Analysis of the pile skin resistance formation

The methods used to describe the relationship between load at the head and settlement of a pile have been analysed by many authors [4, 14, 16]. These works focus mostly on static load test results, which contain the set of load and corresponding settlement values obtained during the test. These sets are presented as load–settlement curves, which can be interpreted using various approaches [3, 15]. The load–settlement curves may be used in a group of piles as well [7]. It has been argued that there is need for additional instrumentation during static load tests in order to obtain additional information about soil–pile interaction [3, 4].

Several papers combine field results and theoretical calculations of the dependency of shear stress distribution along the pile skin and settlement [19]. Others [8, 14, 17] focus on laboratory-based experiments. This allows for analysis in perfect soil environments, which do not appear in field cases. As such, calculations are more precise. However, works that employ the analytical approach, however, and in particular the soil mechanics theory, in particular, are scarce.

The main topic of this paper is the formation of shear stress along the skin of the pile due to soil–pile interaction. In a previous work [12], the authors proposed a definition of shear stress as the result of soil–pile interaction due to the bending of the space of soil around the pile, which is the result of the vertical movement of the pile. Kirchhoff's principle was used as a base for the description of the mentioned mechanism [11]. This theory forms the foundation of this paper. It is used to describe the formation of shear stress while loading a pile vertically with increasing force.

The authors’ intention in this paper is to draw attention to the fact, that in each studied case there is a need to specify the boundary conditions, especially those of the skin of the pile. That is the main goal of this paper.

The model that was analysed is a pile loaded vertically, whose movement is the result of the load put at the head. It is assumed that the cross-section of the pile is circular, and that z-axis, which is the vertical axis of the coordinate system, goes along the pile symmetry axis. The pile diameter is denoted as D, while pile length in the soil is denoted as h. The authors have analysed a concrete pile, which follows the assumption that the concrete is homogeneous and isotropic.

Figure 1

The equations describing the dependence of shear stress and movement of the pile at z depth are based on Kirchhoff's principle as follows: (1) s(z)=τ(z)⋅lG s\left( z \right) = {{\tau \left( z \right) \cdot l} \over G} where: (2) G=Es,ν2(1+ν) G = {{{E_{s,\nu }}} \over {2\left( {1 + \nu } \right)}} It has been verified in experimental analysis [17] that the Equation (2) equation can be applied for practical calculations purposes for the linear part of Q-s graph which can be up to NNgr=0,5 {N \over {{N_{gr}}}} = 0,5 .

For the interior of the pile, along the axis of symmetry, the load at the head changes its value as shown below: (3) N(z)=N2−∫0zπ⋅D⋅τ(z)⋅dz N\left( z \right) = {N_2} - \int\nolimits_0^z {\pi \cdot D \cdot \tau \left( z \right) \cdot dz} In Eqs. (1) and (2), the following symbols are used: E_s,v- modulus of elasticity of soil in vertical direction near pile skin [MPa]; ν – Poisson coefficient of the soil.

Another equation describes the effect of the elastic shortening of a pile, which is due to vertical load. There is: (4) s*(z)=4πD2Ec⋅∫0zN(z)dz where: {s_*}\left( z \right) = {4 \over {\pi {D^2}{E_c}}} \cdot \int\nolimits_0^z {N\left( z \right)dz\,{\rm{where}}:} in which: s_*- pile shortening value at z depth [mm]; E_c- elasticity modulus of concrete [MPa].

Using the above introduced symbols, the relationship between vertical displacement, including the pile shortening effect and the shear stress on its skin is as follows: (5) s(z)−s*(z)=2(1+ν)⋅τ(z)⋅lEt s\left( z \right) - {s_*}\left( z \right) = 2\left( {1 + \nu } \right) \cdot {{\tau \left( z \right) \cdot l} \over {{E_t}}} It is assumed that pile displacement causes no additional skin slip between soil and pile skin.

The next assumption is that in Eq. (5), the following relationships are used: (6) d2s(z)dz2=0 and so {{{d^2}s\left( z \right)} \over {d{z^2}}} = 0\,{\rm{and}}\,{\rm{so}} (7) s(z)=s2−(s2−s1)⋅zh s\left( z \right) = {s_2} - \left( {{s_2} - {s_1}} \right) \cdot {z \over h} It is also assumed that for N₂=0 there is no shear stress on the skin of the pile that may be the result of technology used or the state of tension around the pile.

Under this assumption, combining Eqs. (5), (3) and (4), the following linear equation is obtained: (8) d2τd(a⋅z)2−τ=0 where {{{d^2}\tau } \over {d{{\left( {a \cdot z} \right)}^2}}} - \tau = 0\,{\rm{where}} (9) a=2⋅Et(1+ν)⋅Ec⋅D⋅l a = \sqrt {{{2 \cdot {E_t}} \over {\left( {1 + \nu } \right) \cdot {E_c} \cdot D \cdot l}}} In general, if the requirement in Eq. (6) is not valid, there is shear stress although the pile is unloaded. As such, Eq. (8) is embodied as: (10) d2τd(a⋅z)2−τ=f(z) where {{{d^2}\tau } \over {d{{\left( {a \cdot z} \right)}^2}}} - \tau = f\left( z \right)\,{\rm{where}} F(z) is a given function for additional boundary conditions. As mentioned above, this effect was neglected. The solution of the differential Eq. (8) is as follows: (11) τ(z)=A1⋅sinh(az)+A2⋅cosh(az) \tau \left( z \right) = {A_1} \cdot \sinh \left( {az} \right) + {A_2} \cdot \cosh \left( {az} \right) Boundary conditions for this equation are: (12) For z=0 there is τ=τ2 {\rm{For}}\,z = 0\,{\rm{there}}\,{\rm{is}}\,\tau = {\tau _2} (13) For z=h there is τ=τ1 {\rm{For}}\,z = h\,{\rm{there}}\,{\rm{is}}\,\tau = {\tau _1} (14) A2=τ2 {A_2} = {\tau _2} (15) A1=τ1−τ2⋅cosh(ah)sinh(ah) {A_1} = {{{\tau _1} - {\tau _2} \cdot \cosh \left( {ah} \right)} \over {\sinh \left( {ah} \right)}} Function τ(z) allows to calculate the load from Eq. (3) and the pile shortening from Eq. (4).

After implementation of Eqs. (13) and (14), it follows that: (16) τ(z)=τ1−τ2⋅cosh(ah)sinh(ah)⋅sinh(ah)+τ2⋅cosh(az) \tau \left( z \right) = {{{\tau _1} - {\tau _2} \cdot \cosh\left( {ah} \right)} \over {\sinh \left( {ah} \right)}} \cdot \sinh \left( {ah} \right) + {\tau _2} \cdot \cosh \left( {az} \right) A detailed analysis of constant “a” suggests that, in practical engineering, its value is very small; therefore, a•h≪1 which takes into consideration the asymptotic solution: (17) lima→0(τ,a)=τ(z)=τ2−(τ2−τ1)⋅zh \mathop {\lim }\limits_{a \to 0} \left( {\tau ,a} \right) = \tau \left( z \right) = {\tau _2} - \left( {{\tau _2} - {\tau _1}} \right) \cdot {z \over h} Eq. (17) provides another relationship that includes loads at the head N₂ and base N₁ of the pile: (18) N2−N1πDh=τ2+τ12 and so {{{N_2} - {N_1}} \over {\pi Dh}} = {{{\tau _2} + {\tau _1}} \over 2}\,{\rm{and}}\,{\rm{so}} (19) N(z)=N2−πDh⋅[τ2⋅(zh)−12(τ2−τ1)⋅(zh)2] N\left( z \right) = {N_2} - \pi Dh \cdot \left[ {{\tau _2} \cdot \left( {{z \over h}} \right) - {1 \over 2}\left( {{\tau _2} - {\tau _1}} \right) \cdot {{\left( {{z \over h}} \right)}^2}} \right] The above equation was applied for the estimation of τ₂ based upon field measurements.

Field tests were conducted on real piles. The method of static load test provided sets of data {N;s} for different load stages. Independently, with the use of extensometers, the vertical axial force available for different stages of external load was distributed at the head of the pile N₂. The equation used for the approximation for reference N₂ is as follows: (20) N(zi)=N2−πDh[τ2⋅(zih)−(τ2−N2−N1πDh)⋅(zih)2] N\left( {{z_i}} \right) = {N_2} - \pi Dh\left[ {{\tau _2} \cdot \left( {{{{z_i}} \over h}} \right) - \left( {{\tau _2} - {{{N_2} - {N_1}} \over {\pi Dh}}} \right) \cdot {{\left( {{{{z_i}} \over h}} \right)}^2}} \right] which includes: (21) τ1=N2−N1πDh⋅2−τ2 {\tau _1} = {{{N_2} - {N_1}} \over {\pi Dh}} \cdot 2 - {\tau _2} The value of τ₂ in Eq. (21) was calculated using Eq. (20) for a given load at the head of the pile and the forces obtained along the pile axis. The least squares method was used for the calculations, which can be presented as: (22) δ2=∑[Ni−N(zi;τ2)]2=min {\delta ^2} = \sum {{{\left[ {{N_i} - N\left( {{z_i};{\tau _2}} \right)} \right]}^2} = {min}} in which: N_i – pile skin force measured at “z_i” depth; N(z_i) – load calculated using Eq. (20).

The main aim of the proposed method was to check if the least squares method with Eq. (22) led to normal squared differences distribution. This would prove that the simplification given by Eqs. (7) and (17) can be applied.

Two piles were fully analysed. Their dimensions are: diameter D=0.8m, length L=18.05 m. The piles were drilled using the CFA method. They were used as foundations for overpasses in Gdańsk on the Wysoczyzna Kaszubska territory. The soil was a mix of cohesive and non-cohesive layers.

The most important results were shear stress values. In order to measure them, one approach is to place the steel tube with the extensometers along the pile shaft. The system of extensometers was an addition to the static load test. The obtained results allowed the authors to compare the theoretical analyses they were working on, with actual results from the piles used as a functioning foundation.

Fig. 2 presents the results of a set of extensometers for one of the analysed piles:

Figure 2

The tables below show the main data obtained from the static load test of Pile 1 and Pile 2.

For these piles, as mentioned above, we obtained the CPTu soil investigation results. The graphs are presented in Fig. 3.

Figure 3

The first step of the calculations was to determine if the “a” parameter should be implemented or neglected. Then, the shear stress graph was optimised based on the “a” value. The authors tried to obtain the most probable, and closest to the experiment, graph of shear stress value. Upon this step, calculations were made using the least squares methods. In this way, the “a” parameter value was optimised. It can be proved that parameter “a”, specified by Eq. (9), is a very small value in comparison to the width and length values of the pile. Assuming average conditions E_s,v=10 MPa, ν=0,25, E_con=30 GPa, D=0,8 m, l≅ 3D, the value of a=0,016 [1/m].

Upon calculations for different piles, it was concluded that, for practical purposes, the influence of “a” can be neglected.

The next step was to compare the theoretical calculations of pile loads with the corresponding experimental results. Using the method shown above, for each pile, calculations of shear stress were made.

Once the set of data was obtained, a comparison was made between theoretical and experimental results. For each pile, there was normal distribution of squared differences, which proves the assumption that d2sdz2=0 {{{d^2}s} \over {d{z^2}}} = 0 . In Figs. 4 and 5, we compare the theoretical and experimental values N(z).

Figure 4

Figure 5

The same comparison for Pile 2 is shown in Fig. 5.

In Figs. 6a and 6b we show an example of a normal distribution graph obtained with the least squares method for both analysed piles.

Figure 6a

Figure 6b

The least squares methods, as shown in Eq. (20), allows to optimise the value of shear stress along the skin of the pile. Examples are shown in Figs. 7a, 7b, 8a, and 8b.

Figure 7a

Figure 7b

Figure 8a

Figure 8b

At this stage, conclusions can be formulated based upon the experiments:

The experimental results prove that for practical calculations, the following simplification can be applied: −d2sdz2=0 - {{{d^2}s} \over {d{z^2}}} = 0 ; and a→0, moreover

For a practical approach to combining field investigation data with theoretical calculations, it is essential to estimate value τ₂ using soil investigations and static load tests without the need for extensometers. Kirchhoff's approach can be used for the analysis of soil reaction on pile settlement, Eqs. (1) and (2). This gives the following relationship: (23) τ2=Es,ν2⋅(1+ν)⋅l⋅s2 {\tau _2} = {{{E_{s,\nu }}} \over {2 \cdot \left( {1 + \nu } \right) \cdot l}} \cdot {s_2} In field investigations for each external load, settlement s₂ was measured and τ₂ calculated, which allows to estimate (23). The following relationships are now available: (24) τ2=B2⋅s2 {\tau _2} = {B_2} \cdot {s_2} (25) B2=Es,ν2⋅(1+ν)⋅l {B_2} = {{{E_{s,\nu }}} \over {2 \cdot \left( {1 + \nu } \right) \cdot l}} In order to obtain τ₂ stress values, the M–K method [11] can be used with the static load test results. This method has been described above. The relationships are as follows: (26) s2=τ2⋅lEs,ν⋅2(1+ν) {s_2} = {{{\tau _2} \cdot l} \over {{E_{s,\nu }}}} \cdot 2\left( {1 + \nu } \right) (27) s2=C2Ngr2⋅(1−N2Ngr2)−κ2−1κ2 {s_2} = {C_2}{N_{gr2}} \cdot {{{{\left( {1 - {{{N_2}} \over {{N_{gr2}}}}} \right)}^{ - {\kappa_2}}} - 1} \over {{\kappa_2}}} In these equations: C₂ – aggregated Winkler modulus is introduced in [12]; N_gr2 – limits pile bearing capacity [MN]; κ₂ – non-dimensional parameter, which represents the relation between pile skin and base resistance.

The above parameters may be obtained using statistical methods and the static load test results available as {N;z}. For further analysis, it is convenient to consider the following M–K method outcome:

For N2Ngr2<0,5 {{{N_2}} \over {Ng{r_2}}} < 0,5 simplified expressions can be used: (28) s2=C2⋅N2 and so {s_2} = {C_2} \cdot {N_2}\,{\rm{and}}\,{\rm{so}} (29) τ2=C2⋅N2⋅B2 and furthermore {\tau _2} = {C_2} \cdot {N_2} \cdot {B_2}\,{\rm{and}}\,{\rm{furthermore}} (30) Es,ν=4qc⋅(1+14qc13) [12] and for practical calculations {E_{s,\nu }} = 4{q_c} \cdot \left( {1 + {1 \over 4}q_c^{{1 \over 3}}} \right)\,\left[ {12} \right]\,{\rm{and}}\,{\rm{for}}\,{\rm{practical}}\,{\rm{calculations}} (31) l≅3D[ 17 ] {\rm{l}} \cong 3{\rm{D}}\left[ {17} \right] As a result of the field investigations and the static load test, there is a set of data {N_i;s_i} for each pile obtained, and the following parameters are obtained with the statistical method: C₂, κ₂, N_gr2 [13]. These give the relation N₂= N₂(s₂), as well as the corresponding set of {τ₂;N₂}. With this approach, we can calculate coefficient B₂ and so τ₂(N₂) can be compared with relevant τ₂(s₂).

The last value which remains to be calculated is τ₁, namely, shear stress at the depth of the pile base, which is a reaction on τ₂(N₂) pile settlement. As above, τ₁ is obtained with (21). (32) τ1=N2(s2)−N1(s2)πDh⋅2−τ2(s2)=τ1(s2) {\tau _1} = {{{N_2}\left( {{s_2}} \right) - {N_1}\left( {{s_2}} \right)} \over {\pi Dh}} \cdot 2 - {\tau _2}\left( {{s_2}} \right) = {\tau _1}\left( {{s_2}} \right) With the application of the previous relationships and the M-K method parameters [13, 14, 17], the following relation is obtained: (33) τ1=N2⋅{2πdH⋅(1−N1N2)−B2⋅C2} {\tau _1} = {N_2} \cdot \left\{ {{2 \over {\pi dH}} \cdot \left( {1 - {{{N_1}} \over {{N_2}}}} \right) - {B_2} \cdot {C_2}} \right\} The final version of Eq. (32) with M–K method relationships [17] applied, is as follows: (34) τ1=N2⋅{2πdH⋅κ2⋅(2+κ2)(1+κ2)2−B2⋅C2} {\tau _1} = {N_2} \cdot \left\{ {{2 \over {\pi dH}} \cdot {{{\kappa _2} \cdot \left( {2 + {\kappa _2}} \right)} \over {{{\left( {1 + {\kappa _2}} \right)}^2}}} - {B_2} \cdot {C_2}} \right\} For simplification, the authors described the previous equation as: (35) τ1=N2⋅B1 where B1=2πdH⋅κ2⋅(2+κ2)(1+κ2)2−B2⋅C2 {\tau _1} = {N_2} \cdot {B_1}\,{\rm{where}}\,{B_1} = {2 \over {\pi dH}} \cdot {{{\kappa _2} \cdot \left( {2 + {\kappa _2}} \right)} \over {{{\left( {1 + {\kappa _2}} \right)}^2}}} - {B_2} \cdot {C_2} Figs. 9a and 9b show the linear distribution {τ₂;N₂} obtained using Eqs. (25), (29), (30). C₂ values were optimised using the least squares method. Figs. 10a and 10b show the linear distribution for {τ₁;N₂} obtained with Eq. (34).

Figure 9a

Figure 9b

Figure 10a

Figure 10b