Finite element analysis for soil impedance functions of homogeneous and two-layered half-spaces

Dynamic axial soil impedance function for rigid circular structures buried in elastic half-space

3.4 Finite element analysis for soil impedance functions of homogeneous and two-layered half-spaces

3.4.1 Numerical computation of impedance function

Seylabi et al. (2016) developed a method to extract the impedance functions of a semi- infinite half-space from a FE model. The approach is general enough that can be applied equally well to flexible interfaces as well as 3D problems. This study considers a rigid interface and an anti-plane-strain problem as in Fig.3.8.

Recall in Eq. (3.1), the axial SIF is defined in frequency domain. Normally, the frequency domain FE solver is used to obtain the ratio of reactions to prescribed displacements, which directly yields SIF for a specific frequency. This procedure must be repeated for different frequencies to get the impedance spectra, in which the number of simulations depends on the desired resolution of the spectra. In contrast,Seylabi et al.(2016) showed that only one

soil elements

truncated domain

half-space domain x

y

truncated domain

x y

(a) (b)

z y

F_z(t)

PML elements

soil elements

fixed boundary 1

w(t) a

h

massless rigid

z y

F_z(t) w(t) a massless

rigid

Figure 3.8: Numerical model for the estimation of SIF: (a) infinite half-space FE model;

and (b) truncated half-space FE model using perfectly matched layer (PML) elements.

simulation is required to find SIF in time domain FE analysis. Therefore, we choose the latter approach to perform the SIF calculation. The motion of a rigid cylinder in an anti- plane-strain setting is shown in Figure 3.8. The SIF can be computed using the following procedure:

1. Apply the force time history𝐹_𝑧(𝑡) to the centroid of rigid interface. Except transla- tional motion along the𝑧-axis, all other degrees of freedom are constrained.

2. Record the resulting axial displacement time history𝑤(𝑡).

3. Use Fourier transform to compute ˆ𝐹_𝑧(𝜔). 4. Use Fourier transform to compute ˆ𝑤(𝜔).

5. Compute impedance function𝐾_𝑧 (𝜔) =𝐹ˆ_𝑧 (𝜔) /𝑤ˆ (𝜔).

It should be noted that this is a single-degree-of-freedom problem, therefore there is not much difference between stiffness method (applying unit displacement) and flexibil- ity method (applying unit force) because only a simple division is required instead of taking the inverse of the compliance matrix to get the stiffness matrix. Besides, it may be necessary to apply zero-padding to time signals of the applied force and resulting displacement before performing Fourier transform to increase the resolution of the calculated impedance.

3.4.2 Finite element models 3.4.2.1 Input time signal

To achieve SIF over a wide range of frequency, the energy of the input time history force should be distributed over the corresponding frequency band. A Ricker wavelet was chosen to fulfil this requirement, with the applied force in time𝐹(𝑡) is calculated as

𝐹(𝑡) =𝐹₀ 2𝜋²𝑓²

𝑐 (𝑡−𝑡₀)²−1 𝑒^−𝜋

2𝑓²

𝑐(𝑡−𝑡₀)²

. (3.22)

In Eq (3.22) 𝐹₀ is the amplitude of the applied force, 𝑓_𝑐 is the central frequency of the signal, and𝑡₀is the time when the maximum amplitude occurs. This study used𝐹₀=1 kN, 𝑓_𝑐 =15 Hz, and𝑡₀=0.2 sec. The generated Ricker wavelet using these values is shown in Fig.3.9. As one may see, the signal energy is distributed over a broad band from 0 to 40 Hz, which will finally yield the SIF over this frequency band.

0 0.2 0.4 0.6 0.8 1

-1 -0.5 0 0.5

0 10 20 30 40 50 60

0 1 2 3 10⁶

Figure 3.9: The applied force in time and frequency domain.

3.4.2.2 Spatial-temporal discretization

The FE method was used to discretize the infinite half-space model. The anti-plane- strain soil domain is truncated using absorbing boundaries conditions, provided to limit the occurrence of spurious waves that are reflected from the far-field boundaries. To avoid the fictitious energy bouncing back to the interested domain, we used PML elements (Basu, 2009) as shown in Fig.3.8. The FE analyses were conducted using 3D LS-DYNA R10.0.0.

Element dimensions are chosen based on the highest frequency (𝑓_{𝑚 𝑎𝑥}) and shear wave velocity of soil medium (𝑐_𝛽). Large mesh plays a role of low-pass filter and removes short-wavelength (high-frequency) energy, whereas excessively small mesh can generate numerical instability and require much more computational effort. The approximate dimen- sion of element𝑑_𝑒 is calculated as𝑑_𝑒 =𝜆_𝑚𝑖𝑛/𝑛_{𝑒 𝑝 𝑤}, where𝜆_𝑚𝑖𝑛=𝑐_𝛽/𝑓_{𝑚 𝑎𝑥} is the minimum wavelength , 𝑛_{𝑒 𝑝 𝑤} is the number of elements per wavelength. In wave propagation prob- lem, 𝑛_{𝑒 𝑝 𝑤} is typically chosen from 6 to 10. To ensure high accuracy and computational

efficiency, we used unstructured quadrilateral grid mesh with𝑛_{𝑒 𝑝 𝑤} = 40 near the cylinder circumference and𝑛_{𝑒 𝑝 𝑤} =15 near the boundary of truncated domain.

We used the explicit central difference time integration method, which is conditionally stable. Therefore, the time increment must be chosen carefully to maintain the numerical stability and accuracy. The critical time stepΔ𝑡_𝑐is based on the Courant–Friedrichs–Lewy1 condition and calculated as Δ𝑡_𝑐 = 𝑑_𝑒/𝑐_𝛼, where 𝑐_𝛼 is the compression wave velocity. In this work we used a time step size ofΔ𝑡 = 0.9Δ𝑡_𝑐. Besides, the time step is related to the sampling rate at which the resulting displacement is calculated. This sampling rate must be sufficiently high to adequately reconstruct the highest frequency in the signal. According to Nyquist–Shannon sampling theorem2we enforcedΔ𝑡 < 0.5/𝑓_{𝑚 𝑎𝑥}.

Furthermore, termination time should be chosen to consider the SH waves bouncing back and forth between the free surface and the cylinder. The termination time must be long enough so that the cylinder displacement signal completely returns to zero. In other words, the bouncing waves must vanish, meaning that the amplitude (energy) of the bouncing waves is small enough that it can be negligible. If the termination time is short, the signal (at later time) of the bouncing waves cannot be recorded, some energy of the frequency spectrum is thus removed. That will, in turn, lead to disparities in the frequency content of the displacement signal. Fig.3.10(a) shows an example of a 3.0-second resulting displacement in time domain for ℎ/𝑎 = 16 with the effect of bouncing waves from the free surface.

Whereas, Fig.3.10(b) illustrates the frequency domain of the displacement signals, which are truncated at 1.0, 1.5, and 2.0 sec from the original displacement signal. Obviously, when the termination time is 1.0 sec, some of the displacement time histories and thus the energies are missing, causing discrepancy in frequency domain between the 1.0-second truncated signal and the 3.0-second signal. When the displacement is calculated up to 1.5 sec, the first bouncing wave is fully accounted for, and the frequency spectrum closely approaches that of the 3.0-second signal. If the termination time is 2.0 sec, the second bouncing wave also is included, the frequency domains of 2.0-second and 3.0-second signals are identical.

Apparently, the displacement signal should consist of at least two bouncing waves to achieve adequate results. Moreover, the excitation force needs to be loaded and totally unloaded within termination time, which can be roughly taken as 1.0 sec. To that end, we chose termination time as 5ℎ/𝑐_𝛽or 1.0 sec, whichever is larger.

For homogeneous half-space soil domain, the shear modulus, Poisson’s ratio, and density

1The Courant–Friedrichs–Lewy (CFL) condition is a necessary condition for convergence of hyperbolic PDEs, which arises from studying the numerical domain of dependence.

2The Nyquist–Shannon sampling theorem provides a sufficient condition for the sample rate that allows a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth

0 0.5 1 1.5 2 2.5 3 -0.02

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

1^st bounce 2^nd bounce

(a)

0 5 10 15 20 25 30 35 40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Termination at 1.0s Termination at 1.5s Termination at 2.0s Termination at 3.0s

10.5 11 11.5 0.5

0.55 0.6

(b)

Figure 3.10: Displacement signal for ℎ/𝑎 = 16 in: (a) time domain; and (b) frequency domain.

are 𝜇 = 4.5 MPa, 𝜈 = 0.25, and 𝜌 = 1800 kg/m³, respectively. The outside radius of the cylinder is𝑎 =1.28 m.