Finite Element Models

The auger consists of segments of three metres length each. In the 2D simulations the auger is considered as a beam. Neglecting the helix, the flexural stiffness of the auger is

101

Figure 6.1: Mechanical drawing of the auger.

I_a ≈ 4.7 · 10⁻⁷ m⁴, that of the connector I_c ≈ 1.7 · 10⁻⁷ m⁴ (cf. dimensions in Fig. 6.1).

With these values the buckling load without lateral support of the soil is F_B = 11.1 kN in case of an auger consisting of three segments and a connector length l_c = 10 cm. For l_c = 30 cm the buckling load reduces to F_B = 9.7 kN. The static system as well as the first three eigenmodes of buckling are depicted in Figure 6.2.

In order to study the influence of the soil surrounding the auger on the buckling load, three different FE models were developed:

1. 2D plane strain model including the installation process (cavity expansion) and the granular column around the auger

2. Simplified 2D plane strain model without cavity expansion and granular column 3. 3D model, again without cavity expansion and granular column

In all models undrained conditions are assumed. There is no interface between the auger and the soil, i.e. relative displacements are not possible. The models are described in brief in the following subsections.

2D Plane Strain Model #1

The finite element mesh of this 2D model is shown in Figure 6.3. The lateral boundaries of the calculation domain are 20 m away from the auger and constrained in the horizontal direction whereas the lower boundary is vertically constrained. The model consist of approximately 5400 4-node plane strain continuum elements. The auger is modelled with 120 2-node beam elements. In order to determine the buckling load and to enable

6.1. Installation of Granular Columns in Soft Soils 103

F

Mode 0 1 2 3

Figure 6.2: Static system and eigenmodes of the buckling analysis (orange sections denote connectors).

deformations beyond, it is necessary to apply initial geometrical imperfections to the auger, i.e. small lateral deflections ∆x relative to the reference position in 1-direction.

The simulation is carried out in several steps:

1. Expansion of the cavity from an initial width of two centimetres to 23 cm. The resulting state of the soil (effective stress, excess pore water pressure) is transferred to another (undeformed) mesh and serve as the initial state for the next calculation step.

2. In the new mesh the auger enclosed by a granular column modelled with aid of a hypoplastic constitutive equation is activated inside the cavity.

3. Eventually the displacement-controlled axial loading is applied to the top of the auger.

1 2

3

20,00 m

8,00 m

Figure 6.3: 2D the plane strain model.

2D Plane Strain Model #2

Geometry and boundary conditions are the same as before. However, there is no granular material around the auger and the excess pore water pressures and changes of state due to the expansion of the cavity are not considered. The axial loading of the auger is carried out in the same manner as in model #1.

3D Model

In order to be able to examine how far the plane strain considerations described above are justifiable, a 3D model (Fig. 6.4) was developed. The height of the model is the same as in the preceding models (h = 8 m), the outer radius is r_o = 10 m. The lower boundary is fixed in the vertical direction, the outer one in radial direction. The auger is modelled as a steel tube (linear elastic). In order to avoid a fixed support at the toe of the auger, only one single node is fixed in the vertical direction (cf. Fig. 6.5). The resulting small eccentricity e_x of the support serves as initial imperfection. At the surface the whole section of the auger is forced down uniformly.

Results of the Comparative Calculations

1. The simulations performed with the plane strain model #1 take relatively long and are somewhat pedestrian, since the cavity expansion and the axial loading of the auger are two independent calculations. Fortunately, the comparative calculations with the plane strain model #2 show that there is practically no influence of the change of state due to the cavity expansion and of the granular body around the auger on the buckling load.

6.1. Installation of Granular Columns in Soft Soils 105

Figure 6.4: 3D model.

Figure 6.5: Vertical support of the auger in the 3D model.

2. The calculations with the 3D model yield that when reaching the buckling load in the soil the shape of the auger does not correspond to eigenmode 1 but 3 (Fig. 6.2). The evolution of the deformation of the auger during loading is depicted in Figure 6.7.

Recent model and in situ tests confirm this result (Fig. 6.6) [49].

Figure 6.6: Buckling tests with micro piles embedded in soft soil [49].

3. If the shape of the deformed auger as obtained from the 3D calculations (corre-sponding to eigenmode 3, cf. Fig. 6.2) is applied as initial geometrical imperfection in the 2D simulations, a comparable buckling load results.

Thus we act on the assumption that with the aid of a parametric study using plane strain model #2, it is possible to identify the relevant parameters with respect to the buckling load.

Parametric Study

The influence of the following quantities on the buckling load F_B of the auger was exam-ined in the numerical analyses.

1. Surrounding Soil:

6.1. Installation of Granular Columns in Soft Soils 107

Figure 6.7: 3D calculation – evolution of the deformation of the auger.

• Compressibility: λ = 0.15 / λ = 0.45

• Critical Friction Angle: ϕc = 30^◦ / ϕc= 40^◦

• Void ratio: e0 = 8 / e0 = 18 2. Auger:

• Flexural Stiffness of the connectors: I_c= 1.7 · 10⁻⁷ m⁴ / I_c= 0.5 · 10⁻⁷ m⁴

• Length of the connectors: l_c= 0.1 m / l_c = 0.3 m

• Static system: Toe (= tip) horizontally restrained (fixed) / unrestrained (free)

• Shape of initial imperfections: shape of buckling mode 1 / 3

• Maximum amplitude of initial imperfections: | max (∆x)| = (1, 3, 5) cm

The calculation results are summarized in Table 6.1. In the scope of this study it was not possible to perform calculations for all combinations of the quantities mentioned above.

Based on the numerical results the following relevant quantities with respect to the buckle load can be identified:

Compressibility of the soil: The comparison between #16 and #18 shows the signif-icant influence of the stiffness of the soil, expressed by λ, on F_B.

Geometrical imperfections of the auger: For the same | max (∆x)| the choice of the mode of the initial deflections has a clear influence on F_B. The same holds true for a change in | max (∆x)| for a given mode.

# e₀ = 8 ¯e₀ = 18 Imp. | max(∆x)| I_c l_c ϕ_c Toe λ F_B^∗

mode in m in 10⁻⁷ m⁴ in m in kN

1 x – 1 0,01 1,7 0,1 40^◦ free 0,45 164

2 – x 1 0,01 1,7 0,1 40^◦ free 0,45 165

3 – x – – 1,7 0,1 40^◦ free 0,45 > 400

4 – x 3 0,01 0,5 0,3 40^◦ free 0,45 102

5 – x 1 0,01 0,5 0,3 40^◦ free 0,45 142

13 – x 3 0,03 0,5 0,3 30^◦ free 0,45 71

14 – x 3 0,03 0,5 0,3 30^◦ fixed 0,45 72

15 – x 3 0,05 0,5 0,3 30^◦ fixed 0,45 59

16 – x 3 0,05 0,5 0,3 40^◦ fixed 0,45 60

17 – x 3 0,05 1,7 0,3 40^◦ fixed 0,45 82

18 – x 3 0,05 0,5 0,3 40^◦ fixed 0,15 110

Table 6.1: Calculation Results.

Flexural stiffness of the auger and the connectors: The significant influence of I_a and I_c on F_b is self-explanatory.

The shear strength of the soil expressed as the critical friction angle ϕ_c has little influence on F_B (cf. #15 and #16). The constraint of the toe in horizontal direction played also a minor role (cf. #13 and #14).