Retaining walls of various constructions are often anchored. Anchored walls are usually more economical and of greater heights, unobtainable without reinforcement with anchors, than classical cantilever walls [1,2,3,4]. The anchoring element may be, as often used by engineers, a cylindrical concrete block situated at a certain distance from the wall. The cylindrical concrete block may be constructed from precast concrete tubes filled with concrete.

This paper provides finite element analysis [5] of the ultimate pullout capacity acting at different heights of the cylindrical block. The point of force application significantly influences the geometry of the failure zone in soil surrounding the block and the ultimate pullout force value. The ultimate pullout capacity of the plate anchors, essential in engineering, has been tested both experimentally [6,7] and theoretically [8,9,10,11,12,13]. The tests, however, concerned the ultimate pullout capacity of the plate anchors moving without rotation in the direction of the acting force. The paper presents results of the computational ultimate pullout capacity analyses of the cylindrical block rotating during displacement. The relationships between the anchoring block’s dimensions, depth of embedment, points of pullout force application and pullout capacity have been shown. It has been proved that the location of the force applied to the cylinder has influence on its resistance.

Anchors provide additional support to the upper part of an embedded wall. In that kind of wall, its base is embedded in the natural ground below the excavation bottom or the original ground level. In the most traditional sense, the anchor can be made up of an anchor block attached to a retaining wall with a tendon. After the anchors have been constructed, the ground is re-profiled with backfill. A pre-formed wall where there is no excavation during wall construction is shown in Fig. 1. Such a solution was proposed for the extension of the terrain surrounding the newly constructed warehouse. Fig. 2 presents the natural slope in the vicinity of the warehouse under construction and the completed lower part of the retaining wall. The natural ground in the slope is made of non-cohesive soils like fine and medium sand with local addition of coarse gravel in medium dense and dense state. Determination of pullout resistance was an integral and crucial part of the design.

A cylindrical block of diameter

The horizontal ground surface is not loaded. The elastic – perfectly plastic Mohr-Coulomb model was assumed for the sand surrounding the block and the linear elastic model for concrete of the anchor block. The parameters of both models are listed in Table 1.

Material parameters.

sand | concrete | ||
---|---|---|---|

Young Modulus | 80 | 30000 | |

Poisson’s ratio | 0.25 | 0.20 | |

Cohesion | 0.0 | - | |

Friction angle | ^{o}] | 34 | - |

Unit weight | ^{3}] | 18 | 25 |

Volume changes of soil at yielding are reflected by the value of dilatancy angle ψ. Different behaviours of soil during shearing were taken into account in calculations by assuming incompressible yielding with ψ = 0°, as well as the increase in volume with ψ = 5° and 10°. The influence of dilatancy angle on ultimate pullout capacity was tested.

Initial stress state before the application of the pullout force is given by _{z}_{x}_{y}_{0}σ_{z}_{0}

Two thicknesses of the soil layer over the top of the block z_{0} = 1 m and 2 m were considered in calculations. In FE models the following contact conditions between the block and the soil were taken into account: full contact with no interface elements on the block’s surface as well as sliding and separation due to interface elements [14,15] between the block and the soil, with friction angle related to the internal friction angle of sand Φ = 34°: δ = 0, 1/3Φ (11.3°), 2/3Φ (22.7°), Φ.

The cylindrical block is treated as a very strong element; thus, it is assumed to be described by linear elasticity.

The three-dimensional model of the soil mass in which the concrete block is embedded is shown in Fig. 4. The discretization of the domain considered was done with 3D isoparametric elements with the 1^{st} order interpolation function [5]. The block was located centrally in the analysed soil mass. Nodes of the elements along boundaries of the analysed region were supported in the usual manner to restrain horizontal translations on the vertical boundaries and both horizontal and vertical translations on the bottom boundary plane.

Several variants of the geometric model have been considered. They differed in the size of the whole model as well as the size of elements. To enable any possible failure mechanism in the sand to develop and to avoid any influence of the outer boundary, the models had to be extended beyond the reach of the passive pressure developed in front of the anchor block as the tendon pulls against it. In the case of embedment z_{0} = 2 m and for friction angle 34° the range of the passive zone is 9.40 m (Fig. 5). For a shallower embedment, the range is correspondingly smaller.

In FE model, the mesh refinement is required in zones of high displacement gradients. It has been noticed that the size of elements discretizing zone surrounding the anchor block affects the resulting ultimate resistance. The finer the mesh, the lower the resistance of the block was. After a few trial discretizations, the calculated ultimate resistance converged to a value that did not change any more. This final discretization was chosen for further analysis.

Finally, for the cylindrical block of diameter _{x}_{y}_{z}

The simulation of the process of pulling the block was done by assuming a kinematic boundary condition for one node on the surface of the block. This node represents the point of attachment of the tendon. The horizontal displacement of the node increased to a total value of 0.80 m in increments of 0.01 cm. The nodal force increases after each displacement increment. It was found that the applied displacement of 0.8 m is sufficient for reaching the ultimate value of pulling force (the bearing capacity of the block).

In the analysis, both full contact of sand with the concrete block and sliding on block’s surface was assumed. With the use of computer program Z_Soil, the value of the horizontal ultimate force pulling out the cylindrical concrete block was calculated. The ultimate force was determined based on the diagram of the ‘force–displacement’ relationship. For elastic – perfectly plastic Mohr-Coulomb model the ‘force-displacement’ curve increases monotonically to the ultimate value. In Fig. 6, there are five curves corresponding to the increasing pullout force for the case of the block without contact elements. In Fig. 7, there are curves obtained for the model with contact elements of different friction angles 0°, 1/3Φ, 2/3Φ, Φ and the location of imposing horizontal displacement 2 m below the block’s top.

The computations were carried out for _{0}_{ult}

The value of ultimate pullout force differs significantly depending on the place of its action. Fig. 9 shows the values of ultimate force for the thickness of overburden layer z_{0} = 1 m, different contact conditions and varying dilatancy at yielding. Similar results for z_{0} = 2 m are presented in Fig. 10.

The block embedment significantly affects the value of the ultimate force. For the overburden thickness _{0}_{0}

The ultimate pullout force increases with the increase of the dilatancy angle ψ. This trend explains the increase of the block’s resistance with the compaction of sand surrounding the block. The higher the compaction, the higher is ψ for the same critical value of friction angle [17,18,19,20].

In each considered case, the maximum ultimate value of the horizontal force pulling out the block was achieved for

Influence of tendon attachment location and dilatancy on ultimate force for z_{0} = 1 m.

ψ | contact conditions | F_{ult h = 0.5 m}, kN | F_{ult h = 2 m}, kN | F_{ult h = 0.5}/F_{ult h = 2}×100, % |
---|---|---|---|---|

0° | no contact elements | 977 | 2270 | 43.0% |

δ = 0° | 657 | 1925 | 34.1% | |

5° | no contact elements | 1230 | 2486 | 49.5% |

δ = 0° | 733 | 2053 | 35.7% | |

10° | no contact elements | 1404 | 2643 | 53.1% |

δ = 0° | 789 | 2152 | 36.7% |

Influence of tendon attachment location and dilatancy on ultimate force for z_{0} = 2 m.

ψ | contact conditions | F_{ult h = 0.5 m}, kN | F_{ult h = 2 m}, kN | F_{ult h = 0.5}/F_{ult h = 2}×100, % |
---|---|---|---|---|

0° | no contact elements | 1743 | 3741 | 46.6% |

δ = 0° | 1305 | 3333 | 39.2% | |

5° | no contact elements | 2244 | 4187 | 53.6% |

δ = 0° | 1434 | 3483 | 41.2% | |

10° | no contact elements | 2583 | 4470 | 57.8% |

δ = 0° | 1571 | 3693 | 42.5% |

The highest values of the ultimate force for all

The scheme of horizontal components of the soil pressures acting on the cylindrical block moving in the

The horizontal components of the soil pressures may be calculated from the formulae:

where _{a}_{p}_{an}_{pn}_{a}_{p}

Coefficients _{a}_{p}

0 | 2 | 4 | 6 | 8 | 10 | |
---|---|---|---|---|---|---|

_{a} | 1 | 0.68 | 0.46 | 0.33 | 0.26 | 0.22 |

_{p} | 1 | 1.64 | 2.32 | 2.90 | 3.56 | 4.0 |

For the vertical wall, horizontal ground surface and the angle of friction between the wall and the soil δ

where the angles _{t}_{w}

In (5) – (8) Φ and δ are given in radians. Upper signs should be used for calculating passive pressure and lower signs for calculating active pressure [16].

Values of _{pn}_{an}

Coefficients of earth pressure for different contact friction.

δ | K_{pn} | K_{an} |
---|---|---|

0° | 3.5371 | 0.2827 |

1/3ϕ | 4.8707 | 0.2577 |

2/3ϕ | 6.0623 | 0.2395 |

ϕ | 6.7123 | 0.2280 |

Due to the non-linear relationship between the coefficient _{p}_{ph}_{ph}_{p1} at the top and e_{p2} at the bottom of the block, is the area of a trapezoid of height

Comparison of ultimate forces obtained from the finite element calculations for the most conservative case of incompressible plastic flow (ψ = 0) with analytical results reveals that the analytical method recommended in [22] yields significantly lower forces for all considered contact friction, which makes the analytical prediction very conservative. Detailed comparison is given in Table 8.

Passive earth pressure for different contact friction and embedment z_{0} = 1 m.

δ | e_{p1}, kPa | e_{p2}, kPa | E_{ph}, kN |
---|---|---|---|

0° | 84.04 | 590.84 | 1012.3 |

1/3ϕ | 115.73 | 813.60 | 1394.0 |

2/3ϕ | 144.04 | 1012.65 | 1735.0 |

ϕ | 159.48 | 1121.23 | 1921.1 |

Passive earth pressure for different contact friction and embedment z_{0} = 2 m.

δ | e_{p1}, kPa | e_{p2}, kPa | E_{ph}, kN |
---|---|---|---|

0° | 208.83 | 830.87 | 1559.6 |

1/3Φ | 287.56 | 1144.12 | 2147.5 |

2/3Φ | 357.92 | 2672.94 | 2672.9 |

Φ | 396.30 | 1576.72 | 2959.5 |

Comparison of ultimate pullout forces for different contact friction and embedment z_{0}.

z_{0} = 1m | z_{0} = 2m | |||||
---|---|---|---|---|---|---|

δ | F_{ult anal}, kN | F_{ult FEM}, kN | F_{ult anal}/F_{ult FEM}×100, % | F_{ult FEM}, kN | F_{ult anal}, kN | F_{ult FEM}/F_{ult anal}×100, % |

Φ | 1921.1 | 2221 | 86% | 2959.5 | 3680 | 80% |

2/3Φ | 1750.03 | 2193 | 79% | 2672.9 | 3564 | 75% |

1/3Φ | 1394.0 | 2112 | 66% | 2147.5 | 3448 | 62% |

0° | 1012.3 | 1925 | 53% | 1559.6 | 3333 | 47% |

The resultant passive pressure is situated at the distance of 1.88 m for _{0}_{0}

The ultimate value of the cylindrical pulled out block depends on its dimensions, depth of embedment and points of pullout force application. The maximum resistance of a single block is acquired when the block is displaced horizontally without rotation. Significantly smaller resistance is acquired when the block rotates during displacement. By making use of formulae accessible in the literature, we can only find solution to the cylindrical block displacement without rotation, and the pullout resistance is at least 24% smaller than the resistance calculated numerically.

Having introduced the contact elements with friction angle

The analysis shows that when designing cylindrical concrete anchors, it is absolutely necessary to ensure block displacement without rotation. This kind of displacement is easiest to acquire by the application of two horizontal flexible tendons attached to the block.

#### Influence of tendon attachment location and dilatancy on ultimate force for z0 = 2 m.

ψ | contact conditions | F_{ult h = 0.5 m}, kN | F_{ult h = 2 m}, kN | F_{ult h = 0.5}/F_{ult h = 2}×100, % |
---|---|---|---|---|

0° | no contact elements | 1743 | 3741 | 46.6% |

δ = 0° | 1305 | 3333 | 39.2% | |

5° | no contact elements | 2244 | 4187 | 53.6% |

δ = 0° | 1434 | 3483 | 41.2% | |

10° | no contact elements | 2583 | 4470 | 57.8% |

δ = 0° | 1571 | 3693 | 42.5% |

#### Passive earth pressure for different contact friction and embedment z0 = 1 m.

δ | e_{p1}, kPa | e_{p2}, kPa | E_{ph}, kN |
---|---|---|---|

0° | 84.04 | 590.84 | 1012.3 |

1/3ϕ | 115.73 | 813.60 | 1394.0 |

2/3ϕ | 144.04 | 1012.65 | 1735.0 |

ϕ | 159.48 | 1121.23 | 1921.1 |

#### Passive earth pressure for different contact friction and embedment z0 = 2 m.

δ | e_{p1}, kPa | e_{p2}, kPa | E_{ph}, kN |
---|---|---|---|

0° | 208.83 | 830.87 | 1559.6 |

1/3Φ | 287.56 | 1144.12 | 2147.5 |

2/3Φ | 357.92 | 2672.94 | 2672.9 |

Φ | 396.30 | 1576.72 | 2959.5 |

#### Comparison of ultimate pullout forces for different contact friction and embedment z0.

z_{0} = 1m | z_{0} = 2m | |||||
---|---|---|---|---|---|---|

δ | F_{ult anal}, kN | F_{ult FEM}, kN | F_{ult anal}/F_{ult FEM}×100, % | F_{ult FEM}, kN | F_{ult anal}, kN | F_{ult FEM}/F_{ult anal}×100, % |

Φ | 1921.1 | 2221 | 86% | 2959.5 | 3680 | 80% |

2/3Φ | 1750.03 | 2193 | 79% | 2672.9 | 3564 | 75% |

1/3Φ | 1394.0 | 2112 | 66% | 2147.5 | 3448 | 62% |

0° | 1012.3 | 1925 | 53% | 1559.6 | 3333 | 47% |

#### Influence of tendon attachment location and dilatancy on ultimate force for z0 = 1 m.

ψ | contact conditions | F_{ult h = 0.5 m}, kN | F_{ult h = 2 m}, kN | F_{ult h = 0.5}/F_{ult h = 2}×100, % |
---|---|---|---|---|

0° | no contact elements | 977 | 2270 | 43.0% |

δ = 0° | 657 | 1925 | 34.1% | |

5° | no contact elements | 1230 | 2486 | 49.5% |

δ = 0° | 733 | 2053 | 35.7% | |

10° | no contact elements | 1404 | 2643 | 53.1% |

δ = 0° | 789 | 2152 | 36.7% |

#### Coefficients ma and mp.

0 | 2 | 4 | 6 | 8 | 10 | |
---|---|---|---|---|---|---|

_{a} | 1 | 0.68 | 0.46 | 0.33 | 0.26 | 0.22 |

_{p} | 1 | 1.64 | 2.32 | 2.90 | 3.56 | 4.0 |

#### Material parameters.

sand | concrete | ||
---|---|---|---|

Young Modulus | 80 | 30000 | |

Poisson’s ratio | 0.25 | 0.20 | |

Cohesion | 0.0 | - | |

Friction angle | ^{o}] | 34 | - |

Unit weight | ^{3}] | 18 | 25 |

#### Coefficients of earth pressure for different contact friction.

δ | K_{pn} | K_{an} |
---|---|---|

0° | 3.5371 | 0.2827 |

1/3ϕ | 4.8707 | 0.2577 |

2/3ϕ | 6.0623 | 0.2395 |

ϕ | 6.7123 | 0.2280 |

Reviews on Finite Element Modeling Practices of Stone Columns for Soft Soil Stabilization Beneath an Embankment Dam Proposal of concept for structural modelling of hybrid beams Resonance of a structure with soil elastic waves released in non-linear hysteretic soil upon unloading Settlement Analysis of a Sandy Clay Soil Reinforced with Stone Columns The evolution of the shape of composite dowels Parametric study of the earth dam's behaviour subjected to earthquake Impact of longwall mining on slope stability – A case study