Impedance functions of a strip foundation in presence of a trench

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 $l/B$ are examined, where $l$ is distance between the mid-point of the strip foundation and the mid-point of the trench and $B$ 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.

Keywords:

• Foundation impedance functions
• soil and foundation dynamics
• wave propagation
• trenches

Disclosure statement

No potential conflict of interest was reported by the author(s).

Nomenclature
$l=$	=	distance between the mid-point of the strip foundation and the mid-point of the wave barrier (trench)
$B=$	=	foundation’s half-width
$V=$	=	vertical force
$H=$	=	horizontal force
$M=$	=	moment
$v=$	=	vertical displacement
$u=$	=	horizontal displacement
$\phi =$	=	rocking angle
$IR=$	=	impedance ratio
${\rho }_i=$	=	density of the in-fill material
$V_{si}=$	=	shear wave velocity of the in-fill material
${\rho }_s=$	=	density of the soil material
$V_{ss}=$	=	shear wave velocity of the soil material
$f_{\mathit{max}}=$	=	highest frequency of vibration
$V_j=$	=	propagation velocity of the waves (body and Rayleigh type) in the soil
$d_{\mathit{element}}=$	=	dimension of the finite element
${\lambda }_R=$	=	wavelength of Rayleigh’s waves
${\lambda }_{\mathit{min}}=$	=	shortest wavelength
$k=$	=	coefficient ranging from 4 to 10, according to the type of the finite element and its shape function
$V_R=$	=	Rayleigh waves velocity
$\sigma =$	=	normal stress at the artificial boundaries
$\tau =$	=	shear stress at the artificial boundaries
$V_n=$	=	particle velocity in the normal direction at the artificial boundaries
$V_t=$	=	particle velocity in the tangential direction at the artificial boundaries
$c_d=$	=	longitudinal wave velocity of the transmitting medium (soil)
$c_s=$	=	shear wave velocity of the transmitting medium (soil)
$\left[C\right]=$	=	matrix of Rayleigh damping
$\left[M\right]$	=	mass matrix
$\left[K\right]=$	=	stiffness matrix
$a,\beta$	=	coefficients of Rayleigh’s damping matrix
$\Delta t=$	=	time step
$h=$	=	maximum dimension of the finite element
$C=$	=	Courant number
$d=$	=	depth of trench
$w=$	=	width of trench
$s=$	=	distance of a surface soil particle from the mid-point of the trench
$A_R=$	=	amplitude reduction ratio
$K_{ij}=$	=	term of the impedance matrix
$\omega =$	=	frequency of excitation
$\left[F\right]=$	=	flexibility matrix
$a_o=$	=	normalised frequency of excitation
$m_{\mathit{ratio},ss}=$	=	mass ratio superstructure-soil
$m_{\mathit{ratio},fs}=$	=	mass ratio foundation-superstructure
$J_{\mathit{mratio}}=$	=	mass moment of inertia ratio foundation-superstructure
$h_{\mathit{ratio}}=$	=	slenderness ratio
$H_{\mathit{str}}=$	=	height of the superstructure
$m_2=$	=	mass of the SDOF oscillator
$m_1=$	=	mass of the foundation
$J_{m1}=$	=	mass moment of inertia of the foundation
$u_2=$	=	translational displacement of the SDOF oscillator
$u_1=$	=	translational displacement of the foundation
${\theta }_1=$	=	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.