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Nonlinear seismic analysis of unsymmetric-plan structures retrofitted by hysteretic damped braces

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Abstract

A displacement-based design procedure using hysteretic damped braces (HYDBs) is proposed for the seismic retrofitting of unsymmetric-plan structures. An expression of the viscous damping equivalent to the hysteretic energy dissipated by the damped braced frame is proposed under bidirectional seismic loads, where corrective factors are assumed as a function of design parameters of the HYDBs. To this end, the nonlinear dynamic analysis of an equivalent two degree of freedom system is firstly carried out on seven pairs of real ground motions whose displacement response spectra match, on average, the design spectrum proposed by the Italian seismic code for a high-risk seismic zone and a medium subsoil class. Then, the extended N2 method considered by the European seismic code, which combines the nonlinear static analysis along the in-plan principal directions of the structure with elastic modal analysis, is adopted to evaluate the higher mode torsional effects. The town hall of Spilinga (Italy), a reinforced concrete (r.c.) framed building with an L-shaped plan, is supposed to be retrofitted with HYDBs. Six structural solutions are compared considering two alternative in-plan distributions of the HYDBs, to eliminate (elastic) torsional effects, and three design values of the frame ductility combined with a constant design value of the damper ductility. To check the effectiveness and reliability of the DBD procedure, the nonlinear static analysis of the test structures is carried out, by evaluating the vulnerability index of r.c. frame members and the ductility demand of HYDBs for different in-plan directions of the seismic loads.

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Abbreviations

\(\gamma \in \left( {X,Y} \right)\) :

Principal in-plan direction

\(VI^{(\gamma )}\) :

Vulnerability index

\(d_{D,\gamma }^{(ls)}\) :

Displacement demand at the life-safety limit state

\(d_{C,\gamma }^{(ls)}\) :

Displacement capacity at the life-safety limit state

\(d_{yl,\gamma }^{(F)}\) :

Yield displacement of the frame

\(d_{yl,\gamma }^{(DB)}\) :

Yield displacement of the HYDB

\(d_{p,\gamma }^{{}}\) :

Displacement of the damped braced frame (DBF) for a given performance level

\(K_{F,\gamma }^{{}}\) :

Lateral (elastic) stiffness of the frame

\(K_{DB,\gamma }^{{}}\) :

Lateral (elastic) stiffness of the HYDB

\(K_{\gamma }^{*}\) :

Stiffness ratio

\(d_{\gamma }^{*}\) :

Displacement ratio

r F :

Stiffness hardening ratio of the frame

r F :

Stiffness hardening ratio of the HYD

r DB :

Stiffness hardening ratio of the HYDB

\(K_{e,\gamma }^{(F)}\) :

Effective (secant) stiffness of the frame

\(K_{e,\gamma }^{(DB)}\) :

Effective (secant) stiffness of the HYDB

\(K_{e,\gamma }\) :

Effective (secant) stiffness of the DBF

\(V_{yl,\gamma }^{(F)}\) :

Base shear of the frame at the yielding point

\(V_{p,\gamma }^{(F)}\) :

Base shear of the frame at the performance point

\(V_{yl,\gamma }^{(DB)}\) :

Base shear of the HYDB at the yielding point

\(V_{p,\gamma }^{(DB)}\) :

Base shear of the HYDB at the performance point

\(\mu_{F,\gamma }\) :

Ductility demand of the frame

\(\mu_{DB,\gamma }\) :

Ductility demand of the HYDB

\(m_{e}\) :

Effective mass

\(K_{e,\gamma }\) :

Effective stiffness

\(T_{e,\gamma }\) :

Effective vibration period of the inelastic (trilinear) ETDOF system

\(T_{i,\gamma }\) :

Initial vibration period of the inelastic (trilinear) ETDOF system

\(\xi_{V}\) :

Elastic viscous damping ratio

\(\xi_{DBF,\gamma }^{(h)}\) :

Viscous damping ratio equivalent to the hysteretic energy dissipation of the DBF

\(\xi_{DBF,\gamma }\) :

Viscous damping ratio equivalent to the energy dissipation of the DBF

\(C_{\gamma }\) :

Corrective factor of the viscous damping ratio

ν :

Normalized axial load

\(\bar{\xi }_{DBF,\gamma }\) :

Revised viscous damping ratio equivalent to the energy dissipation of the DBF

\(\xi_{e,DBF}^{{}}\) :

Equivalent biaxial damping ratio of the DBF

\(E_{S,\gamma }\) :

Potential energy of the DBF

\(\phi_{i}\) :

First mode horizontal component at the ith storey

\(K_{i,\gamma }^{(DB)}\) :

Lateral stiffness of the HYDBs at the ith storey

\(V_{yli,\gamma }^{(DB)}\) :

Shear strength of the HYDBs at the ith storey

\(N_{yl,i}^{(DB)}\) :

Yield-load of the HYDBs at the ith storey

C S,i :

Centre of stiffness of the damped braced frame at the ith storey

C M,i :

Centre of mass of the damped braced frame at the ith storey

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Acknowledgments

The present work was financed by Re.L.U.I.S. (Italian network of university laboratories of earthquake engineering), in accordance with the “Convenzione D.P.C.–Re.L.U.I.S. 2014–2016, WPI, Isolation and Dissipation”.

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  1. Dipartimento di Ingegneria Civile, Università della Calabria, Rende, Cosenza, Italy

    Fabio Mazza

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Correspondence to Fabio Mazza.

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Mazza, F. Nonlinear seismic analysis of unsymmetric-plan structures retrofitted by hysteretic damped braces. Bull Earthquake Eng 14, 1311–1331 (2016). https://doi.org/10.1007/s10518-016-9873-z

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