Abstract
This study is devoted to investigate the effects of inertial soil–structure interaction (SSI) on the constantstrength inelastic displacement ratios of elastoplastic singledegreeoffreedom systems using a suit of 91 pulselike ground motions. The soil beneath the foundation is simulated based on the cone model. A local minimum for the inelastic displacement ratios around the interacting systemtopulse period ratio of one is demonstrated. Moreover, the soil flexibility increases the inelastic displacement ratios at all interacting systemtopulse period ratios. However, the aspect ratio has decreasing and increasing effects on the inelastic displacement ratios before and after a threshold interacting systemtopulse period ratio, approximately very close to one. It is confirmed that for slender structures, the SSI effects are the lowest at small interacting systemtopulse period ratios and as this ratio increases, the SSI effects on the inelastic displacement ratios increase. However, for squat structures, the SSI approximately has more significant effects on the inelastic displacement ratios at lower interacting systemtopulse period ratios and the effects decrease for higher interacting systemtopulse period ratios. It is noted that the equal displacement rule is not valid as the SSI effects are taken into account. In addition, a formula is proposed to estimate the inelastic displacement ratios of soil–structure systems using nonlinear regression analysis, which is desirable for the displacement assessment of existing structures. Besides, the mean ratios of approximatetoanalytical values, very close to one, emphasize well accuracy of the proposed formula.
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Abbreviations
 SSI:

Soilstructure interaction
 SDOF:

Singledegreeoffreedom
 \(\hbox {C}_{\upmu }\) :

Constantductility inelastic displacement ratio
 \(\hbox {C}_{\mathrm{R}}\) :

Constantstrength inelastic displacement ratio
 \(\hbox {u}_{\mathrm{in}}\) :

Maximum inelastic lateral displacement
 \(\hbox {u}_{\mathrm{e}}\) :

Maximum elastic lateral displacement
 \(\hbox {u}_{\mathrm{y}}\) :

Maximum yielding lateral displacement
 \(\upmu \) :

Structural ductility ratio
 R:

Strength reduction factor
 \(\hbox {F}_{\mathrm{e}}\) :

Elastic lateral force
 \(\hbox {F}_{\mathrm{y}}\) :

Yielding lateral force
 \(\upalpha \) :

Postelastic stiffness ratio
 \({\upxi }_{\mathrm{o}}\) :

Structural viscous damping
 \(\hbox {T}_{\mathrm{fix}}\) :

Vibration period of fixedbase structure
 \(\hbox {M}^{*}\) :

Effective mass of the structure in the first mode
 \(\hbox {I}^{*}\) :

Effective mass moment of inertia of the structure around its geometric center
 \(\hbox {h}^{*}\) :

Effective height of the structure in the first mode
 N:

Number of stories
 \(\hbox {C}_{\mathrm{m}}\) :

Effective mass coefficient
 \(\hbox {C}_{\mathrm{h}}\) :

Effective height coefficient
 r:

Radius of circular plan and foundation
 \(\hbox {K}_{\mathrm{fix}}^{*}\) :

Elastic stiffness of the fixedbase structure
 \(\hbox {m}_{\mathrm{f}}\) :

Foundation mass
 \(\hbox {I}_{\mathrm{f}}\) :

Mass moment of inertia of the foundation
 \(\hbox {A}_{\mathrm{f}}\) :

Area of the foundation
 \(\hbox {r}_{\mathrm{e}}\) :

Equivalent radius of the noncircular foundation
 DOF:

Degreeoffreedom
 s:

Sway degreeoffreedom of the foundation
 \({\upvarphi }\) :

Rocking degreeoffreedom of the foundation
 \(\uptheta \) :

Additional internal rotational degreeoffreedom
 \(\hbox {m}_{\uptheta }\) :

Polar mass moment of inertia of additional freedom
 \({\upupsilon }\) :

Soil Poisson’s ratio
 \(\hbox {V}_{\mathrm{a}}\) :

Soil axialwave velocity
 \(\hbox {V}_{\mathrm{s}}\) :

Soil shearwave velocity
 \(\Delta \hbox {M}_{{\upvarphi }}\) :

Trapped mass moment of inertia
 \(\hbox {k}_{\mathrm{s}}\) :

Sway stiffness coefficient of soil
 \(\hbox {k}_{{\upvarphi }}\) :

Rocking stiffness coefficient of soil
 \(\hbox {C}_{\mathrm{s}}\) :

Sway damping coefficient of soil
 \(\hbox {C}_{{\upvarphi }}\) :

Rocking damping coefficient of soil
 \(\uprho \) :

Mass density of soil
 \({\upomega }_{\mathrm{fix}}\) :

Circular frequency of fixedbase structure
 G:

Shear modulus of the soil
 \({\upxi }_{\mathrm{M}}\) :

Material damping of the soil
 \(\hbox {a}_{0}\) :

Nondimensional frequency
 M:

Mass matrix of soil–structure system
 C:

Damping matrix of soil–structure system
 K:

Stiffness matrix of soil–structure system
 {U}:

Displacement vector of soil–structure system
 \(\hbox {u}_{\mathrm{g}}\)(t):

Input acceleration time history
 {r}:

Influence vector
 \(\hbox {u}^{*}\) :

Absolute displacement of soil–structure system
 \(\hbox {u}_{\mathrm{s}}\) :

Sway displacement of the foundation
 c:

Structure damping
 \(\hbox {T}_{\mathrm{p}}\) :

Pulse period
 \(\hbox {T}_{\mathrm{ssi}}\) :

Vibration period of soil–structure system
 \({\upbeta }\) :

Ratio of the \(\hbox {C}_{\mathrm{R}}\)values resulting from the soil–structure systems to those obtained from the fixedbase structures
 \({\upbeta }_{\mathrm{ap}}\) :

Approximate value of \({\upbeta }\)
 \({\upbeta }_{\mathrm{an}}\) :

Analytical value of \({\upbeta }\)
 \(\hbox {E}_{\mathrm{m}}\) :

Mean ratio of \({\upbeta }_{\mathrm{ap}}\) to \({\upbeta }_{\mathrm{an}}\)
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Khoshnoudian, F., Ahmadi, E. Effects of inertial soil–structure interaction on inelastic displacement ratios of SDOF oscillators subjected to pulselike ground motions. Bull Earthquake Eng 13, 1809–1833 (2015). https://doi.org/10.1007/s105180149693y
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DOI: https://doi.org/10.1007/s105180149693y