Pile–soil interactions have been discussed since the 1940s [2; 30]. The bearing capacity of piles can be estimated using various methods [1; 3; 8; 26; 32]. The most widely known are based on static load test results [4; 5; 18] and transformational functions [12; 13; 19]. The author of the present study developed a method based on the Meyer–Kowalów (M–K) curve [20], which includes the static load test results and allows the vertical distribution of the skin shear stress as a boundary condition in numerical solutions. In this approach, the following mathematical descriptions are applied:
Pile–soil interaction is described using Kirchhoff's principle [22] in accordance with the graph shown in Fig. 1. The relationship between the pile rigidity and the soil elasticity modulus These assumptions can help determine the linear distribution of the pile skin resistance, which is presented later herein.
Graphical description of the loads acting on a pile in the presented approach.
In contrast to the approaches presented in literature, which discuss, for example, skin resistance calculations based on a centrifuge model [7], the method used in this study describes the mechanism of pile skin resistance mobilization as a result of the soil deformation under the load applied to the head of the pile and the axial force variation in the pile. Author intentionally decline to define the skin resistance as the result of the friction occurring between the pile and the soil. This friction is considered to occur after the boundary skin resistance value is surpassed when the soil starts to slide along the pile skin, which may decrease the actual pile-bearing capacity. It has been suggested that the residual stress should be included when calculating the pile-bearing capacity [6]. The omission of this effect is considered a mistake. This study did not directly use the force of the soil friction. Instead, it considered the mechanism of the bending of the soil space around the pile according to Kirchhoff's principle, which allowed to consider the entire soil body. An analysis of the experimental results showed that the applied mathematical description validates the presented approach [28]. The verification of the static load test results showed a positive correlation between the calculation and the field investigation results [22]. Multiple studies have been conducted on specific soil geotechnical parameters based on static load testing [9; 14; 15; 25; 33]. Piles technology affects the surface of pile skin, which should be considered in the design approach [29; 31].
Because the analysis is based on a set of piles, the research is extended to more field tests, including static load tests equipped with extensometers, to measure the pile-length shortening and verify the possibility of using the method in a wider range of cases. The piles that are considered have been the subjects of two notable studies [16; 17] and appear to be the most reliable for verification.
The additional static load tests conducted were part of a project conducted in Northern Poland, Szczecin, Leona Heyki 3, near the Odra River. The approximate location is shown in Fig. 2.
Location of a static load test project in Northern Poland, Szczecin, Leona Heyki 3 street.
Static load tests were designed to determine the boundary-bearing capacity of the piles with an axial force distribution along the pile shaft. For this project, approximately 1,000 piles were ordered from a local developer [35]. Piles were designed with diameter
The test piles were equipped with extensometers to determine the axial distribution of the force
Pile designed for testing with visible extensometers attached to the reinforcing steel while casting (image captured by the author).
Shown below are part of the results obtained during static load tests. Figure 5 presents the distribution of the axial force along the pile shaft and deformation measured during the static load test. In the scheme, there are placements of extensometers presented, along the reinforcement basket of the analyzed pile. Each of the analyzed pile had prefabricated reinforcement baskets with measurement devices attached, and they were placed in the pile after it was bored.
The author was present during the preparation period and static load process. I believe that performing natural-scale tests in field environments is the best way to verify the theoretical assumptions. Figure 4 shows one of the static load test stations after the casting of necessary piles to perform a single test. All the piles used to perform the tests needed to achieve their design before constructing the stand are shown in Fig. 6.
Part of the site prepared for static load testing (image captured by the author).
Distribution of the axial force along the pile shaft and deformation of the pile shaft measured during the static load test with extensometers [36].
Completed static load test site ready for testing (image obtained from authors' library).
Displacement piles were used to determine whether previous calculations could be applied to other piling technologies, as previous work focused on CFA piles [22]. Tables 1 and 2 present the static load test results. Figures 7 and 8 present the results of the CPTu investigation. Table 3 presents the geotechnical parameters of the soil profile.
Load–settlement values obtained from the static load test; Pile Heyki 1.
s [mm] | 0,00 | 0,57 | 1,14 | 1,86 | 2,64 | 3,69 | 4,82 | 6,54 | 8,38 | 10,83 | 13,38 | 16,41 | 20,28 | 26,08 |
Load–settlement values obtained from static load test for Pile Heyki 2.
s [mm] | 0,00 | 0,71 | 1,27 | 1,88 | 2,57 | 3,49 | 4,48 | 6,04 | 7,77 | 10,24 | 13,24 | 16,13 | 19,59 | 23,20 | 27,14 | 31,73 | 36,42 |
CPTu investigation results for Pile 1 [27].
CPTu investigation results for Pile 2 [27].
Geotechnical parameters of the soil profile [27].
0,0 | |||||||||||||
1,8 | Mg(hgrcFSa) | nN(Pd+H+c+ż) | 2,20 | 35 | 20 | - | 2,19 | 0,02 | - | 29° 30′ | - | - | 9,5 |
2,3 | Mg(grchclSa) | nN(Pg+H+c+ż) | 1,30 | 42 | - | 0,54 | 1,27 | 0,01 | 11,1 | 22° 10′ | 6 | 98 | 9,2 |
2,5 | Mg(grchFSa) | nN(Pd+H+c+ż) | 2,20 | 57 | 15 | - | 2,14 | 0,00 | - | 29° | - | - | 9,4 |
3,8 | Or(Nm) | Nm | 0,45 | 92 | - | 0,50 | 0,38 | 0,04 | 3,7 | 13° 10′ | 3 | 20 | 1,1 |
6,6 | Or(Nm) | Nm | 0,65 | 117 | - | 0,60 | 0,58 | 0,12 | 5,1 | 15° 50′ | 4 | 31 | 2,5 |
7,0 | Or(Nmp)/HFSa | Nmp/PdH | 1,10 | 133 | - | 0,44 | 1,00 | 0,01 | 6,8 | 18° 50′ | 5 | 71 | 7,9 |
8,2 | FSa | Pd | 3,30 | 149 | 20 | - | 3,18 | 0,00 | - | 29° 20′ | - | - | 14,2 |
8,6 | FSa | Pd/Ps | 6,40 | 177 | 40 | - | 6,25 | 0,00 | - | 32° 10′ | - | - | 28,4 |
11,1 | FSa | Pd | 13,20 | 208 | 65 | - | 13,03 | 0,00 | - | 35° 10′ | - | - | 64,9 |
11,8 | FSa | Pd | 7,30 | 220 | 40 | - | 7,12 | 0,00 | - | 32° 20′ | - | - | 32,4 |
12,4 | saSi | Πp | 2,60 | 236 | - | 0,74 | 2,40 | 0,00 | 11,1 | 22° 10′ | 7 | 171 | 21,8 |
13,3 | FSa | Pd | 7,00 | 250 | 35 | - | 6,80 | 0,00 | - | 31° 50′ | - | - | 31,1 |
13,8 | FSa | Pd | 11,10 | 268 | 55 | - | 10,88 | 0,00 | - | 33° 50′ | - | - | 54,6 |
15,2 | saSi/siSa | Πp/Pπ | 3,40 | 292 | - | 0,82 | 3,17 | 0,00 | 12,2 | 22° 50′ | 7 | 226 | 28,5 |
16,2 | FSa/MSa | Pd/Ps | 14,00 | 320 | 60 | - | 13,74 | 0,00 | - | 34° 30′ | - | - | 70,3 |
18,0 | FSa | Pd | 7,00 | 342 | 30 | - | 6,73 | 0,00 | - | 31° | - | - | 31,2 |
18,4 | FSa | Pd | 10,50 | 353 | 45 | - | 10,22 | 0,00 | - | 33° | - | - | 51,8 |
19,1 | FSa/MSa | Pd/Ps | 15,40 | 400 | 60 | - | 15,08 | 0,00 | - | 34° 30′ | - | - | 75,9 |
The Meyer–Kowalów curve concept has been introduced and widely applied [20; 23; 24]. It is based on soil mechanics principles [10]. This method allows the extrapolation of a static load test curve to estimate the boundary-bearing capacity of the pile (
The M–K curve exhibits a vertical asymptote:
It can be proven that:
Figure 9 shows an example of the M–K curve graph.
Example of the M–K curve graph [20].
The M–K curve depicts the physical aspects of pile settlement as an effect of the axial load acting on the pile head. For
The principle behind the M–K curve approximation is the estimation of the
The M–K curve facilitated the estimation of the extreme value of the skin resistance of the pile. Briaud and Tucker [2] concluded that estimating the boundary resistance of a pile yields better results than predicting the settlement.
It is assumed that the skin resistance can be expressed using the shear stress
This is the result of the equilibrium of the axial forces in the pile. The shear stress
Furthermore:
Another phenomenon that must be considered in the calculations is the pile-shortening effect caused by the axial force
It is assumed that pile–soil interaction can occur without soil slipping along the pile skin, which has been verified in previous research [21]:
Pile skin shear stress
Surpassing the maximal static friction between the skin of the pile and the soil results in a terminal skin resistance and soil-slipping effect. The proposed mathematical model is based on the following assumptions:
Skin resistance is presented as the shear stress Shear stress along the pile's skin is described using Kirchhoff's principle. Pile shortening is described as an elastic deformation process. There is no slipping effect of the soil along the pile's skin.
With these assumptions, a differential equation for the axial change in the shear stress against the vertical coordinate “z” can be obtained:
The
The solution to this equation can be expressed as follows:
The boundary conditions of the solution are
The axial change in
This equation was used to verify the
Siemaszko [28] performed an essential analysis of the results obtained from static load testing based on (18). Least-squares graphs were obtained to verify whether the static load test results can be presented as a linear distribution. The general form of the least-squares method is as follows:
In this study, the results of the least-squares method analysis showed extreme values and normal distributions, as shown in Fig. 10.
Least-squares graphs of the results obtained for the analyzed piles.
This analysis allowed to draw graphs of the shear stress distribution using the mathematical approach presented by Siemaszko [28]. Figures 11 and 12 show the graphs obtained for the piles analyzed in this study.
Relationship between the load acting on the head of the pile and the shear stress at the head level (τ2) and base level (τ1) for the pile taken from Heyki 1 [by author].
Relationship between the load acting on the head of the pile and the shear stress at the head level (τ2) and base level (τ1) for the pile taken from Heyki 2 [by author].
The author proposed an equation for the practical calculations of
In this equation,
This analysis showed that influencing the force applied to the soil by the pile base results in changes within its structure, and therefore, its geotechnical properties. The changed properties can be represented as
This yields the following:
For the practical purpose of performing engineering calculations with sufficient accuracy, Equation (22) can be expressed as follows:
For the soil under the pile base, both the internal shear angle and the dynamic friction coefficient vary. With these changes applied, the following equation for the shear stress
The equation that includes both
Equation (25) is the primary relationship, including the phenomena at the base of the pile with varying soil properties, as shown in Fig. 13.
Soil under and around the pile base forming a sphere, where the parameters vary significantly [22, 28].
Another calculated value is
From the linear distributions of the shear stress and pile skin resistance, the following can be assumed:
Using the M–K curve theory, the following relationships can be obtained for rough calculations:
For further analysis, the
The final equations for
Thus, the relationships presented above were verified. Figures 11 and 12 show the values of
The coefficients
M–K curve and static load test results compared for Pile 1 (left) and Pile 2 (right).
A comparison was made between the values
The experimental results can be described as follows:
The author attempted to verify the linear dependency between the head load
Here,
The equations include the linear characteristic of the shear stress acting on the skin of the pile
Therefore:
The values obtained for Piles 1 and 2 are presented in Table 4.
Equation (42) results, left side and right side.
Pile 1 | 0,83 | 0,803 |
Pile 2 | 0,85 | 0,84 |
Using the above equation, the
This study presents an analytical approach verified by conducting field experiments at a natural scale. Studies have focused on describing the mechanisms of the formation of skin resistance. Piles were designed and executed as soil-displacement piles. The study results confirmed that the pile-shortening effect may be applied to calculations, resulting in a linear shear stress distribution, which represents the pile skin resistance. The experimental research and analytical evaluation showed that