In their present form, Vibro piles were implemented as early as in the 1960s. They are installed by driving in a steel pipe with a closed end using a diesel or hydraulic hammer. Then, reinforcing is inserted into the pipe inside and the pipe is filled with concrete. Extraction of the pipe using a vibrator causes compaction and good integration of the pile shaft with the ground. An additional advantage of the technology is the short time necessary for installation in diverse soil conditions, and concreting in a casing pipe ensures pile continuity [1–2].

Pile designing frequently makes use of the results of in situ testing. These are primarily the parameters determined based on a cone penetration test (CPT), dynamic probe testing and testing using dilatometer and pressuremeter [3,4,5]. These methods are among the group of direct methods and have been based on empirical dependencies. The CPT is particularly useful for designing pile foundations [1, 6]. Probing parameters q_{c} and f_{s} obtained from the CPT are directly used to determine unit resistance under the base and on the pile side surface. In most cases, empirical correlations are adopted, confirmed by the results of static pile load tests on a natural scale. Calculation methods of pile load bearing differ in the ways by which q_{c} values are averaged and by correlation coefficients that depend on the type and state of soil and the method of pile installation [7,8,9,10,11]. A special place belongs to methods based on transfer functions. The calculations are based on non-linear load transfer functions of

To sum up, the idea of transformation functions is, therefore, to determine the course of the pile settlement curve to the previously calculated bearing capacity. This curve is usually a hyperbola from the starting point up to a conventional limit value. The new curve prediction method for Vibro piles, presented in this work, is of a completely different nature. Based on the developed correlations, the full load–settlement characteristics for the analysed pile are determined first. The next step is to read off from this curve the appropriate value of bearing capacity which corresponds to the assumed limit settlements. It is also possible to determine the settlement that corresponds to the design loads. The shape of the settlement curve is an important element. For the procedure presented in this article, the curve is described by two functions: linear and exponential. The linear part, which is characteristic of Vibro piles due to the technology and the excellent interaction of the pile with the soil, describes the elastic deformations. This part of the curve is generated when the resistances on the lateral are mobilised. The non-linear part is described by an exponential function. It describes the elastic–plastic phase in which there is a mobilisation of resistance under the pile base up to a boundary value. The combination of these sections, linear and nonlinear functions, gives us a true picture of the pile–soil interaction as the axial vertical load increases.

In many CPT-based methods, use is made of high global safety factors (F_{S}). For the methods applied in Europe, F_{S} ranges from 1.5 to 3.0 (e.g. methods of Bustamante - Gianeselli or Gwizdała -Stęczniewski) [14–15], depending on the soil conditions, technology of piles, type of structure, etc. Another approach to methods based on the soil test results has been included in the standard PN-EN 1997-1:2008. When calculating the characteristic value of load-bearing capacity, use is made of correlation coefficients ξ_{3} and ξ_{4}. Passing on to design values, the characteristic value is divided by the relevant resistant coefficient, which depends on the design approach [16]. In order to determine the computational curves of Vibro pile settlement, which were determined using the presented procedure, an approach according to Eurocode 7 was proposed. The use of a set of correlation coefficients and partial safety factors guarantees safe values of the pile-bearing capacity. The global safety factor Fs = 1.3 was also used for comparison. This is the minimum value of the coefficient, which also provides safe estimate values of the design vertical bearing capacity of Vibro piles.

Based on the existing theoretical models for pile interaction with the soil (Fig. 1) [17,18,19] and 137 settlement curves obtained from static load tests, from various locations in Poland (Table 1), the Q–s characteristics dedicated to the Vibro pile technology have been defined (Fig. 2).

Compilation of data used for correlation analysis.

_{b}^{*} |
|||||||
---|---|---|---|---|---|---|---|

Gdańsk | 5 | 406 | 460 | 9.0 to 13.5 | 6 | 2.0 to 10.0 | FSa |

9 | 508 | 560 | 16.0 to 22.0 | 7 | 2.0 to 10.0 | FSa | |

Gdynia | 14 | 560 | 610 | 14.0 to 18.0 | 10 | 2.0 to 6.0 | FSa |

Grudziądz | 16 | 508 | 560 | 10.6 to 20.0 | 12 | 1.0 to 8.0 | FSa, MSa |

Hajnówek | 4 | 508 | 560 | 7,5 | 4 | 4.0 to 6.0 | CSa, FSa |

Olsztynek | 44 | 610 | 660 | 6.5 to 12.0 | 30 | 1.0 to 6.0 | MSa |

Szczecin | 13 | 408 | 460 | 12.0 to 14.0 | 6 | 2.0 to 10.0 | MSa, FSa |

6 | 457 | 510 | 17.6 to 18.4 | 6 | 1.0 to 8.0 | FSa | |

10 | 457 | 520 | 17.5 to 21.5 | 9 | 1.0 to 6.0 | FSa | |

Wrocław | 4 | 610 | 660 | 8.5 to 11.0 | 4 | 2.0 to 6.0 | CSa, MSa |

12 | 508 | 560 | 6.0 to 18.0 | 8 | 2.0 to 6.0 | CSa, MSa | |

S | 137 | 102 |

The data comes from geotechnical and geological-engineering documentation created for real engineering structures such as road viaducts, bridges, tanks, harbour quays and industrial halls. The analysis concerns pile diameters 408, 457, 508, 560 and 610 mm with lengths from 6 to 22 m. The relationships presented in the next section are based on over 100 pile test results and nearly 90 CPT static soundings. The correlations shown are concerned with piles whose base is in the layer of fine and medium sands. Piles terminated in coarse sands were omitted due to the small set of cases. However, they provide an excellent reference for the relationships obtained and confirm the correctness of the developed method. This is because the curves from static test loads for these piles are better than the curves for piles in medium and fine sands.

The developed procedure and the adopted theoretical assumptions enable determination of the full Q–s curve to identify the value of load-bearing capacity and indirect (Q_{i}, s_{i}) or ultimate (Q_{ult}, s_{ult}) settlements. According to the guidelines of the currently valid standard PN-EN:1997–2008 [16, 20], the load-bearing capacity threshold value is adopted for settlements of 0.1D (10% of the pile diameter).

Pursuant to the adopted assumptions, the elastic stage of pile interaction with the soil has been defined by the following dependence:
_{s} = value of applied load causing settlement s”, load initiating the elastic–plastic stage of pile–soil interaction (kN).

The non-linear elastic phase is determined with the following dependence:

Key elements that affect the form of the Q–s curve and the form of a function that describes this curve are the parameters of the subsoil. As the adopted method is a proposal of the direct method based on CPT probing, it requires appropriate averaging of resistance q_{c} measured on the penetrometer cone. Methods of averaging the resistance q_{c} are presented in Fig. 3. The following were considered as being of importance:

averaged resistance q_{c} along the pile side surface – q_{cs}, within particular soil layers;

averaged resistance q_{c} under the pile base – q_{cp} to the depth of 1D_{b}, where D_{b} is the pile base diameter, for Vibro piles: diameter of steel shoe (the base should be in the soil with qc > 5 MPa) and

determination of pile base penetration in bearing soils _{gn}, measured from the pile base to the lower surface of weak soil.

For the proposed method, of particular importance is the strict empirical correlation between ground susceptibility s“⁄Q_{s}, which determines the form of the Q–s curve, and soil parameters around the pile, that is, the mean value of cone resistance of the CPT along the side surface – q_{cs} and under the pile base – q_{cp}. The dependence has been defined on the basis of an analysis of no-linear regression between soil parameters and those of Q–s characteristics. A graphic representation of the dependencies is presented in Fig. 4.

Further parameters necessary to define the Q–s curve, which fully depend on ground susceptibility, are as follows:

initial point of an exponential function – s’ and

transition point of the linear phase into a non-linear one, that is, the initial point of curvature – s”.

Based on the actual curves obtained from 50 static load tests of the piles, it has been found that the settlement value s’ should be determined on the basis of ground susceptibility s“⁄Q_{s}, allowing for the type of soil under the pile base (Fig. 5).

There is a specific constant linear relation between the settlement values s’ and s” [2] (Fig. 6) and this relationship results from the continuity of the linear and non-linear functions s(Q).

If the values s’ and s” are available, it is possible to set out the value of Q_{s}, load causing settlement s”, which initiates a non-linear elastic phase of the pile–soil interaction.

The advantage of using EC7 is the possibility of applying, for design and verification needs, any method applied in national practice. The consistency of calculations depends to a great extent on the reliable determination of unit resistance under the pile base and on its side surface. Many calculation methods exist that may be adopted to perform practical calculations. The adoption of a reliable method should be based on comparative analyses and the adoption of statistical methods that utilise results obtained from static load tests.

Below is an outline of the determination method of Q–s characteristics in accordance with the developed procedure for a Vibro pile having a diameter of 457/520 mm and length of L = 18.5 m, installed in fine sand.

On the basis of a standard set of data necessary to design a pile foundation (Fig. 7), such as soil parameters (in this case, the cone resistance from CPT) and the geometric parameters of pile, the full settlement curve has been determined. The presented empirical correlations (Figs 4–6) enable the definition of values that characterise particular stages of pile interaction with the soil, and specifically the linear elastic stage (Fig. 8a) and the non-linear one (Fig. 8b), determined by dependencies (1) and (2) and the theoretical point of soil yield under the pile base (Table 2). A synthesis of these relations is a prediction of the pile–soil interaction depending on the applied axial load (Fig. 8c). The best method for verifying the accuracy and reliability of analytical methods includes tests performed on a natural scale in real soil conditions. Fig. 8d presents a comparison of the theoretical settlement curve with the curve from static load test.

Output parameters according to correlations from Fig. 4 to Fig. 6.

_{s} |
_{s} |
||
---|---|---|---|

0.0042 | 1.73 | 4.7 | 1107 |

Below are given compilations of resistance q_{c} (Fig. 9) and curves (Fig. 10) from different projects from Gdansk (Pile No. 1–3), Szczecin (Pile No. 4–6) and Grudziadz (Pile No. 7–9). The piles vary in diameter and length and their bases are in fine or medium sand.

Full agreement of the analysis results with the actual bearing capacity is not obtained in every case. Sometimes, we get slightly higher values, and sometimes slightly lower values.

In order to use the theoretical curve Q–s for design purposes, it is suggested to use the safety factor of the minimum value of Fs = 1.3, taking into account the following:

errors resulting from averages of individual parameters in the data generation phase;

variability of soil conditions in relation to the sounding results;

distance of piles from CPT sounding, which is sometimes equal to approx. 10 m;

technological factors of pile construction;

influence of time on soil condition and bearing capacity increment and

statistical analysis errors.

The factor Fs = 1.3 provides a safe range of values of theoretical bearing capacity for all analysed piles. If the calculation method with too high uncertainty has been adopted, it is allowed to introduce a design model factor into the analyses to correct for systematic errors and to ensure that the estimated pile-bearing capacity is more safety.

The recommended computational path for the proposed method assuring better security is the methodology proposed by EC7 [16]. For piles and pile foundation, the adopted computational approach is DA2. A schematic diagram of the system of calculation curves after application of particular safety factors is shown in Fig. 11.

Values of computational load-bearing capacity Q_{ult,d} (R_{c,d}), according to EC7, are determined by adopting the following:

correlation coefficients ξ_{3} and/or ξ_{4} for determination of load-bearing characteristic values based on the results of soil testing (R_{c,k}) and

partial coefficients for load-bearing capacity γ_{R} (R_{c,d}) (Fig. 11).

In Fig. 12 are presented computed curves with the use of coefficients Fs = 1.3 and according to EC7 and the curves from SPLT tests for the most unfavourable cases from Fig. 10. Theoretical curves run for pile No. 4 and pile No. 7 above the actual curves. In the other cases presented, the predicted curves overlap or are under the curves from the test loads. In both cases (pile No. 4 and pile No. 7), dividing the load value by the factor Fs = 1.3 corrects the overestimated values. The use of the DA2 approach according to EC7 makes the calculation curves determined from the procedure very safe. They will definitely ensure safe cooperation of the pile with soil and fulfilment of the ultimate limit state condition.

Design values Q_{ult,d} are a safe estimation of load bearing, given diverse methods of interpreting static load test. As has been mentioned already, Vibro piles are characterised by very good load-bearing capacities in non-cohesive soils, hence the nature of Q–s characteristics is quite similar to the linear one. Determination of the ultimate value requires a curve extrapolation. Fig. 13 contains a reference to analytical values of Q_{ult,cal} to limit the load-bearing capacity determined on the basis of curve interpretation obtained from SPLT by the Chin's method [21]. The Chin's method is based on the assumption that the Q–s curve for a pile is approximately hyperbolic.

From the above graph, it can be seen that the characteristic values of the limit load based on the proposed method are close to the results obtained by Chin's method. The difference between these values is much smaller than the relationship between the values from Chin's method and those calculated using the direct method according to EC7. It should be emphasised that the method included in EC7 gives higher values and closer to real pile-bearing capacities than many other methods used in Europe and in the rest of the world, for example, LCPC, Gwizdała–Stęczniewski's method, Schmertmann's method, De Ruiter and Beringen's method, Aoki and Velloso's method and Penpile method. These relationships have been proven many times in many publications also by the author of this article [2, 14, 22,23,24].

It should also be borne in mind that the values obtained from the proposed method are more accurate than those obtained from the analytical method (R_{c,d} (EC7/DIN)), proposed in the standard PN-EN 1997-2:2008 Eurocode 7: Geotechnical design. – Part 2: Ground investigation and testing, based on DIN1054 [20].

The graph presented in Fig. 14 proves that using a safety factor Fs = 1.3 gives a safe estimate of the load-bearing capacity.

Not only for the nine examples shown, but also for all 100+ cases considered in the analysis, all theoretical settlement curves run under the curves from the SPLT extrapolated using China's method. They are safe for design purposes. When using coefficients that are consistent with the DA2 approach, according to the current standard, the reserve range increases. However, with respect to the results obtained from the calculation method contained in EC7/DIN1054, these values are higher. This allows Vibro piles to realise their true potential in load transfer.

The ultimate load-bearing capacity from the developed correlations is on average 28% lower than values from the Chin method, based on the actual Q–s curve. This assures sufficient reserve for design needs. However, if the set of coefficients from EC7 is used, it enables a very safe calculation of the load bearing with a reserve up to 39%. Despite such a significant reserve, the results are not overly cautious, which may be proven by differences as compared to the load-bearing capacity values obtained based on the standard method (Fig. 15).

The paper presents a proposal for a prediction method of Q–s curve course. The developed method belongs to a group of direct methods that make use of the results of CPT to determine theoretical load-bearing capacities for Vibro piles with their base situated in non-cohesive soils. It has been defined on the basis of static load tests, and the necessary parameters for determination of full Q–s curve are cone resistance q_{c} and pile geometry. The defined correlations apply to piles whose base is located in fine and medium sands in moderately compacted, compacted and very compacted condition, that is, with a resistance value of q_{c} > 13 kPa [25]. They can be successfully used to calculate Vibro piles with typical diameters ranging from 408 to 610 mm and of various lengths.

The analysis has shown that the obtained values of ultimate load bearing are similar to the results obtained in SPLT interpretation methods. Particular attention should be drawn to the possibility of more precise estimation of pile ultimate load-bearing capacities, as compared to EC7 and other direct methods based on the results of CPT probing. Comparisons were made with the values of ultimate bearing capacity, corresponding to 0.1D settlements, obtained from the method contained in EC7. This is because it is the method whose results are usually the highest and closest to the actual Vibro pile capacities. This is proven by numerous publications [2, 14, 22,23,24]. The differences between the results obtained from the different direct methods based on CPT are due to the adopted criterion of safe bearing capacity, characterised in detail by Gwizdała [9, 23]. Results of calculations performed based on popularly used analytical methods may be as follows:

critical (indirect) bearing capacity for settlement within the range of 3%–5% of the pile diameter D [23];

ultimate bearing capacity Q_{ult}, conforming to conventional settlements s_{ult} equalling to 10% of the pile diameter, as, for example, the methods presented in PN-EN 1997:2008, which have been based on the German standard DIN 1054 and the Dutch standard NEN 6743 [16].

Classical transformation functions, whose idea is to fit the settlement curve to the calculated ultimate bearing capacity, enable to plot the hyperbolic characteristics of settlement [9, 12–13]. The hyperbola does not represent the actual shape of the Q–s curve for Vibro piles. In the proposed new method of predicting the interaction of Vibro piles with the ground, it is possible to determine a two-phase course of the curve: a linear phase describing elastic interaction, characteristic for Vibro piles and confirmed by test pile loads, and a non-linear elastic–plastic phase.

An aspect that distinguishes the developed method is the possibility of determining the full pile load–settlement curve, based on the results of soil testing. If the full curve Q–s is available, it is possible to assume for design needs values of critical or ultimate bearing capacities or any indirect settlement values.

#### Compilation of data used for correlation analysis.

_{b}^{*} |
|||||||
---|---|---|---|---|---|---|---|

Gdańsk | 5 | 406 | 460 | 9.0 to 13.5 | 6 | 2.0 to 10.0 | FSa |

9 | 508 | 560 | 16.0 to 22.0 | 7 | 2.0 to 10.0 | FSa | |

Gdynia | 14 | 560 | 610 | 14.0 to 18.0 | 10 | 2.0 to 6.0 | FSa |

Grudziądz | 16 | 508 | 560 | 10.6 to 20.0 | 12 | 1.0 to 8.0 | FSa, MSa |

Hajnówek | 4 | 508 | 560 | 7,5 | 4 | 4.0 to 6.0 | CSa, FSa |

Olsztynek | 44 | 610 | 660 | 6.5 to 12.0 | 30 | 1.0 to 6.0 | MSa |

Szczecin | 13 | 408 | 460 | 12.0 to 14.0 | 6 | 2.0 to 10.0 | MSa, FSa |

6 | 457 | 510 | 17.6 to 18.4 | 6 | 1.0 to 8.0 | FSa | |

10 | 457 | 520 | 17.5 to 21.5 | 9 | 1.0 to 6.0 | FSa | |

Wrocław | 4 | 610 | 660 | 8.5 to 11.0 | 4 | 2.0 to 6.0 | CSa, MSa |

12 | 508 | 560 | 6.0 to 18.0 | 8 | 2.0 to 6.0 | CSa, MSa | |

S | 137 | 102 |

#### Output parameters according to correlations from Fig. 4 to Fig. 6.

_{s} |
_{s} |
||
---|---|---|---|

0.0042 | 1.73 | 4.7 | 1107 |

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