https://orcid.org/0009-0001-1667-550X
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ABSTRACT
Shear waves in underground longitudinal structures impose deformations that need to be accounted for in structural design. Simplified approaches for such structures typically consider the Winkler foundation model, assuming Euler-Bernoulli beam theory, which neglects shearing-induced distortions, especially significant when shear waves are a dominant component of the seismic motion. In order to overcome these limitations, Timoshenko beam models have been proposed in the literature. These approaches however depend on an appropriate determination of the ground springs. Existing analytical formulations often assume plane-strain conditions, inadequate for representing low frequencies and thus are not directly applicable for pseudo-static interaction analyses. The present paper develops analytical solutions to determine transverse and rotation Winkler springs for structures subjected shear waves. The proposed springs overcome the drawbacks of plane-strain models and can be construed as a generalisation of them. The springs are obtained as a function of the seismic wavelength and the ground-structure stiffness contrast. Results obtained are validated against solutions from the literature and numerical results from a full 3D finite-element model. A non-dimensional parametric study is also presented, that allow an expedited evaluation of ground springs for practical applications.
List of symbols
| As | = | cross-sectional area |
| E | = | elastic modulus of the soil |
| G | = | shear modulus of the soil |
| Es | = | elastic modulus of the structure |
| Gs | = | shear modulus of the structure |
| Is | = | cross-sectional moment of inertia |
| kS | = | transverse spring under S-waves |
| kφ | = | rotation spring |
| L | = | length of the structure |
| R | = | soil reaction factor |
| r0 | = | cross-sectional radii of structure |
| u | = | radial displacement |
| us | = | structural displacement |
| uff | = | free-field motion |
| uff0 | = | amplitude of free-field motion |
| v | = | tangential displacement |
| VP | = | compressional wave velocity of the medium |
| VS | = | shear wave velocity of the medium |
| w | = | longitudinal displacements |
| η | = | compressibility factor |
| θ | = | polar coordinate |
| λS | = | seismic shear wavelength |
| ν | = | Poisson’s coefficient of the soil |
| νs | = | Poisson’s coefficient of the structure |
| σr | = | radial normal stress |
| τrθ | = | tangential shear stress |
| ω | = | frequency of harmonic oscillation |
| ωS | = | equivalent frequency for S-waves |
| φs | = | sectional rotation |
| φs0 | = | amplitude of structural response in rotation |
Disclosure statement
No potential conflict of interest was reported by the author(s).
Correction Statement
This article has been corrected with minor changes. These changes do not impact the academic content of the article.