An Attempt to Analytically Determine Course of the Continuous Q-S Curve in Case of Changed Pile Length or Diameter

After more than 10 years of research conducted at the West Pomeranian University of Technology in Szczecin, it is possible to extensively describe many important aspects of pile-soil interaction. In this paper authors would like to show a practical case of determining a static load test curve in case of changed length or diameter of a pile without the necessity of repeating static load tests. In their analysis, authors used the results of a static load test conducted in a full range of loads by Energopol Szczecin S.A. [1] for CFA piles bored in mostly loam soil conditions. By analysing those results it was possible to determine the previously described relation [1] between pile maximum bearing capacity as a function of pile length, diameter, and CPTU probe cone resistance at the level of pile base. Furthermore, it allows to obtain analytical relations between parameters describing static load test curve and its diameter and length that can be used for practical curve conversion. The parameters of the pile used in the analysis can be seen in Tab.1.

During the process of structural design, it is vital to correctly establish maximum bearing capacity of a pile, which is the most important factor determining safety. Bearing capacity of a pile can be obtained through several methods, as described by Briaud [2], Hansen [3], Chin [4], Coyle and Reese [5], Dung and Kim [6], Gwizdała [7], Meyer and Kowalow [8]. It is also worth noting that the method described by Chin is a special case of the Meyer-Kowalow method for a case when skin resistance is equal to the base resistance of the pile. In the further part of this paper, authors used a method that allows for a full description of the continuous load-settlement curve, which has been proposed by Meyer and Kowalow (M-K method) in 2010 [8]. M-K curves satisfy physical conditions from Boussinesq's theory, as described by equation (1) for values of load closer to maximum bearing capacity tend to asymptote described by formula (2). (1) limN2→0sN2=C2⋅N2 \mathop {\lim }\limits_{{N_2} \to 0} s\left( {{N_2}} \right) = {C_2} \cdot {N_2} (2) limN→Ngr2sN2=∞ \mathop {\lim }\limits_{N \to {N_{gr2}}} s\left( {{N_2}} \right) = \infty

It is possible to write a equation describing a continuous load-settlement curve (M-K curve) in a full range of loads, that is proposed in the M-K method [8]: (3) sN2=C2⋅Ngr2⋅1−N2Ngr2−κ2−1κ2 s\left( {{N_2}} \right) = {C_2} \cdot {N_{gr2}} \cdot {{{{\left( {1 - {{{N_2}} \over {{N_{gr2}}}}} \right)}^{ - {\kappa _2}}} - 1} \over {{\kappa _2}}} where: s

pile settlement [mm],

Parameters describing the M-K curve in the equation can be determined using the least square method directly from set of values {s_i,N_i} obtained as a result of the static load test [1, 8, 9, 10, 11].

After reorganizing equation (3) it is also possible to obtain a formula that describes inverse relation, which can be written as follows: (4) N2s=Ngr2⋅1−1+κ2⋅sC2⋅Ngr2−1κ2 {N_2}\left( s \right) = {N_{gr2}} \cdot \left[ {1 - {{\left( {1 + {{{\kappa _2} \cdot s} \over {{C_2} \cdot {N_{gr2}}}}} \right)}^{ - \left( {{1 \over {{\kappa _2}}}} \right)}}} \right] and by analogy formulas (5) and (6) describing curves regarding skin T(s) and toe N₁(s) resistances in the M-K method can also be written, as shown below: (5) N1s=Ngr1⋅1−1+κ1⋅sC1⋅Ngr1−1κ1 {N_1}\left( s \right) = {N_{gr1}} \cdot \left[ {1 - {{\left( {1 + {{{\kappa _1} \cdot s} \over {{C_1} \cdot {N_{gr1}}}}} \right)}^{ - \left( {{1 \over {{\kappa _1}}}} \right)}}} \right] (6) Ts=N2s−N1s T\left( s \right) = {N_2}\left( s \right) - {N_1}\left( s \right) where:

C₁, N_gr1, κ₁, N₁ – parameters analogical like in equation 3, describing load-settlement relation for a pile toe.

Curves described by equation (4) – (6) can also be schematically shown as in Fig. 1. Value of settlement corresponding with maximum skin friction T_max. is represented as s*.

Figure 1.

As it has been previously stated, the parameters describing a continuous load-settlement curve can be obtained directly from the results of static load tests using the least square method. Results of static load tests conducted in a full range of loads by Energopol Szczecin S.A. [1] allowed for further verification of existing methods of obtaining M-K curve parameters with regard to using those parameters in the method of curve conversion [1, 11].

Assuming that the value of measurement inaccuracy is a constant value for every two consecutive points in a load-settlement relations it is possible to subtract the M-K equation (3) written for two consecutive points and therefore minimize the impact of measurement inaccuracy on a final result of curve approximation.

After subtracting equation (3) written for two consecutive points (i and i+1) of the static load test data set {s_i,N_i} we ultimately obtain for κ₂ ≠ 0: (7) si+1−si=C2Ngr21−Ni+1Ngr2−κ2−1−NiNgr2−κ2κ2 {s_{i + 1}} - {s_i} = {C_2}{N_{gr2}}{{{{\left( {1 - {{{N_{i + 1}}} \over {{N_{gr2}}}}} \right)}^{ - {\kappa _2}}} - {{\left( {1 - {{{N_i}} \over {{N_{gr2}}}}} \right)}^{ - {\kappa _2}}}} \over {{\kappa _2}}} After reorganizing we can use the least square method: (8) δi2=(Yi−AXi)2 \delta _i^2 = {({Y_i} - A{X_i})^2} where: (9) Yi=si+1−si {Y_i} = {s_{i + 1}} - {s_i} (10) A=C2Ngr2→C2=ANgr2 A = {C_2}{N_{gr2}} \to {C_2} = {A \over {{N_{gr2}}}} (11) Xi=1−Ni+1Ngr2−κ2−1−NiNgr2−κ2κ2 {X_i} = {{{{\left( {1 - {{{N_{i + 1}}} \over {{N_{gr2}}}}} \right)}^{ - {\kappa _2}}} - {{\left( {1 - {{{N_i}} \over {{N_{gr2}}}}} \right)}^{ - {\kappa _2}}}} \over {{\kappa _2}}} (12) Yi=AXi→A=∑(XiYi)∑(Xi2) {Y_i} = A{X_i} \to A = {{\sum {({X_i}{Y_i})} } \over {\sum {(X_i^2)} }}

The analysis previously described by the authors [11] shows that the most important factor with regard to pile safety which is pile maximum bearing capacity can be accurately determined from different parts of the data set. For example by only using the first part of data set {s_i,N_i} obtained in case of static load test conducted only in a limited range of loads which is a more common practice. In the case of κ₂ and C₂ conducted research shows that for the analyzed piles most accurate results were obtained by using only the middle part of the data set {s_i,N_i}. Most likely values for small load-settlement have the higher impact of measurement inaccuracy and values close to the vertical asymptote tends to disrupt statistical calculations. Above in Fig. 2 and Fig. 3 we can see an example of the results of the calculation of M-K parameters for one of the analyzed piles and in Tab. 2 we can see a comparison between calculated and measured values of pile settlement [1].

Figure 2.

Figure 3.

M-K method has been chosen as a basis of the proposed method of load-settlement curve conversion due to the relations between parameters describing curves N₁(s), N₂(s), T(s). Those relation has been the subject of extensive work conducted at the Geotechnical Department of West Pomeranian University of Technology in Szczecin [1, 13, 14] and can be written as follows: (13) C1C2=(κ2+1)2 {{{C_1}} \over {{C_2}}} = {({\kappa _2} + 1)^2} (14) κ1=ln1+κ2 {\kappa _1} = {\rm{\;ln\;}}\left( {1 + {\kappa _2}} \right) (15) Ngr1=Ngr2⋅2−κ2 {N_{gr1}} = {N_{gr2}} \cdot {2^{\left( { - {\kappa _2}} \right)}}

The idea of curve conversion can be schematically described, as shown in Fig. 4.

Figure 4.

In previously published paper [15] authors have described the method of Q-s curve conversion based on linear elasticity theory. However, due to the method being based on linear elasticity theory, it applies only to a small range of loads for which load-settlement relation can approximate as a straight line. In practice, most pile loads are in the non-linear part of the load-settlement curve. Other research conducted by the authors published in [1, 16], indicate the possibility of conversion in a full range of loads both in the linear and non-linear part of the Q-s curve.

In order to the describe static load test curve for any given pile firstly it is necessary to determine the parameters of the curve concerning total resistance, which means κ₂, N_gr2, C₂. As has been previously stated in most practical cases those parameters can be established using the least square method. After that it is possible to determine the parameters of the resistance curve N₁(s) from formulas (13) – (15) and lastly from equation (6) we obtain the curve of skin resistance T(s). It is also possible to obtain the relation between CPTU measurements and M-K curve parameters. Relation between κ₂ and CPTU cone resistance has been described by Meyer and Siemaszko [16] and can be written as follows: (16) κ2=4β20.86HD0.785qc¯qb11+14qb1/33/5 {\kappa _2} = {\left[ {{{4\beta } \over {20.86}}{{\left( {{H \over D}} \right)}^{0.785}} {{{\overline{q_c}} \over {{q_b}}}} {1 \over {1 + {1 \over 4}q_b^{1/3}}}} \right]^{3/5}} where: q̅

average resistance along the pile skin [MPa];

Results of static load tests conducted in a full range of loads that have been conducted by Energopol Szczecin S.A. [1] were used to determine the relation that can be used for conversion of a maximum bearing capacity value in case of changed length and diameter: (17) Ngr2=ξHDηqbD2 {N_{gr2}} = \xi {\left( {{H \over D}} \right)^\eta }{q_b}{D^2} ξ and η parameters are determined using statistical mathematics, in the analyzed case they were equal to 4.439 · 10⁻³ and 1.757 respectively.

In Tab. 3 we can see comparison between calculated from equation (17) N_gr2,calc and measured values N_gr2,test of plie maximum bearing capacity [1].

It is also possible to inverse above-mentioned equation (17) and in the case of a known value of the maximum bearing capacity of a pile calculate cone resistance at the level of its base.

The method of static load curve conversion is based on the assumption that the change in pile length or diameter would result in a change of M-K parameters, which represents aggregated soil conditions. Firstly, we has to determine parameters describing pile-soil interaction for a pile that has been tested. As previously described, this can be achieved by using the least square method. Those parameters are called state”0” of a pile which represents the pile for which static load tests were conducted. Using the method of conversion, we obtain state “1” of a pile which represents the pile after curve conversion with calculated parameters. Parameters of state “0” are going to be described by upper index 0 and state “1” with upper index 1.

Using previously mentioned research conducted by Meyer and Siemaszko [17] concerning calculating κ₂ directly from CPTU measurements we can obtain the relation between values κ₂ for piles with different lengths and diameters and in different soil conditions it is possible to write the following relation: (18) κ21=κ20D0H1D1H00.785q¯c1qb11+0.25qb01/3q¯c1qb10+0.25qb11/33/5 \kappa _2^{\left( 1 \right)} = \kappa _2^{\left( 0 \right)}{\left[ {{{\left( {{{{D^{\left( 0 \right)}}{H^{\left( 1 \right)}}} \over {{D^{\left( 1 \right)}}{H^{\left( 0 \right)}}}}} \right)}^{0.785}}{{\bar q_c^{\left( 1 \right)}q_b^{\left( 1 \right)}1 + 0.25{{\left( {q_b^{\left( 0 \right)}} \right)}^{1/3}}} \over {\bar q_c^{\left( 1 \right)}q_{b1}^{\left( 0 \right)} + 0.25{{\left( {q_b^{\left( 1 \right)}} \right)}^{1/3}}}}} \right]^{3/5}}

And in the case when soil conditions do not change, it is possible to simplify formula (18): (19) κ21=κ20D0D1H1H00.471 \kappa _2^{\left( 1 \right)} = \kappa _2^{\left( 0 \right)}{\left( {{{{D^{\left( 0 \right)}}} \over {{D^{\left( 1 \right)}}}}{{{H^{\left( 1 \right)}}} \over {{H^{\left( 0 \right)}}}}} \right)^{0.471}}

Above mentioned equation holds true in the case of the plasticized zone under the pile toe being, as shown in Fig. 5.

Figure 5.

Figure 6.

Pile-soil interaction as well as stresses distribution under the pile base have been the subject of many researches described among others by Dembicki et al. [18] which shows that stresses under the pile base have also an impact along the pile skin. As we can see in Fig. 4 stresses under pile base can have also an impact along the pile skin.

Analysis of those relations has led to the formulation of an equation that can be used for the conversion C₂ parameter [1]: (20) c21=c20D0D11+κ2031+κ213 c_2^{\left( 1 \right)} = c_2^{\left( 0 \right)}{{{D^{\left( 0 \right)}}} \over {{D^{\left( 1 \right)}}}}{{{{\left( {1 + \kappa _2^{\left( 0 \right)}} \right)}^3}} \over {{{\left( {1 + \kappa _2^{\left( 1 \right)}} \right)}^3}}}

Previously mentioned research conducted by the authors [1, 11,19] with regards to the maximum bearing capacity of a pile points to the relation between limit bearing capacity, pile dimensions and stresses at the pile base (17) can be rearranged in order to obtain a relation that can be used for conversion in a full range of loads: (21) Ngr21=Ngr20H1H01,757D1D00,243qb1qb0 N_{gr2}^{\left( 1 \right)} = N_{gr2}^{\left( 0 \right)}{\left( {{{{H^{\left( 1 \right)}}} \over {{H^{\left( 0 \right)}}}}} \right)^{1,757}}{\left( {{{{D^{\left( 1 \right)}}} \over {{D^{\left( 0 \right)}}}}} \right)^{0,243}}{{q_b^{\left( 1 \right)}} \over {q_b^{\left( 0 \right)}}} And in the case when soil conditions do not change, we can write: (22) Ngr21=Ngr20H1H01,757D1D00,243 N_{gr2}^{\left( 1 \right)} = N_{gr2}^{\left( 0 \right)}{\left( {{{{H^{\left( 1 \right)}}} \over {{H^{\left( 0 \right)}}}}} \right)^{1,757}}{\left( {{{{D^{\left( 1 \right)}}} \over {{D^{\left( 0 \right)}}}}} \right)^{0,243}}

Parameters that describe the curve of pile base resistance in state “1” can be calculated directly from formulas (13), (14) and (15). Lastly, after obtaining a full description of curves N₁(s) and N₂(s) we can obtain curve T(s) from equation (6).

For any given data set of static load test results {s_i,N_i} it is possible to obtain parameters describing the M-K curve for the tested pile, from the method described previously in this paper. After that, we can calculate values of M K parameters that will correspond with new pile length and new diameter from equations (18), (19), (20), (21) and (22). For example, if we assume in state “0” parameters of pile 31-10L as shown in tab. 1 that describes load-settlement relation. It is now possible to determine from equations (13), (14) and (15) parameters of base resistance curve N₁(s), which would also allow us to for a description of skin resistance, as a difference between total resistance and base resistance. (23) c10=c20⋅(1+κ20)2=7,7⋅10−4⋅(1+1,4)2= =4.44⋅10−3mm \matrix{ {c_1^{\left( 0 \right)} = c_2^{\left( 0 \right)} \cdot {{(1 + \kappa _2^{\left( 0 \right)})}^2} = 7,7 \cdot {{10}^{ - 4}} \cdot {{(1 + 1,4)}^2} = } \hfill \cr {\;\;\;\;\;\; = 4.44 \cdot {{10}^{ - 3}}mm} \hfill \cr } (24) κ10=ln1+κ20=ln1+1,4=0.88 \kappa _1^{\left( 0 \right)} = {\rm{\;ln\;}}\left( {1 + \kappa _2^{\left( 0 \right)}} \right) = {\rm{\;ln\;}}\left( {1 + 1,4} \right) = 0.88 (25) Ngr10=Ngr202κ20=870021.4=3297kN N_{gr1}^{\left( 0 \right)} = {{N_{gr2}^{\left( 0 \right)}} \over {{2^{\kappa _2^{\left( 0 \right)}}}}} = {{8700} \over {{2^{1.4}}}} = 3297\;kN

In state”1” it is assumed that the length of a pile is changed from 27.5 m to 15 m and its diameter from 2 m to 1.0 m and that the soil condition does not change. From equations (19), (20) and (22) it is possible to determine parameters describing the M-K curve for new pile conversion assumed after conversion. (26) Ngr21=8700⋅1527.51,757⋅1,020,243=2534kN N_{gr2}^{^{\left( 1 \right)}} = 8700 \cdot {\left( {{{15} \over {27.5}}} \right)^{1,757}} \cdot {\left( {{{1,0} \over 2}} \right)^{0,243}} = 253\;4kN (27) κ21=1,4⋅2.01.0⋅1527.50.471=1.46 \kappa _2^{\left( 1 \right)} = 1,4 \cdot {\left( {{{2.0} \over {1.0}} \cdot {{15} \over {27.5}}} \right)^{0.471}} = 1.46 (28) c21=7.7⋅10−4⋅2.01.0⋅(1+1.4)3(1+1.46)3=1.43⋅10−3 c_2^{\left( 1 \right)} = 7.7 \cdot {10^{ - 4}} \cdot {{2.0} \over {1.0}} \cdot {{{{(1 + 1.4)}^3}} \over {{{(1 + 1.46)}^3}}} = 1.43 \cdot {10^{ - 3}} Analogically like in state “0” it is possible to determine parameters describing base resistance curve N₁(s) from equations (13), (14) and (15). (29) c11=c21⋅1+κ212=1,43⋅10−3⋅ (1+1,46)2 =8,66⋅10−3mm \matrix{ {c_1^{\left( 1 \right)} = c_2^{\left( 1 \right)} \cdot {{\left( {1 + \kappa _2^{\left( 1 \right)}} \right)}^2} = 1,43 \cdot {{10}^{ - 3}} \cdot {\rm{ }}{{(1 + 1,46)}^2}} \hfill \cr {\;\;\;\;\;\; = 8,66 \cdot {{10}^{ - 3}}mm} \hfill \cr } (30) κ11=ln1+κ21=ln1+1,46=0,90 \kappa _1^{\left( 1 \right)} = {\rm{\;ln\;}}\left( {1 + \kappa _2^{\left( 1 \right)}} \right) = {\rm{\;ln\;}}\left( {1 + 1,46} \right) = 0,90 (31) Ngr11=Ngr212κ21=253421,46=922 kN N_{gr1}^{\left( 1 \right)} = {{N_{gr2}^{\left( 1 \right)}} \over {{2^{\kappa _2^{\left( 1 \right)}}}}} = {{2534} \over {{2^{1,46}}}} = 922\;kN After conducting above mentioned-calculation it is possible to analyse the change in load-settlement relation described using a continuous curve as shown in Fig. 7

Figure 7.

Another practical application of the proposed is an analysis of a change in safety factor SF defined as the ratio between maximum bearing capacity of a pile and its load N₂: (32) SF=Ngr2N2 SF = {{{N_{gr2}}} \over {{N_2}}}

If we assume the recommended value of settlement s_rec. we can easily calculate the value of safety factor SF for any given pile using the M-K cure. It is also possible to analyze how this value change when we changed pile dimensions using the proposed method of conversion of M-K curve parameters. The equation describing the relation between the safety factor defined in equation (32) and recommended value of the settlement can be written as follows: (33) SFsrec=−srec⋅κ2C2⋅Ngr2+1−1κ2−1−1 SF\left( {{s_{rec}}} \right) = {\left\{ { - \left[ {{{\left( {{{{s_{rec}} \cdot {\kappa _2}} \over {{C_2} \cdot {N_{gr2}}}} + 1} \right)}^{ - {1 \over {{\kappa _2}}}}}} \right] - 1} \right\}^{ - 1}}

Using parameters determined in equations (29), (30) and (31) in the case of conversion of pile 31-10L it is possible to describe the change of safety factor with regards to assumed values o recommended settlement for pile dimensions before and after curve conversion as shown in Fig. 8.

Figure 8.

For any given value of the recommended settlement, it is possible to determine the safety factor SF as described in equation (33). The results of the comparison are stated in Tab. 3.

As we can see both in Fig. 8 and Tab. 4 difference between safety factor became smaller for increased values of recommended settlement Such analysis can be used in pile optimization in order to find optimal pile dimensions that would satisfy recommended conditions in structural design with regard to pile foundations.

The authors have presented a method of static load test curve conversion in a full range of loads. The analysis was based on the Meyer-Kowalow method of static load test curve interpretation [8] The proposed method applies for a total resistance curve conversion N₂(s), as well as base resistance curve N₁(s) and skin resistance curve T(s). This means that after conducting static load tests for a given length and diameter of a pile, it is possible to describe pile-soil interaction for a changed diameter and length, without the necessity of repeating the static load tests. The equation describing the relation between the ultimate bearing capacity of the pile and its geometrical parameters [1, 11, 18] was determined on the basis of the analysis of static pile loads carried out in the full load range by Energopol Szczecin S.A [1] for CFA piles bored mostly in loam soil conditions. Empirically determined constant values of ξ and η may change in different conditions, which requires an analysis of the results of the static load test for piles of different diameters and lengths made in different soil conditions. Taking this into account will be the subject of further research. An important parameter introduced by the M-K method is the value of κ₂ that defines the relations between the skin and base resistance of the pile. The equation used for conversion is based on the relation between the value of κ₂ and CPTU measurements that also considers pile length and diameter, as described by Meyer and Siemaszko [17]. An important aspect of practical application is also the ability to analyze changes in values of safety factors defined as a ratio between pile load and its maximum bearing capacity. Analyzing safety factors as well as changes in pile-soil interaction described by the M-K curve could be used in a process of pile optimization. Further research should expand analysis concerning the impact of pile length and diameter on safety factors, as it is the most important parameter regarding proper construction design. Practical application of the proposed method of conversion requires full-scale confirmation on a large number of piles which will be the subject of further research.