### Abstract

An approximation is developed for obtaining the nonlinear stiffness K_{R} and damping C_{R} of a shallow circular or strip footing undergoing rocking oscillation on a homogeneous but inelastic undrained clayey stratum. Based on the parametric results of 3-D and 2-D finite-element analyses, equivalent-linear K_{R} and C_{R} are expressed in readily usable dimensionless forms. K_{R}, normalized by its linear elastic value, is shown to be a unique function of: (1) the vertical factor of safety F_{s} against static bearing capacity failure, and (2) the angle of rotation θ{symbol} normalized by a characteristic angle θ{symbol}_{s}. The latter is approximately the angle for which uplifting usually initiates at one edge of the foundation. Three sources contribute to the value of the dimensionless damping ratio ξ_{R} (derived from C_{R}): wave radiation, which is a function of frequency but is shown to amount to <3 %; soil inelasticity (hysteresis), for which graphs are developed in terms of θ{symbol}/θ{symbol}_{s} and F_{s}; and energy loss due to impact and the collateral vertical motion when severe uplifting takes place, which is tentatively determined from dynamic M: θ{symbol} loops. Comparative parametric seismic time-history analyses provide an adequate validation of the iterative equivalent-linear approximation which implements the developed equivalent K_{R} and ξ_{R}, but they also highlight its limitations.

Original language | English |
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Pages (from-to) | 1177-1200 |

Number of pages | 24 |

Journal | Bulletin of Earthquake Engineering |

Volume | 12 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jun 2014 |

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### Keywords

- Equivalent linear
- Seismic response
- Shallow foundations
- Soil failure
- Soil-structure interaction
- Uplifting

### Cite this

*Bulletin of Earthquake Engineering*,

*12*(3), 1177-1200. https://doi.org/10.1007/s10518-013-9554-0