Abstract
The influence of a trench in the behaviour of a rigid, strip foundation is numerically investigated under the prism of impedance functions. Both the soil and the foundation are simulated by 2-D plane strain elements. The soil half-space finite element model is truncated using absorbing boundaries. The foundation has three degrees of freedom: vertical and horizontal translation and rotation. Therefore, the foundation is excited by: vertical and horizontal force and moment loading for various frequencies. The corresponding displacements are generated by the finite element code. Finally, a 3 × 3 impedance matrix is calculated in the frequency domain applying FFT (Fast Fourier Transform). The vertical translation is uncoupled, whilst the horizontal translation and rotation are coupled. Various cases of are examined, where
is distance between the mid-point of the strip foundation and the mid-point of the trench and
is the foundation’s half-width. The finite element model is validated for: screening of waves; impedance functions. Parametric analyses are conducted for a three-degree of freedom system. The existence of an open trench significantly affects the behaviour of the foundation. The foundation’s response depends on: the superstructure; the foundation’s inertia; the distance and type of the trench.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Nomenclature | ||
= | distance between the mid-point of the strip foundation and the mid-point of the wave barrier (trench) | |
= | foundation’s half-width | |
= | vertical force | |
= | horizontal force | |
= | moment | |
= | vertical displacement | |
= | horizontal displacement | |
= | rocking angle | |
= | impedance ratio | |
= | density of the in-fill material | |
= | shear wave velocity of the in-fill material | |
= | density of the soil material | |
= | shear wave velocity of the soil material | |
= | highest frequency of vibration | |
= | propagation velocity of the waves (body and Rayleigh type) in the soil | |
= | dimension of the finite element | |
= | wavelength of Rayleigh’s waves | |
= | shortest wavelength | |
= | coefficient ranging from 4 to 10, according to the type of the finite element and its shape function | |
= | Rayleigh waves velocity | |
= | normal stress at the artificial boundaries | |
= | shear stress at the artificial boundaries | |
= | particle velocity in the normal direction at the artificial boundaries | |
= | particle velocity in the tangential direction at the artificial boundaries | |
= | longitudinal wave velocity of the transmitting medium (soil) | |
= | shear wave velocity of the transmitting medium (soil) | |
= | matrix of Rayleigh damping | |
= | mass matrix | |
= | stiffness matrix | |
= | coefficients of Rayleigh’s damping matrix | |
= | time step | |
= | maximum dimension of the finite element | |
= | Courant number | |
= | depth of trench | |
= | width of trench | |
= | distance of a surface soil particle from the mid-point of the trench | |
= | amplitude reduction ratio | |
= | term of the impedance matrix | |
= | frequency of excitation | |
= | flexibility matrix | |
= | normalised frequency of excitation | |
= | mass ratio superstructure-soil | |
= | mass ratio foundation-superstructure | |
= | mass moment of inertia ratio foundation-superstructure | |
= | slenderness ratio | |
= | height of the superstructure | |
= | mass of the SDOF oscillator | |
= | mass of the foundation | |
= | mass moment of inertia of the foundation | |
= | translational displacement of the SDOF oscillator | |
= | translational displacement of the foundation | |
= | rotational displacement of the foundation |
Acknowledgements
The seismogram used is based on the following Joint BG/GR research project: ‘Development of analysis tools & Software for synthesis of seismograms in complex geological regions’, Coordinators: G.D. Manolis (AUTH-GR) and P.S. Divena & T.V. Rangelov (BAS-BG), funded by: General Secretariat for Research & Technology (GR) and National Science Fun (BG). The use of the code for the synthetic seismogram is gratefully acknowledged by the author.