Interpretation of Vibratory Pile Penetration Based on Digital Image Correlation

Abstract

A combined interpretation of force measurements together with the evaluation of dynamic motion around the pile based on digital image correlation (DIC) is performed to identify soil deformation during vibratory pile driving in model tests. The tests are executed under water-saturated 1g-conditions. We prove the occurrence of the so-called cavitational pile driving but without the experimental evidence of the forming of a cavity under the pile tip. Using the DIC results, first attempts are made to evaluate the volumetric cyclic deformation of soil around the pile tip during the vibro-penetration. The results show an alternation of contractancy and dilatancy in proximity of the pile tip with volumetric peak-to-peak strain amplitudes up to 2 %. They indicate drained or at least partially drained conditions. Based on the test results, existing phenomenological interpretations of soil deformation due to pile penetration are reviewed.

Appendices

Appendix 1: Summation of Incremental Displacements and Strain Calculation Procedure

The DIC procedure evaluates a series of images and calculates the incremental displacements $\Delta \mathbf {u}^{(i)}_{DIC}$ (displacements occurring between image i-1 and i) for a fixed grid in eulerian coordinate system. Search patches of 32$\,\times \,$32 pixel that are detected in zones four times as large are used for this evaluation. In order to obtain the total displacements $\mathbf {u}^{(i)}=\mathbf {u}(\mathbf {X},t_i)$ with respect to a reference configuration defined in the first image, the incremental displacements have to be summed. Therefore, material points $\mathbf {X}$ defined in the reference configuration are followed throughout the image series. Since a material point of the reference configuration lies generally not on a mesh point of the current configuration, its incremental displacement $\Delta \mathbf {u}^{(i)}$ has to be interpolated using the information of the four surrounding points, $\Delta \mathbf {u}^{(i)}_{DIC, I-IV}$. Linear interpolation functions are used. The incremental displacements $\Delta \mathbf {u}^{(i)}$ are then added to the total displacement of the preceding step to obtain the current total displacement $\mathbf {u}^{(i)}$. This procedure is illustrated in Fig. 13a.

Similar to [20], for strain calculation a triangular 3-node element with linear interpolation functions is used, Fig. 13b.

Fig. 13

a Interpolation of incremental displacements $\Delta \mathbf {u}$ from DIC results $\Delta \mathbf {u}_{DIC}$ and summation to total displacements $\mathbf {u}$ and b 3-Node element for strain calculation

For this element the deformation gradient $\mathbf {F}$ is calculated as the derivative of the current position $\mathbf {x}^{(i)}=[x^{(i)}, y^{(i)}]^T$ with regard to the reference configuration $\mathbf {X}=\mathbf {x}^{(0)}=[X, Y]^T$. Linear interpolation functions $N_j$ are used for the approximation of the current configuration.

$$\begin{aligned} \mathbf {F} = \frac{\partial \mathbf {x}^{(i)}}{\partial \mathbf {X}}=\sum ^{3}_{j=1} \frac{\partial N_j}{\partial \mathbf {X}} \mathbf {x}^{(i)}_j \end{aligned}$$

(1)

The derivatives of the interpolation functions can be calculated from the nodal coordinates in the reference configuration:

$$\begin{aligned} \frac{\partial N_1}{\partial X}= & {} \frac{1}{2A_e} (Y_{2}-Y_{3});\qquad \frac{\partial N_1}{\partial Y} \,=\, \frac{1}{2A_e} (X_{3}-X_{2})\end{aligned}$$

(2)

$$\begin{aligned} \frac{\partial N_{2}}{\partial X}= & {} \frac{1}{2A_e} (Y_{3}-Y_{1});\qquad \frac{\partial N_{2}}{\partial Y} \,=\, \frac{1}{2A_e} (X_{1}-X_{3})\end{aligned}$$

(3)

$$\begin{aligned} \frac{\partial N_{3}}{\partial X}= & {} \frac{1}{2A_e} (Y_{1}-Y_{2});\qquad \frac{\partial N_{3}}{\partial Y} \,=\, \frac{1}{2A_e} (X_{2}-X_{1}) \end{aligned}$$

(4)

With the element area $A_e$:

$$\begin{aligned} A_e= & {} 1/2[X_1(Y_{2}-Y_{3})+X_{2}(Y_{3}-Y_{1})+X_{3}(Y_{1}-Y_{2})] \end{aligned}$$

(5)

The Right Cauchy-Green-deformation tensor $\mathbf {U}$ is obtained as follows:

$$\begin{aligned} \mathbf {U}= (\mathbf {F}^T\cdot \mathbf {F})^{1/2} \end{aligned}$$

(6)

The principal in-plane (Hencky-)strains and the maximum in-plane shear strain are obtained from the eigenvalues $U_{I/II}$ of $\mathbf {U}$. Therefrom, the volumetric strain $\varepsilon _{\text {vol}}$ is calculated assuming axial symmetry ($u_x\hat{=}u_r$).

$$\begin{aligned} \varepsilon _{I}= \ln U_{I};&\qquad \varepsilon _{II}= \ln U_{II};&\qquad \end{aligned}$$

(7)

$$\begin{aligned} \gamma _{\text {max}}= \varepsilon _{I}-\varepsilon _{II};&\qquad \varepsilon _{\text {vol}}= \varepsilon _{I}+\varepsilon _{II}+\ln (1+u_x/X)&\end{aligned}$$

(8)

Fig. 14

a Grain size distribution of the test sand (Karlsruhe Sand) and b results of permeability tests with constant head for different porosities

Appendix 2: Test Sand, Deposition Method and Uniformity Control

A poorly graded medium quartz sand with sub-rounded grains is used in the tests. A typical grain size distribution and some granulometric properties are given in Fig. 14a. The minimum and maximum void ratios at negligible stress level are $e_{\text {min}}=0.549$ and $e_{\text {max}}=0.851$. From permeability tests with constant head, the dependence of the coefficient of permeability k on the porosity n was evaluated. Results are given in Fig. 14b. For a rough estimation of the permeability k as a function of the porosity n, the well-known Kozeny/Carman-equation [1, 6] was fitted to the test results, Eq. 9:

$$\begin{aligned} k(n)=\frac{1}{308}\cdot \frac{\gamma _w}{\eta _w}\cdot \frac{n^3}{(1-n)^2}\cdot d_w^2 \; \end{aligned}$$

(9)

with the specific weight of water $\gamma _w=10$ kN/m$^3$, the dynamic viscosity of water $\eta _w=1.137\,\times \, 10^{-3}$ kNs/m$^2$ and the effective grain size $d_w=0.5$ mm.

For the preparation of the vibratory pile driving tests, the model pile is fixed in the starting position and approximately half of the test device is filled with deaired water. The dry sand is pluviated onto the water surface using a travelling diffusor which is manually operated in such a way that the sand sediments in horizontal layers. This procedure results in relative densities of about 40 %. Higher densities are achieved by dynamic excitation of the test device, e.g. by applying multiple hammer blows against the base.

For numerical simulation of the tests it is essential to obtain a homogeneous density distribution. In order to control the uniformity of the sample after the described deposition and densification method a series of cone penetration tests (CPT) was conducted. The same set-up like in [18] was used. The results are shown in Fig. 15a and the position of the CPTs in Fig. 15. The same sample was used for all six CPTs. The sample was densified step-wise and after each densification two opposited CPTs with respect to the symmetry plane y–z were performed. As expected, the cone resistance $q_c$ increases for higher relative densities. The three CPT-pairs show very similar results although one could expect that the second CPT for each density is influenced by the preceding. The results indicate that the preparation method provides homogeneous samples and can be used for benchmark experiments.

Fig. 15

CPT for uniformity control: a measurements and b position of the CPTs