal pushover analysis (MPA) proposed by Chopra and Goel (2002), multiple pushover analyses are carried out separately with lateral load pattern corresponding to the considered elastic mode shapes. Then the total seismic response is

calculated by combining the responses due to each modal load. Since in the higher modes the increase of the roof displacement is not proportional to the other storey displacement, and may even be reversals, Hernandez-Montes et al. (2004) have proposed an MPA methodology based on the energy concepts. Furthermore, in order to account the ongoing changes in the modal attribute, adaptive form of the multi-run pushover procedures are also developed (Gupta and Kunnath 2000; Kalkan and Kunnath 2006). In order to improve the performance of the multi-run pushover procedures, in this paper a Weighted Adaptive Modal Combination (WAMC) procedure is proposed. The WAMC procedure implements the force-based adaptive load pattern and the energy concept to calculate the target displacement. Also a novel modal combination rule, based on the effective mass ratio is used to combine the resulted responses of the considered modes. 2. WEIGHTED ADAPTIVE MODAL COMBINATION METHODOLOGY The proposed Weighted Adaptive Modal Combination (WAMC) procedure is developed based on three main features. 1- Implementing energy concept in order to translate the capacity curve coordinates of the multi degree of freedom (MDOF) system into a single degree of freedom (SDOF) system and finding the target displacement. 2- Using adaptive lateral load patterns in each mode. 3- Obtain the final results by using an innovative modal combination rule. The main aspects of the WAMC procedure are described in the following sections. 2.1. Target displacement In the pushover procedures, the seismic demands are obtained when the structure achieve the target displacement. Therefore, like the load pattern, the target displacement is one of the important key elements in the pushover analyses. In the enhanced pushover methodologies which use multi-mode analysis (e.g. MPA) the target displacement in each mode is achieved by transforming the MDOF responses into a bilinear SDOF response (Chopra and Goel 2002). In the higher modes, the increase of the roof displacement is not proportional to the other stories and may even be reversal. In this regard an energy-based method is proposed by Hernandez-Montes et al. (2004) to convert the displacement coordinate from the MDOF system to an equivalent SDOF system. In the WAMC procedure, the capacity curve is also developed based on the energy concept. For more detail, refer to Shakeri et al. (2010) 2.2. Adaptive lateral load pattern Using Adaptive lateral load patterns is the other principal feature of the WAMC method. First of all an incremental base-shear quantity, λ is selected. At each step, the λ, incremental base-shear is multiplied by the mass matrix, m, and the corresponding instantaneous mode shape vector, using Eqn. 2.1. to achieve the incremental applied force in the i-th storey at k-th step, dFi

(k-1)

. The incremental applied forces at each step are added to the previous existing forces using Eqn. 2.2.

)1()1( .. −− = kki mdF φλ (2.1)

)1()1()( −− += k

ik

ik

i FdFF (2.2)

2.3. Modal combination rule based on the modal effective mass The square-root-of-sum-of-the-squares (SRSS) and the complete quadratic combination (CQC) rules commonly are used by the different researchers to combine the responses resulted from the different modes. Also Shakeri and his co-workers (2007 and 2010) have developed an efficient optimal weight (OW) combination rule for using in MPA procedure. In OW the genetic algorithm is used to minimize the error and optimize the weight factors. Here an innovative weighted modal combination rule based on the modal effective mass is proposed to combine the responses resulted from the different individual pushovers with adaptive load pattern corresponding to the considered modes. For a number of the first considered modes, the effective masses are computed, Meff,i

. For each mode the absolute response is multiplied by the ratio of the corresponding effective mass to the effective mass of the first mode and then combined with together using the direct algebraic addition (Eqn. 2.3).

...... 31,

3,2

1,

2,1

1,

1, +

+

+

= r

MM

rMM

rMM

reff

eff

eff

eff

eff

eff (2.3)

Where r1, r2 and r3Assuming the first mode as the main mode in the structural vibration is the principal concept of this method. It means that the first mode effective mass could be accepted as the total mass of the structure.

are the responses resulted from the considered modes. Also r is the final result.

3. VALIDATION OF THE WAMC In order to verify the accuracy of the WAMC methodology a computer code called OpenSees (Mazzoni et al.2007) has been implemented. OpenSees is an object-oriented framework for finite element analysis. Validation studies are verified for two steel moment frame buildings in which higher modes have more effects on the results. Results obtained with the WAMC method are compared with the conventional first-mode lateral load pattern (M1), MPA procedure and the adaptive modal combination (AMC) methodology (Kalkan and kunnath 2006) using the force-based lateral load pattern instead of the displacement-based which is called Force-based Adaptive Modal Combination (FAMC) in this paper. The results of the different pushover analyses are then compared to the benchmark response resulted from the detailed nonlinear time history analysis (NTHA). 3.1. Structural models In this paper, SAC-9 and SAC-20 buildings, perimeter steel moment resistant frame (SMRF) buildings designed by consulting structural engineers for the Phase II of the SAC project, (FEMA-355C, 2000) are selected as the sample buildings. Only one of the five-bay perimeter SMRFs representing half of these buildings in the N-S direction are modelled (two dimensional). The effects of the gravity loads are neglected in the analysis. Hysteretic material and displacement based element, which is a distributed-plasticity displacement beam-column element is used to model the structures. 3.2. Earthquake records In order to assess the accuracy of the WAMC in comparison to the other pushover procedures under different ground motions, seven near fault ground motions were considered in this research. All records are downloaded from the Pacific Earthquake Engineering Research (PEER) site, (http://peer.berkeley.edu/smcat). The earthquake records are summarized in the table 1.

Table 3.1.

The earthquake records

Used name Record/Component HP (Hz) LP (Hz) PGA (g) PGV (cm/s) PGD(cm) Chichi CHICHI/CHY080-V 0.03 50 0.724 49 27.82 Erzican ERZIKAN/ERZ-NS 0.1 Null 0.515 83.9 27.35 Kobe KOBE/KJM000 0.05 Null 0.821 81.3 17.68 Landers LANDERS/LCN275 0.08 60 0.721 97.6 70.31 Lomaperieta LOMAP/LGP000 0.1 Null 0.563 94.8 41.18 Northridge NORTHR/RRS228 Null Null 0.838 166.1 28.78 Tabas TABAS/TAB-LN 0.05 Null 0.836 97.8 36.92 3.3. Comparative assessment The structural models are simulated in the OpenSees and then subjected to the different ground motions listed in Table 3.1. The NTHA procedure is performed for each earthquake records and the maximum responses of the structures are considered as the benchmark responses. Final responses of the different pushover procedures are compared with the benchmark responses. The accuracy of FAMC, MPA, M1 and the proposed WAMC procedures are evaluated in this study. In all of these methods except the conventional pushover procedure based on the first mode, M1 the first three modes are considered. The target displacement has been estimated through the maximum inelastic displacement resulted from the nonlinear dynamic analysis of the equivalent SDOF systems. 3.4. Inter-storey drift prediction To express the success of different pushover procedures, an error index is defined by Eqn. 3.1. as presented by Lopez-Menjivar and Pinho (2004).

2

1

1100(%) ∑= −

−−∆

∆

∆−∆×=

n

i NTHAi

NSPiNTHAi

nError (3.1)

Where, Δ i-NTHA is the peak inter-storey drift at a given level i, resulting from the NTHA procedure, Δi-NSP

is the corresponding inter-storey drift resulted from the considered NSP procedure, and n is the number of stories. Decreasing in the error index means that the responses of the nonlinear static procedure are close to the NTHA responses.

The inter-storey drift profiles resulting from the different pushover procedures and the NTHA procedure are compared for the SAC-9 and SAC-20 buildings subjected to different ground motions respectively in Fig. 3.1 and Fig.3.2. For each considered pushover procedure, the average inter-storey drifts error resulted from seven different ground motions are also presented in these figures. The results obtained from this research validate the accuracy of the WAMC procedure and also express that this methodology result in better prediction of the inter-story drift profiles in comparison to the M1, MPA and FAMC in both SAC-9 and SAC-20 under the most of the ground motions. The FAMC and MPA procedures usually overestimate the inter-story drift, especially the FAMC methodology. While, the WAMC procedures underestimates the inter-story drift profile in most cases but it is more accurate than the other considered methodologies.

Figure 3.1

. SAC-9 building peak inter-story drift profiles resulted from the NTHA and different NSP methods under the different ground motion and their corresponding average drift errors.

Figure 3.2

. SAC-20 building peak inter-story drift profiles resulted from NTHA and different NSP methods under the different ground motion and their corresponding average drift errors.

4. CONCLUSION As a new nonlinear static procedure, a weighted adaptive modal combination (WAMC) method which is based on the energy-concept is proposed. WAMC include the effects of the higher modes and progressive changes in the vibration characteristics during the inelastic response. Also this method considers the period elongation and the frequency content of a design or particular response spectra. Moreover a new modal combination rule is used to combine the modal responses. For each mode the absolute response is multiplied by the ratio of the corresponding effective mass to the effective mass of the first mode and then combined with together using the direct algebraic addition. The WAMC procedure is evaluated through the two nine and twenty story buildings under seven ground motions. Furthermore this method compare with other pushover procedures. This study shows that the modal pushover analysis (MPA) usually overestimate the inter-storey drift of the buildings and it is a reliable method but the accuracy of the WAMC procedure is much better than the MPA in drift prediction. REFERENCES ATC-40 (1996). Seismic Evaluation and Retrofit of Concrete Buildings, Applied Technology Council, Redwood

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