Pile–Soil Interaction during Static Load Test

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 (N_gr). The curve can be described using C, N_gr2, and k. Parameter C is introduced as an aggregation of the toe and skin resistances during pile settlement. The following equation is used to extrapolate the static load test curve in the M–K method: (4) s=C⋅Ngr⋅(1−NNgr)−v−1v s = C \cdot {N_{gr}} \cdot {{{{\left({1 - {N \over {{N_{gr}}}}} \right)}^{- v}} - 1} \over v} s – Settlement of the head of the pile [mm]; C – Reversed aggregated Winkler modulus [m/MN] [21]; N_gr – Axial force, M–K curve vertical asymptote [MN]; k – Parameter showing the proportions of base and shaft resistances [-]

The M–K curve exhibits a vertical asymptote: (5) N=Ngr and diagonal N = {N_{gr}}\,{\rm{and}}\,{\rm{diagonal}} (6) s=C⋅N s = C \cdot N

It can be proven that: (7) limN→0s(N)=C⋅N and \mathop {\lim}\limits_{N \to 0} s(N) = C \cdot N\,\,{\rm{and}} (8) limN→Ngrs(N)=∞ \mathop {\lim}\limits_{N \to {N_{gr}}} s(N) = \infty

Figure 9 shows an example of the M–K curve graph.

Figure 9:

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 N⟶0, a small displacement and linear characteristic of the load–settlement relationship could be observed. For N⟶N_gr, the settlement increased uncontrollably, and the pile lost its bearing capacity. The M–K curve parameters were estimated statistically using a set of load–settlement values {s_i; N_i}.

The principle behind the M–K curve approximation is the estimation of the N_gr value and the prediction of the mobilization of the pile base and shaft capacity. The curve presented by Chin [5] is a variation of the M–K curve, where κ = 1.

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 τ(z), which is the result of the axial force acting on the head of the pile and its variation along the shaft. (9) dN=πD•τdz dN = \pi D \bullet \tau dz

This is the result of the equilibrium of the axial forces in the pile. The shear stress τ(z) in the soil around the pile is the result of bending the soil body in the range of “l” from the pile head caused by its settlement. Kirchhoff's principle can be expressed as follows: (10) G=Es,v2(1+υ) G = {{{E_{s,v}}} \over {2(1 + \upsilon)}} G – Kirchhoff's modulus of the soil along the skin of pile [Pa]; E_c – Elasticity modulus of the concrete piles [Pa]; E_s,v – Soil elasticity modulus along the pile skin [Pa]; ν – Poisson's coefficient for soil along the skin of the pile [-].

Furthermore: (11) s(z)=τ(z)⋅lG s(z) = {{\tau (z) \cdot l} \over G}

Another phenomenon that must be considered in the calculations is the pile-shortening effect caused by the axial force N(z) from a depth z=0 to z=h, assuming an elastic deformation of the pile: (12) s*(z)=4πD2Ec⋅∫0zN(z)dz {s_*}(z) = {4 \over {\pi {D^2}{E_c}}} \cdot \int\limits_0^z {N(z)dz}

It is assumed that pile–soil interaction can occur without soil slipping along the pile skin, which has been verified in previous research [21]: (13) s(z)−s*(z)=0 s(z) - {s_*}(z) = 0

Pile skin shear stress τ(z) is unrelated to the phenomenon of soil slipping along the pile shaft and the soil stress component perpendicular to the shaft. This depends on Kirchhoff's principle of the bending space around the pile.

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 τ(z).

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  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: (14) d2τdz2=4Gl⋅D⋅Ec⋅τ {{{d^2}\tau} \over {{dz}^2}} = {{4G} \over {l \cdot D \cdot {E_c}}} \cdot \tau

The a coefficient, defined in Eq. (2), yields: (15) d2τd(a⋅z)2−τ=0 {{{d^2}\tau} \over {d{{(a \cdot z)}^2}}} - \tau = 0

The solution to this equation can be expressed as follows: (16) τ(z)=A1 sinh(az)+A2 cosh(az) \tau (z) = {A_1}\,\sinh (az) + {A_2}\,\cosh (az)

The boundary conditions of the solution are z=0; τ=τ₂ and z=h; τ=τ₁. A detailed analysis of the solution presented in previous research [22] showed that the influence of the a coefficient, expressed in (2), does not have a carryover effect on the shear stress distribution. This verification confirms the possibility of applying such a simplification. The author estimated the value of constant a in a practical environment, producing results close to 0.05 [1/m]. Considering these results, the linear distribution of τ(z) along the skin of the pile can be obtained as follows: (17) τ(z)=τ2−(τ2−τ1)∨(zh) \tau (z) = {\tau_2} - ({\tau_2} - {\tau_1}) \vee \left({{z \over h}} \right)

The axial change in N(z) can be expressed as follows: (18) N(z)=N2−πDh⋅[τ2⋅(zh)−12(τ2−τ1)⋅(zh)2] N(z) = {N_2} - \pi Dh \cdot \left[ {{\tau_2} \cdot \left({{z \over h}} \right) - {1 \over 2}({\tau_2} - {\tau_1}) \cdot {{\left({{z \over h}} \right)}^2}} \right]

This equation was used to verify the τ(z) axial distribution in the analyzed case and was applied to the experimental analysis of the field experiments. In previous studies [22; 28], it was important to determine if there is a possibility of omitting the influence of the a value and assuming a=const. The verification results showed that including the value constant a produced a good correlation with the graphs, where its influence was omitted. Meyer and Siemaszko [22] and Siemaszko [28] reported this correlation.

The author proposed an equation for the practical calculations of τ₁ and τ₂. Following the shear stress τ₁ calculations conducted in a previous experimental analysis by [11], the following equation is included: (20) τ1=4⋅N2πD2⋅ftan2(45−ϕ2)tan2(45+ϕ2) {\tau_1} = {{4 \cdot {N_2}} \over {\pi {D^2}}} \cdot f{{{{\tan}^2}\left({45 - {\phi \over 2}} \right)} \over {{{\tan}^2}\left({45 + {\phi \over 2}} \right)}}

In this equation, f is the kinematic friction coefficient, and ϕ is the internal friction angle of the soil in its natural state.

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 f⟶f_* and ϕ⟶ϕ_*. (21) tan2(45+ϕ*2)=1+k2 {\tan^2}\left({45 + {{{\phi_*}} \over 2}} \right) = \sqrt {1 + {k_2}}

This yields the following: (22) ϕ*=2⋅[arctan(1+k2)−45°] {\phi_*} = 2 \cdot [\arctan (\sqrt {1 + {k_2}}) - 45^\circ ]

For the practical purpose of performing engineering calculations with sufficient accuracy, Equation (22) can be expressed as follows: (23) ϕ*=18,31⋅k20,715 {\phi_*} = 18,31 \cdot k_2^{0,715} where ϕ_* is in degrees.

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 τ₁ can be used: (24) τ1=4⋅N2πD2⋅f*tan2(45−ϕ*2)tan2(45+ϕ*2) {\tau_1} = {{4 \cdot {N_2}} \over {\pi {D^2}}} \cdot {f_*}{{{{\tan}^2}\left({45 - {{{\phi_*}} \over 2}} \right)} \over {{{\tan}^2}\left({45 + {{{\phi_*}} \over 2}} \right)}}

The equation that includes both f⟶f_* and ϕ⟶ϕ_* can be presented as follows: (25) τ1=4⋅N2πD2⋅f⋅k2(1+k2)4 {\tau_1} = {{4 \cdot {N_2}} \over {\pi {D^2}}} \cdot {{f \cdot {k_2}} \over {{{(1 + {k_2})}^4}}}

Equation (25) is the primary relationship, including the phenomena at the base of the pile with varying soil properties, as shown in Fig. 13.

Figure 13:

Soil under and around the pile base forming a sphere, where the parameters vary significantly [22, 28].

Another calculated value is τ₂, which is the shear stress at the head of the pile. The following boundary conditions were assumed: for z=0, there is τ(z)=τ₂; for z=h, there is τ(z)=τ₁. The vertical force equilibrium is expressed as follows: (26) T=N2−N1 T = {N_2} - {N_1}

From the linear distributions of the shear stress and pile skin resistance, the following can be assumed: (27) T=πDH(τ2+τ1)2 T = {{\pi DH({\tau_2} + {\tau_1})} \over 2}

Using the M–K curve theory, the following relationships can be obtained for rough calculations: (28) N1=N2(1+k2)2 {N_1} = {{{N_2}} \over {{{(1 + {k_2})}^2}}} (29) τ2+τ1=2N2πDH⋅k2(1+k2)2⋅[2+k2−2HfD⋅1(1+k2)2] {\tau_2} + {\tau_1} = {{2{N_2}} \over {\pi DH}} \cdot {{{k_2}} \over {{{(1 + {k_2})}^2}}} \cdot \left[ {2 + {k_2} - {{2Hf} \over D} \cdot {1 \over {{{(1 + {k_2})}^2}}}} \right]

For further analysis, the α coefficient is substituted into the equation: (30) α=2HfD \alpha = {{2Hf} \over D}

The final equations for τ₂ and τ₁ are as follows: (31) τ1=4⋅N2πD2⋅f⋅k2(1+k2)4 {\tau_1} = {{4 \cdot {N_2}} \over {\pi {D^2}}} \cdot {{f \cdot {k_2}} \over {{{(1 + {k_2})}^4}}} (32) τ2=2N2πDH⋅k2(1+k2)2⋅[2+k2−α(1+k2)2] {\tau_2} = {{2{N_2}} \over {\pi DH}} \cdot {{{k_2}} \over {{{(1 + {k_2})}^2}}} \cdot \left[ {2 + {k_2} - {\alpha \over {{{(1 + {k_2})}^2}}}} \right]

Thus, the relationships presented above were verified. Figures 11 and 12 show the values of τ(z)=τ₂ and τ(z)=τ₁. As the distributions presented in the graphs suggest a linear correlation between τ₂(N₂) and τ₁(N₂), the following relationships can be applied: (33) τ1=A1⋅N2 {\tau_1} = {A_1} \cdot {N_2} (34) τ2=A2⋅N2 {\tau_2} = {A_2} \cdot {N_2}

The coefficients A₁ and A₂ were estimated using the least-squares method for both the piles. For both the piles, statistical calculations were performed to obtain the M–K curve parameters N₂, C₂, and κ₂. The results of the Q-s curves obtained using the M–K curve method are shown through the graphs in Fig. 14.

Figure 14:

M–K curve and static load test results compared for Pile 1 (left) and Pile 2 (right).

A comparison was made between the values A₁ and A₂ calculated from the experimental results obtained under natural conditions and the values obtained using the M–K curve method. (35) τ1=2N2πDH⋅2Hf⋅k2D(1+k2)4=2N2πDH⋅2Hf⋅k2D(1+k2)4 {\tau_1} = {{2{N_2}} \over {\pi DH}} \cdot {{2Hf \cdot {k_2}} \over {D{{(1 + {k_2})}^4}}} = {{2{N_2}} \over {\pi DH}} \cdot {{2Hf \cdot {k_2}} \over {D{{(1 + {k_2})}^4}}} (30) α=2HfD \alpha = {{2Hf} \over D} (36) τ2=2N2πDH⋅k2(1+k2)2⋅[2+k2−α(1+k2)2] {\tau_2} = {{2{N_2}} \over {\pi DH}} \cdot {{{k_2}} \over {{{(1 + {k_2})}^2}}} \cdot \left[ {2 + {k_2} - {\alpha \over {{{(1 + {k_2})}^2}}}} \right]

The experimental results can be described as follows: (37) A1=2⋅k2πDH⋅α(1+k2)4 {A_1} = {{2 \cdot {k_2}} \over {\pi DH}} \cdot {\alpha \over {{{(1 + {k_2})}^4}}} (38) A2=2⋅k2πDH⋅1(1+k2)2⋅[2+k2−α(1+k2)4]=A1⋅(1+k2)2[2+k2−α(1+k2)2]α \matrix{{{A_2} = {{2 \cdot {k_2}} \over {\pi DH}} \cdot {1 \over {{{(1 + {k_2})}^2}}} \cdot \left[ {2 + {k_2} - {\alpha \over {{{(1 + {k_2})}^4}}}} \right] =} \cr {{A_1} \cdot {{{{(1 + {k_2})}^2}\left[ {2 + {k_2} - {\alpha \over {{{(1 + {k_2})}^2}}}} \right]} \over \alpha}} \cr}

The author attempted to verify the linear dependency between the head load N₂ and the skin shear stresses τ₁ and τ₂. The following solutions were used based on the experimental field test results: (39) τ1=τ1(N2)=A1⋅N2 {\tau_1} = {\tau_1}({N_2}) = {A_1} \cdot {N_2} (40) τ2=τ2(N2)=A2⋅N2 {\tau_2} = {\tau_2}({N_2}) = {A_2} \cdot {N_2}

Here, A₁; A₁=const. As for the equilibrium of the axial forces, it is: (41) N2−N1=τ1+τ22⋅πDH {N_2} - {N_1} = {{{\tau_1} + {\tau_2}} \over 2} \cdot \pi DH

The equations include the linear characteristic of the shear stress acting on the skin of the pile τ(z), which represents the skin resistance [22]. Based on the analysis previously conducted based on the M–K theory, Eq. (28) is valid. This can be used to obtain the following relationship: (42) N2⋅[1−1(1+k2)2]=A1⋅N2+A2⋅N22⋅πDH {N_2} \cdot \left[ {1 - {1 \over {{{(1 + {k_2})}^2}}}} \right] = {{{A_1} \cdot {N_2} + {A_2} \cdot {N_2}} \over 2} \cdot \pi DH

Therefore: (43) 1−1(1+k2)2=A1+A22⋅πDH 1 - {1 \over {{{(1 + {k_2})}^2}}} = {{{A_1} + {A_2}} \over 2} \cdot \pi DH

The values obtained for Piles 1 and 2 are presented in Table 4.

Using the above equation, the κ₂ parameter value for the results can be verified, including the value of the pile-shortening effect. However, the value of the κ₂ parameter may not be equal to κ₂ derived from {s_i;N_i} obtained using statistical calculations. This suggests that the principles of earlier research [34] require further verification.