On the peak inter-storey drift and peak inter-storey velocity profiles for frame structures

doi:10.1016/j.soildyn.2016.12.009

Soil Dynamics and Earthquake Engineering

Volume 94, March 2017, Pages 18-34

https://doi.org/10.1016/j.soildyn.2016.12.009 Get rights and content

Highlights

•
Analytical predictions of peak inter-storey drifts and velocities for framed structures.
•
The predictions are based on assumed first mode deformed shapes.
•
Validation through parametric study by means of time-history simulations.
•
Correction factors (higher modes) for peak inter-storey velocities.

Abstract

It is well known that peak inter-storey drifts and peak inter-storey velocities are fundamental response quantities to assess the seismic response of a frame structure equipped with added viscous dampers. In the present work, analytical estimations, grounded on the first mode response, of the two response quantities are introduced. Then, a large parametric study is carried out to assess the effectiveness of the proposed predictions. A special attention is devoted to the peak inter-storey velocities and to their sensitivity on the higher modes contribution.

Introduction

Since the 1980s the peak inter-storey drift profile has been used in earthquake engineering to estimate damages in building structures due to the seismic ground shaking (Sozen [20], Moehle [11], Bozorgnia and Bertero [3], Akkar et al. [2]). Iwan [9], by making use of the waves propagation theory through continuum shear beams, introduced the concept of the inter-storey drift spectrum. The main advantage in the use of the inter-storey drift spectrum with respect to the conventional displacement response spectrum is that it accounts for the higher modes contribution, thus resulting in a more precise estimation of the seismic displacement responses. Chopra and Chintanapakdee [7] showed that the inter-storey drift spectrum may also be computed by classical modal analysis. Few years later, Miranda and Akkar [12] generalized the inter-storey drift spectrum to shear-flexural building behaviour accounting for different beam-to-column stiffness ratios.

All the mentioned studies showed that the predictions of the peak inter-storey drifts based on the first mode response only are, in general, relatively accurate for structures characterized by fundamental periods ranging in the so-called acceleration sensitive spectral region, while they may become quite un-conservative for structures with larger fundamental periods. Nonetheless, despite the potential advantages of the inter-storey drift spectrum, it remained quite confined among the academia and practically unknown to professional engineers. A possible reason is the absence of schematized (code-like) expressions of the inter-storey drift spectrum which could be used by professional engineers in the practical design.

In case of structures equipped with added inter-storey viscous dampers, in addition to the peak inter-storey drifts, also the peak inter-storey velocities play an important role in the evaluation of the structural response, given that the forces in the viscous dampers directly depend on the inter-storey velocities (Christopolous and Filiatrault [5]). Adachi et al. [1] recently showed that, for high-rise buildings, the peak inter-storey velocities due to earthquake ground motions may become very large, especially at the lower storeys, due to the higher modes effects. In a recent work, the authors [15] introduced the so-called Seismic Modal Contribution Factors (SMCFs) to estimate the contribution of the higher modes to a given response quantity (e.g. inter-storey drifts, inter-storey velocities) due to the seismic input. SMCFs represent an improvement of the well-known Modal Contribution Factors (MCFs, [6]) which are grounded on the concept of modal static response and do not account for the dynamic nature of the seismic input.

The present paper focuses on the study of peak inter-storey drifts and peak inter-storey velocities developed under seismic excitation in frame structures equipped with inter-storey viscous dampers. The main aim is to assess the effectiveness of simple analytical predictions which could be useful for professional engineers, especially in the preliminary design phase.

Section snippets

An analytical estimation of peak inter-storey drifts and velocities

The analytical formulations which are here used for the prediction of peak inter-storey drifts and peak inter-storey velocities developed by multi-storey frame structures during an earthquake are based on an assumed analytical first mode shape and have been first derived by the authors in a recent work [16] with the purpose of simplifying a procedure for the sizing of viscous dampers inserted in frame structures (known as the five-step procedure, [19]).

In detail, two analytical first mode

The objectives of the study

With the purpose of verifying the effectiveness of the analytical estimations of the peak inter-storey drift and velocities recently proposed by the authors [16], a comprehensive parametric study has been developed. In more detail:

•
A first parametric study (Section 4) is developed with reference to idealized structures (i.e. uniform shear type structures) subjected to a reference seismic input (an ensemble of 10 artificial earthquakes with average spectrum compatible with the design spectrum)

The uniform shear-type structures

Shear-type frame structures with uniform mass and stiffness distribution (along the height) and equipped with uniform (along the height) inter-storey viscous dampers are investigated in this section.

The structures are characterized by:

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uniform floor mass m =100 t;
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uniform storey stiffness $ρ \cdot k$ with k=1000000 kN/m; $ρ$ values are set equal to 0.5, 1.0, 2.0 and 5.0 in order to cover a wide range of fundamental periods T₁ between 0.1 and 5.0 s;
•
storey damping coefficient c leading to damping ratios ξ=0.05,

Correction factors for the actual peak inter-storey velocity at the ground floor

In the present section the attention is devoted to the peak inter-storey velocity at the first inter-storey, where, according to the results of the previous section, the largest magnifications are expected. With specific reference to the peak responses at the ground floor, a further investigation on the trends and correlations between total peak inter-storey velocity ( $v_{\max}^{t o t}$ ), first mode peak inter-storey velocity ( $v_{\max}^{1}$ ) and the predictions according to the assumed first mode shapes ( ${v^{1}}_{\max, A}$

G1 and G2 real ground motions ensembles

It is well known that the characteristics of the seismic input may significantly affect the dynamic response of building structures. In a recent work aimed at an engineering characterization of earthquake ground motions [14], the authors highlighted that the ground motion at a specific site can be adequately described in terms of the three peak ground parameters (namely the peak ground acceleration, PGA, the peak ground velocity, PGV, and the peak ground displacement, PGD). As such, a ground

The parametric study

In this section the influence of the along-the-height lateral stiffness distribution and damper placement on the peak velocity profiles and correction factors is investigated. The lateral stiffness k_i (shear-type behaviour is considered) at the i-th storey is assumed to gradually decrease from the bottom to the top: $k_{i} = λ \cdot \frac{N (N + 1) - i (i - 1)}{2}$

It can be shown that such stiffness variation leads to an almost linear first mode shape ( $λ$ is a constant given that Eq. (9) represents a shape).

The other main

RC moment resisting frames

In this final section, four realistic reinforced-concrete (RC) planar moment-resisting frames equipped with added viscous dampers are considered. The main objective is to evaluate possible discrepancies in the trends of the peak inter-storey velocity profiles with respect to those observed for the shear-type frame structures analysed in the previous sections. The influence of the beam-to-column stiffness ratio ρ= EJ_beam/EJ_column has been deeply investigated by Miranda and Akkar [12]. Large

Conclusions

The paper presents the main results of a study focused on the assessment of peak inter-storey drifts and peak inter-storey velocities developed in frame structures equipped with added viscos dampers during earthquakes. Simple analytical formulas for their predictions have been introduced.

The effectiveness of the prediction formulas, which are grounded on assumed first mode shapes and targeted to professional engineers for the preliminary design phase, has been evaluated by performing a large

Acknowledgements

Financial supports of Department of Civil Protection (DPC-Reluis 2014–2018 Grant – Research line 6: ‘‘Seismic isolation and dissipation’’) is gratefully acknowledged.

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View full text

On the peak inter-storey drift and peak inter-storey velocity profiles for frame structures

Highlights

Abstract

Introduction

Section snippets

An analytical estimation of peak inter-storey drifts and velocities

The objectives of the study

The uniform shear-type structures

Correction factors for the actual peak inter-storey velocity at the ground floor

G1 and G2 real ground motions ensembles

The parametric study

RC moment resisting frames

Conclusions

Acknowledgements

Eng Struct

Eng Struct

Drift estimates in frame buildings subjected to near-fault ground motions

J Struct Eng

Principles of passive supplemental damping and seismic isolation

Dynamics of structures. Theory and applications to earthquake engineering

Drift spectrum versus modal analysis of structural response to near-fault ground motions

Earthq Spectra

Improved numerical dissipation for time integration algorithms in structural dynamics

Earthq Eng Struct Dyn

Drift spectrum: measure of demand for earthquake ground motions

J. struct. Eng.