Extreme scour effects on the buckling of bridge piles considering the stress history of soft clay

Abstract

Scour is a natural phenomenon caused by the erosion or removal of streambed or bank material from bridge foundations due to flowing water. As the buckling capacity of bridge piles varies inversely with the square of the unsupported pile height, an extreme scour at the pile caused by floodwater could result in a buckling failure of the pile and collapse of the bridge. The common way to analyze the scour-affected buckling stability of bridge piles is to remove the scoured soil layers while possible changes in stress history of the remaining soils are ignored. In reality, however, the remaining soils undergo an unloading process due to scour, and its overconsolidation ratios are increased. In this study, an analytical model with modified lateral subgrade modulus is presented to investigate the extreme scour effect on the buckling of bridge piles in soft clay considering the stress history of the remaining soils. A case study is used to compare the calculated results by considering and ignoring the stress history effect. The results show that ignoring the stress history of the soft clay will overestimate the buckling capacity of bridge pile foundation under scour.

1 Introduction

The US Federal Highway Administration (FHWA) defines scour as “erosion or removal of streambed or bank material from bridge foundations due to flowing water, usually considered as long-term bed degradation, contraction, and local scour” (Richardson and Davis 2001). It should be noted that an extreme scour at the bridge pile caused by floodwaters could result in a buckling failure of the pile and collapse of the bridge, as the buckling capacity of the pile varies inversely with the square of the unsupported pile. Therefore, extreme scour at bridge piles is a potential safety hazard to the travelling public and has caused many casualties as reported by Su and Lu (2013) and Hong et al. (2012). Although several approaches are currently available for analyzing buckling of axially loaded piles (Golder and Skipp 1957; Brandtzaeg and Harboe 1957; Davisson 1963; Davisson and Robinson 1965; Reddy and Valsangkar 1970; Prakash 1987; Gabr et al. 1997; Heelis et al. 2004), only a few studies have been conducted to investigate extreme scour effects on the buckling stability of bridge piles. Hughes et al. (2007) presented an investigation of the effects of the extreme scour and the soil subgrade modulus on the buckling capacity of a bridge pile bent. Schambeau et al. (2014) developed a screening tool to evaluate the buckling failure of the bridge timber pile bent in extreme scour events. Lin et al. (2012) employed the integrated analysis process to evaluate the lateral response and the buckling capacity of a bridge pile group. However, the effect of soil stress history after scour was not considered in these studies. In reality, the remaining soils around a foundation after scour event undergo an unloading process (Brown and Castelli 2010; Lin et al. 2010, 2014). In other words, the overconsolidation ratios (OCRs) of the remaining soils are increased due to scour. Moreover, as the OCR is the major factor influencing the ultimate soil resistance of soft clay (Lin et al. 2014), the effect of stress history of soft clay on bridge pile buckling under extreme scour conditions should be further examined.

In this paper, a variational approach for pile buckling in uniform soil (Gabr et al. 1997) was improved to evaluate the extreme scour effect on the buckling of bridge piles in layered soft clay. In this analytical model, the lateral subgrade modulus that was correlated to the undrained shear strength of soft clay was modified to account for the stress history effect after scour. The procedures of the modification are similar to those presented by Lin et al. (2014), while the latter work focused on the scour effect on the behavior of laterally loaded piles in soft clay. In general, the present work has been developed on the basis of the previous studies (Gabr et al. 1997; Lin et al. 2014) to improve the analyses for bridge pile buckling under different scour conditions. The analysis procedures from the previous studies are firstly reviewed. An improved analytical approach is then developed for the analysis of pile buckling considering stress history of soft clay, which has not been reported before. With the improved approach, a case of bridge pile buckling under scour conditions is studied. The calculated results considering and ignoring the stress history effect are compared and discussed.

2 Analysis method

The pinned and 50 % fixity boundary conditions at the pile head, which were employed by Hughes et al. (2007) for bridge pile bent, are used in this study for the demonstration purposes as shown in Fig. 1. It should be noted that the only restraint for either lateral translation or rotation at the embedded end of the pile is provided by the soil (Heelis et al. 2004), so that for the pile embedded in the soft clay, “free- end” condition is applied at the pile tip to give a reasonably conservative buckling capacity for a safe design. Furthermore, the improved approach can also be used for other boundary conditions, e.g., the fixed-sway top-free tip boundary condition. In Fig. 1, the coordinates system for length and deflection is also presented, where L and h are the total length and embedded length of the pile, respectively; S _d is the scour depth at the bridge pile; P is the axial load acting at the pile head; q is the lateral soil reaction per unit length of pile (kN/m).

Fig. 1

Buckling modes of bridge bent piles in a longitudinal direction and b transverse direction under extreme scour

In this study, the pile is assumed to be embedded in layered soft clay modeled by an elastic Winkler foundation. Hence, the lateral soil reaction q for the layered soil system (Fig. 2) can be expressed as

$$q(z) = k_{i} y(z)$$

(1)

where k _i is the lateral subgrade modulus for each particular layer, such as the ith layer (Fig. 2), and is described in a subsequent section of the paper (note that k _i has units of force/length² and is computed by multiplying the coefficient of subgrade reaction k _hi , force/length³, by the pile diameter); y is the lateral deflection due to pile buckling; z is the distance from the pile tip. In addition, the effect of the skin friction is neglected here, and this simplification is acceptable for general cases (Reddy and Valsangkar 1970; Budkowska and Szymczak 1997).

Fig. 2

Notations for soil–pile system

2.1 Potential energy of the pile–soil system

The total potential energy π _p of the scoured single pile subjected to axial load, as depicted in Figs. 1 and 2, can be written as

$$\pi_{\text{p}} = \frac{\text{EI}}{2}\int\limits_{0}^{L} {\left( {y^{{\prime \prime }} } \right)^{2} {\text{d}}z} + \frac{1}{2}\sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {q(z)y{\text{d}}z} } - \frac{P}{2}\int\limits_{0}^{L} {\left( {y^{{\prime }} } \right)^{2} {\text{d}}z}$$

(2)

In this equation, the first and second terms on the right-hand side give the strain energy of the system due to the bending of the pile and elastic deformation of soil, where EI is the flexural stiffness of pile; h _i is the elevation of the ith soil layer from the pile tip (h ₀ = 0) (Fig. 2); m is the total number of the soil layers. The third term on the right-hand side of Eq. (2) corresponds to the work done by the external axial load P at the pile head. Substituting Eq. (1) into (2) yields

$$\pi_{\text{p}} = \frac{\text{EI}}{2}\int\limits_{0}^{L} {\left( {y^{{\prime \prime }} } \right)^{2} {\text{d}}z} + \frac{1}{2}\sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} y^{2} {\text{d}}z} } - \frac{1}{2}\int\limits_{0}^{L} {P\left( {y^{\prime } } \right)^{2} {\text{d}}z}$$

(3)

2.2 Deflection functions

Two boundary condition cases, pinned top-free tip (0 % rotational fixity with translation restrained at the pile top) and fixed top-free tip (100 % rotational fixity with translation restrained at the pile top), are selected for modeling the two typical buckling modes as described previously. Note that for the bent pile buckling in the transverse direction, its top fixity is between a pinned and a fixed condition. For this case, Hughes et al. (2007) averaged the results calculated by an analytical method (Granholm 1929) for the pinned and fixed condition, and gave a reasonably similar buckling capacity that calculated by the approximate equations (such as P _cr = 2π ²EI/L ²). Therefore, when elastic buckling occurs, the averages are also used to estimate the partial fixity condition (50 % rotational fixity with translation restrained at the pile top) in this study as shown in Fig. 1b. Additionally, another boundary condition case (fixed-sway top-free tip) can also be adopted to model the possible buckling mode as described previously. The deflection functions for the three boundary conditions, as Gabr et al. (1997) proposed, are chosen using the Rayleigh–Ritz method to satisfy geometric requirements. In these deflection functions, considering the pinned top-free tip boundary condition, we have

$$y = c_{0} \left( {1 - \frac{z}{L}} \right) + \sum\limits_{n = 1}^{\infty } {c_{n} } \sin \frac{n\pi }{L}z$$

(4)

Similarly, the fixed top-free tip boundary condition can be expressed as

$$y = \sum\limits_{n = 1}^{\infty } {c_{n} } \left[ {1 - \cos \frac{2n - 1}{2L}\pi (L - z)} \right]$$

(5)

Furthermore, the fixed-sway top-free tip boundary condition can be expressed as

$$y = c_{0} + \sum\limits_{n = 1}^{\infty } {c_{n} } \sin \frac{(2n - 1)\pi }{2L}z$$

(6)

where c _n is the unknown constant, and these deflection functions are validated in the subsequent sections.

2.3 Minimization of the potential energy

The principle of minimum potential energy requires that π _p be an extremum with respect to the admissible deflection functions characterized by the undetermined coefficient. Hence,

$$\frac{{\partial \pi_{\text{p}} }}{{\partial c_{\text{j}} }} = 0\quad \left( {j = 1, 2, 3, \ldots , n} \right)$$

(7)

where c _j and n denote the undetermined coefficients of deflection functions and the number of half-wave of deflection functions, respectively. Substituting the total potential energy given by Eq. (3) into (7) yields

$${\text{EI}}\int\limits_{0}^{L} {y^{{\prime \prime }} \frac{{\partial y^{{\prime \prime }} }}{{\partial c_{j} }}{\text{d}}z} + \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} y\frac{\partial y}{{\partial c_{j} }}{\text{d}}z} } - P\int\limits_{0}^{L} {y^{{\prime }} \frac{{\partial y^{{\prime }} }}{{\partial c_{j} }}{\text{d}}z = 0}$$

(8)

2.4 Lateral subgrade modulus for soft clay

For clays, empirical correlations with other soil properties are usually used to determine the lateral subgrade modulus. For example, Terzaghi (1955) recommended that the lateral subgrade modulus k of 7.16, 14.3, and 28.6 MPa could be used for stiff (c _u = 50–100 kPa), very stiff (c _u = 100–200 kPa), and hard (c _u = 200 kPa) overconsolidated clay (note that c _u is the undrained shear strength).

For layered soft clay, the lateral subgrade modulus k _i of each particular layer can be assumed to be constant with depth, and only a limited number of values of the lateral subgrade moduli have been related to c _u. Nevertheless, two alternative empirical correlations for the lateral subgrade modulus are adopted from Poulos and Davis (1980) and Davisson (1970), respectively; the former one ranged between 80c _u and 320c _u (the lower values tend to be associated with very soft clays and the higher values with stiff clays) and gave a median value of 120c _u for design, while the latter suggested a more conservative and more satisfactory value of 67c _u.

2.5 Modified lateral subgrade modulus after scour

The empirical correlations (Poulos and Davis 1980; Davisson 1970) indicate that the key soil parameter for the lateral subgrade modulus of soft clay is undrained shear strength. To address the stress history effect on the buckling stability of scoured piles, this key parameter after scour should be examined. As the normally consolidated soil after scour becomes overconsolidated, the undrained shear strengths after scour can be evaluated using Eqs. (9) and (10), which were derived based on the modified Cam-clay model (Kulhawy and Mayne 1990; Muir Wood 1990)

$$\frac{{\left( {{{c_{\text{u}} } \mathord{\left/ {\vphantom {{c_{\text{u}} } {\sigma_{\text{v}}^{\prime } }}} \right. \kern-0pt} {\sigma_{\text{v}}^{\prime } }}} \right)_{\text{sc}} }}{{\left( {{{c_{\text{u}} } \mathord{\left/ {\vphantom {{c_{\text{u}} } {\sigma_{\text{v}}^{\prime } }}} \right. \kern-0pt} {\sigma_{\text{v}}^{\prime } }}} \right)_{\text{int}} }} = \frac{{\left( {{{c_{\text{u}} } \mathord{\left/ {\vphantom {{c_{\text{u}} } {\sigma_{\text{v}}^{\prime } }}} \right. \kern-0pt} {\sigma_{\text{v}}^{\prime } }}} \right)_{\text{OC}} }}{{\left( {{{c_{\text{u}} } \mathord{\left/ {\vphantom {{c_{\text{u}} } {\sigma_{\text{v}}^{\prime } }}} \right. \kern-0pt} {\sigma_{\text{v}}^{\prime } }}} \right)_{\text{NC}} }} = {\text{OCR}}^{\varLambda }$$

(9)

$$\varLambda = \frac{\lambda - \kappa }{\lambda } = 1 - \frac{{C_{\text{ur}} }}{{C_{\text{c}} }}\quad \left( {{\text{approximately}}\;0.8} \right)$$

(10)

where ${\sigma_{\text{v}}^{\prime }}$ = vertical effective stress (note that ${\sigma_{\text{v}}^{\prime }}$ has units of force/length² and is computed by multiplying the effective unit weight, force/length³, by the depth of the point of interest measured from the mud line); λ and κ = compression and swelling indexes from isotropic consolidation tests; and C _c and C _ur = compression and swelling indexes obtained from the oedometer tests.

To facilitate the use of Eqs. (9) and (10), another parameter (the effective unit weight) after scour should be determined by Eq. (11) as Lin et al. (2014) presented

$$\gamma_{\text{sc}}^{{\prime }} = \frac{{1 + e_{\text{int}} }}{{1 + e_{\text{int}} + C_{\text{ur}} \log \left[ {\frac{{\gamma_{\text{int}}^{{\prime }} (z \, + \, S_{\text{d}} )}}{{\gamma_{\text{sc}}^{\prime } z}}} \right]}}\gamma_{\text{int}}^{\prime }$$

(11)

where $\gamma_{\text{sc}}^{{\prime }}$ = soil effective unit weight after scour (kN/m³); $\gamma_{\text{int}}^{\prime }$ = soil effective unit weight before scour (kN/m³); z = depth of the point of interest measured from the mud line after scour (m); S _d = scour depth (m); and $e_{\text{int}}$ = soil void ratio before scour, which can be determined using Eq. (12) if the soil is saturated

$$e_{\text{int}} = \frac{{\left( {\gamma_{\text{w}} + \gamma_{\text{int}}^{\prime } } \right)w_{\text{int}} }}{{\gamma_{\text{w}} - \gamma_{\text{int}}^{\prime } w_{\text{int}} }}$$

(12)

where γ _w = unit weight of water (kN/m³); and $w_{\text{int}}$ = soil moisture content.

By substituting Eq. (9) into the empirical correlations (Davisson 1970), the lateral subgrade modulus of the soft clay after scour, k _sc, can be rewritten as

$$k_{\text{sc}} = 67({\text{OCR}})^{\varLambda } \gamma_{\text{sc}}^{\prime } z\left[ {\frac{{(C_{\text{u}} )_{\text{int}} }}{{\gamma_{\text{int}}^{\prime } (z + S_{\text{d}} )}}} \right]$$

(13)

where $\gamma_{\text{sc}}^{{\prime }}$ can be obtained by solving Eq. (11) through iterations, and the OCR can be determined using Eq. (14)

$${\text{OCR}} = \frac{{\gamma_{\text{int}}^{\prime } (z + S_{\text{d}} )}}{{\gamma_{\text{sc}}^{\prime } z}}$$

(14)

Once the soil parameters before scour (e.g., moisture content or specific gravity, effective unit weight, and undrained shear strength) and the scour depth are known, the lateral subgrade modulus after scour can be determined by applying Eqs. (11) and (13).

2.6 Solution for critical buckling load

Substituting the deflection functions [Eqs. (4), (5), and (6), respectively] and the modified lateral subgrade modulus into Eq. (8) and performing the integration, a set of homogeneous linear equations in terms of c _j can be obtained. This system possesses nonzero solutions only if the determinant of the coefficient matrix is equal to zero. The determinants of the coefficient matrices for the buckling modes (see “Appendix 1”) have been coded into MATLAB. The smallest root of the solution provides the critical buckling load, P _cr, for the model.

3 Verification

To verify the reasonableness and accuracy of the analytical model before the case study, the results from the present method are compared with the results that have been reported in the published literature. The published results include a model test (Gouvenot 1975) under an initial condition (before scour) and a sensitivity study on the pile buckling capacity to lateral subgrade modulus (Hughes et al. 2007) under a scour condition. The number of terms (n values in the deflection functions) used for all calculations in this paper is 50. Details of the verification are provided below.

3.1 Model test (Gouvenot 1975)

Gouvenot (1975) instrumented and load-tested three piles, two in peat and the third in soft clay. The concrete piles were 60 mm in diameter with a central core of 20-mm-diameter steel reinforcement, giving a flexural stiffness, EI, of 8.0 × 10⁹ N mm². They were 4,000 mm in length. Considering the pinned top-pinned tip boundary condition (see “Appendix 2”) as was used by Gouvenot (based on Hetenyi 1946) and the pinned top-free tip boundary condition, the experimental buckling loads for the three piles are compared with the theoretical ones calculated by the present method in Table 1. It can be seen from the table that the theoretical solution with a pinned embedded end condition considerably overestimated the actual buckling load, while that with a free embedded end condition underestimated the actual load. However, the difference between the predicted and measured results is acceptable, and the present model with free embedded end may give a conservative buckling capacity.

Table 1 Theoretical values compared with Gouvenot’s experimental buckling loads (Gouvenot 1975)

3.2 Sensitivity study (Hughes et al. 2007)

Hughes et al. (2007) developed an analytical model to investigate the sensitivity of bridge bent pile buckling capacity to lateral subgrade modulus under extreme scour condition. ALDOT’s most widely used bent piles, i.e., HP10 × 42 (EI _x  = 17,480 kNm², EI _y  = 5,965 kNm²) with various lateral subgrade modulus, are used in this sensitivity analysis as shown in Fig. 3. Buckling in the bridge longitudinal direction (about pile x–x axis) and transverse direction (about pile y–y axis) is considered. The corresponding critical buckling loads calculated by Hughes et al. (2007) are compared with the results from present method in Figs. 4 and 5, respectively. A reasonable agreement can be seen in these figures, although an acceptable difference (within 1.5 %) is found in Fig. 5. Moreover, it should be noted that the critical buckling loads increase with the increasing lateral subgrade modulus, especially for soft clay with low lateral subgrade modulus. Therefore, the stress history effect (change in the lateral subgrade modulus) on the buckling stability of scoured piles will be further investigated through a case study in the following section.

Fig. 3

Pile–soil conditions for sensitivity analysis

Fig. 4

Buckling load P _cr versus k for bridge pile buckling in longitudinal direction

Fig. 5

Buckling load P _cr versus k for bridge pile buckling in transverse direction

4 Case study

4.1 Site conditions

To investigate the effect of soil stress history on the pile buckling under extreme scour, a site condition near Lake Austin, Texas (Matlock 1970), is used in this study. The soil is a fat clay (unified soil classification system classification of CH) with its properties summarized in Table 2. Some of these properties are estimated based on the empirical relationships for demonstration purposes. They can be more accurately determined by laboratory or in situ tests. The compression index, C _c, is estimated based on the moisture content (Djoenaidi 1985), and the swelling index, C _ur, is taken as (1/5) C _c (Kulhawy and Mayne 1990). The undrained shear strengths at different depths are measured with a field vane (Reese and Van Impe 2001), and the results are plotted in Fig. 6. According to the undrained shear strengths shown in Fig. 6, layered soft clay can be assumed for demonstration purpose in this study. The soil initial void ratio, e _int, is estimated to be 1.60 in Eq. (12) based on the given $\gamma_{\text{int}}^{\prime }$ and $w_{\text{int}}$. A typical pile is used for demonstration purpose in this study with its parameters provided in Table 3. The water table is kept above the ground surface. Four scour depths of 5D, 10D, 15D, and 20D (where D is the pile diameter) and two typical end conditions (as described previously) are investigated as shown in Fig. 7.

Table 2 Properties of soft clay

Fig. 6

Distribution of undrained shear strength of soft clay (data from Reese and Van Impe 2001)

Table 3 Pile parameters

Fig. 7

Soil and pile profile for studies under scour conditions

4.2 Modified lateral subgrade modulus after scour

Based on the procedures discussed earlier, the properties of the remaining soft clay and the modified lateral subgrade modulus (Davisson 1970) are determined at scour depths of 5D, 10D, 15D, and 20D as presented in Tables 4, 5, 6, and 7. The results show that the lateral subgrade modulus after scour decreases significantly when compared with those before scour (especially for extreme scour condition), while the changes in the effective unit weight values are negligible during scour. It is worth noting that the remaining soils undergo an unloading process due to scour, and thus, high OCR can be obtained near the soil surface. To avoid a passive earth failure, however, a limited OCR should be determined by using Eq. (15) (Mayne and Kulhawy 1982), which is based on the K _o–OCR relationship in soils

$${\text{OCR}}_{\text{limit}} = \left[ {\frac{{1 + \sin \phi^{\prime}}}{{\left( {1 - \sin \phi^{\prime}} \right)^{2} }}} \right]^{{\left( {{1 \mathord{\left/ {\vphantom {1 {\sin \phi^{\prime}}}} \right. \kern-0pt} {\sin \phi^{\prime}}}} \right)}}$$

(15)

where $\phi ^{\prime}$ is the effective friction angle of the clay under a drained condition. The limited OCR of 27 is thus calculated by assuming a typical friction angle of a soft clay ($\phi ^{\prime}$ = 20°) for this case study. Accordingly, the critical depth where the limited OCR is reached, i.e., above which the lateral subgrade modulus is not valid, is calculated to be 0.06, 0.13, 0.20, and 0.27 m, respectively, for the scour depth of 5D, 10D, 15D, and 20D.

Table 4 Soil properties and modified lateral subgrade modulus considering stress history effect (S _d = 5D)

Table 5 Soil properties and modified lateral subgrade modulus considering stress history effect (S _d = 10D)

Table 6 Soil properties and modified lateral subgrade modulus considering stress history effect (S _d = 15D)

Table 7 Soil properties and modified lateral subgrade modulus considering stress history effect (S _d = 20D)

4.3 Critical buckling load

By using the present model, the critical buckling loads are calculated with the unmodified and modified lateral subgrade modulus as shown in Figs. 8 and 9 (note that the buckling critical load, P _cr, is expressed in dimensionless form as a ratio of the Euler load P _E = π² EI/L ²). It can be seen that the critical buckling loads decrease significantly with an increase in the scour depth. Moreover, for the extreme scour condition, considering the stress history effect results in 12–14 % lower critical buckling load compared with the case in which stress history effects are neglected.

Fig. 8

Effects of extreme scour and stress history of soft clay on the buckling load for pinned restraint at the pile head

Fig. 9

Effects of extreme scour and stress history of soft clay on the buckling load for 50 % fixity restraint at the pile head

5 Conclusions

An analytical model based on variational approach and modified lateral subgrade modulus is proposed in this paper to analyze the effect of extreme scour on the buckling stability of bridge piles in layered soft clay. This model can consider the stress history effect of the remaining layered soft clay after scour. A few conclusions can be drawn based on this study such as

• 1. Deflection functions can be used to approximate the buckling stability of a bridge pile under extreme scour condition. This has been verified by reasonable agreements of the present solution with the model tests and existing solutions.

• 2. Extreme scour causes a significant change in the undrained shear strength (related to the lateral subgrade modulus) of the remaining soils, and this change contributes to the difference in the pile buckling load when the stress history is considered or neglected.

• 3. When the scour depth is increased, stress history effect is found to result in a 12–14 % lower critical buckling load than the case in which stress history effects are neglected. Therefore, ignoring the stress history of the soft clay will overestimate the analysis of the buckling capacity of bridge pile foundation under scour.

Appendices

Appendix 1

For a pinned top-free tip boundary condition, the coefficient determinant is expressed as:

$$\Delta = \left| {\begin{array}{*{20}c} {b_{1,1} - {P \mathord{\left/ {\vphantom {P L}} \right. \kern-0pt} L}} & {b_{1,2} } & {b_{1,3} } & \cdots & {b_{1,n} } \\ {b_{2,1} } & {b_{2,2} - {{\pi^{2} P} \mathord{\left/ {\vphantom {{\pi^{2} P} {2L}}} \right. \kern-0pt} {2L}}} & {b_{2,3} } & \cdots & {b_{2,n} } \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {b_{n,1} } & {b_{n,2} } & {b_{n,3} } & \cdots & {b_{n,n} - {{(n - 1)^{2} \pi^{2} P} \mathord{\left/ {\vphantom {{(n - 1)^{2} \pi^{2} P} {2L}}} \right. \kern-0pt} {2L}}} \\ \end{array} } \right| = 0$$

(A1)

where b _i,j are the intermediate parameters for calculation and can further be described as:

$$b_{1,1} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \left( {1 - \frac{z}{L}} \right)}^{2} } {\text{d}}z$$

(A2)

$$b_{1,jj} = b_{jj,1} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \left( {1 - \frac{z}{L}} \right)} } \sin \frac{(jj - 1)\pi z}{L}{\text{d}}z\quad (jj = 2, \ldots ,\, n)$$

(A3)

$$b_{ii,jj} = \frac{{(jj - 1)^{4} {\text{EI}}\pi^{4} }}{{2L^{3} }} + \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \sin \frac{(ii - 1)\pi z}{L}} } \, \sin \frac{(jj - 1)\pi z}{L}{\text{d}}z\quad (ii = jj = 2, \ldots ,\,n)$$

(A4)

$$b_{ii,jj} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \sin \frac{(ii - 1)\pi z}{L}} } \, \sin \frac{(jj - 1)\pi z}{L}{\text{d}}z\quad (ii = 2, \ldots ,\,n) \, (jj = 2, \ldots ,\,n)$$

(A5)

For a fixed top-free tip boundary condition, the coefficient determinant is expressed as:

$$\Delta = \left| {\begin{array}{*{20}c} {b_{1,1} - {{\pi^{2} P} \mathord{\left/ {\vphantom {{\pi^{2} P} {8L}}} \right. \kern-0pt} {8L}}} & {b_{1,2} } & {b_{1,3} } & \cdots & {b_{1,n} } \\ {b_{2,1} } & {b_{2,2} - {{9\pi^{2} P} \mathord{\left/ {\vphantom {{9\pi^{2} P} {8L}}} \right. \kern-0pt} {8L}}} & {b_{2,3} } & \cdots & {b_{2,n} } \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {b_{n,1} } & {b_{n,2} } & {b_{n,3} } & \cdots & {b_{n,n} - {{(2n - 1)^{2} \pi^{2} P} \mathord{\left/ {\vphantom {{(2n - 1)^{2} \pi^{2} P} {8L}}} \right. \kern-0pt} {8L}}} \\ \end{array} } \right| = 0$$

(A6)

where b _i,j are the intermediate parameters for calculation and can further be described as:

$$b_{ii,jj} = \frac{{(2jj - 1)^{4} {\text{EI}}\pi^{4} }}{{32L^{3} }} + \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \left( {1 - \cos \frac{(2ii - 1)\pi (L - z)}{2L}} \right)} } \left( {1 - \cos \frac{(2jj - 1)\pi (L - z)}{2L}} \right){\text{d}}z\quad (ii = jj = 1, \ldots ,n)$$

(A7)

$$b_{ii,jj} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \left( {1 - \cos \frac{(2ii - 1)\pi (L - z)}{2L}} \right)} } \left( {1 - \cos \frac{(2jj - 1)\pi (L - z)}{2L}} \right){\text{d}}z\quad (ii = 1, \ldots ,n) \, (jj = 1, \ldots ,n)$$

(A8)

For a fixed-sway top-free tip boundary condition, the coefficient determinant is expressed as:

$$\Delta { = }\left| {\begin{array}{*{20}c} {b_{1,1} } & {b_{1,2} } & {b_{1,3} } & \cdots & {b_{1,n} } \\ {b_{2,1} } & {b_{2,2} - {{\pi^{2} P} \mathord{\left/ {\vphantom {{\pi^{2} P} {8L}}} \right. \kern-0pt} {8L}}} & {b_{2,3} } & \cdots & {b_{2,n} } \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {b_{n,1} } & {b_{n,2} } & {b_{n,3} } & \cdots & {b_{n,n} - {{(2n - 3)^{2} \pi^{2} P} \mathord{\left/ {\vphantom {{(2n - 3)^{2} \pi^{2} P} {8L}}} \right. \kern-0pt} {8L}}} \\ \end{array} } \right| = 0$$

(A9)

where b _i,j are the intermediate parameters for calculation and can further be described as:

$$b_{1,1} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} } } {\text{d}}z$$

(A10)

$$b_{1,jj} = b_{jj,1} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} } } \, \sin \frac{(2jj - 3)\pi z}{2L}{\text{d}}z\quad (jj = 2, \ldots ,n)$$

(A11)

$$b_{ii,jj} = \frac{{(2jj - 3)^{4} {\text{EI}}\pi^{4} }}{{32L^{3} }} + \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \, \sin \frac{(2ii - 3)\pi z}{2L}} } \sin \frac{(2jj - 3)\pi z}{2L}{\text{d}}z\quad (ii = jj = 2, \ldots , \, n)$$

(A12)

$$b_{ii,jj} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \sin \frac{(2ii - 3)\pi z}{2L}} } \sin \frac{(2jj - 3)\pi z}{2L}{\text{d}}z\quad (ii = 2, \ldots ,n) \, (jj = 2, \ldots , \, n)$$

(A13)

Appendix 2

The deflection function for the pinned top-pinned tip boundary condition is given as:

$$y = \sum\limits_{n = 1}^{\infty } {c_{n} } \sin \frac{n\pi }{L}z$$

(B1)

where c _n is the unknown constant.

Similarly, for the buckling load of a pile in a homogeneous soil with pinned top-pinned tip boundary condition, the coefficient determinant is expressed as follows:

$$\Delta { = }\left| {\begin{array}{*{20}c} {b_{1,1} - {{\pi^{2} P} \mathord{\left/ {\vphantom {{\pi^{2} P} {2L}}} \right. \kern-0pt} {2L}}} & {b_{1,2} } & {b_{1,3} } & \cdots & {b_{1,n} } \\ {b_{2,1} } & {b_{2,2} - {{4\pi^{2} P} \mathord{\left/ {\vphantom {{4\pi^{2} P} {2L}}} \right. \kern-0pt} {2L}}} & {b_{2,3} } & \cdots & {b_{2,n} } \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {b_{n,1} } & {b_{n,2} } & {b_{n,3} } & \cdots & {b_{n,n} - {{n^{2} \pi^{2} P} \mathord{\left/ {\vphantom {{n^{2} \pi^{2} P} {2L}}} \right. \kern-0pt} {2L}}} \\ \end{array} } \right| = 0$$

(B2)

where b _i,j are the intermediate parameters for calculation and can further be described as:

$$b_{ii,jj} = \frac{{jj^{4} \, {\text{EI}}\pi^{4} }}{{2L^{3} }} + \frac{Lk}{2}\quad \left( {ii = jj = 1, \ldots ,n} \right)$$

(B3)

$$b_{ii,jj} = 0\quad \left( {ii = 1, \ldots ,n} \right)\quad \left( {jj = 1, \ldots ,n} \right)$$

(B4)