# Approximate Multimodal Dynamic Analysis to Estimate the Seismic Demands of Structures

## Abstract

Methods for the seismic demands evaluation of structures require iterative procedures. Many studies dealt with the development of different inelastic spectra with the aim to simplify the evaluation of inelastic deformations and performance of structures. Recently, the concept of inelastic spectra has been adopted in the global scheme of the Performance-Based Seismic Design (PBSD) through Capacity-Spectrum Method (CSM). For instance, the Modal Pushover Analysis (MPA) has been proved to provide accurate results for inelastic buildings to a similar degree of accuracy than the Response Spectrum Analysis (RSA) in estimating peak response for elastic buildings. In this paper, a simplified nonlinear procedure for evaluation of the seismic demand of structures is proposed with its applicability to multi-degree-of-freedom (MDOF) systems. The basic concept is to write the equation of motion of (MDOF) system into series of normal modes based on an inelastic modal decomposition in terms of ductility factor. The accuracy of the proposed procedure is verified against the Nonlinear Time History Analysis (NL-THA) results and Uncoupled Modal Response History Analysis (UMRHA) of a 9-story steel building subjected to El-Centro 1940 (N/S) as a first application. The comparison shows that the new theoretical approach is capable to provide accurate peak response with those obtained when using the NL-THA analysis. After that, a simplified nonlinear spectral analysis is proposed and illustrated by examples in order to describe inelastic response spectra and to relate it to the capacity curve (Pushover curve) by a new parameter of control, called normalized yield strength coefficient (\( {\varvec{\upeta}} \)). In the second application, the proposed procedure is verified against the NL-THA analysis results of two buildings for 80 selected real ground motions.

## List of Symbols

- \( {\text{M}} \)
Mass matrice

- \( {\text{C}} \)
Damping matrice

- \( {\text{F}} \)
Resisting force vector

- \( {\ddot{\text{x}}}_{\text{g}} \left( {\text{t}} \right) \)
Earthquake acceleration

- \( {\text{m}}_{\text{i}} \)
Mass of the ith level

- \( {\text{F}}_{\text{i}} \)
Resisting force of the ith level

- \( {\text{k}}_{{{\text{e}},{\text{i}}}} \)
Elastic stiffness of the ith level

- \( {\text{k}}_{{{\text{p}},{\text{i}}}} \)
Postyield stiffness of the ith level

- \( {\text{Q}}_{\text{i}} \)
Yield strength of the ith level

- \( {\text{x}}_{{{\text{y}},{\text{i}}}} \)
Yield displacement of the ith level

- \( {\text{K}}_{\text{p}} \)
Postyield stiffness matrix

- \( {\text{Q}} \)
Yield strength vector

- \( {\text{z}} \)
Dimensionless variable

- A = 1
\(\phantom{0} \)

- B = 0.1
\(\phantom{0} \)

- λ = 0.9
\(\phantom{0} \)

- β = 6
\(\phantom{0} \)

- \( \upgamma_{\text{n}} \left( {\text{t}} \right) \)
Modal coordinate

- \( \upphi_{\text{n}} \)
nth natural vibration mode of the structure

- \( \upomega_{\text{n}} \)
Natural vibration frequency

- \( \upxi_{\text{n}} \)
Damping ratio

- \( \upalpha_{\text{n}} \)
Post-to-preyield stiffness ratio

- \( {\text{Q}}_{\text{n}} =\upphi_{\text{n}}^{\text{t}} {\text{Q}} \)
Yield strength

- \( {\text{M}}_{\text{n}}^{ *} = \frac{{{\text{L}}_{\text{n}} }}{{\Gamma _{\text{n}} }} \)
Effective mass

- \( \Gamma _{\text{n}} =\upphi_{\text{n}}^{\text{T}} {\text{m }}\upiota/\upphi_{\text{n}}^{\text{T}} {\text{m}}\upphi_{\text{n}} \)
Modal participation factor

- \( {\text{L}}_{\text{n}} =\upphi_{\text{n}}^{\text{T}} {\text{m}}\,\upiota \)
\(\phantom{0} \)

- \( \mu_{n} \)
Ductility demand

- \( {\text{D}}_{{{\text{n}},{\text{m}}}} \)
Peak displacement

- \( {\text{D}}_{{{\text{n}},{\text{y}}}} \)
Yield displacement

- \( {\text{q}}_{\text{n}} \)
Yield strength coefficient

- \( {\text{S}}_{\text{an}} \)
Spectral acceleration

- \( {\text{V}}_{\text{bn}} \)
Base shear

- \( \upphi_{\text{rn}} \)
Amplitude of \( \upphi_{\text{n}} \)

- \( {\text{x}}_{\text{rn}} \)
Roof displacement

## 1 Introduction

Several simple evaluation methods have been proposed as an alternative to the complex nonlinear dynamic analysis to estimate the seismic demands of structures (Gülkan and Sözen 1974; Freeman et al. 1975; Newmark and Hall 1982; Fajfar and Fischinger 1988; Kowalsky 1994; Sasaki et al. 1998; Fajfar 1999; Gupta and Kunnath 2000; Albanesi et al. 2000; Priestley and Kowalsky 2000; Miranda 2001; Chopra and Goel 2001; Lin and Chang 2003; Maja and Fajfar 2012; Chikh et al. 2014; Zerbin and Aprile 2015; Kazaz 2016). The basic idea of these methods is to relate the structural capacity to the physical basis of elastic or inelastic demand spectra, as the Capacity Spectrum Method (ATC-40 1996) and its different implementations.

The seismic demands assessment methods are generally based on the nonlinear static analysis, where the structure is subjected to lateral loads increasing monotonically over the entire height until a predetermined target displacement. The distribution of these forces and the target displacement are based on the assumption that the response is controlled only by the fundamental mode, knowing that constant distribution of forces will not capture the contribution of higher modes in the overall structural response. Several researchers have proposed adaptive force distributions that attempt to follow more closely the distribution of inertial forces over time (Fajfar and Fischinger 1988; Baracci et al. 1997; Gupta and Kunnath 2000). Attempts have also been made to consider more than the fundamental mode of vibration in the Pushover analysis (Paret et al. 1996; Sasaki et al. 1998; Gupta and Kunnath 2000; Matsumori et al. 1999; Chopra and Goel 2001).

In this paper an inelastic equation of motion of MDOF system will be rewritten in terms of the ductility to obtain an approximate multimodal dynamic analysis (AMDA) that consider the ductility factor as the inelastic response of the system.

## 2 Approximate Multimodal Dynamic Analysis

### 2.1 Inelastic Modal Decomposition in Terms of Ductility

It can be observed from Eq. (15) that for a given ground acceleration, \( \upmu_{\text{n}} \left( {\text{t}} \right) \) depends on \( \upxi_{\text{n}} ,\upomega_{\text{n}} ,\upalpha_{\text{n}} \) and \( {\text{q}}_{\text{n}} \) of the nth natural vibration mode.

### 2.2 Application

In recent years Chopra and Goel (2002) assessed the strength variation of several procedures including the modal Pushover analysis (MPA), that they developed. The MPA analysis is based on structural dynamics theory. Its accuracy and reliability in estimating the peak response of inelastic MDOF systems has been evaluated extensively by the authors. Goel and Chopra (2004) analyzed and evaluated the response of several procedures for nonlinear static analysis, including Pushover analysis where only fundamental mode was taken into account.

The accurate of the proposed procedure is evaluated for a 9-story SAC steel building (Chopra and Goel 2001). The ‘exact’ response of a rigorous nonlinear time history analysis (NL-THA) is compared with the response obtained by the approximate multimodal dynamic analysis (AMDA).

Properties of modal inelastic SDOF systems

Properties | Mode 1 | Mode 2 | Mode 3 |
---|---|---|---|

\( L_{n} \left( {kg} \right) \) | 2736789 | −920860 | 696400 |

\( \varGamma_{n} \) | 1.36 | −0.5309 | 0.2406 |

\( M_{n}^{*} \left( {kg} \right) \) | 3740189 | 488839.1 | 167531.5 |

\( D_{n,y} \left( {cm} \right) \) | 26.51 | 18.65 | 19.12 |

\( T_{n} \left( {sec} \right) \) | 2.2671 | 0.8525 | 0.4927 |

\( \alpha_{n} \) | 0.19 | 0.13 | 0.14 |

\( k_{n} \left( {kN/cm} \right) \) | 210.3867 | 500.2020 | 1132.6086 |

\( \xi_{n} \left( \% \right) \) | 1.948 | 1.103 | 1.136 |

\( Q_{n} \left( {kN} \right) \) | 6168.977 | 4374.343 | 4414.347 |

\( q_{n} \left( g \right) \) | 0.168 | 0.912 | 2.685 |

In which \( {\text{S}}_{\text{an}} \) is the spectral acceleration, \( {\text{V}}_{\text{bn}} \) the base shear, \( \upphi_{\text{rn}} \) is the amplitude of \( \upphi_{\text{n}} \) and \( {\text{x}}_{\text{rn}} \) the roof displacement.

Following the AMDA procedure aforementioned, the total response is determined using the UMRHA and NL-THA (“exact”). Figure 4 shows the ductility demand, also is shown in the same figure the roof displacements time histories. It is clear from the comparison shown in Figs. 4 and 5 that the AMDA gives results in good agreement with the NL-THA.

## 3 Conclusion

An approximate procedure for seismic demands assessment of MDOF system has been developed and its accuracy was verified by examples. An inelastic modal decomposition in terms of ductility has been developed to construct the Approximate Multimodal Dynamic Analysis. That was verified using the seismic response of an example steel frame structure for which capacity curve data is available. The results indicated that more reliable displacement predictions are obtained from the proposed method.

The base shear-roof displacement (\( {\text{V}}_{\text{bn}} - {\text{x}}_{\text{rn}} \)) curve is developed from a Pushover analysis. This Pushover curve is idealized as a bilinear force-deformation relation for the nth mode of inelastic SDOF system. This idealization is used to determine the yield strength coefficient \( {\text{q}}_{\text{n}} \) and the post-to-preyield stiffness ratio \( \upalpha_{\text{n}} \) to estimate the ductility demand. The peak deformation of this SDOF system, determined by the Approximate Multimodal Dynamic Analysis, is used to determine the target value of roof displacement at which the seismic response is determined by the Pushover analysis. The total demand is determined by the sum of responses of the first three modes.

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